Scuola di Ingegneria Industriale e dell’Informazione Laurea Magistrale in Ingegneria Meccanica
Flywheel energy storage for vehicle applications
Ettore Rasca 841979
Supervisor: prof. Francesco Braghin
Academic Year 2016-2017
Esprimo il mio ringraziamento a Stefano Sorti per tutto il supporto fornito.
Abstract 1
Abstract
In recent years, a significant increase in the market share of electric vehicles was observed. Most of these vehicles are meant for private use and are equipped with chemical batteries. Despite the huge improvements made on the capacity of the new generation lithium ion batteries, the long charging time remains a main drawback of this technology and opens the possibility for alternative solutions. The present work describes a preliminary study aimed at investigating the possibility to realize an electric vehicle relying on the flywheel energy storage technology as a primary energy source. First, a numerical and an analytical model of such a system are proposed and evaluated. Next, two sets of optimizations are performed on these models. Through the first optimization set, the optimal geometry for the rotors in the energy storage system is identified. This first process is repeated several times considering different alternatives for the rotors material, maximum rotational speed and basic geometry. Through the second optimization set, the ideal displacement and orientation of the rotors on the vehicle frame, as well as the total number of rotors, are investigated. Finally, three multi-rotor configurations for the energy storage system are proposed and described. The data collected after performing simulations on the dynamics of these systems are then studied. In conclusion, after presenting observations on the feasibility of such a technical solution, a set of future steps for the development of the flywheel energy storage technology for vehicle applications are proposed.
Flywheel energy storage 2
Contents
Abstract ...... 2 Contents ...... 2 1 Introduction and aims of the work ...... 7 1.1 Energy storing in flywheel-based devices ...... 8 1.1.1 Historical overlook ...... 8 1.1.2 High performance flywheels ...... 10 1.1.3 Characteristics of flywheel energy storage devices ...... 10 1.1.4 Kinetical energy storage systems applications ...... 12 1.2 Vehicle applications ...... 12 1.2.1 Kinetic energy recovery system ...... 13 1.2.2 Oerlikon Gyrobus ...... 14 1.3 Objectives of the study ...... 15 2 Analytical model of the rotor-frame system ...... 17 2.1 Introduction ...... 18 2.1.1 Degrees of freedom for the model ...... 18 2.1.2 Multibody system ...... 20 2.1.3 Inertial and non-inertial reference frames ...... 22 2.1.4 Cardan angles and their properties ...... 24 2.2 General procedure...... 26 2.2.1 Motion equation for the two subsystems ...... 28 2.2.2 Preliminary observations for subsystem coupling ...... 28 2.2.3 Analysis of the rotor motion equations ...... 29 2.2.4 Preparing the rotor motion equations for coupling ...... 29 2.2.5 Analysis of the frame motion equations ...... 31 2.2.6 Preparing the frame motion equations for coupling ...... 31 2.2.7 Motion equations coupling ...... 31 2.3 Rotor motion equation ...... 32 2.3.1 Rotor subsystem overlook ...... 32 2.3.2 Lagrange equation components ...... 33 2.3.3 Solving the rotor motion equation ...... 36 Contents 3
2.3.4 Simulations and results ...... 36 2.4 Preparing the rotor motion equation for coupling ...... 38 2.4.1 Variable change procedure ...... 38 2.4.2 Rotor subsystem boundary displacements ...... 43 2.5 Frame motion equation ...... 44 2.5.1 Frame subsystem overlook ...... 44 2.5.2 Lagrange equation components ...... 45 2.5.3 Solving the frame motion equation ...... 48 2.5.4 Simulations and results ...... 48 2.6 Equations coupling ...... 48 2.7 Final observations and SimMechanics models ...... 50 3 Single rotor optimization ...... 55 3.1 SimMechanics models ...... 56 3.2 Analysis of the problem ...... 56 3.2.1 General variables of the optimization ...... 56 3.2.2 System models for the optimization ...... 57 3.2.3 System forcing ...... 59 3.2.4 Limits of the optimization field ...... 59 3.2.5 Additional constrains of the optimization ...... 60 3.2.6 Elementary cost functions ...... 61 3.3 Definition of the optimization procedure for the rotor geometry ...... 62 3.3.1 Preliminary evaluation 1 ...... 63 3.3.2 Preliminary evaluation 2 ...... 65 3.3.3 Preliminary evaluation 3 ...... 66 3.3.4 Introduction to the rotor geometry optimization ...... 69 3.3.5 The selected optimization procedure...... 69 3.4 The optimal rotor geometry and orientation ...... 72 3.4.1 First attempt optimal geometry ...... 72 3.4.2 Perfecting the results...... 75 3.5 Orientation independent geometry ...... 84 3.5.1 General procedure ...... 84 3.5.2 Results averaging ...... 86 4 Advanced single-rotor optimization ...... 89 4.1 Evaluation of the results of chapter 3 ...... 90
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4.2 Improvements on the optimization process ...... 91 4.2.1 Low density flywheel ...... 92 4.2.2 Low density flywheel at higher velocity ...... 94 4.2.3 Geometry improvement ...... 94 4.3 Evaluation of the results ...... 97 4.3.1 Rotor Jtb-2 ...... 99 4.3.2 Rotor Ltb-1 ...... 100 4.3.3 Rotor Ntb-5 ...... 101 4.3.4 Rotor Ptb-3 ...... 102 4.3.5 Rotor Rtb-2 ...... 103 4.4 Final rotor selection ...... 104 5 Multi-rotor optimization ...... 107 5.1 Rotor positioning ...... 108 5.1.1 Selection of the rotation direction ...... 109 5.2 Final solutions ...... 111 5.2.1 Solution 1: from rotor Ntb-5 ...... 112 5.2.2 Solution 2: from rotor Ptb-3 ...... 113 5.2.3 Solution 3: from rotor Rtb-5 ...... 114 6 Conclusions ...... 117 6.1 Analysis of the results ...... 118 6.1.1 Mass and mass-based energy density ...... 118 6.1.2 Encumbrance and volume-based energy density ...... 119 6.1.3 Self-discharge ...... 119 6.1.4 Effects on the vehicle dynamics ...... 121 6.2 Final conclusions and future developments ...... 121 7 Bibliography ...... 125 8 Appendix ...... 126 8.1 Appendix A...... 126 8.2 Appendix B ...... 131 8.3 Appendix C ...... 132 8.4 Appendix D ...... 134 8.5 Appendix E ...... 139 8.6 Appendix F ...... 145
1 Introduction and aims of the work
In this first chapter a general introduction on the flywheel-based energy storage systems is provided. In the initial sections a historical overlook on this technology is presented, with particular focus on how the energy storage systems of this nature have always played a role in human technical development. A discussion on the more modern use of these devices is then provided together with a description of their main characteristics and properties. It follows an analysis of some of the most interesting implementations related to the vehicle sector, and some significant examples are also presented.
In the closing section of this chapter the aims of this work are detailed. Furthermore, an overlook to the general procedure is provided with regard for the different areas of the study.
Flywheel energy storage 8
1.1 Energy storing in flywheel-based devices
1.1.1 Historical overlook The first flywheel-like devices where developed far before the kinetic energy conservation principle was clearly understood. In fact, the failure of ancient thought to recognize the basic principles of motion did not impede the development of devices which exploit the inertia of bodies. It must be considered that in the ancient world, science and technology, were far more separated than in modern times. Beside this, devices exploiting inertial properties were developed and established thousands of years before the first attempts were made to explain the energy conservation of a rotating object. [1]
The first ever use through history of tools that makes use of the principle of inertia, however, was not related with rotational motions. Linear motion, in fact, can be investigated in a much easier way and with more basic tools. The utensils making use of this principle are all the hammer like devices. However, hundreds of thousands of years were to elapse before rotary inertia could play a role in human life. The first tool using rotary motion was probably the drill, with the main application of lighting fires. But the hand drill, like its immediate successors, did not need the addition of a flywheel to work properly. [2]
The first technically advanced application of the flywheel to appear in very ancient times is the potter's wheel. With these devices some major difficulties are to be found both with the significant size and energy stored and the need for some sort of bearing. This last problematic is due to the fact that potters' wheels must rotate about a fixed axis. Since most of these machines where almost entirely built of wood, not many archaeological evidences of them are available. For this, it is not easy to evaluate the exact development age of the first devices. Perhaps the potter’s wheel preceded the wheeled vehicle. The first findings are from Sumer and Susiana, and date from between 3500 and 3000 BC. [1] [2]
With these devices, the wheel needs to maintain its rotation for long enough to complete the operation of shaping the jar, whether in one or more stages. It is therefore a true flywheel energy storage system. It is assumed that the wheel should maintain its rotation for five to seven minutes with an initial angular velocity of 100 푟푒푣/푚𝑖푛.
Thousands of years later the first mechanical prime mover appeared. These were mainly the water mill, introduced in the 1st century AD and in widespread use by the 4th century, and the windmill. This last one appeared in Western Europe about a thousand years later. One of the first and most common use of mills was corn grinding. For this application the inertia of the water wheel and of the grinders, combined with the high torque provided by the prime mover, was sufficient for operating. Even for other more demanding applications an additional flywheel was not required and rarely introduced. This does not mean that a high inertia for the system is not required, on the contrary this keeps playing a fundamental role in providing a smooth and affective operation, but in this case, this is provided by the rotating elements composing the machine. [1] [3]
During the middle ages gearwheels started to appear with increasing frequency. A high moment of inertia was often added to all the shafts because of the poor kinematic characteristics of the gear wheels of various types then in use. An exception is for the shafts which carried the grinders and the water wheel. The high inertia was usually introduced by building heavy gear wheels. Anyhow in some machines flywheels were purposely added, at least on some shafts. [2] Introduction and aims of the work 9
In the 18th century, with the industrial revolution, the steam engine started its expansion as a new prime mover. Furthermore, the drop in the production costs of iron, and particularly of cast iron and later of steel, and the need for metal parts of the complex steam engines, resulted in the use of iron initially in some parts of the machines and later in the machines as a whole. Two important developments related to this age affected the evolution of flywheels technology. In first place flywheels where extensively applied and studied for the realization of the new steam engines and, finally, the widespread use of metal in the construction of machines instead of wood allowed for the obtainment of iron flywheels. The first flywheels to be realized of a single iron piece could be built. Because of the greater density of iron with respect to wood these devices could embody a greater moment of inertia in the same space. Most of the practical applications of the age operated at low speed and so the use of high density and low resistance materials like cast iron was of no disadvantage. [1] [2]
In the years following the Industrial Revolution, the steam engine was developed to give higher power and efficiency. Very large flywheels were built for the largest engines.
From soon after the industrial revolution new engines operating at much higher rotational speed were developed. Among these new prime movers, the internal combustion engines, like the petrol and diesel engines, became widely used for both ground and naval operations. To cope with the higher centrifugal forces, new designs for the flywheel were developed. The solid disc-type flywheel started to become popular after it was first introduced in 1889 on the V-type twin cylinder power plant for automotive use, a small internal combustion engine capable of very high rotational speed for the time. [1] [2] [4]
Figure 1: V-type twin cylinder power plant for automotive use
Nowadays flywheels are employed in a wide variety of applications, ranging from slow to very fast rotational speed operations. These devices are designed with very different characteristics on the base of the application and of the amount of energy they are intended to store. Other than steel, new and more advanced materials are commonly applied, like compound materials for high rotational speed operations.
Flywheel energy storage 10
1.1.2 High performance flywheels A notable development in the flywheel technology occurred in two phases at the end of the ‘60s and at the beginning of the ‘70s. Two factors led to this breakthrough. Firstly, advances in the field of high-strength composite materials made it possible to build flywheels which could store much more energy for the same mass than conventional ones. Secondly, the growing concerns of ecological problems pushed the development for energy storage devices for a wide range of applications. These clean and cheap energy storages were meant to be coupled with internal combustion engines – mainly on road vehicles – to reduce the overall pollutants production. [1]
Many important achievements in this field were obtained at the beginning of the ‘70s after many governments provided substantial research funding. Many successful applications where realize employing a 'conventional' technology. The flywheels, in fact, were steel discs and were of 'conventional', although often very good, design.
The research was also dedicated to the development of a new generation of flywheels, or the super- flywheels. The main difference between these and conventional flywheels can be assessed in terms of the energy density accumulated, a parameter obtained as the ratio between energy stored and mass of the system. This is often achieved with an increase of the nominal rotational speed. A second peculiar feature of these new generation flywheels is that in case of anomaly the failure is much less destructive than with conventional ones. Despite all the efforts made in this direction, currently no advanced flywheel design which is successful enough to be mass produced is yet achieved.
1.1.3 Characteristics of flywheel energy storage devices A flywheel can be introduced in a mechanical system for two different purposes. The first reason is to provide more stability to an otherwise not steady state system by waring off oscillations in the angular velocity of the rotating machine. This result is achieved by introducing such a high inertial element on the fast-moving shaft. In this case the flywheel acts as a short-term energy storage system and so it is not necessary to have much energy accumulated. On the other hand the second reason why a flywheel can be introduced in a mechanical system is to store kinetic energy and to slowly providing it to the user load. In this second case the flywheel acts as a long-term energy storage system; it is now of greater importance that the device is capable of storing significant amounts of energy. The latter case is the one that is analysed in detail in this work.
In both cases the flywheel is a device introduced in a mechanical system for its capability of storing and releasing energy. A parameter that evaluates the quality with which the energy is accumulated is the energy density, which is defined as the ratio between the useful energy stored over the flywheel mass. With useful energy it is to be intended only the amount of energy that can actually be extracted from the system during nominal operation. Usually this parameter is related to the critical failure speed, the rotational speed at which the centrifugal forces are sufficiently high to cause a main failure in the device. In this case the following expression for the energy density is defined.
푒 휎푢 ( ) = 퐾 (1) 푚 푢.푓. 휌
Where 푒⁄푚 is the flywheel energy density evaluated with respect to the ultimate failure stress, 휎푢 is the ultimate tensile stress, 휌 is the flywheel density and 퐾 is a parameter called shape factor. In case the rotor is composed of an isotropic material, the shape factor is function of the flywheel geometry only. Introduction and aims of the work 11
Of course, during nominal operation, the flywheel speed does not work between null and critical failure speed and this must be considered while evaluating the energy density. In particular, three factors are introduced:
- Safety factor 훼′ . This factor represents the ratio between the energy stored in the flywheel at operational conditions and the energy stored ad failure. - Depth of discharge 훼′′. This is the factor that considers the ratio between the stored energy and the useful one and it can be calculated in the following way.
2 ′′ 휔푚푖푛 훼 = (1 − 2 ) (2) 휔푚푎푥
An excessive reduction in the rotor rotational speed leads to technological issues and so, for this reason, it is often imposed that the minimum rotational speed is equal to half of the maximum. - Mass factor 훼′′′. This factor is defined as the ratio between the flywheel mass and the total mass of the energy storage device.
With these parameters the energy density of the flywheel during nominal operation can be stated. This is defined as the overall energy density.
푒 ′ ′′ ′′′ 푒 ′ ′′ ′′′ 휎푢 ( ) = 훼 훼 훼 ( ) = 훼 훼 훼 퐾 (3) 푚 표.표. 푚 푢.푓. 휌
Flywheels can be classified in three categories on the base of the energy density.
- Low energy density: when the energy stored is lower than 36 퐾퐽/퐾푔. - Medium energy density: when the energy stored is between 36 and 90 퐾퐽/퐾푔. - High energy density: when the energy stored is higher than 90 퐾퐽/퐾푔.
The high energy density is one of the greatest advantages of kinetic energy storage devices over other energy accumulators. Moreover, the speed up phase in a flywheel energy storage can be performed in a limited amount of time and so the loading phases of such devices is far faster than the loading of chemical batteries. Finally, when energy is stored in the form of kinetical energy this can be extracted at very fast rates providing a high-power output to the user load. [5] [6]
The storage efficiency of flywheel-based energy storage systems – or the capability of preserving the energy stored for long periods with little losses – is typically very high, usually even higher than the efficiency of chemical based battery. However, this parameter generally decreases when the accumulation time gets longer. High performances can be achieved with magnetically suspended rotors even if, for this kind of applications, still more development is needed before the magnetic bearing technology can be wildly used. It has also to be considered that the transmission system has a crucial relevance on the overall efficiency of the system, especially because a continuously variable ratio transmission is often needed to couple the rotor to the generator/motor. [1]
The main disadvantages of kinetic energy storage systems are related with the intrinsically problematic nature of a big and heavy object rotating at fast angular speed. First, it must be considered that a catastrophic failure may occur and so the rotor design must be developed in order to reduce all the possible risks to a minimum. Besides also other complications related to vibrations, noise, wear and fatigue may arise. However, the fatigue problem is usually far more relevant when considering chemical-based batteries.
Flywheel energy storage 12
Finally, the design of a continuously variable transmission usually causes a heavy penalty on the overall system efficiency, to the point that constant speed and variable inertia flywheels designs have recently been considered. [1] [7]
1.1.4 Kinetical energy storage systems applications On the base of the application the flywheel energy storage systems can be categorized in two groups: the storage systems operating on stationary machines and the storage systems operating on vehicles. In this work the second class is studied in detail and, since a closer look on this is provided in the following section, a brief introduction on the stationary applications is now presented.
Most of the stationary application of the flywheel energy storage systems are related to low power applications. These are usually emergency devices mainly used to provide load levelling for uninterruptible power supply like, for example, for data centres, as they save a considerable amount of space compared to battery systems. [8]
Flywheels are sometimes used as short term spinning reserve for momentary grid frequency regulation and balancing sudden changes between supply and consumption. No carbon emissions, faster response times and ability to buy power at off-peak hours are among the advantages of using flywheels instead of traditional sources of energy like natural gas turbines. Operation is very similar to batteries in the same application, their differences are primarily economic. [9]
Flywheel based high power energy storages are technically possible but currently these are not economically convenient, unless for the case in which other traditional energy storage systems are not feasible. [1]
1.2 Vehicle applications The idea of introducing a flywheel energy storage system on a vehicle is not new and many different applications have already been investigated. From an energetic point of view two main different configurations can be considered. Firstly, the flywheel can be coupled with a prime mover – like, for example, an internal combustion engine – to recover part of the energy that would be lost during the deceleration of the vehicle. This energy is then stored and used to reduce the prime mover energy needed during acceleration. Secondly, the flywheel energy storage system can be employed as the main energy source of the vehicle. Anyhow the use of a fast-rotating element on a moving machine could lead to gyroscopic effects that must be carefully evaluated. In the following paragraphs an overlook is given on these two main applications with respect to the case of road vehicles.
It is estimated that on a road vehicle almost 60% of the energy is used for the acceleration, and later lost during breaking. This is a particularly critical problem for the mobility of private vehicles in cities and for public transportation, where a high number of acceleration-deceleration cycles are usually needed. Many different options have been considered to recover the lost energy. Nowadays hybrid vehicles with internal combustion and electrical engines are one of the most popular alternatives to address this problem. Flywheel energy storage systems could also be a valid alternative in that a high energy density can be achieved. More importantly this kind of energy storage allows for fast charging and fast discharging, yielding high-power input and output energy fluxes. Technical problems, mainly related to the design of a continuously varying transmission held back the spreading of such a device. Also other applications, like the use of a flywheel to help with restarting an internal combustion engine, are currently under development. [1] Introduction and aims of the work 13
Nowadays electrical vehicles are rapidly getting more popular. Despite the many advantages of an electric vehicle, some main disadvantages – mostly related to the battery technology – affect them. A battery storage system, in fact, needs long time to be recharged and consists of a heavy and bulky device. Finally, it has also to be considered that with current technology the lifespan of chemical energy storage systems is rather short. All three of these problems could be addressed by using a flywheel-based energy storage system. Some prototypes and working models of vehicles with this kind of energy storage have already been developed, however no one successful enough to be mass produced.
In the following subsections one significant example for each of the two applications now introduced is presented.
Figure 2: flywheel and motor-generator of the Gyrobus
1.2.1 Kinetic energy recovery system A kinetic energy recovery system (KERS) is an example of vehicle application for flywheel energy storage systems, designed to recover the energy lost during breaking. As it was previously stated not all the kinetic energy recovery systems are flywheel based: some of them use a high voltage battery instead.
One of the most interesting and advance applications of KERSs is with Formula One race cars. In 2009 FIA allowed the use of a KERS in the regulations for the 2009 Formula One season to push for the development of a responsible solutions to the world environmental challenges. This application is particularly fascinating because the regulation allows for the use of either a mechanical energy storage via flywheel or an electrical storage via battery or supercapacitors. These three technologies have thus been developed and are currently competing to provide the best possible solution.
As typically happens for the use of a flywheel energy storage system on vehicles, one of the biggest challenges related to the development of a flywheel based KERS for Formula One applications was related to the design of a continuous variable transmission.
The kinetic energy recovery system allowed from the FIA regulation was a 60 KW device in 2009 and the power was later increased to 120 KW. According to rules, the stored power can be used for not more than 6.6 seconds and the total energy that can be accumulated is 400.000 KJ. Finally, the angular speed of flywheels based KERSs is of 60.000 rpm. [10]
Flywheel energy storage 14
Even if the use of this device in the Formula One competitions had ups and downs in the last years, this led to important developments in the technology. Many other automotive competitions are also currently pushing for the development of such a knowledge and ultimately contribute to the spread of such systems in private vehicles, and in the transfer of this technology to road cars. [11]
1.2.2 Oerlikon Gyrobus An interesting vehicle application of flywheels as prime mover are gyrobuses, electrical buses that uses flywheel-based energy storages. With comparison to trolleybuses, gyrobuses don’t need to be continuously connected to the grid by wires. While there are no gyrobuses currently in use commercially, development in this area continues. [1]
Figure 3: Maschinenfabrik Oerlikon Gyrobus, 1953
The concept of a flywheel-powered bus was first developed and realized during the ‘40s by the swiss company Oerlikon, with the aim of creating a feasible alternative to battery energy storage in road vehicles, that didn’t involve any overhead-wire electrification. The flywheel designed to store energy was a large metal disc spun at up to 3.000 rpm. During the charging phase, this was accelerated by an electrical motor powered by means of three booms mounted on the vehicle roof, which contacted charging points located as required or where appropriate. Natural charging position were at passenger stops and at terminals. During movement, the energy to power the vehicle electric motor was obtained by using the flywheel charging motor as a generator and gradually slowing down the metal disk. In this way the kinetic energy of the flywheel was once more converted into electricity. Finally, since the bus breaking was electric, part of the breaking energy was recovered and used to speed up the flywheel and increase the overall range. [12] [13]
A gyrobus could typically travel as far as 6 km on a level route at speeds of up to 50 to 60 km/h, depending on vehicle load. Charging a flywheel took between 30 seconds and 3 minutes and, in order to reduce this Introduction and aims of the work 15 time, the option of increasing the charging voltage was considered. Given the relatively restricted energy stored, issues related to the vehicle range could arise in case of dense urban traffic. [14]
Some of the greatest advantages of this kind of vehicle are that it is quiet, pollution-free (or at least pollution confined to generators on electric power grid) and that it can operate flexibly at varying distances. The main disadvantages are however related to the great weight introduced by the flywheel, the risks related to having a heavy fast-moving object on board, the short range that can be achieved and the additional complexities in driving a vehicle affected by strong gyroscopic effects. However, it must be considered that with the use of modern technology most of these drawbacks could be solved or at least reduced.
1.3 Objectives of the study In this work a preliminary study to the design of an electric vehicle based on the flywheel energy storage technology is performed. With this, a feasible alternative to the more popular chemical battery energy storage for road vehicles is investigated. A comparison between these two alternatives is made in terms of both the encumbrance and weight of the system, of the storable energy and of the achievable range in practical use. This study is limited to a configuration in which the rotor support system is obtained by mechanical bearings. Advantages and disadvantages of the two energy storage systems are analysed, and particular attention is put on the effects that can arise from the accelerating and decelerating dynamics of a big and heavy rotating object mounted onto a moving vehicle. This study is split in four main areas that are now introduced.
The first step is the development of a working model for the rotor-frame system. This model is initially developed in an analytical form with Matlab and later evaluated using a numerical model obtained with SimMechanics. Comparable results are expected. This model is first introduced for a single rotor configuration but the possibility of later considering other flywheels on the same frame must be left open. All the following optimization processes are performed on these models.
An optimization on the geometry of the rotor for a single flywheel configuration is performed. In this phase the cost functions to be minimized are carefully selected together with the variables to be optimized and the boundary to be imposed. The possibility of optimizing the rotor geometry independently from its displacement and orientation on the frame is investigated and outcomes for both cases are provided. The obtained results are evaluated and discussed. Since the goal of this work is to perform a feasibility analysis, the structural assessment of the solutions obtained is not analyzed in detail. However, evaluations on the viability of these results are performed. The identified optimal geometry for the single rotor is later used as the starting point – in that of the geometry for the rotors in the vehicle energy storage system – for the multirotor optimization.
After the single rotor geometry optimization, a new optimization on a multirotor system is performed. The optimal multi-rotor configuration is found imposing the constraint that this solution must be capable of storing a sufficient amount of energy for the vehicle on road use. The volume occupied by the rotors and the mass of the whole system are carefully evaluated and compared to those that would be found using chemical battery for storing the same amount of energy.
A detailed study of the final configuration obtained by performing the two optimization processes is carried out. General conclusions are drawn on the possibility of realizing a road vehicle, for private or public use, that uses a flywheel based energy storage system with mechanical support system for the rotors.
2 Analytical model of the rotor-frame system
In this chapter the procedure to develop an analytical model for the rotor-frame system is detailed. This mathematical work frame is developed using Matlab, and later compared with a numerical model gained with SimMechanics. Equal results are expected.
The model is initially build for a single rotor configuration since this is needed for the rotor geometry optimization described in chapter 3. Subsequently, a multi-rotor model is built to perform the multi-rotor optimization of chapter 5. This latter is developed on the single rotor model, which is designed to be easily modified to account for more than one flywheel.
The general procedure is here introduced together with a description of the complications that were encountered during this development. Crucial information on how the Matlab function s are written are also presented. Finally, the SimMechanics model is explained and advantages and disadvantages related to the use of this latter option are discussed.
Flywheel energy storage 18
2.1 Introduction
2.1.1 Degrees of freedom for the model As previously stated, the analytical model for the rotor-frame system is developed for the purpose of performing an optimization process, aimed at finding the best rotor geometry. In a second moment – in chapter 5 – this same model is also used to identify the optimal multi-rotor configuration. Finally, this same mathematical framework allows to simulate the behaviour of the system with respect to time in any given configuration and for any relevant external excitation. For this reason, it is with this model that the system dynamics is simulated.
In the model development, to completely describe the system, the appropriate number of degrees of freedom must be selected. The set of degrees of freedom must include enough coordinates to express the system dynamics of interest, plus several others to be used for introducing the external excitations. On the base of the selected degrees of freedom the system model is represented, and the equation of motion developed. For the sake of simplicity, the system is split into two subsystems which are analysed separately. These are the rotor subsystem and the frame subsystem.
Figure 4: the rotor subsystem (11 degrees of freedom)
When it comes to the rotor, it is clear that a set of six degrees of freedom is needed to describe its dynamics. Three of them represent the rotor displacement on the frame, while the other three the rotor orientation. Moreover, five additional degrees are introduced for considering the boundary imposed displacements, and this is done in accordance with the modelling choices detailed in the following sections. Now that the subsystem degrees are selected, its model can be established. A representation of the rotor subsystem model is provided in Figure 4.
For what concernes the frame, instead, the choice on the degrees of freedom number is not that obvious, and many different alternatives can be evaluated. On the base of how many degrees of freedom are considered for the frame subsystem, different models can be defined for it. Among all these possibilities, three are selected and used to define the frame subsystem models, and these are represented from Figure 5 to Figure 7. Analytical model of the rotor-frame system 19
Figure 5: Option 1 for the frame subsystem (7 degrees of freedom)
Figure 6: Option 2 for the frame subsystem (11 degrees of freedom)
An increasing complexity characterizes the three models. Further details on these are provided in the following subsections.
All three of these system representations have been developed, each into a different analytical model composed of a set of differential equations. For the sake of simplicity, in this chapter we focus on the development of the motion equations for the system composed of the rotor subsystem combined with the first option for the frame subsystem. Anyway, the method changes only slightly when it comes to obtaining the motion equations for a system defined with one of the other two alternatives for the frame representation.
Flywheel energy storage 20
Figure 7: Option 3 for the frame subsystem (15degrees of freedom)
2.1.2 Multibody system As it can be denoted from the representations in the preceding subsection the system objective of the study is a multibody system characterized by multiple degrees of freedom. In this subsection the main elements appearing in the models are described. The focus is put on both the rigid bodies and on the stiffness and dumping elements. Some of these elements are common to all three of the models, other are present only when the second or the third alternatives for the frame subsystem are considered.
In the description of the model elements, the parameters are presented without detailing any numerical value. Reason of this stays in the fact that the model is developed in a completely analytical way, and no numerical values are introduced. In this phase of the study, in fact, it is not considered which of these parameters are to be imposed as constant and which are the variable of the optimization process.
Quantity Unit Symbol
Mass 퐾푔 푚푟표푡 2 푥푥 Moment of inertia (x-axis) 퐾푔 ∙ 푚 퐽푟표푡 2 푦푦 Moment of inertia (y-axis) 퐾푔 ∙ 푚 퐽푟표푡 2 푧푧 Moment of inertia (z-axis) 퐾푔 ∙ 푚 퐽푟표푡 3 Rotor density 퐾푔/푚 휌푑푒푛푠,푟표푡
Rotor eccentricity 푚 푥푂 Table 1: rotor characteristics
2.1.2.1 Rotor The rotor is a rigid body composed of three cylindrical elements. The rotation axis is the symmetry axis common to the three elements. The rotor mass centre is placed outside from the rotation axis. The distance
between this axis and the mass centre is denoted as 푥푂.
Table 1 is provided to summarize the main geometrical and physical characteristics of this rigid body. Analytical model of the rotor-frame system 21
2.1.2.2 Frame The frame represents the main vehicle body. It is modelled by a single homogenous brick. Table 2 summarizes the frame characteristics.
Quantity Unit Symbol
Mass 퐾푔 풎풕풆풍 2 풙풙 Moment of inertia (x-axis) 퐾푔 ∙ 푚 푱풕풆풍 2 풚풚 Moment of inertia (y-axis) 퐾푔 ∙ 푚 푱풕풆풍 2 풛풛 Moment of inertia (z-axis) 퐾푔 ∙ 푚 푱풕풆풍 3 Rotor density 퐾푔/푚 𝝆풅풆풏풔,풕풆풍 Table 2: frame characteristics
2.1.2.3 Rotor support system The rotor support system is the physical linking element between rotor and frame. It is composed of a pair of roller bearings and one axial bearing. The roller bearings are placed at the very end of the rotor shaft. The axial bearing is placed in such a way that the whole axial load acts on the central element of the rotor. For what concerns the model of this latter element, we can consider as its load was directly applied to the central point of the symmetry axis of the rotor central element.
In the next chapter also a two-roller bearing and a three-roller bearing per side options are investigated. In these cases, the load of any additional bearing is applied to the shaft at a certain distance from the shaft end. This distance is equal to the clearance needed to place the more external bearing on the same part of the shaft.
Every roller bearing is represented in the model as two spring-dumper subsystems placed in the two directions the bearing can handle the load. The axial bearing is represented as a spring-damper subsystem. The subsystems are composed of a spring in parallel with a dumper.
The variables of the rotor suspension system are listed in table 3.
Quantity Unit Symbol(s)
Roller bearing stiffness 푁/푚 퐾퐴,푥 퐾퐵,푥 퐾퐴,푦 퐾퐵,푦
Roller bearing dumping (푁 푠)/푚 푟퐴,푥 푟퐵,푥 푟퐴,푦 푟퐵,푦
Axial bearing stiffness 푁/푚 퐾푂,푧
Axial bearing dumping (푁 푠)/푚 푟푂,푧 Table 3: rotor support system characteristics
2.1.2.4 Vehicle support system The vehicle support system is composed of the vehicle suspension system and of the vehicle wheels. It is, therefore, the mechanical subsystem linking the vehicle to the ground. With the three alternatives for the frame subsystem introduced in subsection 2.1.1, three different representations of the vehicle support are presented. It can be noted that the three options have an increasing level of detail. It follows a short description of these alternatives.
With the first option, a very simple modeling of the vehicle support system is provided. In this case, the whole system is represented with four spring-dumper subsystems placed vertically on the vehicle corners. The spring-damper couples should have such characteristics to represent the assembly composed of both wheel and suspension.
Flywheel energy storage 22
The second alternative is a further development of the first option. At each corner of the vehicle two spring- dumper subsystems are placed. The wheel is modelled as a suspended mass in between these two. The spring-damper connecting the suspended mass to the ground represents the wheel elastic and dumping characteristics, while the spring-damper connecting the suspended mass to the frame models the suspension properties. The additional inertia that the suspended mass provides accounts for the effects that the wheel introduces on the system dynamics.
With the third model, also the wheel transversal and longitudinal stiffness and dumping are considered. With this option it is also possible to simulate the behaviour of the vehicle during acceleration and deceleration phases. Moreover, with this model, also the dynamics of the vehicle during curves can be evaluated.
It is important to notice that with the first and second alternatives the overall system is not bounded in the longitudinal and transversal directions, and according to rotation along the vertical axis. To prevent this problem, during the development of the motion equation, some additional constraints are introduced along these directions. Finally, to gain consistency whit the selected model, the equations for the unconstrained degrees of freedom are simply neglected.
The main characteristics of the vehicle support system are listed in table 4.
Quantity Unit Symbol(s)
Suspension stiffness 푁/푚 퐾푎,푠푥 퐾푎,푑푥 퐾푝,푠푥 퐾푝,푑푥
Suspension damping (푁 푠)/푚 푟푎,푠푥 푟푎,푑푥 푟푝,푠푥 푟푝,푑푥
Wheel radial stiffness 푁/푚 퐾푤,푟
Wheel radial damping (푁 푠)/푚 푟푤,푟
Wheel transversal stiffness 푁/푚 퐾푤,푡
Wheel transversal damping (푁 푠)/푚 푟푤,푡
Wheel longitudinal stiffness 푁/푚 퐾푤,푙
Wheel longitudinal damping (푁 푠)/푚 푟푤,푙 Table 4: frame support system characteristics
2.1.3 Inertial and non-inertial reference frames To develop the motion equations for the overall system some reference frames should be introduced. It is reminded that the equations of motion must be defined in an inertial reference system. With this in mind, the reference system selected for defining the motion equations is a reference placed on the ground. Since the overall rotor-frame assembly is rather complicated, some additional reference systems are defined to keep all the procedure steps as most clear and simple as possible.
Before introducing the references that is used, a convention to identify position and orientation of a body in these references must be defined. First it is reminded that at least six coordinates are needed to fully describe the position and the orientation of a rigid body in any given 3D reference. The set of coordinates that was chosen is composed of three displacements and three rotations. The displacements are defined as cartesian coordinates while for the rotations cardan angles are used. This choice is common to every reference system adopted.
The details of the chosen convention are now provided; it is shown how the six coordinates are used to identify the body position into space. The body is initially considered to be placed in the origin of the reference system with its director axis lying on the z direction. The following transformations are applied to move from such position to the one identified by the set of coordinates: Analytical model of the rotor-frame system 23
- Three cardan rotations are applied in sequence. The magnitude of these rotations is expressed by the three angles in the coordinates vector. - Only after the final orientation for the body is identified, it is displaced in space according to the three translations in the coordinates vector.
From here to the end of this subsection the reference systems are introduced and briefly described. More details on these frames and their use are given as the motion equation development method is described in detail.
2.1.3.1 Local reference frame {푥퐿, 푦퐿, 푧퐿} This reference system is fixed on the rotor. Its origin is placed on the rotation axis of the rotor at the minimum distance point from the rotor centre of mass. The 푧퐿 axis lies on the rotor rotation axis and the 푥퐿 axis is placed in such a way that the rotor centre of mass is a point on it.
This reference system is introduced to define the moments of inertia of the rotor along the three axes in the simplest way. Reason of this is that volume integrals defining the moments and products of inertia are independent from time when they are evaluated in the local reference. Moreover, in this reference system the characteristic of eccentricity of the rotor is introduced.
The local reference is denoted with letter L in Figure 8.
푂 푂 푂 2.1.3.2 Global reference frame {푥퐺 , 푦퐺 , 푧퐺 } This reference system is fixed to the vehicle frame. When the rotor is in its neutral position the origin of the global reference and of the local reference, together with the 푧퐿 and the 푧퐺 axis, coincide. Moreover when the rotor lies in its initial position the two reference frames overlap.
The global reference system is introduced to study the dynamics of the rotor around its equilibrium position. In fact, in this system six degrees of freedom can be introduced to study the rotor behavior. The motion equations for the rotor are first written in this reference, given the many advantages that this provides for the rotor dynamics analysis.
The global reference is denoted with letter G in Figure 8.
푂 푂 푂 2.1.3.3 Absolute reference frame {푥퐴 , 푦퐴 , 푧퐴 } As for the global reference frame, also the absolute frame is fixed to the vehicle chassis. However, the origin and the orientation of these two reference systems are generally not the same. The origin of this latter one is placed in the centre point of the lower face of the parallelepiped representing the frame. The 푧퐴 axis is placed along the longitudinal direction while the 푥퐴 axis is in the vertical one.
This frame is introduced to create a reference on the chassis that is oriented with the vehicle direction, and it is also particularly useful when studying a multi-rotor configuration.
The absolute reference is denoted with letter A in Figure 8.
2.1.3.4 Ground reference frame {푥푇, 푦푇, 푧푇} The ground reference is the only inertial reference system introduced. It is the reference system the motion equations are to be written in. The ground reference is fixed to the ground and its origin is placed vertically under the absolute reference system origin. The 푥푡푒푟푟 axis is the vertical one while the 푧푡푒푟푟 axis is in the longitudinal vehicle direction.
Flywheel energy storage 24
The ground reference is denoted with letter T in Figure 8.
Figure 8: the reference frames
The motion equations for the frame can be simply written in this reference system, while the motion equation for the rotor can be easily evaluated in the global reference. A procedure must be developed in order to couple these two set of motion equations, and this is detailed within this chapter.
2.1.4 Cardan angles and their properties In this subsection some important properties of the reference systems are described. These properties are later used to define the system motion equations. First, it is reminded that rotations in space are not commutative. This makes it much harder to describe the dynamics of a system in space with respect to the dynamics on a flat surface. For this reason, a convention for defining the orientation of a body in a three- dimensional space must be introduced. As already stated in the preceding section, the chosen convention is the cardan angle convention.
Let’s consider the example in which the cardan angles 휎, 훽 and 휌 are used to define the local reference system with respect to the global one. In this case the cardan angles consist of three consecutive rotations that are applied in a determined sequence to transform the directions of the global reference axes into the local reference axes ones. The three rotations are:
1. A first rotation of magnitude 휎 performed around the 푥퐺 axis, of the global reference. This transforms the global reference in the axes of an intermediate reference {푥퐼, 푦퐼, 푧퐼}. 2. A rotation of magnitude 훽 performed around the 푦퐼 axis. This transforms the axes of the intermediate reference {푥퐼, 푦퐼, 푧퐼} in the axes of a second intermediate reference {푥퐼퐼, 푦퐼퐼, 푧퐼퐼}. 3. A final rotation of magnitude 휌 performed around the 푧퐼퐼 axis. This transforms the axes of the
intermediate reference {푥퐼퐼, 푦퐼퐼, 푧퐼퐼} in the axes of the local reference {푥퐿, 푦퐿, 푧퐿}.
Since the rotations in space are not commutative the order in which they are performed is relevant. Different application orders provide different results. Analytical model of the rotor-frame system 25
Figure 9: cardan angles application
2.1.4.1 The rotation matrix The example in which the cardan angles 휎, 훽 and 휌 are used to define the local reference system with respect to the global one is analyzed. In this case, the rotation matrix is a matrix that allows to transform a vector 푎퐺, defined in the global reference, into the corresponding vector 푎퐿 in the local reference. This matrix is indicated as follows:
푎퐿 = [Λ퐿퐺]푎퐺 (4)
This matrix depends on the orientation of the local reference with respect to the global one and therefore, ultimately, from the value of the Cardan angles. To define the expression of the rotation matrix as function of the Cardan angles, we proceed according to the three successive rotations defined above. Therefore, the matrixes that define the three intermediate transformations are introduced. This are planar rotation matrixes.
1 0 0 [ΛI−G] = [0 cos(휎) sin (휎)] (5) 0 −sin (휎) cos(휎)
1 0 0 [ΛI−G] = [0 cos(휎) sin (휎)] (6) 0 −sin (휎) cos(휎)
cos(휌) sin(휌) 0 [ΛL−II] = [− sin(휌) cos(휌) 0] (7) 0 0 1 Finally, the rotation matrix is obtained as the product of these three matrixes.
[Λ퐿퐺] = [ΛL−II][ΛII−I][ΛI−G] (8)
It is to be noted that the planar rotation matrices are orthogonal, so that their inverse matrix coincides with the transpose. Since a matrix obtained as a product of orthogonal matrices is orthogonal, it follows that the rotation matrix is also orthogonal.
−1 푇 [Λ퐺퐿] = [Λ퐿퐺] = [Λ퐿퐺] (9)
Where:
푎퐺 = [Λ퐺퐿]푎퐿 (10)
Flywheel energy storage 26
2.1.4.2 Rigid body angular speed with cardan angles The angular speed vector of a body in space is defined as the ratio between the infinitesimal rotation over the infinitesimal time in which the rotation takes place. Since the cardan angles are used to define the rotation of the body, the angular speed vector is obtained as the sum of three vectors:
1. A first vector with modulus 휎̇ pointing along the 푥퐺 axis.
2. A second vector with modulus 훽̇ pointing along the 푦퐼 axis of the first intermediate reference.
3. A final vector with modulus 휌̇ pointing along the 푧퐼퐼 axis of the second intermediate reference.
The angular speed vector can now be defined in the global or in the local references.
1. In the global reference the speed vector is defined as follows.
휎̇ 0 0 휔퐺 = {0} + [ΛG−I] ({훽̇} + [Λ퐼−퐼퐼] {0}) (11) 0 0 휌̇ The expressions for the matrixes, reported in equations from (5) to (7), are substituted, and the equation (11) is rearranged as follows.
1 0 sin(훽) 휎̇ ̇ 휔퐺 = [0 cos(휎) − sin(휎) cos(훽)] {훽} = [퐴퐺]{휗̇} (12) 0 sin(휎) cos(휎) cos (훽) 휌̇
2. In the local reference the speed vector is defined as follows.
0 0 휎̇ 휔퐿 = {0} + [ΛL−II] ({훽̇} + [Λ퐼퐼−퐼] {0}) (13) 휌̇ 0 0 The expressions for the matrixes, reported in equations from (5) to (7), are substituted, and the equation (13) is rearranged as follows.
cos(휌) cos (훽) sin(휌) 0 휎̇ ̇ 휔퐿 = [− sin(휌) cos(훽) cos(휌) 0] {훽} = [퐴퐿]{휗̇} (14) sin(훽) 0 1 휌̇
2.2 General procedure In this subsection the overall approach that has been used to develop the equations of motion of the whole system is described in detail. The one now introduced is just one out of many different available alternatives.
As briefly stated in the last section, the basic concept of this whole process is to develop the equations of motion for the frame separately from the equations of motion for the rotor. In fact, in a first moment the motion equations for the rotor in the global reference frame and the motion equation for the frame in the ground reference frame are evaluated. Subsequently the two sets of equations must be coupled. Before coupling, the rotor equations must be defined in the ground reference frame. This means that a variable changing procedure needs to be defined.
In the following subsections details on this procedure are provided. To make the explanation easier to understand, a simplified version of the model is used through the whole section for the purpose of illustration. This model is based on the coupling of the rotor subsystem together with the first alternative for Analytical model of the rotor-frame system 27 the frame subsystem. The simplified model used for illustration differs from the real one because of its two- dimensional nature. Of course, it does not mean that this procedure applies to a 2D model only, but this is just a way of making the problem easier to describe and to understand. It follows a representation of the simplified model.
Figure 10: The simplified rotor-frame model
It is to be noted that in the figure the points 푂′– 푂′′, 퐴′– 퐴′′ and 퐵′– 퐵′′ do not coincide. This is always the case if the rotor is not in its initial position. However, the represented configuration is the initial – or equilibrium – one, and the points are not illustrated as overlapped only in the interest of clarity.
Figure 11: The rotor subsystem
As briefly introduced in the preceding sections, this system can be split in to two subsystems: the frame subsystem and the rotor subsystem. In Figure 11 and Figure 12 these two are shown. From these, the selected
Flywheel energy storage 28
boundaries between the two subsystems can be observed. Obviously, this is not the only possible choice for tracing the boundary, and other options can also be considered.
It is already possible to note that the two subsystems exchanges forces in the points 퐴′, 퐵′, and 푂′. Moreover, it is clear that, once the spatial position of the frame is known, the boundary movement acting on the rotor can be calculated.
From now on, with ‘motion equation’ the motion equations of the whole system are intended. That is the motion equations of the frame combined to the motion equations of the rotor.
Figure 12: The frame subsystem
2.2.1 Motion equation for the two subsystems In this step the development of the motion equation for the two separate subsystems is considered. No particular challenge is found in this phase, if the rotor motion equations are developed in the global reference and the frame equations in the ground one. Since with this section only an overlook on the general procedure is provided, the details of this phase – and all the others – are presented in the following sections.
Results of this initial stage are the motion equations for the rotor in the global reference and the equations for the frame in the ground one.
2.2.2 Preliminary observations for subsystem coupling Let’s now suppose that the motion equations for the rotor in the global reference and the motion equations for the frame in the ground reference are already available. To couple the motion equations of the two subsystems, two fundamental steps are required:
Step 1: Write both sets of equations in the same reference. This means that in both sets of equations the same variable should be used to express quantities that are not independent.
Step 2: Identify and consider all the effects (in terms of displacements and forces) that the two systems mutually exchange. This means that, for example, the boundary movement acting on the rotor subsystem must be expressed as a function of the frame degrees of freedom. Analytical model of the rotor-frame system 29
Summarizing, to couple the two sets of equations these must be expressed as functions of the least number of variables. No variable that is a combination of others should be present. Moreover, all the exchanged forces should be considered.
2.2.3 Analysis of the rotor motion equations As previously stated the rotor motion equation are obtained according to the global reference frame. This reference is fixed with the frame. This means that it is not an inertial reference.
The rotor motion equations in this form are a function of the rotor degrees of freedom in the global reference frame and of the imposed boundary movement. In this case the boundary movement is represented by the motion of the spring-damper anchor point on the frame (points 푂′, A′ and 퐵′ on the picture). If we now consider the 2D model of the system that is used as an example, the motion equations for the rotor are in the following form:
푔푙표푏푎푙 (푚표푡𝑖표푛 푒푞. )푟표푡표푟 = 푓(푥퐺, 푧퐺, 훽퐺, 푥퐴′, 푥퐵′, 푧푂′) (15)
2D model: 6 equations and 6 degrees of freedom. 3D model: 11 equations end 11 degrees of freedom.
It is once more reminded that the 3-dimensional model is the one which was object of the motion equation investigation, while the 2- dimensional one is introduced only for the sake of making the explanation more understandable.
In the preceding expression, the rotor degrees of freedom in the global reference frame are indicated in blue. These are two displacements and one rotation. In red are represented the boundary imposed motions, each for one of the spring-damper systems. Of course, if we now consider the 3-dimensional model, the number of degrees of freedom for the rotor in the global reference would be six (three displacements and three rotations). Moreover, we would also have five imposed boundary movements (푥 and 푦 displacements for points 퐴′ and 퐵′ and 푧 displacement for point 푂′).
2.2.4 Preparing the rotor motion equations for coupling The motion equations for the rotor in the current form are correct and sufficient for the study of the isolated rotor dynamics, and the same can be said for the frame equations. However, since the final goal is to define the motion equation for the whole system, coupling of these two sets of equations must be performed. To prepare these sets for coupling, the operations summarized in the previously introduced steps – step 1 and step 2 – should be done. In this subsection, a detailed description of these steps for the rotor motion equations is provided.
With step 1, the coordinates representing the rotor displacement and orientation in the global reference frame (indicated in red in expression 15) are substituted with a new set of coordinates expressing the rotor displacement and orientation, but this time defined in a reference system consistent with the one used to describe the chassis equations. As stated before, this last reference is the ground reference, and this is an inertial reference system. This variable change is performed into two different sub-steps that are now presented. Since the goal of this section is to provide an overall presentation to the motion equation development method, the details of the variable change procedure are overlooked. These are considered in the following section.
Flywheel energy storage 30
In the first sub-step a variable change between global coordinates and absolute ones is performed.1 Once this first sub-step is performed the motion equations for the rotor in the absolute reference are obtained. These are represented in equation 16.
푎푏푠표푙푢푡푒 (푚표푡𝑖표푛 푒푞. )푟표푡표푟 = 푓(푥퐴, 푧퐴, 훽퐴, 푥퐴′, 푥퐵′, 푧푂′) (16)
2D model: 6 equations and 6 degrees of freedom. 3D model: 11 equations end 11 degrees of freedom.
The second sub-step involves a further variable change. In this case the variables in the absolute reference are substituted with those in the ground one. This new variable change differs from the preceding in that the two reference systems involved are not fixed together. This means that some terms, keeping into account that the absolute reference system is dragged by the frame motion, are to be introduced. These are the frame – and so the absolute reference system – displacements and rotations evaluated with respect to the ground reference. These new coordinates are denoted in green and are the frame degrees of freedom in the ground reference.
As for the previews variable change the details are shown in the following chapter. The new rotor motion equations are the one represented in equation 17.
푔푟표푢푛푑 (푚표푡𝑖표푛 푒푞. )푟표푡표푟 = 푓(푥푇, 푧푇, 훽푇, 푥푡푒푙, 푧푡푒푙, 훽푡푒푙, 푥퐴′, 푥퐵′, 푧푂′) (17)
2D model: 6 equations and 9 degrees of freedom. 3D model: 11 equations end 17 degrees of freedom.
It is interesting to note that, with this new expression for the rotor motion equation, also the frame degrees of freedom appear. This underlines the physical connection between rotor and frame.
The variable change of step 1 is now completed, and the rotor motion equations are expressed in the ground reference frame. In accordance with Figure 10, that is used to remind which are the boundaries traced between the two subsystems, step 2 is performed. Whit this step, forces and displacements acting on these boundaries are considered.
The forces that are applied by the frame to the rotor are exerted on points 푂′, 퐴′ and 퐵′. These forces have already been considered in the Lagrange component while writing the rotor motion equation, however the virtual displacements associated to these forces must be expressed in accordance to the variables used to
describe the frame dynamics. In fact, it can be noted that the terms 푥퐴′, 푥퐵′ 푎푛푑 푧푂′ (represented in red in equation 17) can be expressed as a function of the frame degrees of freedom. However, it is of crucial importance to understand that the displacements of points A’, B’ and O’ that appears in the rotor motion
1 It is to be noted that both the absolute and the global reference frame are fixed with the chassis. This means that they are also fixed together. With this, the two references differ of a quantity that is constant with time. The reasons why the absolute reference is introduced, and this first sub-step is performed are now listed.
- With this it is easier to consider that the resting position for the rotor has a certain displacement and orientation on the frame. - Since more rotors are later considered on the same frame, it is possible to define different placements and different orientations for them.
Both of this reference systems are non-inertial. Analytical model of the rotor-frame system 31 equations is not a generic movement of those points. Instead only the components of this displacements in the correct directions should be considered. The directions are defined according to the rotor point of view, and so according to the global reference frame.
Summarizing it is possible to define a relation in the form of equation 18. Like for the preceding variable change all the details are shown in the following section.
푥퐴′ = 푓(푥푡푒푙, 푧푡푒푙, 훽푡푒푙) (18)
Now this substitution is applied to rotor motion equation in the ground reference frame.
푔푟표푢푛푑 (푚표푡𝑖표푛 푒푞. )푟표푡표푟 = 푓(푥푇, 푧푇, 훽푇, 푥푡푒푙, 푧푡푒푙, 훽푡푒푙) (19)
2D model: 6 equations and 6 degrees of freedom. 3D model: 11 equations end 11 degrees of freedom.
2.2.5 Analysis of the frame motion equations As previously stated the frame equations are written in the ground reference. These have the following form:
푔푟표푢푛푑 ( 푚표푡𝑖표푛 푒푞. )푓푟푎푖푚 = 푓(푥푡푒푙, 푦푡푒푙, 훽푡푒푙, 푥푎푛푡, 푥푝표푠푡) (20)
2D model: 4 equations and 4 degrees of freedom. 3D model: 7 equations end 7 degrees of freedom.
The terms in orange represent the constraint imposed motion. These are the degrees of freedom of the points where the spring-dampers subsystems, representing the frame support system, are linked to the ground. In the final model these are to be the inputs that the ground provides to the system.
2.2.6 Preparing the frame motion equations for coupling In this subsection, the procedure related to the implementation of steps 1 and 2 are presented for the frame motion equations.
It can be noted that the frame motion equations are written in the ground reference system. The objectives of step 1 are therefore already achieved. For what concerns the first step, no changes are performed on the equations. When considering step 2, however, it can be noted that some forces are exchanged between rotor and frame. More in detail these forces are exerted on points 푂′, 퐴′ and 퐵′. These forces should appear into the Lagrange component of the frame motion equations. The virtual displacements that regard these forces must be expressed as a function of the frame degrees of freedom. No further variable is introduced.
2.2.7 Motion equations coupling Let’s consider what was achieved;
- Both the rotor and the frame motion equations are written within the same reference. - The used reference system is inertial. - All the effect mutually exchanged between the two systems have been considered.
It is now possible to couple the two set of equation to obtain the dynamics of the whole system.
Flywheel energy storage 32
푔푟표푢푛푑 (푚표푡𝑖표푛 푒푞. )푟표푡표푟 = 푓(푥푇, 푧푇, 훽푇, 푥푡푒푙, 푦푡푒푙, 훽푡푒푙) { 푔푟표푢푛푑 (21) (푚표푡𝑖표푛 푒푞. )푓푟푎푚푒 = 푓(푥푡푒푙, 푦푡푒푙, 훽푡푒푙, 푥푎푛푡, 푥푝표푠푡)
2.3 Rotor motion equation Development in the global reference system
There are many different options to obtain the rotor motion equation. It has been decided to use the Lagrange equation in the form reported in equation 22.
푑 휕푇 휕푇 휕퐷 휕푉 − + + = 푄푖 (22) 푑푡 휕푥̇푖 휕푥푖 휕푥̇푖 휕푥 Where:
- 푇 stands for the system kinetic energy. - 퐷 stands for the system dissipative term. - 푉 stands for the system potential energy. - 푄 is the Lagrange component of external forces.
In the following subsections all these terms are analyzed and computed.
2.3.1 Rotor subsystem overlook The rotor subsystem is composed of one rigid body – the rotor – and the spring-damper subsystems representing the rotor support system.
Figure 13: the rotor subsystem
The rotor itself is characterized by six degrees of freedom. This are three displacements and three rotations. As previously stated we consider these degrees of freedom in the global reference during the motion equation development phase. The boundary displacement provides other five degrees of freedom. Table 5 summarizes the subsystem degrees of freedom.
Degree of freedom Symbol 푂 푂 푂 Rotor displacement 푥퐺 , 푦퐺 , 푧퐺
Rotor rotation 휎 , 훽 , 휌
Displacement of the boundary in O’ 푧푂 Analytical model of the rotor-frame system 33
Displacement of the boundary in 푨’ 푥퐴, 푦퐴
Displacement of the boundary in 푩’ 푥퐵, 푦퐵 Total number of degrees of freedom 11 Table 5: the rotor subsystem degrees of freedom
These eleven degrees of freedom are arranged into a vector. This is referred to as the degrees of freedom vector for the rotor in the global reference frame. The same order for the degrees of freedom has been adopted in the writing of the Lagrange equation. It follows, in equation 23, the degrees of freedom vector.
푂 푥퐺 푂 {푦퐺 } 푂 푧퐺 푂 휎 푋̅퐺 푋̅퐺 = {훽 } = { 휗̅ } (23) 휌 푋̅푏 푥퐴 푦 { 퐴} 푥퐵 { 푦퐵 } In the subsystem, the rotor is the only element with no neglectable mass. The gravitational effects on the potential term appearing in the Lagrange equations are ignored, given their minor influence on the system dynamics. The five spring-damper subsystems are responsible for the whole subsystem stiffness and dumping. Finally, five external forces act on the subsystem according to the boundary displacements.
2.3.2 Lagrange equation components In the following subsections the components appearing in the Lagrange equation are evaluated.
2.3.2.1 Rotor kinetic energy T In this subsection, the analytical expression for the rotor subsystem kinetic energy is obtained. This expression must be derived as a function of the system independent coordinates. The kinetic energy expression is later introduced in the Lagrange equation, allowing to obtain the inertial term in the body motion equations.
The formula for the kinetical energy of a rigid body moving in a tridimensional space is reported in equation (24).
1 1 푇 = (푣푐.표.푚.)푇[푚]푣퐺 + (휔 )푇[퐽 ]휔 2⏟ 퐺 퐺 2⏟ 퐿 퐿 퐿 (24) 푡푟푎푛푠푙푎푡푖표푛푎푙 푟표푡푎푡푖표푛푎푙 Where:
- Matrix [푚] is a diagonal matrix where every diagonal element is equal to the rotor mass 푚푟표푡.
- Matrix [퐽퐿] is the rotor tensor of inertia defined in the local reference frame and evaluated on the rotor center of mass.
For the translational component, the velocity of the center of mass (c.o.m.) defined in the global reference is used. For the rotational component, the angular speed must be defined in the local reference frame. Only in this way, in fact, it is possible to use the rotor tensor of inertia evaluated in the local reference, which elements are constant with time, and depend only on the rotor mass distribution.
Flywheel energy storage 34
Expression (24) can be rewritten as follows.
1 푣푐.표.푚. 푇 [푚] [0] 푣푐.표.푚. 푇 = { 퐺 } [ ] { 퐺 } (25) 2 휔퐿 [0] [퐽퐿] 휔퐿
Since the origin of the local reference frame 푂 does not coincide with the rotor center of mass, the following relation can be considered.
푐.표.푚. 푂 푂−푐.표.푚. 푂 푂−푐.표.푚. ̇ 푣퐺 = 푣퐺 + [ΛGL] [푥̃퐿 ] 휔퐿 = 푥̇퐺 + [ΛGL] [푥̃퐿 ] [퐴퐿]{휗} (26)
Now, the relation between the kinematic quantities and the rotor independent coordinates can be written.
푂−푐.표.푚. 푣푐.표.푚. [퐼] [Λ ] [푥̃ ] [퐴 ] 푋̅푂 { 퐺 } = [ GL 퐿 퐿 ] { 퐺 } (27) 휔퐿 ̅ [0] [퐴퐿] 휗 Expression (27) is now substituted into equation (25). By rearranging, the following formulation for the rotor kinetic energy is obtained.
푂−푐.표.푚. 푇 [ ] [ ] [ ] 1 ̅푂 푚 [푚] ΛGL [푥̃퐿 ] 퐴퐿 ̅푂 푋퐺 푋퐺 푇 = { } [ 푇 푇 ] { } (28) 2 휗̅ 푂−푐.표.푚. 푇 푂−푐.표.푚. 푂−푐.표.푚. 휗̅ ([푚][ΛGL] [푥̃퐿 ] [퐴퐿]) [퐴퐿] ([푥̃퐿 ] [푚] [푥̃퐿 ] + [퐽퐿]) [퐴퐿]
It is reminded that the degrees of freedom vector for the rotor subsystem does not include only the rotor independent coordinates. In fact, also the boundary displacements should be considered. However, these have no effects on the subsystem kinetic energy. For this reason, a null term is placed in correspondence with them in the system mass matrix.
푂−푐.표.푚. 푇 [ ] [ ] [ ] ̅푂 푚 푚] ΛGL [푥̃퐿 ] 퐴퐿 [0] ̅푂 푋퐺 푋퐺 1 푇 푇 푇 = { 휗̅ } 푂−푐.표.푚. 푇 푂−푐.표.푚. 푂−푐.표.푚. { 휗̅ } (29) 2 ([푚][ΛGL] [푥̃퐿 ] [퐴퐿]) [퐴퐿] ([푥̃퐿 ] [푚] [푥̃퐿 ] + [퐽퐿]) [퐴퐿] [0] 푋̅푏 푋̅푏 [ [0] [0] [0]]
Finally, it can be noted that the term appearing in the 3 × 3 central block of the mass matrix can be interpreted as the tensor of inertia of the body evaluated with respect to the local reference system origin 푂.
푇 푐푒푛푡푒푟푑 푖푛 푂 푂−푐.표.푚. 푂−푐.표.푚. (30) [퐽퐿 ] = [푥̃퐿 ] [푚] [푥̃퐿 ] + [퐽퐿]
Figure 14: rotor kinematic analysis
2.3.2.2 Rotor potential energy V The subsystem potential energy term is related to the five spring elements supporting the rotor. The spring position is evaluated by defining the kinematic linkages. These can be obtained using the graphical representation of Figure 14. Analytical model of the rotor-frame system 35
From simple trigonometrical observation the following relations can be defined.
푂 ∆푙퐴,푥 = 푥퐺 − 푑1 sin 훽 − 푥퐴
푂 ∆푙퐴,푦 = 푦퐺 + 푑1 sin 휎 cos 훽 − 푦퐴
푂 ∆푙퐵,푥 = 푥퐺 + 푑2 sin 훽 − 푥퐵 (31)
푂 ∆푙퐵,푦 = 푦퐺 − 푑2 sin 휎 cos 훽 − 푦퐵
푂 ∆푙푂,푧 = 푧퐺 − 푧푂
Where 푑1 and 푑2 are defined as follows.
푙2 푑 = 푙 + 1 1 2 (32) 푙2 푑 = + 푙 2 2 3 Now the mathematical relations between spring length and degrees of freedom are known. Using these relations, the potential energy is written according to equation (33).
1 2 1 2 1 2 1 2 1 2 푉 = 푘 (∆푙 ) + 푘 (∆푙 ) + 푘 (∆푙 ) + 푘 (∆푙 ) + 푘 (∆푙 ) (33) 2 퐴,푥 퐴,푥 2 퐴,푦 퐴,푦 2 퐵,푥 퐵,푥 2 퐵,푦 퐵,푦 2 푂,푧 푂,푧
2.3.2.3 Rotor dissipative energy D The elements responsible for the energy dissipation in the system are the five dampers. It can be noted that these elements are placed in parallel to the springs. Therefore, the damper motion speed can be calculated deriving equations (31) with respect to time. Therefore, the following mathematical relations between dampers speed and degrees of freedom are obtained.
̇ 푂 ∆푙퐴,푥 = 푥̇퐺 − 휔퐺,2 푑1 cos 훽 − 푥퐴̇ ̇ 푂 ∆푙퐴,푦 = 푦̇퐺 + 푑1 (−휔퐺,2 sin 훽 sin 휎 + 휔퐺,1 cos 훽 cos 휎) − 푦퐴̇
̇ 푂 ∆푙퐵,푥 = 푥̇퐺 + 휔퐺,2 푑2 cos 훽 − 푥̇퐵 (34) ̇ 푂 ∆푙퐵,푦 = 푦̇퐺 − 푑2 (−휔퐺,2 sin 훽 sin 휎 + 휔퐺,1 cos 훽 cos 휎) − 푦̇퐵
̇ 푂 ∆푙푂,푧 = 푧퐺̇ − 푧푂̇
퐺 Where 휔퐺,1 and 휔퐺,2 are defined as in equation (35), where the matrix [퐴 ] is obtained according to equation (12).
휔퐺,1 휎̇ 퐺 {휔퐺,2} = [퐴 ] {훽̇} (35) 휔퐺,3 휌̇
Now, by substituting equations (34) into equation (36), the dissipation term in the Lagrange equation can be written.
Flywheel energy storage 36
1 2 1 2 1 2 1 2 1 2 퐷 = 푟 (∆푙̇ ) + 푟 (∆푙̇ ) + 푟 (∆푙̇ ) + 푟 (∆푙̇ ) + 푟 (∆푙̇ ) (36) 2 퐴,푥 퐴,푥 2 퐴,푦 퐴,푦 2 퐵,푥 퐵,푥 2 퐵,푦 퐵,푦 2 푂,푧 푂,푧
2.3.2.4 Lagrange component As previously stated, the forces acting on the subsystems are the one exchanged between the rotor bearing and the frame. Since these five forces act on five degrees of freedom, it is easy to define the Lagrange component vector.
푇 푄̅ = {0 0 0 0 0 0 퐹퐴,푥 퐹퐴,푦 퐹퐵,푥 퐹퐵,푦} (37)
This completes what is required for the subsystem motion equation.
2.3.3 Solving the rotor motion equation By substituting the components introduced in subsection 2.3.2 into the Lagrange equation, the rotor motion equations in the global reference are computed. These are a set of eleven second-order differential equation in eleven time-variating variables.
These equations can be used to simulate the dynamic behavior of the isolated rotor. To do so, these must be rearranged. First, it must be noted that the first six equations are related to the rotor dynamics while the last five equations refer to the force equilibrium on the boundaries. Now, the first six equations must be rearranged in order to extract the second-order derivative of the rotor degrees of freedom. Moreover, the last five equations can be rearranged to obtain an expression for the force acting on the boundary of the system.2
2.3.4 Simulations and results In this subsection two evaluations are performed on the rotor model. With the first one the system poles are calculated end plotted in a real-imaginary plane. With the second one some simulations of the rotor dynamics are performed. Both the pole plotting, and the dynamics simulation are computed using a set of first attempt data for the system physical properties. Reason of these evaluations is to give a rough estimation of the correctness of the model.
Figure 15: the rotor subsystem poles
2 To perform these rearrangements in the motion equations the Matlab command ‘solve’ is used. Analytical model of the rotor-frame system 37
The rotor subsystem poles are represented in Figure 15. From this representation some observations can be made:
- There are no poles with positive real part. This means that the model is not unstable. - There is one pole with real part equal to zero. This means that the system presents lability.
These two characteristics are consistent with what expected for the rotor subsystem. The lability is related to the possibility for the rotor to rotate without friction or other stopping forces.
The dynamic simulation of the model is performed via a Simulink model. A representation of this is reported in Figure 16. The simulation is based on the time evolution of a state vector. This is composed of the eleven degrees of freedom and their first-order derivative, as shown in equation (38).
푂 푂 푂 푂 푂 푂 ̇ 푋̅푠푡푎푡푒 = {푥̇퐺 푥퐺 푦̇퐺 푦퐺 푧퐺̇ 푧퐺 휎̇ 휎 훽 훽 휌̇ 휌} (38)
In the scheme, the two blocks ‘Rotor dynamics’ and ‘Force evaluation’ take as input the state vector and the vector of the boundary displacements, and provides as output the time derivative of the state vector. This last is referred to the following simulation step. These two blocks are defined as a Simulink user-defined functions.
The six expressions that were obtained in subsection 2.3.3, solving the first six motion equations, are introduced in the ‘Rotor dynamics’ block. In the ‘Force evaluation’ block, instead, the five expressions that were obtained solving the last five motion equations are introduced.
A set of five simulations are performed to assess the correctness of the model. These are defined in such a way that the results can be predicted and evaluated. Whit such a procedure, conclusions on the correctness of the model can be drawn.
Figure 16: Simulink model for the rotor subsystem simulation
In particular, the five simulations evaluate the response of the subsystem to an initial displacement of the rotor off its equilibrium point. For these simulations a rotor with null eccentricity is considered, together with an initial rotational speed of 6000 푟푝푚. The off equilibrium initial displacements for the rotor are presented in table 6.
Flywheel energy storage 38
Simulation 풙푮 [m] 풚푮 [m] 풛푮 [m] 𝝈푮 [rad] 휷푮 [rad] 𝝆푮 [rad] 1 0,001 0 0 0 0 0 2 0 0,001 0 0 0 0 3 0 0 0,001 0 0 0 4 0 0 0 0,01 0 0 5 0 0 0 0 0,01 0 Table 6: initial configurations for the rotor subsystem simulations
For these five cases, the expected results are those of a damped second order system. This system must be characterized by a mass, a stiffness and a damping that can be easily evaluated on the base of the data introduced for the rotor subsystem. More details on these evaluations are presented in appendix A. The results obtained are in good accordance with the expectations. The correctness of the rotor subsystem motion equations is therefore assessed.
2.4 Preparing the rotor motion equation for coupling In this section the procedure introduced in the subsection 2.2.4 is detailed. It is to be reminded that the procedure involves two distinct tasks. The first task deals with changing the reference frame for the motion equation: from the global reference to the absolute one, first, and from the absolute reference to the ground one, later. The second task aims at defining and substituting into the motion equation an expression for the boundary displacements. This expression should relate the boundary displacements to the frame degrees of freedom.
2.4.1 Variable change procedure In this subsection the variable change procedure is detailed. This procedure was developed into a Matlab function and applied for both the global-to-absolute and the absolute-to-ground variable changes. Before getting into details the rotor degrees of freedom in the three relevant reference systems are listed in table 7.
Reference frame Degrees of freedom 푂 푂 푂 Global reference frame 푥퐺 푦퐺 푧퐺 휎 훽 휌 푂 푂 푂 Absolute reference frame 푥퐴 푦퐴 푧퐴 휎퐴 훽퐴 휌퐴
Ground reference frame 푥푇 푦푇 푧푇 휎푇 훽푇 휌푇 Table 7: the rotor degrees of freedom
Moreover, the position and orientation of the global reference with respect to the absolute one and the position and orientation of the absolute reference with respect to the ground one are listed in the following table. These also represent how the rotor was placed on the frame and how the frame is located into the ground reference.
Item Positioning
Global reference with respect to absolute reference 푥푟표푡 푦푟표푡 푧푟표푡 휎푟표푡 훽푟표푡 휌푟표푡
Absolute reference with respect to ground reference 푥푡푒푙 푦푡푒푙 푧푡푒푙 휎푡푒푙 훽푡푒푙 휌푡푒푙 Table 8: reciprocal displacement of reference systems
These six coordinates vectors are defined according to the convention introduced in section 2.3.1 for identifying the position and orientation of a rigid body in space. Analytical model of the rotor-frame system 39
As previously stated, the absolute reference is fixed with the global one, and so the coordinates identified with subscript 푟표푡 are constant with time. On the other hand, the absolute reference is not fixed to the ground one, and so the coordinates with subscript 푡푒푙 are variables.
The task that the Matlab function must perform is to express the rotor degrees of freedom in the global reference as a function of the rotor degrees of freedom in the absolute one, and express the rotor degrees of freedom in the absolute reference as a function of the rotor degrees of freedom in the ground one. These two tasks are performed into two different stages each. In the first stage a relation between the displacement quantities is found. In the second stage the relation between the orientation angles is investigated. The procedure is presented for the global-to-absolute variable change, but it does apply also to the absolute-to- local variable change.
2.4.1.1 Displacement variable change
Figure 17: displacement, from absolute to global reference
To perform this variable change some initial observations must be done. It can be noted from Figure 17 that the vector identifying a generic position of point O in the absolute reference is composed by the sum of two different vectors. These are the vector finding the global reference origin and the vector that locates point O starting from the global reference origin. These two vectors are defined as follows.
퐺 푥푟표푡 푥푂 퐺 {푦푟표푡} {푦푂 } (39) 푧푟표푡 퐺 푧푂 However, the first vector is defined in the absolute reference while the second one in the global reference. Before summing them they must be referred to the same reference. This is done by using the transformation matrix previously introduced. By doing this we gain the mathematical relation between displacement degrees of freedom in the absolute and in the global reference.
퐴 퐺 푥푂 푥푂 푥푟표푡 퐴 퐺 {푦푂 } = [Λ퐴퐺] {푦푂 } + {푦푟표푡} (40) 퐴 퐺 푧푟표푡 푧푂 푧푂
퐺 퐴 퐴 푥푂 푥푂 푥푟표푡 푥푂 푥푟표푡 퐺 −1 퐴 퐴 {푦푂 } = [Λ퐴퐺] ({푦푂 } − {푦푟표푡}) = [Λ퐺퐴] ({푦푂 } − {푦푟표푡}) (41) 퐺 퐴 푧푟표푡 퐴 푧푟표푡 푧푂 푧푂 푧푂
Flywheel energy storage 40
Note that for the transformation from global to absolute reference the 푟표푡-vector, representing the position of the global reference with respect to the absolute reference, is constant. The 푟표푡-vector is substituted with the 푡푒푙-vector in case the transformation from the absolute to the ground reference is considered. In this second case, this vector is not constant.
2.4.1.2 Orientation variable change The orientation variable change is less intuitive. The procedure that now is introduced is one out of the many developed for solving this task. This is the most reliable and the easiest to understand. The crucial idea of this procedure is to convert the cardan angles into quaternions to make the variable change easier. Finally, the quaternions are transformed back to cardan angles in the new reference system.
To make the explanation more understandable the representation in Figure 18 is presented. In this representation with the vector 푅 is illustrated the rotor orientation. The vector 푅′ represents the neutral orientation for the rotor in the global reference while the vector 푅′′ represents the neutral orientation in the absolute reference.
Figure 18: orientation variable change
By looking at the picture the following observations can be made:
- The orientation of the rotor in the global reference system is defined with the three cardan angles
that represent the transformation from 푅′ to 푅. This transformation is specified as 푇푅−푅′. - The orientation of the rotor in the absolute reference system is define with the three cardan angles
that represent the transformation from 푅′′ to 푅. This transformation is specified as 푇푅−푅′′. - The orientation of the global reference in the absolute frame is define with the three cardan angles
that represent the transformation from 푅′′ to 푅′. This transformation is specified as 푇푅′−푅′′.
- Finally, the transformation 푇푅−푅′′ can be expressed as the sum of transformation 푇푅−푅′ and a
transformation 푇푅′−푅′′.
푇푅−푅′′ = 푇푅−푅′ + 푇푅′−푅′′ (42)
It follows that:
푇 ′ = 푇 ′′ − 푇 ′ ′′ 푅−푅 ⏟푅 −푅 ⏟ 푅 − 푅 (43) 푒푥푡 푖푛푡
To make the notation easier to understand the two transformations on the right-hand side of the previous expression are indicated as external (푒푥푡) and internal (𝑖푛푡). Analytical model of the rotor-frame system 41
This means that the transformation that identifies the rotor orientation in the global reference can be obtained as the difference between the transformation that identifies the rotor orientation in the absolute frame and the transformation that identifies the global reference orientation in the absolute frame. These observations are developed into an algorithm first, and into a Matlab function later, according to the following eight steps.
1. The cardan angles representing the 푇푅−푅′′ transformation are converted into quaternions. This
operation is performed as expressed in equation (44). Since the cardan angles of the 푇푅−푅′′ transformation where defined in the absolute reference, also the quaternion now obtained is defined in the absolute reference.
1 1 1 1 1 1 −푠𝑖푛 ( 휎 ) 푠𝑖푛 ( 훽 ) 푠𝑖푛 ( 휌 ) + 푐표푠 ( 휎 ) 푐표푠 ( 훽 ) 푐표푠 ( 휌 ) 2 퐴 2 퐴 2 퐴 2 퐴 2 퐴 2 퐴 ext 푞1 1 1 1 1 1 1 A ext +sin ( σA) cos ( βA) cos ( ρA) + sin ( βA) sin ( ρA) cos ( σA) 푞2A 2 2 2 2 2 2 ext = (44) 푞3A 1 1 1 1 1 1 ext −sin ( σA) sin ( ρA) cos ( βA) + sin ( βA) cos ( σA) cos ( ρA) {푞4A } 2 2 2 2 2 2 1 1 1 1 1 1 +sin ( σ ) sin ( β ) cos ( ρ ) + sin ( ρ ) cos ( σ ) cos ( β ) { 2 A 2 A 2 A 2 A 2 A 2 A }
2. The cardan angles representing the 푇푅′−푅′′ transformation are converted into quaternions. This
operation is performed as follows. Since also the cardan angles of the 푇푅′−푅′′ transformation where defined in the absolute reference, also the quaternion now obtained is defined in the absolute reference.
1 1 1 1 1 1 −푠𝑖푛 ( 휎 ) 푠𝑖푛 ( 훽 ) 푠𝑖푛 ( 휌 ) + 푐표푠 ( 휎 ) 푐표푠 ( 훽 ) 푐표푠 ( 휌 ) 2 푟표푡 2 푟표푡 2 푟표푡 2 푟표푡 2 푟표푡 2 푟표푡 int 푞1A 1 1 1 1 1 1
int +sin ( σrot) cos ( β푟표푡) cos ( ρrot) + sin ( βrot) sin ( ρ푟표푡) cos ( σ푟표푡) 푞2A 2 2 2 2 2 2 = (45) 푞3int 1 1 1 1 1 1 A −sin ( σ ) sin ( ρ ) cos ( β ) + sin ( β ) cos ( σ ) cos ( ρ ) int 2 rot 2 푟표푡 2 푟표푡 2 rot 2 푟표푡 2 rot {푞4A } 1 1 1 1 1 1 +sin ( σ ) sin ( β ) cos ( ρ ) + sin ( ρ ) cos ( σ ) cos ( β ) { 2 푟표푡 2 푟표푡 2 rot 2 rot 2 rot 2 rot }
3. Now that the transformations 푇푅−푅′′ and 푇푅′−푅′′ have been defined with their quaternion representation, these can be subtracted according to the expression in equation (43). The result is expressed in the absolute reference. The algebraic sum between quaternion is performed as reported in the following equation.
푡표푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푞1퐴 푞1퐴 푞1퐴 − 푞2퐴 푞2퐴 − 푞3퐴 푞3퐴 − 푞4퐴 푞4퐴 푡표푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푞2퐴 푞1퐴 푞2퐴 + 푞2퐴 푞1퐴 + 푞3퐴 푞4퐴 − 푞4퐴 푞3퐴 푡표푡 = 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 (46) 푞3퐴 푞1퐴 푞3퐴 + 푞3퐴 푞1퐴 + 푞4퐴 푞2퐴 − 푞2퐴 푞4퐴 푡표푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 푖푛푡 푒푥푡 {푞4퐴 } {푞1퐴 푞4퐴 + 푞4퐴 푞1퐴 + 푞2퐴 푞3퐴 − 푞3퐴 푞2퐴 }
4. The quaternion that was obtained in step 3 represents the transformation 푇푅−푅′ . This is the transformation connected to the rotor orientation in the global reference. However, the transformation as obtained in step 3 is defined in the absolute reference. To adapt this to the global reference the vector defining this transformation is introduced. To understand the transformation vector physical meaning it must be reminded that the transformation is a rigid rotation. The vector has a direction lying on the rotation axis and a modulus equal to the rotation magnitude. Once the transformation vector is defined, this can easily be obtained also in a new reference system. The transformation vector in the absolute reference is calculated as follows.
Flywheel energy storage 42
Modulus:
푡표푡 푡표푡 휔퐴 = 2 acos(푞1퐴 ) (47)
Direction vector:
ωtot q2tot⁄sin ( A ) A 2 푡표푡 푎푥퐴 tot 푡표푡 tot ωA {푎푦퐴 } = q3A ⁄sin ( ) (48) 2 푎푧푡표푡 퐴 ωtot q4tot⁄sin ( A ) { A 2 }
5. The transformation vector is now converted from the absolute reference to the global one. This task is performed using the transformation matrix already introduced.
푡표푡 푡표푡 푎푥퐺 푎푥퐴 푡표푡 푡표푡 {푎푦퐺 } = [Λ퐺퐴] {푎푦퐴 } (49) 푡표푡 푡표푡 푎푧퐺 푎푧퐴
6. From this vector the quaternion for the same transformation in the global reference frame is obtained.
휔푡표푡 푐표푠 ( 퐺 ) 2 푞1푡표푡 휔푡표푡 퐺 푎푥푡표푡 sin ( 퐺 ) 푡표푡 퐺 푞2퐺 2 푡표푡 = 푡표푡 (50) 푞3퐺 푡표푡 휔퐺 푡표푡 푎푦퐺 sin ( ) {푞4퐺 } 2
휔푡표푡 푎푧푡표푡 sin ( 퐺 ) { 퐺 2 }
7. To go back to the cardan angles representation of this transformation, first the transformation matrix must be calculated. This is done according to the following formulation.
푡표푡 2 푡표푡 2 푡표푡 2 푡표푡 2 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 (푞1퐺 ) + (푞2퐺 ) − (푞3퐺 ) − (푞4퐺 ) 2(푞2퐺 푞3퐺 − 푞1퐺 푞4퐺 ) 2(푞2퐺 푞4퐺 + 푞1퐺 푞3퐺 ) 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 2 푡표푡 2 푡표푡 2 푡표푡 2 푡표푡 푡표푡 푡표푡 푡표푡 [푀] = [ 2(푞2퐺 푞3퐺 + 푞1퐺 푞4퐺 ) (푞1퐺 ) − (푞2퐺 ) + (푞3퐺 ) − (푞4퐺 ) 2(푞3퐺 푞4퐺 − 푞1퐺 푞2퐺 ) ] (51) 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 푡표푡 2 푡표푡 2 푡표푡 2 푡표푡 2 2(푞2퐺 푞4퐺 − 푞1퐺 푞3퐺 ) 2(푞3퐺 푞4퐺 + 푞1퐺 푞2퐺 ) (푞1퐺 ) − (푞2퐺 ) − (푞3퐺 ) + (푞4퐺 ) 8. From the transformation matrix, it is possible to define the rotor orientation, expressed as cardan angles in the global reference system. In this way a relation between cardan angles in the global reference and cardan angles in the absolute one is obtained.
푀(2,3) 휎 = 푎푡푎푛 (− ) ; 퐺 푀(3,3)
푀(1,3) 훽퐺 = 푎푡푎푛 ( 2 ) ; (52) 푠푞푟푡 (1 − (푀(1,3)) ) 푀(1,2) 휌 = 푎푡푎푛 (− ) ; 퐺 푀(1,1) Analytical model of the rotor-frame system 43
2.4.2 Rotor subsystem boundary displacements As it was previously introduced, the boundary displacement of the rotor subsystem can be related to the frame degrees of freedom. In this subsection the mathematical relation between these two quantities is investigated. As for the variable change, a procedure is developed and implemented into a Matlab function. It is reminded that the boundary displacements are not any generic movements of the points 푂′, 퐴′ and 퐵′. Instead they are only the displacements of these points in the specific directions defined in the global reference frame.
To make the procedure description easier to read, the following 2D scheme of the system is presented. Here the frame is represented with the three points linking it to the rotor.
Figure 19: imposed boundary displacements on the rotor
The procedure to define the mathematical relation between boundary displacements and frame degrees of freedom is explained in the following steps.
1. The three displacements for the three points are defined. These are evaluated as the difference between the final and initial positions of these points. The mathematical development for 퐴′ is hereafter illustrated. This same procedure applies also to points 푂′ and 퐵′. Position of point 퐴′ in the absolute reference:
퐴 푥퐴′ 0 푥푟표푡 퐴 푦 {푦퐴′} = [Λ퐴퐺] { 0 } + { 푟표푡} (53) 퐴 −푑1 푧푟표푡 푧퐴′ Final position of point 퐴′ in the ground reference:
푇 퐴 푥퐴′ 푥퐴′ 푥푡푒푙 푇 퐴 푦 {푦퐴′} = [Λ푇퐴] {푦퐴′ } + { 푡푒푙} (54) 푇 퐴 푧푡푒푙 푧퐴′ 푧퐴′ Initial position of point 퐴′ in the ground reference:
푇 퐴 퐴 푥퐴′ 푥퐴′ 푥푡푒푙 푥퐴′ 푇 퐴 푦 퐴 {푦퐴′} = [Λ푇퐴]|𝜎푇=0 {푦퐴′} + { 푡푒푙}| = {푦퐴′} (55) 푇 훽푇=0 퐴 푧푡푒푙 퐴 푧 ′ 푧 ′ 푡=0 푧 ′ 퐴 푡=0 𝜌푇=0 퐴 퐴
Displacement for point 퐴′ in the ground reference:
Flywheel energy storage 44
푇 퐴 퐴 푑𝑖푠푝 푥 ′ 푥 ′ 푥 ′ ( 퐴 ) 퐴 푥푡푒푙 퐴 푇 퐴 푦 퐴 {푑𝑖푠푝 (푦퐴′)} = [Λ푇퐴] {푦퐴′} + { 푡푒푙} − {푦퐴′} (56) 푇 퐴 푧푡푒푙 퐴 푑𝑖푠푝 (푧퐴′) 푧퐴′ 푧퐴′ 2. These displacements are brake down into the three components along the global reference. This is performed with the following mathematical procedure. As for the first step, the procedure is reported for point 퐴′, but this can also be applied to points 푂′ and 퐵′.
푇 푑𝑖푠푝 (푥퐴′) 푑𝑖푠푝 (푥퐴′) 푇 {푑𝑖푠푝 (푦퐴′)} = [Λ퐺퐴][Λ퐴푇] {푑𝑖푠푝 (푦퐴′)} (57) 푇 푑𝑖푠푝 (푧퐴′) 푑𝑖푠푝 (푧퐴′) 3. The six desired displacements are selected between the ones calculated. These are the 푥 and 푦 displacements for points 퐴′ and 퐵′ and 푧 for point 푂′.
2.5 Frame motion equation Development in the ground reference
There are many different options to obtain the frame motion equation. It has been decided to use the Lagrange equation in the following form.
푑 휕푇 휕푇 휕퐷 휕푉 − + + = 푄푖 (58) 푑푡 휕푥̇푖 휕푥푖 휕푥̇푖 휕푥 Where
- 푇 stands for the system kinetic energy. - 퐷 stands for the system dissipative term. - 푉 stands for the system potential energy. - 푄 is the Lagrange component of external forces.
The motion equations for the frame are directly obtained in the ground reference frame. Other for this the overall procedure is similar to the one already described for the rotor. In the following subsection all these terms are analyzed and computed. Fewer details are provided given the similarities between this method and the one already described for the rotor.
2.5.1 Frame subsystem overlook The frame subsystem is composed of one rigid body – the frame itself – and the frame support system. As previously introduced three different alternatives for the model of the support system are considered. For this description of the development of the analytical model the first and simplest model is studied. Only slight changes must be introduced to the method if one of the other two models is analyzed.
Seven degrees of freedom characterize the first alternative of the frame subsystem. Three degrees of freedom are associated to the frame 3 . This can move vertically and rotate along the longitudinal and
3 It is to be noted that, even if in the first model the frame degrees of freedom are only three, no boundaries are put on this while creating the analytical model. Doing this a system with a six degrees of freedom frame is obtained. The equations related to the
Analytical model of the rotor-frame system 45 transversal axis. The boundary displacement provides other four degrees of freedom to the frame subsystem. In this case, the boundary displacement represents the effects of the road roughness on the vehicle. The subsystem degrees of freedom are listed in the following table.
Degree of freedom Symbol
Frame displacement 푧푡푒푙
Frame rotation 훽푟표푡, 휌푟표푡 Displacement of the boundaries 푥푎,푠푥, 푥푎,푑푥, 푥푝,푠푥, 푥푝,푑푥 Total number of degrees of freedom 7 Table 9: the frame subsystem degrees of freedom
The seven degrees of freedom, plus the three that are later ignored, are arranged into a vector. This is referred to as the degrees of freedom vector for the frame in the ground reference system. It is introduced now to underline in which order the ten degrees of freedom of the system are considered in the Lagrange equation.
̅ 푥 푦 푧 푇 푋푡푒푙 = { 푡푒푙 푡푒푙 푡푒푙 휎푡푒푙 훽푡푒푙 휌푡푒푙 } (59)
Figure 20: the frame subsystem
2.5.2 Lagrange equation components The terms appearing in the Lagrange equation are now computed.
2.5.2.1 Frame kinetic energy T The only element with non-neglectable mass in this subsystem is the frame. The mass matrix appearing in the Lagrange equation is computed with the same procedure already described for the rotor. In this case the eccentricity of the frame is linked to the fact that it was chosen to put the absolute reference frame origin on the bottom of the chassis.
푂−푐.표.푚. [푚푡푒푙] [푚푡푒푙][Λ푇퐴][푥̃퐴 ][퐴퐴] [푀] = [ 푇 푇 ] (60) 푂−푐.표.푚. 푇 푂−푐.표.푚. 푂−푐.표.푚. ([푚푡푒푙][ΛTA][푥̃퐴 ][퐴퐴]) [퐴퐴] ([푥̃퐴 ] [푚푡푒푙][푥̃퐴 ] + [퐽퐴]) [퐴퐴]
undesired degrees of freedom are simply ignored to gain consistency with the frame subsystem first model. This can be done because no cross-effects are produced by longitudinal and transversal displacements of the frame and by the rotation along the vertical axis.
Flywheel energy storage 46
2.5.2.2 Frame potential energy V Like for the rotor, the potential term connected to the gravitational field is neglected. In fact, the static spring preload counterbalances this. Only the four spring-damper subsystems, responsible of supporting the frame, account for the potential energy term. To evaluate these the new kinematic linkages are obtained. The following illustration helps with this task.
Figure 21: frame kinematics analysis
The kinematic linkages are now reported.
푙푦 푙푧 ∆푙 = 푥 + 푠𝑖푛(휌 )푐표푠(훽 ) + 푠𝑖푛(훽 ) − 푥 푎,푠푥 푡푒푙 2 푡푒푙 푡푒푙 2 푡푒푙 푎,푠푥
푙푦 푙푧 ∆푙 = 푥 − 푠𝑖푛(휌 )푐표푠(훽 ) + 푠𝑖푛(훽 ) − 푥 푎,푑푥 푡푒푙 2 푡푒푙 푡푒푙 2 푡푒푙 푎,푠푥 (61) 푙푦 푙푧 ∆푙 = 푥 + 푠𝑖푛(휌 )푐표푠(훽 ) − 푠𝑖푛(훽 ) − 푥 푝,푠푥 푡푒푙 2 푡푒푙 푡푒푙 2 푡푒푙 푎,푠푥
푙푦 푙푧 ∆푙 = 푥 − 푠𝑖푛(휌 )푐표푠(훽 ) − 푠𝑖푛(훽 ) − 푥 푝,푑푥 푡푒푙 2 푡푒푙 푡푒푙 2 푡푒푙 푎,푠푥 Now the mathematical relations between spring length and degrees of freedom are known. Substituting these relations into equation (62), the potential energy is written.
1 2 1 2 1 2 1 2 푉 = 푘 (∆푙 ) + 푘 (∆푙 ) + 푘 (∆푙 ) + 푘 (∆푙 ) (62) 2 푎,푠푥 푎,푠푥 2 푎,푑푥 푎,푑푥 2 푝,푠푥 푝,푠푥 2 푝,푑푥 푝,푑푥
2.5.2.3 Frame dissipative energy D The elements responsible for the energy dissipation in the system are the five dampers. It can be noted that the dampers are placed in parallel to the springs. Therefore, the damper motion speed can be calculated deriving the five expressions obtained for the springs position with respect to time. Therefore, the following mathematical relations between dampers speed and degrees of freedom are obtained.
푙푦 ∆푙̇ 푎,푠푥 = 푥̇푡푒푙 + (휔푡푒푙,3 푐표푠(휌푡푒푙) 푐표푠(훽푡푒푙) − 휔푡푒푙,2 푠𝑖푛(훽푡푒푙) 푠𝑖푛(휌푡푒푙)) 2 (63) 푙푧 + 휔 푐표푠(훽 ) − 푥̇ 2 푡푒푙,2 푡푒푙 푎,푠푥 Analytical model of the rotor-frame system 47
푙푦 ∆푙̇ = 푥̇ − (휔 푐표푠(휌 ) 푐표푠(훽 ) − 휔 푠𝑖푛(훽 ) 푠𝑖푛(휌 )) 푎,푑푥 푡푒푙 2 푡푒푙,3 푡푒푙 푡푒푙 푡푒푙,2 푡푒푙 푡푒푙 푙푧 + 휔 푐표푠(훽 ) − 푥̇ 2 푡푒푙,2 푡푒푙 푎,푠푥
푙푦 ∆푙̇ = 푥̇ + (휔 푐표푠(휌 ) 푐표푠(훽 ) − 휔 푠𝑖푛(훽 ) 푠𝑖푛(휌 )) 푝,푠푥 푡푒푙 2 푡푒푙,3 푡푒푙 푡푒푙 푡푒푙,2 푡푒푙 푡푒푙 푙푧 − 휔 푐표푠(훽 ) − 푥̇ 2 푡푒푙,2 푡푒푙 푎,푠푥
푙푦 ∆푙̇ = 푥̇ − (휔 푐표푠(휌 ) 푐표푠(훽 ) − 휔 푠𝑖푛(훽 ) 푠𝑖푛(휌 )) 푝,푑푥 푡푒푙 2 푡푒푙,3 푡푒푙 푡푒푙 푡푒푙,2 푡푒푙 푡푒푙 푙푧 − 휔 푐표푠(훽 ) − 푥̇ 2 푡푒푙,2 푡푒푙 푎,푠푥
Where 휔푡푒푙,2 and 휔푡푒푙,3 are defined as previously introduced.
휔푡푒푙,1 휎̇푡푒푙 휔 푇 { 푡푒푙,2} = [퐴 ] {훽푡푒푙̇ } (64) 휔 푡푒푙,3 휌̇푡푒푙 Now the dissipation term in the Lagrange equation can be written.
1 2 1 2 1 2 1 2 퐷 = 푟 (∆푙̇ ) + 푟 (∆푙̇ ) + 푟 (∆푙̇ ) + 푟 (∆푙̇ ) (65) 2 푎,푠푥 푎,푠푥 2 푎,푑푥 푎,푑푥 2 푝,푠푥 푝,푠푥 2 푝,푑푥 푝,푑푥
2.5.2.4 Lagrange component The Lagrange component in the frame motion equation can be split into two main elements. The first one is the one provided by the boundary imposed displacements. It is referred to this as the Lagrange component for the external forces. A second component is related to the effects of the forces exerted by the rotor onto the frame. This is the internal forces component. These two elements are developed separately.
The external forces component is obtained in a similar way to that it is done for the rotor Lagrange component. This is as follows4. ̅퐹푒푥푡 푇 푄푡푒푙 = {0 0 0 0 0 0 퐹푎,푠푥 퐹푎,푑푥 퐹푝,푠푥 퐹푝,푑푥} (66)
The internal forces component is developed considering the mathematical relation between rotor degrees of freedom and displacement of point 퐴′, 퐵′ and 푂′. This relation was already developed. The procedure consists in calculating the virtual work for the forces exchanged between frame and rotor first, and the Lagrange component later.
The virtual work of these forces is computed as follows, where the displacements are calculated in accordance with equation (57).
푉𝑖푟푡푢푎푙 푤표푟푘 = 퐹퐴,푥 푑𝑖푠푝 (푥퐴′) + 퐹퐴,푦 푑𝑖푠푝 (푦퐴′) + 퐹퐵,푥 푑𝑖푠푝 (푥퐵′) + 퐹퐵,푦 푑𝑖푠푝 (푦퐵′) (67) + 퐹푂,푧 푑𝑖푠푝 (푧푂′)
Finally, the Lagrange component of the internal forces is calculated.
4 Note how in this formulation six degrees of freedom for the rotor are considered. Reason of this is what was stated in section 2.5.2.
Flywheel energy storage 48
̅퐹푖푛푡 푄푡푒푙 = ∇(푉𝑖푟푡푢푎푙 푤표푟푘) (68) Where with ∇ is intended the Jacobean, with respect to the frame subsystem degrees of freedom vector.
2.5.3 Solving the frame motion equation By substituting the components introduced in the last subsections into the Lagrange equation, the motion equations for the frame in the ground reference are computed. These are a set of ten second-order differential equation in ten time-variating variables.
These equations can be used to simulate the dynamic behavior of the isolated frame. To do this they must be rearranged. First, it must be noted that the first six equations are related to the frame dynamics while the last four equations refer to the force equilibrium on the boundaries. To gain consistence with the frame model represented in Figure 20, among the first six equations, the three representing the dynamics of the constrained degrees of freedom are neglected. Now, the remaining equations must be rearranged in order to extract the second-order derivative of the rotor degrees of freedom. Moreover, the last four equations can be rearranged to obtain an expression for the force acting on the boundary of the system. To perform these rearrangements in the motion equations the Matlab command solve is used.
2.5.4 Simulations and results In this subsection a set of simulations on the isolated frame model dynamics are performed. These are done using the first attempt data previously introduced. Reason of this evaluations is to give a rough estimation of the correctness of the model. The dynamic simulation is performed via the Simulink model reported in Figure 22. As it was done for the rotor, the simulation is based on the time evolution of a state vector. This is composed of the eleven degrees of freedom and their first-order derivative as shown in equation (69).
̇ 푋̅푠푡푎푡푒 = {푥̇푡푒푙 푥푡푒푙 푦̇푡푒푙 푦푡푒푙 푧푡푒푙̇ 푧푡푒푙 휎̇푡푒푙 휎푡푒푙 훽푡푒푙 훽푡푒푙 휌̇푡푒푙 휌푡푒푙} (69)
The system dynamics and the force measurement blocks are designed, as it was done for the rotor Simulink model, by introducing the correct set of equations evaluated in section 2.5.3. Now the set of simulations, performed with the aim of evaluating the motion equations correctness, are defined with the same basic concepts that were used for assessing the rotor motion equations. These, in fact, are defined in such a way that the results can be predicted. The predicted results are introduced together with the actual results from the system simulation.
It is observed that the results are in good accordance with what was expected. More information on these evaluations are provided in appendix B.
2.6 Equations coupling In this chapter the motion equations for the two subsystems were achieved. A series of steps were performed to write both the two sets of equations in the same reference, and considering all the mutually exchanged effects. Now the two sets can be coupled without performing any further operation. The motion equation for the whole system are therefore achieved.
Also the two Simulink models for the two subsystems can be coupled to obtain a model capable of simulating the whole system dynamics. Instead of creating a whole new Simulink model, where in a single block all the equations are introduced, the two models are linked together. It is believed, in fact, that with this procedure Analytical model of the rotor-frame system 49
Figure 22: Simulink model for the frame motion equation evaluation
Figure 23: Simulink model for the rotor-frame motion equation evaluation
Flywheel energy storage 50
a more intuitive representation on how the two subsystems exchanges forces and displacements is provided. The model is represented in Figure 23.
Finally, with a similar procedure to the one here presented, the motion equations for the whole system, composed of rotor subsystem and the two other alternatives for the frame subsystem, are obtained.
2.7 Final observations and SimMechanics models To evaluate the correctness of the motion equations for the whole system, three SimMechanics models are developed, each for one of the different alternatives for the frame subsystem. For the sake of brevity, the procedure to obtain the SimMechanics models is overlooked. A representation of the three models block schemes is presented in figures from 24 to 27.
A set of simulations, with random initial conditions, are performed on both the analytical and the SimMechanics models. In both cases, results are expected to be equal. It is to be noted that the execution time and the amount of Ram used for the obtainment of the analytical model exceeded the available resources. For this reason, in order to compare the analytical results to the numerical ones, and in the attempt of solving these problems, it was decided to introduce numerically some quantities before the execution of the script. These parameters are the geometrical and physical characteristics of the system, including the rotor positioning and orientation. Furthermore, some additional simplifications must be introduced to achieve the motion equation for the whole system. It is, in fact, necessary to impose the rotor eccentricity equal to zero, and a two-step process is applied to assess the correctness of the equations for any given eccentricity. In a first phase, the analytical equation for the whole system characterized by a rotor of null eccentricity is compared with the numerical equation of the same system. In the second phase, the analytical equation for the isolated rotor with non-null eccentricity is evaluated.
Under this hypothesis good accordance was observed between the results of the analytical and numerical models. The long execution time and the heavy use of resources associated to the execution of the script developing the analytical motion equation do not come unexpected. The main reason for these problems, in fact, is related to the use of the cardan angles, especially when it comes to the variable changes associated to the two reference systems changes performed. Indeed, the summation of two sets of cardan angles is known to be a particularly challenging task from the analytical point of view. For this reason, in the reference changes, every degree of freedom representing the rotor orientation, and its derivatives, are substituted with a complicated expression, leading to the mentioned computational problems. These problematics are better analysed in section 2.4.1.2 and are known in literature as reported in [15].
Figure 24: SimMechanics graphical representation of the system Analytical model of the rotor-frame system 51
Figure 25: SimMechanics model – first alternative for the frame subsystem
Flywheel energy storage 52
Figure 26: SimMechanics model – second alternative for the frame subsystem Analytical model of the rotor-frame system 53
Figure 27: SimMechanics model – third alternative for the frame subsystem
3 Single rotor optimization
In this chapter the rotor geometry is defined through an optimization process. The research of the optimal geometry is performed on the model of a system composed of the frame and a single flywheel. The flywheel is positioned in the frame origin and different alternatives for its orientation are considered.
An important preliminary study is performed to address which of the many problem variables are to be imposed as constants, which other to vary and which to be selected as objects of the optimization. Moreover, an analysis on which boundary to impose to the optimization problem must be considered. Finally, the cost functions to be minimized must be carefully selected according to the aims of the optimization.
The possibility of optimizing the rotor geometry independently from its displacement and orientation on the frame is investigated. Outcomes for this option, and the one of optimizing together both the rotor geometry and orientation, are presented. A conclusion is drawn on whether an optimal geometry for every given rotor orientation exists.
The rotor geometry that minimizes the previously defined cost functions is obtained; this result is evaluated and discussed. Finally, it must be reminded that the optimal geometry for the single rotor identified in this chapter is later used as the starting point – in that as the geometry for every rotor in the vehicle energy storage system – for the multirotor optimizations described in chapter 5.
Flywheel energy storage 56
3.1 SimMechanics models In chapter 2 both an analytical and a SimMechanics numerical model where developed. This is done for the three systems corresponding to the three alternatives for the frame subsystem. Finally, the results from both alternatives were evaluated and compared. Since these are in good agreement, during the optimization phases of this chapter and the two following ones, it can be decided to work with the numerical or the analytical model. It was decided to use the numerical model for the following reasons:
- The running times for the script developing the analytical model are extremely long. For this reason, in the attempt of reducing the needed resources and the execution times, it was decided to modify the script, introducing the rotor positioning and orientation numerically before the execution of the program. In this way, the produced system motion equations regard a specific displacement an orientation of the rotor. Even if this solution has drastically reduced the needed running times, still these are considerably long and, moreover, now it is necessary to run the script for every new rotor placement. Finally, as stated in section 2.7, additional simplifications must be introduced to achieve the motion equations in an analytical form with the available resources. All these makes the analytical model inconvenient for optimization purposes. - With the SimMechanics numerical model it is very fast and easy to introduce little changes in the system. Moreover, the simulation execution times are short enough to perform an optimization process. Finally, with the SimMechanics model comes a graphical representation of the system that helps understanding the simulated dynamics.
3.2 Analysis of the problem Given the complexity of the problem, before starting with the optimization process, a study to categorize the wide number of variables is performed. The goal is to identify which variables can be independently selected, which can be measured and which are the objects of the optimization process. Moreover, by creating a table with all the variable figuring in the system, it is easier to develop an effective procedure to identify the optimal rotor geometry and not getting lost in a huge choice of possible tests and optimizations.
In the following subsections the list of the variables featuring the system is provided. This are grouped according on the effects that they are performing on the model. Not all the variables are listed but only those which have a crucial influence on the setting of the optimization process. Finally, five cost functions are presented, each representing one of the main negative aspects related to the flywheel dynamics that must be minimized.
3.2.1 General variables of the optimization The variables now listed are the main parameters of the optimization process. A short explanation is provided together with the broad use that is for them intended. A numerical value is given for the ones that are not going to change in every optimization.
Variable 1 (1) Variable representing the rotor geometry. This variable is composed of the six geometrical dimensions of the solid rotor, and so of three diameters and three lengths. Since the final goal of the process is finding the optimal rotor geometry, this variable is left free to change during the optimization. Single rotor optimization 57
Variable 2 (2) Initial angular speed of the rotor. In a first phase the rotor angular velocity is imposed equal to 60.000 rpm. This is the maximum nominal speed of a KERS operating on Formula One cars, which is used as a first attempt guideline. In a second phase it is also possible to consider different values of this parameter.
Variable 3 (3) Kinetic energy initially stored in the flywheel. In a first phase the rotor initial kinetic energy is imposed equal to 400.000 J, this is the energy stored in a Formula One KERS. In a successive phase it is also possible to consider different values of this parameter.
Variable 4 (4) Rotor geometry before optimization. This variable represents the initial geometry of the rotor before the optimization process is performed. Reason for this parameter to be introduced is to evaluate the convergence of the solution on the same results, independently of the initial conditions. Three initial configurations are defined and these are reported in the following table.
Configuration 풍ퟏ 풍ퟐ 풍ퟑ 푫ퟏ 푫ퟐ 푫ퟑ
휶ퟏ 0,01 0,05 0,03 0,07 0,15 0,07
휶ퟐ 0,03 0,03 0,03 0,08 0,08 0,08
휶ퟑ 0,1 0 0,30 0,10 0,04 0,40 0,04 Table 10: initial rotor geometry [m]
Variable 5 (5) Rotor density. This is a parameter that represents the rotor material. An initial value of 7.000 푘푔/푚3 is considerd. In a successive phase it is also possible to consider different values of this parameter.
Variable 6 (6) Percental eccentricity of the rotor. This parameter expresses – as a percentage based on the rotor average diameter – the distance between the rotor rotation axis and the rotor centre of mass. The average diameter is calculated as a weighted average on the three diameters of the solids composing the rotor. A realistic initial value of 0,2% is considered.
Variable 7 (7) Rotor rotational axis. This variable is defined by two angles representing the orientation of the rotor with respect to the frame. In some first evaluations this parameter is imposed, while, later, it is considered as one of the optimization variables of the problem.
Variable 8 (8) Number of bearings. This variable represents the number of bearings per side of the rotor rotation axis. By considering more than one bearing per side it is possible to increase the rotor support system stiffness beyond the limits imposed by mechanical bearings.
Variable 9 (9) Vehicle velocity. This parameter represents the speed at which the vehicle goes on a rough terrain. A first attempt value of 20 푚/푠 is considered.
3.2.2 System models for the optimization In this subsection the models employed during the optimization process are illustrated. Out of the three that were developed, only the following two are actually used for this task. Since these models differs by the frame representation only, the rotor is not considered in the illustrations and in the count of the degrees of freedom.
Flywheel energy storage 58
Model 1 (4) 15 degrees of freedom model. This model allows for better results and doesn’t add as much extra numerical stiffness to the system as the 11 degrees of freedom one does, however the simulation times are longer. Finally, with this model it is also possible to simulate the vehicle dynamics during turning.
Figure 28: 11 degrees of freedom model for the frame
Model 2 (5) 11 degrees of freedom model. The results obtained with this model are expected to be rougher, while the running times shorter. In any case the exact amount in which the results from the two models differ can only be assessed by running a multiple number of simulations.
Figure 29: 15 degrees of freedom model for the frame Single rotor optimization 59
3.2.3 System forcing The factors providing the excitation to the frame, and responsible for the dynamics of the system, are now listed. A short explanation is provided. Moreover, information related on which model must be used to introduce these excitations is specified.
Excitation 1 (1) Excitation provided by the rotor dynamics on the frame. Given the nature and the goals of the optimization process, this excitation is considered in every test and optimization. All three of the models previously introduced can consider this kind of forcing.
Excitation 2 (2) Excitation provided by the road roughness and acting through the vehicle tires and suspensions. In general, this excitation is always considered unless some reason for neglecting is found. The road roughness can be considered in all three of the models previously introduced.
Excitation 3 (3) Excitation provided by the vehicle dynamics during turning or in acceleration and deceleration phases. This forcing can be introduced only with the 15 degrees of freedom model, because of its additional degrees of freedom.
3.2.3.1 Road roughness from legislation The road roughness is produced by means of a random based process, according to the specifications provided by legislation. For the sake of brevity, the details related to the legislation and to the procedure that was implied in developing the road roughness profile are overlooked. However, a graphical representation of a road profile obtained with this method is presented in Figure 30. Moreover, since the legislation imposes constraints on this profile power spectral density, also a representation of this quantity is presented with Figure 31.
Figure 30: road roughness profile
3.2.4 Limits of the optimization field In the optimization process, a set of constraints must be considered on the optimization variables. This is particularly true for the rotor geometrical characteristics that, as previously stated, are always introduced in the list of the optimization variables. The main reasons why these constrains are considered are now listed:
Flywheel energy storage 60
- Degenerated configurations, characterized for example by geometrical structures with negative dimensions, can be avoided. - Unlikely structures are excluded from the results of the optimization process. These are those structures characterized by extremely thin elements, and that would not be able to bear loads or be manufactured. Reasonable boundary must be selected.
It is desired that the final results of the optimization process to not lie on the boundary for any optimization variable. If this is the case, a careful evaluation of the results must be performed.
Figure 31: road profile power spectral density
3.2.5 Additional constrains of the optimization Additional constrains are imposed to the system to grant a faster convergence to the solution. These constraints are selected with the intent of not influencing the solver results, but only of making it easier to get to them. These are described in the following table.
Extra 1 (6) Imposed rotor symmetry. Given the symmetrical nature of the system it is reasonable to expect that also the optimal solution would have some kind of symmetry. A series of preliminary optimizations confirms this hypothesis. By the imposition of a symmetry constraint, the dimension variables characterizing the rotor size are reduced from six to four.
Extra 2 (7) Imposition of a correlation between diameter and length for the two external cylindrical elements, composing the rotor body. With this constraint it is considered that these elements should have the right size to fit a bearing having the inner diameter equal to the cylinder one5. The relation between diameter a length of these elements is obtained by extracting data from the bearing manufacturers catalogues. The points obtained are fitted by an order three polynomial. With this it is possible to further reduce the number of variables characterizing the rotor geometry from four to three.
5 It was decided to consider a not very tight fit for the bearing in the axial direction. Reason of this is that in the final assembly also some washers wold be probably introduced. Single rotor optimization 61
Figure 32: relation between the element diameter and length to provide the bearing fit
3.2.6 Elementary cost functions In this subsection the elementary cost functions for the system optimization are introduced. Each of these functions represents one of the main negative aspects related to the flywheel dynamics, and these must be minimized. The exact procedure that is used to minimize these cost functions is analysed only in the following section. In any case, these elementary cost function are not minimized individually, during the optimization process, for evaluating the final rotor geometry. In fact, the function to be minimized is a composed function that is introduced in the following sections.
Cost C1 (1) Cost function that represents the speed of decay of the kinetic energy stored in the rotor. This function is obtained by drawing the time evolution of the kinetic energy, calculating the angular coefficient of the tangential line to this curve at a given time instance and, finally, calculating the square of this quantity. This cost function is of great value to define the performances of an energy storage system. A representation of the kinetic energy evolution and its tangent line are represented in Figure 33.
2 퐶1 = 푞 (70)
Cost C2 (2) This second elementary cost function is equal to the rotor mass.
퐶2 = 푚푟표푡 (71)
Cost C3 (3) Cost function representing the rotor vibration. It is defined by calculating the mean value
of the rotor acceleration in the three – 푥퐺, 푦퐺 and 푧퐺 – directions, obtaining the square of these three values and finally by summing them together. An example of the rotor vibration during time is represented in Figure 34.
2 2 2 퐶3 = (푚푒푎푛(푥̈푟표푡(푡))) + (푚푒푎푛(푦̈푟표푡(푡))) + (푚푒푎푛(푧푟표푡̈ (푡))) (72)
Cost C4 (4) Cost function representing the frame vibration. It is defined by calculating the mean value
of the frame acceleration in the three – 푥푇, 푦푇 and 푧푇 – directions, obtaining the square of these three values and finally by summing them together. An example of the frame vibration during time is represented in Figure 34.
Flywheel energy storage 62
2 2 2 퐶4 = (푚푒푎푛(푥̈푡푒푙(푡))) + (푚푒푎푛(푦̈푡푒푙(푡))) + (푚푒푎푛(푧푡푒푙̈ (푡))) (73)
Cost C5 (5) Cost function that represents the encumbrance of the rotor. This function is identified by the volume of the smallest parallelepiped that can contain the rotor. The volume of this can be evaluated as the sum of the lengths of the three cylindrical elements composing the rotor, multiplied by the square of the largest diameter.
2 퐶5 = (푙1 + 푙2 + 푙3)퐷2 (74)
A fast-transitory phase can be observed at the beginning of every simulation. For this reason, it is decided to start to collect the data to evaluate the five cost functions only after this first phase wears off. This usually happens in a fraction of a second. It is believed that this transitory phase has a numerical cause. Reason of this is that every spring element has been introduced in the model with an initial position equal to the equilibrium one.
Figure 33:example of the kinetic energy time evolution
3.3 Definition of the optimization procedure for the rotor geometry In this section a series of preliminary evaluations are performed before starting with the optimization process. Goal of these evaluations is to gather enough information to set the overall optimization process by defining a roadmap of the procedure. In other words, the problem that is to be addressed in this section is the one of creating a clear step-by-step technique to identify the rotor optimal geometry, and so to define how to proceed in the following sections.
Some of the problems that are faced are concerned with defining the sequence of optimizations between the multitude of possible alternatives. Moreover, for each of them, which parameters to fix, which to change and on which parameter to perform the optimization on.
In the following subsections some initial evaluations are performed and, finally, the last subsection presents the overall procedure developed. Single rotor optimization 63
Figure 34: example of the rotor vibration
Figure 35: example of the frame vibration
3.3.1 Preliminary evaluation 1
Before getting into the optimization process, an evaluation on how the elementary cost functions 퐶1, 퐶3 and
퐶4 changes is performed. During this process all the variables are kept constant with except for the rotor support bearing stiffness and the percent eccentricity of the rotor. The elementary cost functions 퐶2 and 퐶5 are not considered in this phase since they represent the rotor mass and the rotor encumbrance, and these are independent from the parameters that vary during the evaluation. For this initial optimization the rotor geometry to be used is the one identified as 훼1.
Flywheel energy storage 64
In this phase only one bearing for rotor axis side is considered. For what concerns the stiffness of these elements, a range of values that goes beyond what is normally achievable with mechanical bearings is considered. Reason for this is that the goal is to obtain an overall trend for the elementary cost functions as the rotor support stiffness and the eccentricity vary, and this without considering the boundary imposed by manufacturing problems.
In the graphs from Figure 36 to Figure 38 the results obtained using the 11 degrees of freedom model, and for the rotor with the rotation axis in the vehicle progression directions, are reported. On the vertical axis, the logarithm of the elementary cost functions is presented.
Figure 36: cost function 퐶1, 11 degrees of freedom model, 훼1 rotor
Figure 37: cost function 퐶3, 11 degrees of freedom model, 훼1 rotor
Figure 38: cost function 퐶4, 11 degrees of freedom model, 훼1 rotor Single rotor optimization 65
It can be noted from these results that an increase of the bearing stiffness has a positive effect on all three of the considered elementary cost functions. For this reason, very high stiffness bearings are selected among those available on the market. Moreover, to achieve higher stiffness, the opportunity of building a rotor with more than one bearing for side of the rotation axis is investigated.
It can also be noted that the effects of the percent eccentricity on the cost functions are rather small with respect to those of the bearing stiffness. In any case a reduction in the eccentricity leads to better results. It has also to be considered that, while it is possible to select the desired stiffness for the rotor support system, it is not easy to reduce the eccentricity under a certain amount. This means that while the stiffness is a parameter that can be freely selected, the eccentricity is a consequence of the system imperfections.
Additional evaluations are reported in appendix C.
3.3.2 Preliminary evaluation 2 Given the results of the preliminary evaluation 1, some more assessments on the possibility of introducing more than one bearing per side of the rotor axis are conducted. Getting into detail, since from the previous evaluation it was clear that the stiffer the rotor support system the better, the stiffest mechanical bearings are selected. These are roller bearings. The trends for the elementary cost functions 퐶1, 퐶3 and 퐶4 at the change of the eccentricity are obtained for the configuration having one, two or three bearing per part, and these are reported in graphs from Figure 39 to Figure 41.
Figure 39: cost function 퐶1, 11 degrees of freedom model, 훼1 rotor
Figure 40: cost function 퐶3, 11 degrees of freedom model, 훼1 rotor
Flywheel energy storage 66
Figure 41: cost function 퐶4, 11 degrees of freedom model, 훼1 rotor
As for the previous case, this evaluation is conducted on the 11 degrees of freedom model, with the rotor having the rotation axis along the vehicle progression directions. The rotor configuration used to obtain these
results is 훼1.
It can be stated that the options with two and three bearings per side give better results with respect to the one with a single bearing per side. These alternatives are considered while the optimization process is conducted.
Additional evaluations are reported in appendix C.
3.3.3 Preliminary evaluation 3
The effects that the rotor orientation has on the elementary cost functions 퐶1 , 퐶3 and 퐶4 are now
investigated. As it was for the two previous cases the cost function 퐶2 and 퐶5 are not considered in this analysis since they represent the mass and the bulkiness of the rotor, which are not influenced by the rotor orientation. Goal of this analysis is to define if – and in which measure – the cost functions are related to the orientation of the rotor axis. In case a strong correlation is observed, the option of considering also the angles expressing the rotor orientation as optimization variables must be considered. In this case finding an optimal geometry for the rotor independently from its orientation won’t be an easy task. On the contrary, in case no correlation – or weak correlation – between cost functions and rotor orientation is found, the optimal geometry found is probably always be the same, independently from which orientation is imposed.
Figure 42: cost function 퐶1, 11 degrees of freedom model, 훼1 rotor Single rotor optimization 67
The angles defining the rotor orientation inside the frame are 휎푟표푡 and 훽푟표푡. The elementary cost functions are evaluated for both the 훼1 and 훼2 configurations, varying these orientation angles between 0 and 휋/2. The results are obtained using the 11 degrees of freedom model and the cost functions are represented on a linear scale.
The results for 훼1 configuration are reported in graphs from Figure 42 to Figure 45. The only reported result for the system with 훼2 rotor is presented in Figure 46Figure 46: cost function 퐶1, 11 degrees of freedom model, 훼2 rotor; this is for the elementary cost function 퐶1. The reason is that this is the only cost function with notable differences from the case with 훼1 rotor.
From the representations it can be noted that all three of the elementary cost functions are significantly influenced by the rotor orientation. From this it can be concluded that, in order to optimize the rotor geometry, also the rotor orientation must be kept into account. Moreover, the possibility of obtaining a geometry for the rotor that is optimal independently from the rotor orientation must be carefully investigated. This possibility, in fact, must not be considered as granted since different optima rotor geometries could exist for different rotor orientations.
A particular in-depth analysis must be done on the trend of the 퐶4 cost function, representing the overall vibrations of the frame in the 푧푇, 푦푇 and 푧푇 directions. It is reminded that the model used for this analysis is the 11 degrees of freedom one, in which the frame is bounded to move – and vibrate – only on the vertical axis 푥푇 . Keeping this in mind it is clear why an alignment of the rotor axis whit the 푧푇 one produces a minimum – almost equal to zero – in the 퐶4 cost function. In this case, in fact, the rotor dynamics produces an excitation in the 푧푇- 푦푇 plane, that are constrained degrees of freedom for the frame. Therefore, it might be think that the 퐶4 cost function trend is completely wrong and caused only by the extra numerical stiffness that the model has in the 푧푇 and 푦푇 directions. However, even if the 11 degrees of freedom model introduces such a numerical stiffness to the system, also with a model that allows the frame motion on the 푧푇- 푦푇 plane, these directions would result with a much higher stiffness, with respect to the 푥푇 direction. This means that a similar trend for the 퐶4 function is expected also with the 15 degrees of freedom model, even if the minimum is probably much higher. This trend is calculated and displayed in the following picture for the 훼1 geometry.
Figure 43: cost function 퐶4, 15 degrees of freedom model, 훼1 rotor
Flywheel energy storage 68
Figure 44: cost function 퐶3, 11 degrees of freedom model, 훼1 rotor
Figure 45: cost function 퐶4, 11 degrees of freedom model, 훼1 rotor
Figure 46: cost function 퐶1, 11 degrees of freedom model, 훼2 rotor
Even if the trend is consistent with the one expected, it is believed that, after a first optimization performed with the 11 degrees of freedom model, a second optimization must be performed. This is conducted with the 15 degrees of freedom model, to perfect the results and assess if the extra numerical stiffness introduced by the 11 degrees of freedom model is not affecting the correctness the results. Single rotor optimization 69
3.3.4 Introduction to the rotor geometry optimization The first step to perform, in order to set an optimization process, is to define an overall cost function. This cost function is now introduced as the algebraic sum of the five elementary cost functions, presented in the preceding section.
퐶표푠푡 = 푃1퐶1 + 푃2퐶2 + 푃3퐶3 + 푃4퐶4 + 푃5퐶5 (75)
Where 퐶푖 are the elementary cost functions, while 푃푖 are the weight that each of the elementary cost functions have on the overall cost. One of the most crucial steps in the whole optimization process consists in defining reasonable values of the weights. In the following subsections the selected procedure is described and analysed in detail. A first optimization process is then presented together with the results achieved.
3.3.4.1 Rotor geometry optimization with rotation axis along 푧푇 direction Before going into details with the research of the rotor optimal geometry, a set of optimizations with rotor axis imposed in the 푧푇 directions are performed. It is reminded that the optimal rotor geometry depends on the orientation of the rotor itself, so the results that are now achieved are related to a rotor with axis along the 푧푇 direction only. Anyway, with this first optimization, some interesting results are expected. Firstly, the possibility of introducing more than one bearing per side of the rotor is analysed, secondly, an overlook at the method applicability is made.
In table 11 a schematic representation of the optimizations performed, and of the parameters for them introduced is provided. It can be noted that for the optimizations made, the weights are selected all to be null whit except for one. Finally, in the last four rows of the table the results are presented, consisting of both the optimal geometrical characteristics and of the optimal cost.
From an analysis of these data the following observations are drown:
1. The validity of the constructive alternatives characterized by more than one bearing for side of the
rotor axis are confirmed. For the cost functions 퐶1, 퐶3 and 퐶4 an improvement can be noted with
increasing the number of bearings, while, for the cost functions 퐶2 and 퐶5 – representing weight and bulkiness of the rotor – the better results are obtained with fewer bearings.
2. It can be noted that for what concerns the solutions obtained with the cost functions 퐶1−4, these are obtained whit a rotor constituted by a disk of the biggest diameter possible and as thin as possible, whit the smallest supports. A different result is only achieved by minimizing the elementary cost
function 퐶5. 3. Finally, it can be noted that the same results are obtained in terms of geometry – and similar in terms of costs – by starting the optimization from two different first-attempt solutions. This can be noted by the pairs of optimizations [A1a, A1d], [A1b, A1e], [A1c, A1f] and others not reported. For this
reason, from now on the same first-attempt solution geometry 훼1 is used for all the tests.
3.3.5 The selected optimization procedure From the experience made with the preliminary evaluations and with the first attempt optimization two procedure are developed. The first one aims at optimizing the rotor geometry together with the rotor orientation to find the best single rotor configuration. Whit the second one the opportunity of defining a rotor geometry that is optimal, no matter for the orientation, is investigated. These procedures are now discussed and in the following sections are performed.
Flywheel energy storage 70
1
v
v
x
v
x
v
v
z
3
1
0
0
0
0
20
a
0,5
A5c
7.000
2,5:10
60.000
2,5:100
2,0:100
3,86E-03
400.000
variabili
1,346E-01
6,413E-02
8,312E-02
1
v
v
x
v
x
v
v
z
2
1
0
0
0
0
20
a
0,5
A5b
7.000
2,5:10
60.000
2,5:100
2,0:100
3,10E-03
400.000
variabili
1,313E-01
5,923E-02
9,567E-02
1
v
v
x
v
x
v
v
z
1
1
0
0
0
0
20
a
0,5
A5a
7.000
2,5:10
60.000
2,5:100
2,0:100
2,22E-03
400.000
variabili
1,467E-01
5,272E-02
6,295E-02
1
v
v
x
v
x
v
v
z
3
0
1
0
0
0
20
a
0,5
A4c
7.000
2,5:10
60.000
2,5:100
2,0:100
1,21E+03
400.000
variabili
1,959E-01
2,528E-02
2,000E-02
1
v
v
x
v
x
v
v
z
2
0
1
0
0
0
20
a
0,5
A4b
7.000
2,5:10
60.000
2,5:100
2,0:100
1,28E+03
400.000
variabili
1,959E-01
2,501E-02
2,000E-02
1
v
v
x
v
x
v
v
z
1
0
1
0
0
0
20
a
0,5
A4a
7.000
2,5:10
60.000
2,5:100
2,0:100
1,74E+03
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
3
0
0
1
0
0
20
a
0,5
A3c
7.000
2,5:10
60.000
2,5:100
2,0:100
1,28E+06
400.000
variabili
1,959E-01
2,502E-02
2,000E-02
1
v
v
x
v
x
v
v
z
2
0
0
1
0
0
20
a
0,5
A3b
7.000
2,5:10
60.000
2,5:100
2,0:100
3,11E+06
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
1
0
0
1
0
0
20
a
0,5
A3a
7.000
2,5:10
60.000
2,5:100
2,0:100
1,72E+07
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
3
0
0
0
1
0
20
a
0,5
A2c
7.000
2,5:10
60.000
2,5:100
2,0:100
4,52E+00
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
2
0
0
0
1
0
20
a
0,5
A2b
7.000
2,5:10
60.000
2,5:100
2,0:100
4,42E+00
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
1
0
0
0
1
0
20
a
0,5
A2a
7.000
2,5:10
60.000
2,5:100
2,0:100
4,32E+00
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
3
0
0
0
0
1
20
b
0,5
A1f
7.000
2,5:10
60.000
2,5:100
2,0:100
5,45E+03
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
2
0
0
0
0
1
20
b
0,5
A1e
7.000
2,5:10
60.000
2,5:100
2,0:100
1,38E+04
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
1
0
0
0
0
1
20
b
0,5
A1d
7.000
2,5:10
60.000
2,5:100
2,0:100
1,01E+05
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
3
0
0
0
0
1
20
a
0,5
A1c
7.000
2,5:10
60.000
2,5:100
2,0:100
4,22E+04
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
2
0
0
0
0
1
20
a
0,5
A1b
7.000
2,5:10
60.000
2,5:100
2,0:100
1,06E+05
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
1
v
v
x
v
x
v
v
z
1
0
0
0
0
1
20
a
0,5
A1a
7.000
2,5:10
60.000
2,5:100
2,0:100
7,82E+05
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
Results
Optimization code Optimization
Forcingandsolver
General parametersGeneral
optimal diameter D2 [m] diameter optimal
optimal diameter D1 [m] diameter optimal
optimal length l2length [m] optimal
total cost total function
diameter D2 [cm] diameter
diameter D1 [cm] diameter
length l2length [cm]
simplified rotor geometry simplified rotor
imposed rotor symmetry rotor imposed
15 dof model 15 dof
11 dof model 11 dof
forcing for vehicle accelerationfor forcing
forcing for ground roughness ground for forcing
forcing for rotor dynamics rotor for forcing
vehicle speed [m/s]
number of bearingsof side) (per number
rotation axis rotation
perceptual eccentricity perceptual
rotor density [kg/m^3] density rotor
initial rotor geometry initial rotor
stored energy [J] energy stored
initial rotor velocity [rpm] velocity initial rotor
rotor geometry rotor
rotor encumbrance rotor
frame vibration frame
rotor vibration rotor
rotor mass rotor
kinetic energy decade rate kinetic energy
Cost function weights function Cost
Boundaries of the field ofthe Boundaries
4
3
2
1
3
2
1
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
5
4
3
2 1
Table 11: first attempts in optimizing the rotor geometry Single rotor optimization 71
Let’s start with the geometry and orientation optimization. First a rough optimization is performed by considering the 11 degrees of freedom model and by optimizing the three variables representing the rotor geometry and the two indicating the rotor orientation. When the optimal solution is identified, the results are evaluated with the 15 degrees of freedom model. The goal in this phase is to assess whether the minimum in the cost function that was found with the 11 degrees of freedom model exists also with the 15 degrees of freedom one, for a similar configuration. Main reason of these concern is what was already discussed for the effects of rotor orientation on the 11 and on the 15 degrees of freedom models when considering the elementary cos function 퐶4. If in this phase a preferential direction for the rotation axis is observed, this is imposed in the following optimization stage.
Finally, a fine optimization using the 15 degrees of freedom model is performed to perfect the results obtained in the previous phase. In this last optimization, in case a preferential rotation axis was identified, the optimization variables wold be only the three quantities representing the rotor geometry. On the contrary, in case no preferential direction axis was identify, the optimization is once more performed on the five variables representing geometry and orientation.
Figure 47: optimization processes
The second optimization process aims at finding the optimal solution for the rotor geometry, independently from the rotor orientation. This possibility must be carefully evaluated and must not be given for granted. The procedure that has been selected consists in defining a set of optimizations using the solution identified in the previous case as first attempt solution. This time, different orientations are imposed and the results are evaluated. If a divergence in the rotor geometry is observed it is concluded that no configuration that is optimal – no matter the rotor orientation – can be found. On the other hand, in case only slight changes are observed, the optimal solution can be identified with a weighted average between these results. Of course,
Flywheel energy storage 72
the newly identified optimal configuration is not as good as the one achieved by the first optimization process, but it is independent from orientation.
In Figure 47 the two optimization processes are summarized.
3.4 The optimal rotor geometry and orientation
3.4.1 First attempt optimal geometry Solution for the configuration having two bearings per side of the rotor axis
In this section the procedure used to define the rotor optimal geometry is presented and the obtained results are reported and discussed. As indicated, the two bearings per side solution has been selected. Reason on this stays in the fact that this option provides better results than the single bearing one and is responsible for less encumbrance with respect to the three bearings case. In this section the optimizations are performed on the 11 degrees of freedom model. Finally, because of the correlation between cost functions and rotor orientation, the variables of the optimization process are both the rotor geometry and the rotor orientation.
A set of different solutions is reported. For these a dynamic simulation of the system behaviour is performed and analysed.
3.4.1.1 The minimum costs In order to identify the weights to introduce in the overall cost function, a set of optimizations with elementary cost functions is performed. The aim of these optimisations is that of evaluating the minimum cost that can be achieved for each of the elementary cost functions. As previously stated, the number of optimization variables are now five, the three parameters defining the rotor dimensions and the two indicating the rotor axis orientation. Given the newly introduced degrees of freedom for the optimizer, it is reasonable to expect that the new results have a lower – or equal – cost respect to what was obtained in the
previous section, where the rotor axis was imposed in the 푧푇 direction.
In table 12 the data related to the optimizations performed are presented together with the results obtained.
As it can be noted the forecasted decrease in the costs with respect to the corresponding cases in the previous section is confirmed. The minimum obtainable values of the elementary cost functions are obtained.
3.4.1.2 Weights and rough optimization The weights in the overall cost functions are now defined as the products of two factors according to the following expression.
푃푖 = (퐹푠푐푎푙푒)푖 ∙ 푄푖 (76)
Where 퐹푠푐푎푙푒 is a scaling factor for the costs. This is introduced to make comparable costs that differ for orders of magnitude. It is defined as one over the minimum value that was obtained of each elementary cost function.
1 (퐹푠푐푎푙푒)푖 = (77) (퐶푚푖푛)푖 Single rotor optimization 73
Optimization code B1b B2b B3b B4b B5b Cost function weights 1 kinetic energy decade rate 1 0 0 0 0 2 rotor mass 0 1 0 0 0 3 rotor vibration 0 0 1 0 0 4 frame vibration 0 0 0 1 0 5 rotor encumbrance 0 0 0 0 1 General parameters 1 rotor geometry variable variable variable variable variable 2 initial rotor velocity [rpm] 60.000 60.000 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 400.000 400.000 4 initial rotor geometry a1 a1 a1 a1 a1 5 rotor density [kg/m^3] 7.000 7.000 7.000 7.000 7.000 6 perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis variable variable variable variable variable 8 number of bearings (per side) 2 2 2 2 2 9 vehicle speed [m/s] 20 20 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v v v 2 forcing for ground roughness v v v v v 3 forcing for vehicle acceleration x x x x x 4 11 dof model v v v v v 5 15 dof model x x x x x 6 imposed rotor symmetry v v v v v 7 simplified rotor geometry v v v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 8,99E+04 4,42E+00 3,13E+06 3,85E-02 3,10E-03 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,000E-02 2,005E-02 9,567E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,505E-02 5,923E-02 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,959E-01 1,958E-01 1,313E-01 5 σ_rot [rad] 1,360E+00 --- 1,761E-05 1,396E+00 --- 6 β_rot [rad] 2,995E-02 --- 3,477E-01 1,571E+00 ---
Table 12: minimum cost evaluation
The factor 푄푖, on the other hand, is a parameter arbitrary selected. It is introduced to specify the emphasis given to each of the elementary cost functions. In the first case all of these factors are set equal to one, in a latter moment also different values are considered.
Now that the weighs are defined, a series of optimizations on the rotor geometry and on the rotor axis orientation are set and performed. For these optimizations the scale factors are calculated with the newly introduced formula together with the results achieved in the previous section, while the selected factors 푄푖 are reported in table 13. In the same table are also presented all the other variables to set the optimization problem, together with the results obtained.
A few words are now spent to explain the choice made for the factors 푄푖 . It can be noted that three alternatives are presented. In the first one all five of these parameters are set equal to one. In this the same importance is given to the five elementary cost functions, and this can be considered as the base case. In the second alternative it was decided to give particular emphasis to the 퐶5 cost function. This can be easily justified from the fact that one of the main characteristics that an energy storage device for on vehicle use
Flywheel energy storage 74
should have is that of high volumetric energy density. It can be noted how the results obtained in this second case differ from those obtained in the first one. Finally, a third scenario is introduced in which high emphasis
is put on both the 퐶1 and 퐶5 cost functions. This is because a second really important factor for an energy storage devise for use on vehicles is that of storing energy without losses for the longest time possible. In this last scenario the results are once more different from the previous cases.
Optimization code Ctb-1 Ctb-2 Ctb-3
Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 1,11E-05 1 1,11E-05 2 1,11E-05 2 rotor mass 1 5,00E+01 1 5,00E+01 1 5,00E+01 3 rotor vibration 1 4,00E+01 1 4,00E+01 1 4,00E+01 4 frame vibration 1 5,10E+00 1 5,10E+00 1 5,10E+00 5 rotor encumbrance 1 7,35E-01 2 7,35E-01 2 7,35E-01 General parameters 1 rotor geometry variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 4 initial rotor geometry a1 a1 a1 5 rotor density [kg/m^3] 7.000 7.000 7.000 6 perceptual eccentricity 0,5 0,5 0,5 7 rotation axis variabile variabile variabile 8 number of bearings (per side) 2 2 2 9 vehicle speed [m/s] 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v 2 forcing for ground roughness v v v 3 forcing for vehicle acceleration x x x 4 11 dof model v v v 5 15 dof model x x x 6 imposed rotor symmetry v v v 7 simplified rotor geometry v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 Results 1 total cost function 5,55E+00 1,40E+01 8,22E+00 2 optimal length l2 [m] 2,002E-02 2,158E-02 2,000E-02 3 optimal diameter D1 [m] 2,502E-02 3,825E-02 2,521E-02 4 optimal diameter D2 [m] 1,959E-01 1,920E-01 1,959E-01 5 σ_rot [rad] 1,546E+00 8,751E-01 1,571E+00 6 β_rot [rad] 1,570E+00 1,571E+00 1,571E+00
Table 13: final rotor geometry optimization, 11 degrees of freedom model
It is interesting to note that the solutions obtained are characterized by a rotor having the rotation axis in
the vertical 푥푇 direction. Since in this section only the 11 degrees of freedom model is used, these results can be justified from the observations made in subsection 3.3.3. As previously stated, while the 15 degrees of freedom model is used to perfect these results, it must be investigated whether the vertical orientation of the rotation axis remains an optimal solution.
In appendix D a fast analysis of the obtained results is performed. A more in-depth study of the results is reported for the configurations obtained for the optimizations on the 15 degrees of freedom model. Single rotor optimization 75
Figure 48: graphical representation of the optimal configuration Ctb-1
3.4.2 Perfecting the results The 15 degrees of freedom model is now considered with the aim of refining the results for the rotor geometry. These new optimizations are set considering a first attempt geometry equal to the one that was obtained with the optimizations on the 11 degrees of freedom model, and considering a range for the optimization variables around this result.
As it was already pointed out in the previous subsection, all the solutions obtained till now have a rotation axis almost vertical (푥푇 direction). Given the advantages related to designing a vehicle having a perfectly vertical rotor axis, both for the constructive ease and for problems related to the rotor encumbrance, this solution would be desirable. Moreover, by imposing the rotor axis to be along the 푥푇 direction, there would be also advantage in the optimization process. In this case, in fact, the variables to be optimized would be only the three related to the rotor dimensions while the two variables representing the rotor orientation wold simply be imposed.
Before imposing this condition on the rotor orientation, it must be assured that – also with the 15 degrees of freedom model – the optimal rotor solution is with rotation axis in the 푥푇 direction. This evaluation is performed in the following subsection.
3.4.2.1 Evaluations of the rotor orientation axis The observations performed in subsection 3.3.3, “Preliminary evaluation 3”, are reminded. With these it was stated that an obvious optimal solution was to be found – when using the 11 degrees of freedom model – by placing the rotor whit rotation axis in the vertical direction. This is related to the fact that whit this orientation the dynamic of the rotor excites two constrained frame degrees of freedom – the longitudinal and the transversal one – and so the cost function 퐶4 tends to zero. With the 15 degrees of freedom model, however, these degrees of freedom are not constrained, even if these are characterized by a higher stiffness with respect to the vertical direction. In this section it is evaluated if also with the 15 degrees of freedom model the optimal solution has vertical rotor axis.
In the representations – from Figure 49 to Figure 51– the trends for the 퐶1, 퐶3 and 퐶4 elementary cost functions are represented. These trends are obtained with the 15 degrees of freedom model and for the 훼1 rotor configuration.
Now, a set of optimizations on the rotor geometry and on the rotor orientation is performed to address this doubt. These optimizations are done on the 15 degrees of freedom model considering the elementary cost
Flywheel energy storage 76
functions. If from these optimizations it is found that the vertical rotation axis is still the optimal solution, then this is imposed in the refined optimizations performed with the 15 degrees of freedom model. Results of this assessment are presented in table 14.
From these optimizations it is clear that – also with the 15 degrees of freedom model – the vertical
orientation of the rotation axis is the optimal solution that minimizes the elementary cost functions 퐶1, 퐶3
and 퐶4. However, it must be pointed out that the effects of the rotor orientation, when using the 15 degrees of freedom model, are not as severe on the cost functions as when the 11 degrees of freedom model is used. This can be noted from the graphs from Figure 49 to Figure 51, representing the trend of the three cost functions with the change of the rotor orientation. This is a consequence of the fact that the 11 degrees of freedom model was overestimating the system stiffness in the longitudinal and transversal directions.
In a latter study, the fact that the rotor orientation does not affect as much the cost function as it was previously estimated, could make it easier to consider an optimal geometry for the rotor independently from its orientation. Other data from these optimizations are neglected since the minimal costs and the rotor optimal geometry are evaluated imposing the rotation axis in the vertical direction.
Figure 49: cost function 퐶1, 15 degrees of freedom model, 훼1 rotor
Figure 50: cost function 퐶3, 15 degrees of freedom model, 훼1 rotor Single rotor optimization 77
Figure 51: cost function 퐶4, 15 degrees of freedom model, 훼1 rotor
Optimization code D1b-v D2b-v D3b-v D4b-v D5b-v Cost function weights 1 kinetic energy decade rate 1 0 0 0 0 2 rotor mass 0 1 0 0 0 3 rotor vibration 0 0 1 0 0 4 frame vibration 0 0 0 1 0 5 rotor encumbrance 0 0 0 0 1 General parameters 1 rotor geometry variabili variabili variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 400.000 400.000 4 initial rotor geometry a1 a1 a1 a1 a1 5 rotor density [kg/m^3] 7.000 7.000 7.000 7.000 7.000 6 perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis variabili variabili variabili variabili variabili 8 number of bearings (per side) 2 2 2 2 2 9 vehicle speed [m/s] 20 20 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v v v 2 forcing for ground roughness v v v v v 3 forcing for vehicle acceleration x x x x x 4 11 dof model x x x x x 5 15 dof model v v v v v 6 imposed rotor symmetry v v v v v 7 simplified rotor geometry v v v v v Boundaries of the field 1 length l2 [cm] ±10% ±10% ±10% ±10% ±10% 2 diameter D1 [cm] ±10% ±10% ±10% ±10% ±10% 3 diameter D2 [cm] ±10% ±10% ±10% ±10% ±10% Results 1 total cost function 1,06E+05 4,42E+00 2,76E+07 4,46E+03 3,10E-03 2 optimal length l2 [m] 2,000E-02 2,000E-02 4,400E-02 3,770E-02 9,567E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 9,980E-02 2,540E-02 5,923E-02 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,452E-01 1,672E-01 1,313E-01 5 σ_rot [rad] 1,571E+00 --- 1,571E+00 1,571E+00 --- 6 β_rot [rad] 1,571E+00 --- 1,571E+00 1,571E+00 ---
Table 14: rotor vertical axis evaluation with the 15 degrees of freedom model
Flywheel energy storage 78
3.4.2.2 The new optimal costs The new minimum costs for the elementary cost functions are now evaluated. In this case the 15 degrees of freedom model is used, and the rotation axis is imposed in the vertical direction. The procedure is the same that was previously adopted for the first research of the minimum costs. In this case, however, the first attempt rotor geometry is imposed equal to the optimal geometry obtain from the previous research of the minimum costs, moreover, the interval where the optimization variables can vary is set as a ±10% variation on these values.
Optimization code D1b D2b D3b D4b D5b Cost function weights 1 kinetic energy decade rate 1 0 0 0 0 2 rotor mass 0 1 0 0 0 3 rotor vibration 0 0 1 0 0 4 frame vibration 0 0 0 1 0 5 rotor encumbrance 0 0 0 0 1 General parameters 1 rotor geometry variabili variabili variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 400.000 400.000 4 initial rotor geometry ott ott ott ott ott 5 rotor density [kg/m^3] 7.000 7.000 7.000 7.000 7.000 6 perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x 8 number of bearings (per side) 2 2 2 2 2 9 vehicle speed [m/s] 20 20 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v v v 2 forcing for ground roughness v v v v v 3 forcing for vehicle acceleration x x x x x 4 11 dof model x x x x x 5 15 dof model v v v v v 6 imposed rotor symmetry v v v v v 7 simplified rotor geometry v v v v v Boundaries of the field 1 length l2 [cm] ±10% ±10% ±10% ±10% ±10% 2 diameter D1 [cm] ±10% ±10% ±10% ±10% ±10% 3 diameter D2 [cm] ±10% ±10% ±10% ±10% ±10% Results 1 total cost function 1,05E+05 4,42E+00 3,03E+06 2,17E+03 3,10E-03 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,000E-02 2,000E-02 9,567E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,500E-02 5,923E-02 - 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,313E-01 - Table 15: research of the minimum costs, 15 degrees of freedom model
The tests now performed are described in table 15, together with their results.
In the table, the rotor geometrical configurations indicated with ‘ott’ are defined as followings:
- For the simulation D1b, as the results of optimization B1b. - For the simulation D2b, as the results of optimization B2b. - For the simulation D3b, as the results of optimization B3b. - For the simulation D4b, as the results of optimization B4b. - For the simulation D5b, as the results of optimization B5b. Single rotor optimization 79
It can be noted that the results achieved with the 15 degrees of freedom model are only slightly different from those obtained with the 11 degrees of freedom one. For this reason, it can be stated that an 11 degrees of freedom model might be sufficient to define the rotor optimal geometry. In any case, it is decided to proceed with the last phase of the analysis with also the 15 degrees of freedom model, even if only slight changes to the already obtained results are expected.
3.4.2.3 Weight definition and precise optimization The weights are defined with the same method that was introduced for the 11 degrees of freedom model.
푃푖 = (퐹푠푐푎푙푒)푖 ∙ 푄푖 (78)
These new weights, however, are obtained on the base of the newly calculated minimum costs. The new optimizations are performed using the same 푄푖 factors that were earlier introduced. The first attempt rotor geometry is set equal to the optimal geometry obtained with the 11 degrees of freedom system. A range of the optimizations variables of ±10% around these values is considered. The optimizations performed are presented in the table 16.
Optimization code Etb-1 Etb-2 Etb-3
Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 9,57E-06 1 9,57E-06 2 9,57E-06 2 rotor mass 1 2,26E-01 1 2,26E-01 1 2,26E-01 3 rotor vibration 1 3,30E-07 1 3,30E-07 1 3,30E-07 4 frame vibration 1 4,61E-04 1 4,61E-04 1 4,61E-04 5 rotor encumbrance 1 3,23E+02 2 3,23E+02 2 3,23E+02 General parameters 1 rotor geometry variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 4 initial rotor geometry ott ott ott 5 rotor density [kg/m^3] 7.000 7.000 7.000 6 perceptual eccentricity 0,5 0,5 0,5 7 rotation axis x x x 8 number of bearings (per side) 2 2 2 9 vehicle speed [m/s] 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v 2 forcing for ground roughness v v v 3 forcing for vehicle acceleration x x x 4 11 dof model x x x 5 15 dof model v v v 6 imposed rotor symmetry v v v 7 simplified rotor geometry v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 Results 1 total cost function 4,98E+00 5,69E+00 6,46E+00 2 optimal length l2 [m] 2,000E-02 2,074E-02 2,122E-02 3 optimal diameter D1 [m] 2,503E-02 2,859E-02 2,751E-02 4 optimal diameter D2 [m] 1,959E-01 1,941E-01 1,930E-01
Table 16: final rotor geometry optimization, 15 degrees of freedom model
Flywheel energy storage 80
As it was predicted, only a slight change from the previous results can be noted.
In the following sections some data on the optimal configurations are presented. In detail, these are information related to the physical properties of the optimal rotors, to the dynamics of a single rotor configuration and to the basic physical characteristics of a multi-rotor configuration based on these optimal solutions. For the multirotor configuration, the constraint of storing 20 퐾푊ℎ is set.
Additional data on these configurations is provided in appendix E.
Single rotor optimization 81
3.4.2.3.1 Optimal solution Etb-1 General information is here listed for the optimal rotor configuration Etb-1.
Figure 52: graphical representation of optimal rotor Etb-1
Quantity Unit Value Energy stored per rotor 퐽 400.000 Number of rotors (multirotor) --- 180 Kinetical energy decay rate 퐽/푠 -247,74 Kinetical energy decay rate %/푠 -0,0619% Single rotor mass 퐾푔 4,3241 System mass (multirotor) 퐾푔 778,3302 Single rotor volume 푚3 0,0019 System volume (multirotor) 푚3 0,3425 Table 17: optimal rotor Etb-1 characteristics
Figure 53: time evolution of the kinetical energy stored in the rotor, configuration Etb-1
Flywheel energy storage 82
3.4.2.3.2 Optimal solution Etb-2 General information is here listed for the optimal rotor configuration Etb-2.
Figure 54: graphical representation of optimal rotor Etb-2
Quantity Unit Value Energy stored per rotor 퐽 400.000 Number of rotors (multirotor) --- 180 Kinetical energy decay rate 퐽/푠 -268,67 Kinetical energy decay rate %/푠 -0,0672% Single rotor mass 퐾푔 4,6664 System mass (multirotor) 퐾푔 839,9594 Single rotor volume 푚3 0,0021 System volume (multirotor) 푚3 0,3757 Table 18: optimal rotor Etb-2 characteristics
Figure 55: time evolution of the kinetical energy stored in the rotor, configuration Etb-2
Single rotor optimization 83
3.4.2.3.3 Optimal solution Etb-3 General information is here listed for the optimal rotor configuration Etb-3.
Figure 56: graphical representation of optimal rotor Etb-3
Quantity Unit Value Energy stored per rotor 퐽 400.000 Number of rotors (multirotor) --- 180 Kinetical energy decay rate 퐽/푠 -247,89 Kinetical energy decay rate %/푠 -0,0620% Single rotor mass 퐾푔 4,3250 System mass (multirotor) 퐾푔 778,4970 Single rotor volume 푚3 0,0019 System volume (multirotor) 푚3 0,3433 Table 19: optimal rotor Etb-3 characteristics
Figure 57: time evolution of the kinetical energy stored in the rotor, configuration Etb-3
Flywheel energy storage 84
3.5 Orientation independent geometry
3.5.1 General procedure The solution that is chose to perform this second optimization process is the one identified with the name
Etb-3. This solution is characterized by a rotor with rotation axis in the vertical 푥푇 direction. In this section the opportunity of obtaining – starting from this result – a solution that is optimal independently from the rotor orientation is investigated. To do this the Etb-3 solution is used as the first attempt solution of a new optimization process, where the rotor axis is imposed in different directions. If the results of these new optimizations are only slightly different, then an average solution between these can be considered.
Before proceeding with the newly mentioned optimizations, a new evaluation of the scale factors must be performed. This is conducted considering a first attempt geometry equal to the optimal geometry obtained
from the previous research of the scale factors. On the contrary the same values for the 푄푖 factors that were used in the Etb-3 optimization are now considered. The axis on which to perform this optimization are the
푥푇 -axis – for which the results are already available – the 푦푇 and 푧푇 axis and other two axes randomly selected. These are defined as in the following table.
Axis 𝝈풓풐풕 휷풓풐풕
풙푻 0 휋/2
풚푻 휋/2 0
풛푻 0 0
푹ퟏ 0,1547 0,014
푹ퟐ 1,658 0.941 Table 20a: imposed rotor orientation axis
In table 20 the settings and the results of the series of optimizations, performed imposing the rotor axes in
the 풙푻, 풚푻 and 풛푻 directions, are presented. For each of these axes, the procedure is divided into two steps: a weight evaluation phase and a final optimization phase. The geometrical characteristics of the optimal rotors obtained with these settings are presented. It can be noted that these results differ significantly depending on the rotor axis orientation. These differences can be easily noted with the graph in Figure 58Figure 58: optimal configuration comparison.
Gtb
Ftb
Atb-r
1,0E-02 2,0E-02 3,0E-02 4,0E-02 5,0E-02 6,0E-02 7,0E-02 8,0E-02 9,0E-02 1,0E-01 1,1E-01 1,2E-01 1,3E-01 1,4E-01 1,5E-01 1,6E-01 1,7E-01 1,8E-01 1,9E-01 2,0E-01 0,0E+00
optimal diameter D2 [m] optimal diameter D1 [m] optimal length l2 [m]
Figure 58: optimal configuration comparison Single rotor optimization 85
v
v
v
x
x
v
v
z
2
20
ott
0,5
F_scala
Gtb
3,31E-07
2,26E-01
3,84E-05
3,23E+02
0,000424
7.000
2,5:10
60.000
2,5:100
2,0:102
7,39E+00
400.000
variabili
1,947E-01
3,008E-02
2,046E-02
2
1
1
1
2
C
v
v
v
x
x
v
v
z
2
1
0
0
0
0
20
ott
0,5
G5b
±10%
±10%
±10%
7.000
60.000
3,10E-03
400.000
variabili
1,313E-01
5,923E-02
9,567E-02
v
v
v
x
x
v
v
z
2
0
1
0
0
0
20
ott
0,5
G4b
±10%
±10%
±10%
7.000
60.000
2,36E+03
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
v
x
x
v
v
z
2
0
0
1
0
0
20
ott
0,5
G3b
±10%
±10%
±10%
7.000
60.000
3,02E+06
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
v
x
x
v
v
z
2
0
0
0
1
0
20
ott
0,5
G2b
±10%
±10%
±10%
7.000
60.000
4,42E+00
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
v
x
x
v
v
z
2
0
0
0
0
1
20
ott
0,5
G1b
±10%
±10%
±10%
7.000
60.000
2,61E+04
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
v
x
x
v
v
y
2
20
ott
0,5
F_scala
Ftb
3,21E-07
2,26E-01
8,81E-06
3,23E+02
0,003523
7.000
2,5:10
60.000
2,5:100
2,0:102
1,01E+01
400.000
variabili
1,940E-01
2,500E-02
2,081E-02
2
1
1
1
2
C
v
v
v
x
x
v
v
y
2
1
0
0
0
0
20
ott
0,5
F5b
±10%
±10%
±10%
7.000
60.000
3,10E-03
400.000
variabili
1,313E-01
5,923E-02
9,567E-02
v
v
v
x
x
v
v
y
2
0
1
0
0
0
20
ott
0,5
F4b
±10%
±10%
±10%
7.000
60.000
2,84E+02
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
v
x
x
v
v
y
2
0
0
1
0
0
20
ott
0,5
F3b
±10%
±10%
±10%
7.000
60.000
3,12E+06
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
v
x
x
v
v
y
2
0
0
0
1
0
20
ott
0,5
F2b
±10%
±10%
±10%
7.000
60.000
4,42E+00
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
v
x
x
v
v
y
2
0
0
0
0
1
20
ott
0,5
F1b
±10%
±10%
±10%
7.000
60.000
1,13E+05
400.000
variabili
1,958E-01
2,910E-02
2,000E-02
v
v
x
v
x
v
v
z
2
20
ott
0,5
F_scala
7,35E-01
1,11E-05
5,10E+00
4,00E+01
5,00E+01
7.000
Atb-r
2,5:10
60.000
2,5:100
2,0:102
8,22E+00
400.000
variabili
1,959E-01
2,521E-02
2,000E-02
2
1
1
1
2
C
v
v
x
v
x
v
v
z
2
1
0
0
0
0
20
ott
0,5
±10%
±10%
±10%
7.000
A5b-r
60.000
3,10E-03
400.000
variabili
1,313E-01
5,923E-02
9,567E-02
v
v
x
v
x
v
v
z
2
0
1
0
0
0
20
ott
0,5
±10%
±10%
±10%
7.000
A4b-r
60.000
1,28E+03
400.000
variabili
1,959E-01
2,501E-02
2,000E-02
v
v
x
v
x
v
v
z
2
0
0
1
0
0
20
ott
0,5
±10%
±10%
±10%
7.000
A3b-r
60.000
3,11E+06
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
x
v
x
v
v
z
2
0
0
0
1
0
20
ott
0,5
±10%
±10%
±10%
7.000
A2b-r
60.000
4,42E+00
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
v
v
x
v
x
v
v
z
2
0
0
0
0
1
20
ott
0,5
±10%
±10%
±10%
7.000
A1b-r
60.000
1,06E+05
400.000
variabili
1,959E-01
2,500E-02
2,000E-02
Results
Optimization code Optimization
Forcingandsolver
General parametersGeneral
optimal diameter D2 [m] diameter optimal
optimal diameter D1 [m] diameter optimal
optimal length l2length [m] optimal
total cost total function
diameter D2 [cm] diameter
diameter D1 [cm] diameter
length l2length [cm]
simplified rotor geometry simplified rotor
imposed rotor symmetry rotor imposed
15 dof model 15 dof
11 dof model 11 dof
forcing for vehicle accelerationfor forcing
forcing for ground roughness ground for forcing
forcing for rotor dynamics rotor for forcing
vehicle speed [m/s]
number of bearingsside) (per number
rotation axis rotation
perceptual eccentricity perceptual
rotor density [kg/m^3] density rotor
initial rotor geometry initial rotor
stored energy [J] energy stored
initial rotor velocity [rpm] velocity initial rotor
rotor geometry rotor
rotor encumbrance rotor
frame vibration frame
rotor vibration rotor
rotor mass rotor
kinetic energy decade rate kinetic energy
Cost function weights function Cost
Boundaries of the field ofthe Boundaries
4
3
2
1
3
2
1
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
5
4
3 2 1
Table 20b: research for the optimal geometry, independently from the rotor orientation
Flywheel energy storage 86
3.5.2 Results averaging From the optimizations performed in the preceding subsection it can be noted that the optimal geometry for the rotor differs significantly, depending on the rotor orientation. The variances are too important to consider an optimal configuration defined as a weighted average between the results for the different orientations. However, a procedure for obtaining an optimal solution – no matter the rotor orientation – could still be considered.
The base idea of this procedure is to implement an optimization on the rotor geometry where a new multi- directional cost function is adopted. This new cost is defined as the sum of the cost functions – of the same nature of the one introduced in the preceding chapter – for a set of different rotor orientations. The result obtained would be the rotor geometry that statistically minimizes the costs if the rotor orientation is not known in advance. In any case this result would have higher costs if compared with the result obtained optimizing the rotor on the specific direction of interest. Furthermore, an even better configuration is represented by the rotor having the rotation axis in the vertical direction.
Since, as stated in this chapter, the best rotor configuration is the one with vertical rotation axis, and since this configuration is particularly convenient when designing a flywheel energy storage system for on vehicle use, this solution is adopted for the multi-rotor optimizations that are performed in the next chapter.
4 Advanced single-rotor optimization
In this chapter the results and the procedure developed in chapter 3 are analyzed to evaluate the possibility of obtaining an energy storage unit for on vehicle applications. In particular, the opportunity of realizing a light and compact system that can act as primary energy storage on road vehicles is investigated. The focus is put on not exceeding what are the natural boundaries of weight and bulkiness for such a system with, at the same time, keeping good results for the cost functions that were previously introduced. It must be underlined, though, that in this chapter it is not performed a multirotor optimization, but simple quantities – like the overall system mass and encumbrance – are considered.
The study is conducted according to the following steps. In a first section the optimal rotor configuration defined in the previous chapter is analyzed with the aim of using this solution as a rotor in the energy storage system. Pros and cons are listed and analyzed and the possibility of improving these results with changes in the rotor material or rotor basic geometry are investigated. In the following sections, if an improvement is considered possible with such changes, a new series of optimizations is performed with the same procedure that was used for the already obtained results. Finally, a selection of possible rotors for the energy storage system is presented and studied.
Flywheel energy storage 90
4.1 Evaluation of the results of chapter 3 In this first section the rotor characterized by the geometry obtained with the optimization process of chapter 3 is analyzed. As previously stated, the focus of this study is to define weather an energy storage system based on such a flywheel would have a reasonable weight and encumbrance, making it suitable for the use on a road vehicle. If from this study it results that said flywheel is not compatible with this use, additional optimization processes are defined to solve these problems. In particular, in the following sections, new possibilities to address the problematics related to the fitting of a multirotor energy storage system on vehicle mast be considered.
Optimization code Etb-1 Etb-2 Etb-3
Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 9,57E-06 1 9,57E-06 2 9,57E-06 2 rotor mass 1 2,26E-01 1 2,26E-01 1 2,26E-01 3 rotor vibration 1 3,30E-07 1 3,30E-07 1 3,30E-07 4 frame vibration 1 4,61E-04 1 4,61E-04 1 4,61E-04 5 rotor encumbrance 1 3,23E+02 2 3,23E+02 2 3,23E+02 General parameters 1 rotor geometry variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 4 initial rotor geometry ott ott ott 5 rotor density [kg/m^3] 7.000 7.000 7.000 6 perceptual eccentricity 0,5 0,5 0,5 7 rotation axis x x x 8 number of bearings (per side) 2 2 2 9 vehicle speed [m/s] 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v 2 forcing for ground roughness v v v 3 forcing for vehicle acceleration x x x 4 11 dof model x x x 5 15 dof model v v v 6 imposed rotor symmetry v v v 7 simplified rotor geometry v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 Results 1 total cost function 4,98E+00 5,69E+00 6,46E+00 2 optimal length l2 [m] 2,000E-02 2,074E-02 2,122E-02 3 optimal diameter D1 [m] 2,503E-02 2,859E-02 2,751E-02 4 optimal diameter D2 [m] 1,959E-01 1,941E-01 1,930E-01 Additional results 1 volume [m^3] 3,04E-03 3,13E-03 3,07E-03 2 mass [Kg] 4,424E+00 4,575E+00 4,601E+00
3 rotor number [---] 180 180 180 4 total volume[m^3] (20 kWh) 0,547 0,563 0,553 5 total mass [kg] (20 kWh) 796,3 823,5 828,2 6 maximum stress [MPa] 1,094E+03 1,074E+03 1,062E+03
Table 21: final rotor geometry optimization with additional results, 15 degrees of freedom model With a dot it is specified how the results compare with the others, obtain across the whole work: green dot stands for good results, orange for average and red for poor results Advanced single-rotor optimization 91
To define and evaluate a flywheel based energy storage system a target energy capacity should be selected. A value sufficient for short-middle range vehicle operation is chosen. If it is found that this energy storage capacity is easily obtained with such a system, also improved capacity energy storage systems are considered.
퐸푡표푡 = 20 푘푊ℎ = 72.000.000 퐽 (79)
Now that a target energy to store is defined, the number of rotors necessary in the energy storage system can be calculated and, therefore, also the total mass related to the rotor and the total volume occupied. These mass and encumbrance parameters are introduced in table 21. This is an updated version of the already presented table 16. From this, it can be noted that, even if the volume required for the system is not much, it is easy to understand that its weight is excessive, especially considering the limited energy that the system is capable of storing.
Even if these solutions are to be discarded, it is interesting to evaluate the maximum stresses the rotating disk is subject to during rotation. It is now reminded that it is not the goal of this work to provide a final solution for the geometry of a flywheel based energy storage system, but only to perform a feasibility analysis. Consequently, the structural assessment of the solutions provided is not analyzed in detail. However, it is important to evaluate whether the provided results are feasible. With this in mind, the maximum tensile stresses the rotating disk is subject to for centrifugal force, are evaluated with the following simplified formula.
3 + 휈 휎 = 휎 = 휌휔2푟2 (80) 푟 휗 8 푒푥푡 Where 휈 and 휌 are respectively the material Poisson’s coefficient and the material density, while 휔 is the disk rotational speed and 푟푒푥푡 is the radius of the disk. With this formula it is easy to conclude that the loads produced exceed what most steel alloys are capable to bare. The value calculated for the maximum stresses is reported in table 21.
For these two reasons (overall system weight and maximum material stresses) additional optimizations are performed in the following section, considering lighter weight materials.
4.2 Improvements on the optimization process Given the drawbacks identified for the rotor solution of the optimization process performed in chapter 3, a new set of optimizations is performed. Goal of these is to define the ideal geometry and characteristics for a flywheel to be used for an on-vehicle energy storage system. This implies to identify a solution characterized by low weight and encumbrance, and that produces reasonable stresses on the rotor disk, but that at the same time provides good results in terms of the elementary cost functions. In particular, it is desired that the solution is capable of preserving the energy stored for a reasonably long time.
These new optimization processes are performed by changing the values assigned to some of the variables set as constants in the optimization process detailed in chapter 3. These changes are related to the rotor material characteristics, to the flywheel initial speed and overall geometry and to the amount of energy stored in each single rotor. A comparison between the results is then provided.
Flywheel energy storage 92
Optimization code Jtb-1 Jtb-2 Jtb-3
Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 1,86E-05 1 1,86E-05 2 1,86E-05 2 rotor mass 1 3,85E-01 1 3,85E-01 1 3,85E-01 3 rotor vibration 1 4,92E-07 1 4,92E-07 1 4,92E-07 4 frame vibration 1 6,08E-04 1 6,08E-04 1 6,08E-04 5 rotor encumbrance 1 1,80E+02 2 1,80E+02 2 1,80E+02 General parameters 1 rotor geometry variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 4 initial rotor geometry ott nov dic 5 rotor density [kg/m^3] 2.500 2.500 2.500 6 perceptual eccentricity 0,5 0,5 0,5 7 rotation axis x x x 8 number of bearings (per side) 2 2 2 9 vehicle speed [m/s] 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v 2 forcing for ground roughness v v v 3 forcing for vehicle acceleration x x x 4 11 dof model x x x 5 15 dof model v v v 6 imposed rotor symmetry v v v 7 simplified rotor geometry v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 Results 1 total cost function 6,78E+00 8,25E+00 2,20E+01 2 optimal length l2 [m] 2,016E-02 6,494E-02 1,004E-01 3 optimal diameter D1 [m] 9,518E-02 2,524E-02 2,500E-02 4 optimal diameter D2 [m] 2,465E-01 1,888E-01 1,693E-01 Additional results 1 volume [m^3] 7,22E-03 4,43E-03 4,57E-03 2 mass [Kg] 4,160E+00 4,620E+00 5,724E+00
3 rotor number [---] 180 180 180 4 total volume[m^3] (20 kWh) 1,300 0,798 0,823 5 total mass [kg] (20 kWh) 748,8 831,5 1030,4 6 maximum stress [MPa] 5,923E+02 3,474E+02 2,794E+02
Table 22: final rotor geometry optimization, low weight material, 15 degrees of freedom model
4.2.1 Low density flywheel The most natural change in the system that is to be evaluated is that of considering a low-density flywheel. Firstly, a low-density flywheel could provide better results in terms of the overall weight and encumbrance for the energy storage system, secondly, it has also to be considered that all flywheel based system with such a high rotational speed are realized in aluminum and composed materials such as carbon fiber. The reason of this stays in the fact that, with high density, huge stresses arise from the centrifugal load the system is subject to. As previously stated, since this aims to be a preliminary study, a detailed stress analysis on the rotor is neglected. Anyway, since realistic results are to be obtained, the iron flywheel is not considered as a viable possibility. Advanced single-rotor optimization 93
The new flywheel density is set to 2500 퐾푔/푚3. The optimization procedure that is detailed in chapter 3 is implemented. For brevity only the most significant results are presented in table 23, and for these the overall system weight and encumbrance are evaluated.
Optimization code Ltb-1 Ltb-2 Ltb-3 Ltb-4 Ltb-5 Ltb-5
Cost function weights C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 1,86E-05 1 1,86E-05 2 1,18E-05 2 1,86E-05 2 1,86E-05 2 1,86E-05 2 rotor mass 1 3,85E-01 1 3,85E-01 1 0,509088 2 3,85E-01 4 3,85E-01 8 3,85E-01 3 rotor vibration 1 4,92E-07 1 4,92E-07 1 7,61E-08 1 4,92E-07 1 4,92E-07 1 4,92E-07 4 frame vibration 1 6,08E-04 1 6,08E-04 1 0,000873 1 6,08E-04 1 6,08E-04 1 6,08E-04 5 rotor encumbrance 1 1,80E+02 2 1,80E+02 2 195,7916 1 1,80E+02 1 1,80E+02 1 1,80E+02 General parameters 1 rotor geometry variabili variabili variabili variabili variabili variabili 2 initial rotor velocity [rpm] 80.000 80.000 80.000 80.000 80.000 80.000 3 stored energy [J] 400.000 400.000 400.000 400.000 400.000 400.000 4 initial rotor geometry ott ott ott ott ott ott 5 rotor density [kg/m^3] 2.500 2.500 2.500 2.500 2.500 2.500 6 perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x x 8 number of bearings (per side) 2 2 2 2 2 2 9 vehicle speed [m/s] 20 20 20 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v v v v 2 forcing for ground roughness v v v v v v 3 forcing for vehicle acceleration x x x x x x 4 11 dof model x x x x x x 5 15 dof model v v v v v v 6 imposed rotor symmetry v v v v v v 7 simplified rotor geometry v v v v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2,0:101 2,0:101 2,0:101 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 2,50E+01 2,95E+01 8,09E+01 8,83E+01 9,55E+01 1,42E+01 2 optimal length l2 [m] 6,131E-02 6,15E-02 8,169E-02 6,14E-02 6,49E-02 2,00E-02 3 optimal diameter D1 [m] 2,500E-02 3,760E-02 2,500E-02 6,512E-02 6,236E-02 2,001E-02 4 optimal diameter D2 [m] 1,659E-01 1,656E-01 1,544E-01 1,644E-01 1,624E-01 2,192E-01 Additional results 1 volume [m^3] 3,31E-03 3,59E-03 3,36E-03 4,01E-03 3,97E-03 4,40E-03 2 mass [Kg] 3,385E+00 3,507E+00 3,896E+00 3,984E+00 4,013E+00 2,118E+00
3 rotor number [---] 180 180 180 180 180 180 4 total volume[m^3] (20 kWh) 0,597 0,647 0,604 0,722 0,714 0,792 5 total mass [kg] (20 kWh) 609,2 631,3 701,2 717,2 722,4 381,2 6 maximum stress [MPa] 4,767E+02 4,752E+02 4,130E+02 4,683E+02 4,571E+02 8,329E+02
Table 23: final rotor geometry optimization, low weight material and faster rotation speed, 15 degrees of freedom model
It can be noted that, even if the rotor density is reduced, the overall energy storage system mass increases. Moreover, the lower rotor density affects negatively also the overall volume needed. In any case, because of the much better results when it comes to maximum stresses, a low-density flywheel is the only option considered. It is now clear that some alternative methods for reducing the energy system weight and bulkiness must be considered.
Flywheel energy storage 94
4.2.2 Low density flywheel at higher velocity An option to increase the energy storage density is that of increasing the maximum – or initial – angular speed of the rotor. An optimization process is performed keeping the rotor density at 2500 퐾푔/푚3 and increasing the initial rotor velocity to 80.000 푟푝푚. The results are presented in table 23.
A considerable reduction in the overall system mass is obtained together with positive effects also on the system encumbrance. Even if this seems like a promising method, the technical difficulties of realizing such a fast spinning device are of fundamental importance. For this reason, starting from the next optimization, the initial rotor angular speed is set once more to 60.000 푟푝푚. The quality of the just obtained results is considered as a target to seek with the reduced rotor angular speed, and by considering alternative methods.
It is to be noted that, despite the high rotational velocity of the rotor in these configurations, the maximum stresses on the rotor axis are never exceeding the limits for a carbon fiber flywheel.
4.2.3 Geometry improvement In this subsection an improvement on the low-density flywheel geometry is introduced. The rotor geometry, in fact, is not identified by three cylindrical elements any more, but now also a forth solid part is introduced.
This element is a cylindrical ring, with inner diameter equal to the diameter 퐷2. The difference between inner
and outer radius is indicated by 푠1, and the cylinder height by 푙4. The element is positioned symmetrically
with respect to the disk of diameter 퐷2, and its symmetry axis lies on the rotor rotation axis.
Now the optimization process is performed on five degrees of freedom instead of three. This provides some additional difficulties to achieve a good convergence of the results on the best possible solution. The problem is addressed by repeating every optimization many times, setting for each of them a different first attempt solution. Finally, the best result is selected as solution of the optimization process. The final results are presented in table 24.
The results of the optimization process performed improving the rotor geometry are noticeably better that those previously obtained, both in terms of the overall bulkiness and overall mass. In fact, the advantages related to considering such a geometry overcame those of considering a higher rotational speed. In the following sections an in-depth analysis is performed to assess whether such an improved geometry provides also good results with regard to rotor and frame vibrations and to kinetic energy decay rate.
With this new geometry it is much more complicated to evaluate the maximum stress the rotor is prone to, and the simplified formula (80) introduced in the preceding section is no longer viable. In this case, in fact, to obtain the maximum stresses the rotor is subject to a finite elements method must be implemented. Reasons for this are the complicated geometry of the rotor and the raised notch effects. Anyway, given to low stress values that where obtained in the previous cases, it can be stated with high confidence that a rotor with such a geometry can be realized. The only cases that are to be carefully evaluated are those where a thin disk is coupled with a big cylindrical ring.
Given the good quality of the results obtained with the improved rotor geometry, some additional cases are evaluated. With these new optimizations the same improved geometry is considered together with a higher value for the energy stored in the single rotor. The results are reported in the following tables 25 and 26.
Advanced single-rotor optimization 95
Optimization code Ntb-1 Ntb-2 Ntb-3 Ntb-4 Ntb-5 Ntb-6
Cost function weights C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 2,36E-05 1 2,36E-05 2 2,36E-05 2 2,36E-05 2 2,36E-05 2 2,36E-05 2 rotor mass 1 3,85E-01 1 3,85E-01 1 3,85E-01 2 3,85E-01 4 3,85E-01 6 3,85E-01 3 rotor vibration 1 6,03E-07 1 6,03E-07 1 6,03E-07 1 6,03E-07 1 6,03E-07 1 6,03E-07 4 frame vibration 1 0,013166 1 0,013166 1 0,013166 1 0,013166 1 0,013166 1 0,013166 5 rotor encumbrance 1 3,10E+02 2 3,10E+02 2 3,10E+02 1 3,10E+02 1 3,10E+02 1 3,10E+02 General parameters 1 rotor geometry variabili variabili variabili variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 60.000 60.000 60.000 3 stored energy [J] 400.000 400.000 400.000 400.000 400.000 400.000 4 initial rotor geometry ott ott ott ott ott ott 5 rotor density [kg/m^3] 2.500 2.500 2.500 2.500 2.500 2.500 6 perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x x 8 number of bearings (per side) 2 2 2 2 2 2 9 vehicle speed [m/s] 20 20 20 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v v v v 2 forcing for ground roughness v v v v v v 3 forcing for vehicle acceleration x x x x x x 4 11 dof model x x x x x x 5 15 dof model v v v v v v 6 imposed rotor symmetry v v v v v v 7 simplified rotor geometry v v v v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2,0:101 2,0:101 2,0:101 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 3,58E+00 3,96E+00 4,12E+00 4,44E+00 6,86E+00 9,59E+00 2 optimal length l2 [m] 2,000E-02 2,01E-02 2,000E-02 2,00E-02 2,000E-02 2,00E-02 3 optimal diameter D1 [m] 4,063E-02 2,789E-02 3,382E-02 2,856E-02 2,512E-02 2,500E-02 4 optimal diameter D2 [m] 1,272E-01 1,241E-01 1,082E-01 1,296E-01 1,383E-01 1,390E-01 5 optimal length l4 [m] 5,676E-02 7,074E-02 5,048E-02 5,067E-02 4,433E-02 5,948E-02 6 optimal length s1 [m] 3,671E-02 3,355E-02 4,769E-02 3,818E-02 3,743E-02 3,103E-02 Additional results 1 volume [m^3] 3,692E-03 2,987E-03 3,58E-03 3,49E-03 3,60E-03 3,20E-03 2 mass [Kg] 3,55E+00 3,64E+00 3,556E+00 3,309E+00 3,113E+00 3,297E+00
3 rotor number [---] 180 180 180 180 180 180 4 total volume[m^3] (20 kWh) 0,665 0,538 0,645 0,628 0,648 0,576 5 total mass [kg] (20 kWh) 639,2 655,1 640,1 595,6 560,4 593,5 6 maximum stress [MPa] 1,578E+02 1,501E+02 1,140E+02 1,637E+02 1,863E+02 1,884E+02
Table 24: final rotor geometry optimization, low weight material and improved geometry, 15 degrees of freedom model
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Optimization code Ptb-1 Ptb-2 Ptb-3 Ptb-4 Ptb-5 Ptb-6 Pb-7
Cost function weights C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 2,19E-03 1 2,19E-03 2 2,19E-03 2 2,19E-03 2 2,19E-03 2 2,19E-03 2 2,19E-03 2 rotor mass 1 3,16E-01 1 3,16E-01 1 3,16E-01 2 3,16E-01 4 3,16E-01 6 3,16E-01 8 3,16E-01 3 rotor vibration 1 2,65E-05 1 2,65E-05 1 2,65E-05 1 2,65E-05 1 2,65E-05 1 2,65E-05 1 2,65E-05 4 frame vibration 1 0,017389 1 0,017389 1 0,017389 1 0,017389 1 0,017389 1 0,017389 1 0,017389 5 rotor encumbrance 1 8,50E+02 2 8,50E+02 2 8,50E+02 1 8,50E+02 1 8,50E+02 1 8,50E+02 1 8,50E+02 General parameters 1 rotor geometry variabili variabili variabili variabili variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 60.000 60.000 60.000 60.000 3 stored energy [J] 600.000 600.000 600.000 600.000 600.000 600.000 600.000 4 initial rotor geometry ott ott ott ott ott ott ott 5 rotor density [kg/m^3] 2.500 2.500 2.500 2.500 2.500 2.500 2.500 6 perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x x x 8 number of bearings (per side) 2 2 2 2 2 2 2 9 vehicle speed [m/s] 20 20 20 20 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v v v v v 2 forcing for ground roughness v v v v v v v 3 forcing for vehicle acceleration x x x x x x x 4 11 dof model x x x x x x x 5 15 dof model v v v v v v v 6 imposed rotor symmetry v v v v v v v 7 simplified rotor geometry v v v v v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2,0:101 2,0:101 2,0:101 2,0:101 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 7,00E+00 4,12E+00 7,70E+00 4,23E+00 1,11E+01 9,95E+00 1,28E+01 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,00E-02 2,03E-02 2,00E-02 2,02E-02 2,00E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,530E-02 2,530E-02 4 optimal diameter D2 [m] 1,484E-01 1,039E-01 1,479E-01 9,616E-02 1,479E-01 9,221E-02 1,102E-01 5 optimal length l4 [m] 5,113E-02 1,143E-01 5,113E-02 4,403E-02 5,113E-02 5,042E-02 4,364E-02 6 optimal length s1 [m] 3,994E-02 4,072E-02 4,015E-02 6,753E-02 4,015E-02 6,569E-02 6,108E-02 Additional results 1 volume [m^3] 4,127E-03 3,925E-03 4,123E-03 4,246E-03 4,123E-03 3,98E-03 4,36E-03 2 mass [Kg] 3,96E+00 5,78E+00 3,96E+00 4,26E+00 3,96E+00 4,520E+00 4,150E+00
3 rotor number [---] 120 120 120 120 120 120 120 4 total volume[m^3] (20 kWh) 0,495 0,471 0,495 0,510 0,495 0,478 0,523 5 total mass [kg] (20 kWh) 475,1 693,9 475,7 511,5 475,7 542,4 498,0 6 maximum stress [MPa] 2,147E+02 1,051E+02 2,133E+02 9,013E+01 2,133E+02 8,288E+01 1,185E+02
Table 25: final rotor geometry optimization, low weight material and improved geometry, higher energy stored 1, 15 degrees of freedom model Advanced single-rotor optimization 97
Optimization code Rtb-1 Rtb-2 Rb-3 Rtb-4 Rtb-5
Cost function weights C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 2,19E-03 1 2,19E-03 2 2,19E-03 2 2,19E-03 2 2,19E-03 2 rotor mass 1 3,16E-01 1 3,16E-01 1 3,16E-01 2 3,16E-01 6 3,16E-01 3 rotor vibration 1 2,65E-05 1 2,65E-05 1 2,65E-05 1 2,65E-05 1 2,65E-05 4 frame vibration 1 0,017389 1 0,017389 1 0,017389 1 0,017389 1 0,017389 5 rotor encumbrance 1 8,50E+02 2 8,50E+02 2 8,50E+02 1 8,50E+02 1 8,50E+02 General parameters 1 rotor geometry variabili variabili variabili variabili variabili 2 initial rotor velocity [rpm] 60.000 60.000 60.000 60.000 60.000 3 stored energy [J] 800.000 800.000 800.000 800.000 800.000 4 initial rotor geometry ott ott ott ott ott 5 rotor density [kg/m^3] 2.500 2.500 2.500 2.500 2.500 6 perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x 8 number of bearings (per side) 2 2 2 2 2 9 vehicle speed [m/s] 20 20 20 20 20 Forcing and solver 1 forcing for rotor dynamics v v v v v 2 forcing for ground roughness v v v v v 3 forcing for vehicle acceleration x x x x x 4 11 dof model x x x x x 5 15 dof model v v v v v 6 imposed rotor symmetry v v v v v 7 simplified rotor geometry v v v v v Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2,0:101 2,0:101 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 8,27E+00 9,66E+00 9,16E+00 6,23E+00 1,23E+01 2 optimal length l2 [m] 2,002E-02 2,00E-02 2,000E-02 2,02E-02 2,00E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,551E-02 2,501E-02 4 optimal diameter D2 [m] 1,912E-01 1,912E-01 1,799E-01 1,474E-01 1,733E-01 5 optimal length l4 [m] 5,060E-02 5,060E-02 9,074E-02 7,302E-02 2,992E-02 6 optimal length s1 [m] 3,071E-02 3,071E-02 2,333E-02 3,924E-02 5,143E-02 Additional results 1 volume [m^3] 5,051E-03 5,051E-03 4,66E-03 4,072E-03 6,04E-03 2 mass [Kg] 4,217E+00 4,22E+00 4,724E+00 5,14E+00 3,968E+00
3 rotor number [---] 90 90 90 90 90 4 total volume[m^3] (20 kWh) 0,455 0,455 0,419 0,366 0,543 5 total mass [kg] (20 kWh) 379,5 379,5 425,2 462,5 357,1 6 maximum stress [MPa] 3,561E+02 3,561E+02 3,156E+02 2,117E+02 2,928E+02
Table 26: final rotor geometry optimization, low weight material and improved geometry, higher energy stored 2, 15 degrees of freedom model
4.3 Evaluation of the results In this section an in-depth analysis of some of the most significant cases between those introduced is provided. The analysis of at least one rotor resulting from every optimization setting is reported, and this is selected as the one producing the better results. The configurations at issue are the one highlighted in blue in the results tables.
Flywheel energy storage 98
Two kinds of parameters are investigated: firstly, some quantities related to the dynamics of the single rotor system are analyze and plot with respect to time. These quantities are the kinetical energy stored in the system, the rotor vibration, and the frame vibration. Secondly, parameters related to the mass end encumbrance of the single rotor and multirotor solution are illustrated together with the number of rotors necessary to store 20 퐾푊ℎ.
It is to be noted that in this section a linear kinetical energy decay model is used. This model tents to overestimate the overall kinetical energy decay rate, and a more proper model to depict this trend would be an exponential one. However, with the introduction of the kinetical energy decay rate, it is not intended to forecast how rapidly the system dissipates energy, but it is used only for comparing the different results.
Additional data on these optimal configurations are presented in appendix F.
Advanced single-rotor optimization 99
4.3.1 Rotor Jtb-2 General information is here listed for the optimal rotor configuration Jtb-2.
Figure 59: graphical representation of optimal rotor Jtb-2
Quantity Unit Value Energy stored per rotor 퐽 400.000 Number of rotors (multirotor) --- 180 Kinetical energy decay rate 퐽/푠 -270,68 Kinetical energy decay rate %/푠 -0,0677% Single rotor mass 퐾푔 4,5825 System mass (multirotor) 퐾푔 824,8412 Single rotor volume 푚3 0,0034 System volume (multirotor) 푚3 0,6072 Table 27: optimal rotor Jtb-2 characteristics
Figure 60: time evolution of the kinetical energy stored in the rotor, configuration Jtb-2
Flywheel energy storage 100
4.3.2 Rotor Ltb-1 General information is here listed for the optimal rotor configuration Ltb-1.
Figure 61: graphical representation of optimal rotor Ltb-1
Quantity Unit Value Energy stored per rotor 퐽 400.000 Number of rotors (multirotor) --- 180 Kinetical energy decay rate 퐽/푠 -585,74 Kinetical energy decay rate %/푠 -0,1464% Single rotor mass 퐾푔 3,0767 System mass (multirotor) 퐾푔 553,8126 Single rotor volume 푚3 0,0009 System volume (multirotor) 푚3 0,1708 Table 28: optimal rotor Ltb-1 characteristics
Figure 62: time evolution of the kinetical energy stored in the rotor, configuration Ltb-1 Advanced single-rotor optimization 101
4.3.3 Rotor Ntb-5 General information is here listed for the optimal rotor configuration Ntb-5.
Figure 63: graphical representation of optimal rotor Ntb-5
Quantity Unit Value Energy stored per rotor 퐽 400.000 Number of rotors (multirotor) --- 180 Kinetical energy decay rate 퐽/푠 -0,2375 Kinetical energy decay rate %/푠 -0,0000594% Single rotor mass 퐾푔 3,1132 System mass (multirotor) 퐾푔 560,3780 Single rotor volume 푚3 0,0036 System volume (multirotor) 푚3 0,6480 Table 29: optimal rotor Ntb-5 characteristics
Figure 64: time evolution of the kinetical energy stored in the rotor, configuration Ntb-5
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4.3.4 Rotor Ptb-3 General information is here listed for the optimal rotor configuration Ptb-3.
Figure 65: graphical representation of optimal rotor Ptb-3
Quantity Unit Value Energy stored per rotor 퐽 600.000 Number of rotors (multirotor) --- 120 Kinetical energy decay rate 퐽/푠 -0,3003 Kinetical energy decay rate %/푠 -0,0000500% Single rotor mass 퐾푔 3,9286 System mass (multirotor) 퐾푔 471,4317 Single rotor volume 푚3 0,0011 System volume (multirotor) 푚3 0,1302 Table 30: optimal rotor Ptb-3 characteristics
Figure 66: time evolution of the kinetical energy stored in the rotor, configuration Ptb-3 Advanced single-rotor optimization 103
4.3.5 Rotor Rtb-2 General information is here listed for the optimal rotor configuration Rtb-2.
Figure 67: graphical representation of optimal rotor Rtb-2
Quantity Unit Value Energy stored per rotor 퐽 800.000 Number of rotors (multirotor) --- 90 Kinetical energy decay rate 퐽/푠 -1,1265 Kinetical energy decay rate %/푠 -0,0001408% Single rotor mass 퐾푔 4,1808 System mass (multirotor) 퐾푔 376,2698 Single rotor volume 푚3 0,0018 System volume (multirotor) 푚3 0,1631 Table 31: optimal rotor Rtb-2 characteristics
Figure 68: time evolution of the kinetical energy stored in the rotor, configuration Rtb-2
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4.4 Final rotor selection From the data related to the rotor characteristics and dynamics reported in the preceding section, and from the data related to other rotors that for brevity are not here reported, the following conclusion can be drawn.
1. The selected material for the rotors in the flywheel energy storage system is a low-density composite material. Reason of this is mainly related to the high stresses that the system must bare as a consequence of the centrifugal load. A high-density material flywheel – like an iron flywheel for instance – not only generates higher stresses on the rotor axis, but also is capable of a lower ultimate tensile stress. Finally, a low-density flywheel is the best alternative for safety reasons. 2. For what concerns the rotor mass and encumbrance, the best solutions are those obtained with the improved rotor geometry. At equal mass, or equal volume, and considering all other parameters unchanged, these are the solutions that ensure the higher energy stored. It is also stated that the advantages related to considering such a geometry overcame those of considering a higher rotational speed of 80.000 푟푝푚. 3. The vibrations of both the frame and the rotor are considerably reduced when the improved rotor geometry is implemented. Consequently, less energy is dissipated on the rotor and the frame suspension systems, leading to a drastic reduction of the rotor kinetic energy decay rate. The decay rate is reduced of two orders of magnitude when considering similar optimal configurations for the simple disc and the improved geometry rotors. 4. Additional advantages, both in the reduction for the overall mass and encumbrance, and in the reduction of the kinetic energy decay rate, can be found when considering a higher energy stored in a single rotor. These advantages, however, are not as critical as those related to the improved geometry, moreover, they come at the cost of making a bigger rotor, and so more difficult to manufacture and to control.
For these reasons the rotors selected for the multi-rotor optimization in the following chapter are all characterized by the improved geometry. Moreover, the three alternatives selected differ on the base of the energy stored. Finally, to choose one option over the others in the same energy stored category, the best solution in terms of overall mass for the 20 퐾푊ℎ system is considered. Therefore, the selected rotors for the multi-rotor optimization are Ntb-5, Ptb-3 and Rtb-2.
5 Multi-rotor optimization
In this chapter a multi-rotor model representing a vehicle equipped with a flywheel-based energy storage system is defined. For this system the three different rotors that were selected in chapter 4 are considered. A set of different configurations for every rotor choice is simulated and studied to obtain the best multi-rotor configuration. The possibility of performing the rotor positioning task with an optimization process is analyzed.
The dynamic behavior of the selected configurations is simulated, and the results are presented. Finally, the data collected in this chapter, together with those from the preceding ones, are used in chapter 6 to draw final conclusions on the feasibility of such a solution.
Flywheel energy storage 108
5.1 Rotor positioning The rotor positioning task is the last problem to be addressed in order to define a flywheel based energy storage system. Before developing a procedure to define the rotor positioning, the main characteristics that are desired for the solution are now listed.
1. The solution must provide good results in terms of the elementary cost functions previously introduced, in that the kinetical energy decay rate must be minimized together with the frame and rotor vibrations. Moreover, the total volume occupied by the multirotor energy storage system must be as low as possible. 2. During the rotor acceleration and deceleration phases – and so during the loading and unloading of the energy storage system – the pitch, roll and yaw torques must be minimized. This characteristic is of crucial importance to guarantee good dynamics for the moving vehicle. 3. The overall system center of mass must be as low as possible and, moreover, the solution must have encumbrance characteristics that are compatible with the internal volume of a private road car.
In general, the solution of this problem is not an easy task because of the many alternatives available, however, by analyzing the characteristics of the rotors selected in chapter 4, some simplifications can be introduced. Firstly, a vertical rotation axis characterizes these rotors, and so for every rotor only four coordinates must be selected. These are the three displacements for the rotor global reference origin, and one coordinate that represents if the rotation axis is facing upwards or downwards, and so if the rotor is rotating clockwise or counterclockwise. Secondly, these solutions are already designed to minimize the elementary cost functions, and so this task should not be the main priority in this phase.
In accordance to these observations, the multi-rotor evaluation procedure illustrated in the scheme of Figure 69 is introduced.
Figure 69: multi-rotor evaluation procedure
With the first step the rotors global reference origins are positioned, this is done according with the third point of the desired solution characteristics. This means that a small fraction of the frame volume is assigned to the flywheel energy storage system. This region is a thin parallelepiped-shaped layer that lies at the bottom of the frame and stretches until the very end of it. The layer thickness is enough to accommodate the rotors with vertical rotation axis. In this volume the largest possible number of rotors are placed symmetrically with respect to the vehicle symmetry planes. A minimum distance between two rotors and between rotor and frame boundaries is imposed. No constraints on the total energy this system must store – and so on the Multi-rotor optimization 109 number of rotors – is set. On the contrary, the total energy such a system, composed of a single layer of rotor, can store is one of the parameters that are to be considered once the positioning phase is completed.
The result of this phase is the number of rotors that can fit in the assigned volume and a table containing the coordinates of the origin of each rotor global reference, and so the position of every rotor in the frame. As an example, it is reported in Figure 70Figure 70: rotor positioning example the plot of 푦푟표푡 and 푧푟표푡 coordinates for the case of rotor Ntb-5.
Figure 70: rotor positioning example
In the second step the direction of the rotors rotation axis must be selected, and so it must be decided if the single rotor is rotating clockwise or counterclockwise. The rotation direction of every single rotor is determined in order to minimize the pitch, roll and yaw torques that arise during the flywheel acceleration and deceleration phases. Since all three of the selected rotors are characterized by having a vertical rotation axis, the pitch and roll torques are expected to be null, while the yaw torque must be minimized. In other words, with this step the desired characteristic of the solutions listed at point 2 are meet. The details on how the procedure is performed are provide in the following subsection.
Finally, as a third step, the model with the just obtained characteristics is created and a simulation is run. The elementary cost functions are evaluated and the accordance between these results and the solution desired characteristics listed at point 1 is evaluated. This phase is performed in the next section together with all the details regarding the multirotor system.
5.1.1 Selection of the rotation direction As previously stated, the rotation direction of the rotors is set to minimize the yaw torque that arises from the flywheel acceleration and deceleration phases. For this reason, a model capable of measuring the yow torque during the flywheel acceleration phase is developed.
All the different combinations of rotation directions are possible solution for this problem. Despite the huge number of alternatives, and before developing an optimization process to assess this problem, some peculiar configurations are evaluated. These are the ones where the rotation direction is the same for all the rotors on a line perpendicular to the vehicle progress direction, and where the rotation direction is the same for all
Flywheel energy storage 110
rotors on transverse lines. These configurations are represented in the following plots of Figure 71 for rotor Ptb-3, and are referred to as the horizontal and the transversal configurations.
Figure 71: the two alternatives initially considered for the rotors rotation direction. In the first one, the setup is composed of alternated rows of rotors rotating in the opposite direction, arranged diagonally. In the second one the rows are arranged horizontally.
The flywheel acceleration for this test is set to 2000 푟푝푚/푠. The yaw torque that arises from the acceleration of the energy storage systems obtained with rotor Ptb-3, in the two previously represented configurations, is represented in the two following graphs of Figure 72 and Figure 73.
It can be noted that in both cases the yaw torque is very little and of not particular importance for the system dynamics. In particular the configuration where the rotation direction is the same for all rotors on transverse lines – the transversal configuration – provides the best results. Given the high quality of the results, an optimization process to obtain better ones is considered not necessary, and the transversal configuration is selected as the solution of the rotor rotation problem.
It is to be noted that the results here reported are for the Ptb-3 rotor, because for this an uneven number of flywheel can fit in the bottom layer of the frame. In fact, for cases where the number of rotors is even, the same results are achieved but with an even lower torque magnitude.
Figure 72: yow torque during rotors acceleration for the transversal configuration Multi-rotor optimization 111
Figure 73: yow torque during rotors acceleration for the horizontal configuration
5.2 Final solutions In this section the results obtained by applying the previously defined procedure are listed. Two kinds of parameters are investigated: firstly, some quantities related to the dynamics of the multi rotor system are analyze and plot with respect to time. These quantities are the kinetical energy stored in the system and the frame vibration. Secondly, parameters related to the mass end encumbrance of the multirotor solution are presented together with the number of rotors that can fit in a single layer. Moreover, also the system representations are presented. With these, different rotor colour represents the different rotation directions.
As it was stated for the result presented in the previous chapter, it is to be noted that in this section a linear kinetical energy decay model is used. This model tents to overestimate the overall kinetical energy decay rate, and a more proper model to depict this trend would be an exponential one. However, with the introduction of the kinetical energy decay rate, it is not intended to forecast how rapidly the system dissipates energy, but it is used only for comparing the different results.
A further analysis of these data is provided in the next chapter together with a comparison between this technology and other kind of energy storage systems.
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5.2.1 Solution 1: from rotor Ntb-5
Figure 74: graphical representation of the multirotor system based on the Ntb-5 rotor
Quantity Unit Value Stored energy per rotor 퐽 400.000 Total stored energy 퐽 39.200.000 Number of rotors --- 98 Decay rate 퐽/푠 -24,5741 Decay rate %/푠 -0,0000627% Rotor mass 퐾푔 3,1132 System mass 퐾푔 305,0947 System encumbrance 푚3 0,4043 Energy density 퐾퐽/퐾푔 128,5 Energy density 푀퐽/푚3 97,0 Table 32: characteristics of the multi-rotor configuration base on rotor Ntb-5
Figure 75: frame vibration FFT, 푥푇 direction Multi-rotor optimization 113
5.2.2 Solution 2: from rotor Ptb-3
Figure 76: graphical representation of the multirotor system based on the Ptb-3 rotor
Quantity Unit Value Stored energy per rotor 퐽 600.000 Total stored energy 퐽 54.600.000 Number of rotors --- 91 Decay rate 퐽/푠 -32,0551 Decay rate %/푠 -0,0000587% Rotor mass 퐾푔 3,9645 System mass 퐾푔 360,7722 System encumbrance 푚3 0,4037 Energy density 퐾퐽/퐾푔 151,3 Energy density 푀퐽/푚3 135,2 Table 33: characteristics of the multi-rotor configuration base on rotor Ptb-3
Figure 77: frame vibration FFT, 푥푇 direction
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5.2.3 Solution 3: from rotor Rtb-5
Figure 78: graphical representation of the multirotor system based on the Rtb-5 rotor
Quantity Unit Value Stored energy per rotor 퐽 800.000 Total stored energy 퐽 52.800.000 Number of rotors --- 66 Decay rate 퐽/푠 -65,6273 Decay rate %/푠 -0,0001243% Rotor mass 퐾푔 4,2167 System mass 퐾푔 278,3018 System encumbrance 푚3 0,4038 Energy density 퐾퐽/퐾푔 189,7 Energy density 푀퐽/푚3 130,8 Table 34: characteristics of the multi-rotor configuration base on rotor Rtb-5
Figure 79: frame vibration FFT, 푥푇 direction
6 Conclusions
In this last chapter, a detailed study of the final configurations defined in chapter 5 is performed. Therefore, the solutions here analyzed are results of the two optimization processes. To give an overall evaluation of these results, a comparison between the characteristics of the obtained vehicle and the ones related to an electrical vehicle – selected between those currently available on the market – is presented. In this evaluation, five main characteristics are analyzed. These are: the overall storage system energy density – both in terms of energy over mass, and energy over volume – the encumbrance and the weight of the storage system and, finally, the energy decay rate. Moreover, additional thoughts regarding the effects that such a flywheel-based energy storage system cause on the vehicle during motion are presented.
General conclusions are drawn on the possibility of realizing a road automobile, for private or public use, that relies on a flywheel based energy storage system with mechanical support for the rotors as primary energy source.
Flywheel energy storage 118
6.1 Analysis of the results
6.1.1 Mass and mass-based energy density The overall mass of the constructive solutions for the flywheel-based energy storage systems identified in chapter 5 is analyzed in this section. A comparative representation of the weight of these systems is presented in Figure 80. However, it must be considered that the relevant mass related parameter in the construction of an energy storage system for vehicle use is not the overall mass, but its mass energy density. A representation of this quantity for the three solutions is provided in Figure 81. In this same picture, the energy density of the identified solutions is compared with the energy density of some of the most significant lithium-ion batteries of comparable sizes. These are the Tesla Powerwall 2 – a domestic energy storage device – and the battery developed for Tesla Model S6, and so an energy storage for on vehicle use.
As it can be noted from the graph, increasing the energy stored in each individual rotor the overall energy density of the system is improved. In any case, some additional data must be collected to fully assess whether such a trend could be identified. However, when it comes to energy density, much better results are obtained with the two lithium-ion batteries with respect to those obtained with a flywheel based system.
Rtb-2 based system
Ptb-3 based system
Ntb-5 based system
0 50 100 150 200 250 300 350 400 mass [Kg]
Figure 80: overall mass of the energy storage systems
Tesla Model S Tesla Powerwall
Rtb-2 based system Ptb-3 based system Ntb-5 based system
0 100 200 300 400 500 600 Energy density [KJ/kg]
Figure 81: mass energy density of the energy storage systems
6 The data presented are related to Tesla Model S 75D. Conclusions 119
6.1.2 Encumbrance and volume-based energy density In this section the volume occupied by the energy storage systems defined in chapter 5 is evaluated. A comparative representation of the volume of these systems is provided in Figure 82. As it was done for the mass, in Figure 83 the volume energy density of the obtained solutions is presented, together with the same data for Tesla Powerwall 2 and Tesla Model S.
The three flywheel-based solutions do not present any particular trend for what concerns the volumetric energy density at the change of the amount of energy stored in each individual rotor. In any case, some additional data must be collected to fully assess whether such a trend could be identified. Finally, from the same graph, it must also be noted that the values for the volume energy density of the flywheel energy storage systems are much smaller than the those obtained with the two lithium ion batteries. Even if this difference is more significant than the one observed for the mass energy density, this latter one represents a more crucial problem in the development of a flywheel-based energy storage system.
Rtb-2 based system
Ptb-3 based system
Ntb-5 based system
Volume [m3]
Figure 82: overall volume of the energy storage systems
Tesla Model S Tesla Powerwall
Rtb-2 based system Ptb-3 based system Ntb-5 based system
0 50 100 150 200 250 300 350 400 450 Energy density [MJ/m3]
Figure 83: volume energy density of the energy storage systems
6.1.3 Self-discharge In this section the storing efficiency of the flywheel-based systems is analyzed and compared with the results for the already presented lithium ion batteries solutions. Two different decay rates are considered: the first
Flywheel energy storage 120
one is the kinetical energy decay rate observed during the vehicle motion at the constant speed of 20 푚/푠, and so subject to the forcing introduced by the ground roughness, the second one is the kinetical energy decay rate that can be evaluated for the vehicle at rest. Both of these quantities are presented in Figure 84. In Figure 85 and Figure 86 the decay of the energy stored in the system is forecasted on the base of the data in Figure 84. These trends are related to a hypothetical case where the initial energy stored for each system is equal to 50 푀퐽. In the first figure, the data related to the vehicle during motion are presented while, in the second one, the once for the vehicle at rest are considered. In the same figures, the time evolution of the energy stored for the flywheel based solutions is compared with the one from a Tesla Model S. In this latter case, the self-discharge effect is considered for both the stationary and the dynamic evaluations.
It can be noted that, for the vehicle at rest, a much better storing efficiency is obtained with a flywheel-based system.
Ptb-3 based system
Ptb-3 based system
Ntb-5 based system
-1,E-06 -1,E-06 -1,E-06 -8,E-07 -6,E-07 -4,E-07 -2,E-07 0,E+00 Kinetical energy daced rate [%/s]
Figure 84: kinetical energy decay rate for the vehicle during motion (in orange) and at rest (in blue)
6,E+07
5,E+07
4,E+07
3,E+07 Ntb-5
Energy stored [J] stored Energy 2,E+07 Ptb-3 Rtb-2 1,E+07 Tesla Model S
0,E+00 0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 Time [h]
Figure 85: forecasted evolution of the energy stored, vehicle in motion Conclusions 121
6,E+07
5,E+07
5,E+07
Ntb-5 4,E+07 Energy stored [J] stored Energy Ptb-3
4,E+07 Rtb-2
Tesla Model S
3,E+07 0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 Time [h]
Figure 86: forecasted evolution of the energy stored, vehicle at rest
6.1.4 Effects on the vehicle dynamics From the evaluations performed in section 5.1.1, it can be noted that, for the rotor configuration proposed in the same section, the effects of the flywheel energy storage system charging and discharging phases on the vehicle dynamics are negligible. Moreover, a further reduction of these effects can be achieved selecting an even number of rotors, and imposing for them the rotation directions detailed in the same section. A second improvement could also be obtained by stacking two layers of rotors one over the other. In this case the two rotors sharing the same rotations axis must be designed to rotate in opposite directions.
6.2 Final conclusions and future developments As it is detailed in the preceding sections, the mass and the volume energy density for the developed flywheel-based energy storage systems are considerably lower than what is obtained for some commercially available lithium ion battery solutions. Moreover, it has also to be considered that the mass and the volume of the complete kinetical energy storage systems are negatively affected by all necessary secondary systems. These include a continuously variable transmission, a control unit for the energy storing and many others.
To improve the mass energy density, a higher rotational speed for the flywheels can be investigated. This opportunity must be carefully evaluated because of the problems arising from imposing an even higher rotational speed for the rotors. In this case, in fact, it might be hard to achieve a good dynamic for the flywheel without reducing at really low values the percental eccentricity, and without introducing some additional stiffening element in parallel to the bearings supporting the rotor. If these precautions are not taken, high vibrations might arise, leading to a high energy density decay rate and to a short life for the bearings. Finally, additional problems connected to a higher rotational speed of the rotor regard the structural stress of the rotor itself. In these case, a in depth structural analysis of the spinning rotor must be performed.
A second alternative for improving the mass energy density of the flywheel based energy storage is to investigate the trend that can be noted in Figure 81. In this graph, in fact, an increase in the mass energy density is observed when a solution characterized by higher energy per single rotor is considered. Anyway,
Flywheel energy storage 122
further data must be collected in order to check this trend, since the three values presented in this figure are not sufficient to draw final conclusions. In case the opportunity of realising such a solution is exploited, a compromise between improving the mass energy density and not reducing excessively the kinetical energy decay rate must be done. In fact, from Figure 84 it is noted that with a solution characterized by higher energy stored in the single rotor, a higher kinetical energy decay rate is obtained.
To improve the volume energy density, the opportunity of increasing the flywheel speed can be once more investigated. The observations introduced when dealing with the mass energy density must be considered. Anyway, it is here reminded that the problem of reducing the mass energy density is more critical for vehicle application of energy storage systems of any kind, with respect to the reduction in the volume energy density.
When it comes to the storing efficiency, very good results can be observed from Figure 86, especially when these are compared with those for a lithium ion battery. However, it has to be reminded that the model used for developing these results do not take into account of all the effects that might influence the rotor dynamics. A more complete model could be developed for these purpose, even if very good results are expected. To further improved these, low damping bearings can be introduced.
Additional development must be performed before drawing final conclusions on the feasibility of a road vehicle equipped whit a kinetical primary energy storage. These developments must mainly deal with the problematics related to the design of a continuously variable transmission and with an overall cost analysis. Even if the results obtained in terms of mass energy density might seem disappointing, it is considered possible to perfect these with the techniques yet introduced. Moreover, these results were compared with some of the best lithium ion battery solutions available on the market in 2017. This technology had some major developments in the past 10 years, and if the results for the flywheel based solutions were compared with the best batteries available on the market only 10 years ago, these would look much more encouraging. [16]
800
700
600
500
400
300
200
Mass energy density [KJ/Kg] energydensity Mass 100
0 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
Figure 87: history of development of secondary batteries in view of energy density [16]
Furthermore, for the sake of completeness, one additional study that must be performed to assess the feasibility of such a solution is the one related to the gyroscopic effects arising from the rotors during turning. It must be evaluated weather the gyroscopic torques arising in this phase are significant enough to affect the vehicle dynamics. Conclusions 123
Finally, given the excellent loading and unloading properties of a flywheel energy storage, it is believed that such a solution could provide good functionality when coupled with other main energy sources in hybrid vehicles. These energy sources could be both a chemical battery or an internal combustion engine. The kinetical energy of a vehicle could efficiently be recovered during breaking and, as shown in these work, solutions that minimize the effects of the rotor acceleration and deceleration on the vehicle dynamics can be obtained.
Bibliography 125
7 Bibliography
[1] G. Genta, Politecnico di Torino, Kinetic energy storage, Butterworth & Co. (Publishers) Ltd., 1985.
[2] E. J. H. a. A. R. Hall, History of Technology, C. Singer, 1954.
[3] G. N.V., Flywheel Engines, Mashinostroyeniye Press, 1976.
[4] C. Posthumus, Fathers of Invention, Hamlyn / Phoebus, 1977.
[5] D. Castelvecchi, Spinning into control: High-tech reincarnations of an ancient way of storing energy, Science News, 2007.
[6] E. Beedham, Flywheel Generators for the JET Experiment, GEC Journ. of Sc. and Tech., 1980.
[7] K. I. e. a. T. Hattori, Rotating Strength of Glass-Carbon Fiber Reinforced Hybrid Composite Discs, Bull, of Japan Soc. of Mech. Eng., 1982.
[8] R. Miller, “Flywheels Gain as Alternative to Batteries,” DataCenter Knowledge, 2007. [Online]. Available: http://www.datacenterknowledge.com/.
[9] Beacon Power Corporatio, “Flywheel-based Solutions for Grid Reliability,” Wayback Machine, 2006.
[10] “FIA, Federation Internationale de l'Automobile,” 2009. [Online].
[11] “Formula One race technology to power buses in Oxford,” BBC News, Oxford, 2014. [Online]. Available: http://www.bbc.com/news/uk-england-oxfordshire-29030173.
[12] Oerlikon, The Oerlikon Electrogyro, Automobile Engineer, 1955.
[13] M. Trend, “The GYROBUS: Something New Under the Sun?,” 1952.
[14] M. Mboka, “Leopoldville 1954 – Transports en Commun de Leopoldville hits the streets,” 2011. [Online]. Available: http://kosubaawate.blogspot.it/2011/10/leopoldville-1954-transports-en- commun.html.
[15] D. M. Henderson, “Euler Angles, Quaternions and Transformation Matrices,” McDonnel Douglas Technical Services Co. - NASA, 1977.
[16] H. Kawamoto, “Trends of R&D on Materials for High-power and Large-capacity Lithium-ionBatteries for Vehicles Applications,” Quarterly Review , no. 36, 2010.
Flywheel energy storage 126
8 Appendix
8.1 Appendix A The results from the simulations of the system dynamics performed imposing the initial conditions reported in table 6 are here presented. The units used in these plots are those of the international system. The simulations are numbered from one to six according to the numbering introduced in the same table. The accordance with the expected results is discussed.
Simulation 1
Appendix 127
The results correspond to the expected one of a second order system in the 푥퐺 direction, characterized by a stiffness equal to two times the bearing stiffness and a damping equal to two times the bearing damping. All the other degrees of freedom were expected not to be excited.
Simulation 2
The results correspond to the expected one of a second order system in the 푦퐺 direction, characterized by a stiffness equal to two times the bearing stiffness and a damping equal to two times the bearing damping. All the other degrees of freedom were expected not to be excited.
Flywheel energy storage 128
Simulation 3
The results correspond to the expected one of a second order system in the 푧퐺 direction, characterized by a stiffness equal to the bearing stiffness and a damping equal to the bearing damping. All the other degrees of freedom were expected not to be excited.
Appendix 129
Simulation 4
The results correspond to the expected one. No excitation was expected on the three displacement degrees of freedom, while a transitory vibration was expected on the two angles representing the rotor orientation.
Flywheel energy storage 130
Simulation 5
The results correspond to the expected one. No excitation was expected on the three displacement degrees of freedom, while a transitory vibration was expected on the two angles representing the rotor orientation.
Appendix 131
8.2 Appendix B
The dynamics of the frame, after an initial displacement off its equilibrium point of 1 cm in the 푥푇 direction, is here represented. The results correspond to the expected one of a second order system in the 푥푇 direction, characterized by a stiffness equal to four times the suspensions stiffness and a damping equal to four times the suspensions damping. All the other degrees of freedom were expected not to be excited.
Flywheel energy storage 132
8.3 Appendix C For completeness, some of the results of the preliminary observations are here reported also for a new rotor configuration.
The results obtained in section 3.3.1, for preliminary observation 1, are evaluated for a different rotor
configuration. In this case configuration 훼2 is selected.
Appendix 133
This new evaluation provides similar trends to the one previously obtained, and, for this reason, the conclusions already drawn are confirmed.
The results obtained in section 3.3.1, for preliminary observation 2, are evaluated for a different rotor configuration. In this case configuration 훼2 is selected.
The same results can be stated from this second configuration.
Flywheel energy storage 134
8.4 Appendix D General information is here listed for the optimal rotor configuration Ctb-1.
configuration Ctb-1 representation
Evolution of the kinetical energy stored in the rotor over time, configuration Ctb-1
Rotor vibration over time, configuration Ctb-1 Appendix 135
Frame vibration over time, configuration Ctb-1
The rotor mass and the used volume – objects of cost functions 퐶2 and 퐶5 – are reported.
푚푟표푡 = 7,801 푘푔
3 Vrot = 0,00303 m
General information is here listed for the optimal rotor configuration Ctb-2.
Evolution of the kinetical energy stored in the rotor over time, configuration Ctb-2
Flywheel energy storage 136
Rotor vibration over time, configuration Ctb-2
Frame vibration over time, configuration Ctb-2
The rotor mass and the used volume – objects of cost functions 퐶2 and 퐶5 – are reported.
푚푟표푡 = 7,801 푘푔
3 Vrot = 0,00337 m
General information is here listed for the optimal rotor configuration Ctb-3. Appendix 137
Evolution of the kinetical energy stored in the rotor over time, configuration Ctb-3
Rotor vibration over time, configuration Ctb-3
Frame vibration over time, configuration Ctb-3
Flywheel energy storage 138
The rotor mass and the used volume – objects of cost functions 퐶2 and 퐶5 – are reported.
푚푟표푡 = 7,801 푘푔
3 Vrot = 0, 00304 m
Appendix 139
8.5 Appendix E Additional information on optimal configuration Etb-1.
Rotor vibration over time, configuration Etb-1
Rotor vibration FFT, configuration Etb-1
Flywheel energy storage 140
Frame vibration over time, configuration Etb-1
Frame vibration FFT, configuration Etb-1 Appendix 141
Additional information on optimal configuration Etb-2.
Rotor vibration over time, configuration Etb-2
Rotor vibration FFT, configuration Etb-2
Flywheel energy storage 142
Frame vibration over time, configuration Etb-2
Frame vibration FFT, configuration Etb-2
Appendix 143
Additional information on optimal configuration Etb-3.
Rotor vibration over time, configuration Etb-3
Rotor vibration FFT, configuration Etb-3
Flywheel energy storage 144
Frame vibration over time, configuration Etb-3
Frame vibration FFT, configuration Etb-3
Appendix 145
8.6 Appendix F Additional information on optimal configuration Jtb-2.
Rotor vibration over time, configuration Jtb-2
Rotor vibration FFT, configuration Jtb-2
Flywheel energy storage 146
Frame vibration over time, configuration Jtb-2
Frame vibration FFT, configuration Jtb-2
Appendix 147
Additional information on optimal configuration Ltb-1.
Rotor vibration over time, configuration Ltb-1
Rotor vibration FFT, configuration Ltb-1
Flywheel energy storage 148
Frame vibration over time, configuration Ltb-1
Frame vibration FFT, configuration Ltb-1
Appendix 149
Additional information on optimal configuration Ntb-5.
Rotor vibration over time, configuration Ntb-5
Rotor vibration FFT, configuration Ntb-5
Flywheel energy storage 150
Frame vibration over time, configuration Ntb-5
Frame vibration FFT, configuration Ntb-5
Appendix 151
Additional information on optimal configuration Ptb-3.
Rotor vibration over time, configuration Ptb-3
Rotor vibration FFT, configuration Ptb-3
Flywheel energy storage 152
Frame vibration over time, configuration Ptb-3
Frame vibration FFT, configuration Ptb-3
Appendix 153
Additional information on optimal configuration Rtb-2.
Rotor vibration over time, configuration Rtb-2
Rotor vibration FFT, configuration Rtb-2
Flywheel energy storage 154
Frame vibration over time, configuration Rtb-2
Frame vibration FFT, configuration Rtb-2