Tyre non-uniformities and their eect on chassis vibrations

J.Sueleers

DC 2010.058

Master's thesis

Coach: Dr. Ir. I.J.M. Besselink (TU/e) Supervisor: Prof. Dr. H. Nijmeijer (TU/e) Committee member: Ir. J. de Gier (Vredestein) Committee member: Dr.Ir. Th. Hofman (TU/e) Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Group Eindhoven, October, 2010 Contents

1 Introduction 6

2 Literature survey 8

3 Full vehicle multibody model development 21 3.1 Introduction ...... 21 3.2 Front axle ...... 23 3.3 Rear axle ...... 30 3.4 Full vehicle model validation ...... 34

4 Modeling tyre non-uniformities 41 4.1 Full non-linear model ...... 41 4.2 Study with a nonlinear tyre model in the time domain ...... 46 4.3 Linearisation and transformation to the frequency domain ...... 49 4.4 Analytical study in the frequency domain ...... 51 4.5 Concluding remarks ...... 55

5 Simulation study on tyre non-uniformities 56 5.1 Radial runout ...... 58 5.2 Radial stiness variation ...... 60 5.3 Measured tyre non-uniformity with a peak-characteristic ...... 62 5.4 Measured tyre non-uniformity with a harmonic-characteristic ...... 64 5.5 Run-out and stiness variation combined ...... 65 5.6 Concluding remarks ...... 67

6 Conclusions and recommendations 68 6.1 Conclusions ...... 68 6.2 Recommendations ...... 69

Bibliography 69

Appendices 71

A Vehicle model parameters 72

B Additional vehicle model tests 80

C Tyre model parameters 87

1 Summary

During the production of a tyre several components have to be assembled. Parts of the carcass and tread material, available as a linear strip, are wound around the circumference of already assembled parts, cut and both ends welded together. It is not hard to imagine the possibility of overlaps or gaps when the belts are wound around the other components. It is also possible that the belts are not exactly positioned at the centre of a tyre (i.e. deviations in lateral direction). Within certain production tolerances all these things will happen to all the single components used in a tyre and this will o course have its inuence on the symmetry and uniformity of a tyre.

Tyre non-uniformities, as these imperfections are usually called, is the main subject of this report. The scope of this research and report is to give some basic insight into the inuence of some of these known non-uniformities. In particular the force generation characteristics of a non-uniform tyre and the vehicle interaction are studied. Since this is a rst step into modeling tyre non-uniformities and its inuence on chassis vibrations, the analysed non-uniformities will be restricted to 2D in-plane and the vehicle model used will be equipped with only a single non-uniform tyre. Lateral force generation and lateral non-uniformities of a tyre and their inuence on chassis vibrations is not part of this research and may be the subject of future research.

To analyse the eect of tyre non-uniformities on chassis vibrations, although only in-plane non-uniformities are used in this report, a full vehicle model with detailed suspension layout and compliance has been modeled in cooperation with a fellow student.

A nonlinear non-uniform tyre model has been developed and validated with research per- formed by prof. H.B. Pacejka. This tyre model is used to simulate force variations resulting from radial runout, radial stiness variation and two types of measured non-uniformities and their eect on vehicle vibrations.

2 Samenvatting

Tijdens de productie van een band worden verschillende componenten geassembleerd. Gedeeltes van het carcass en proel materiaal, aangeleverd als een rechte strip, worden rond de reeds geassembleerde onderdelen gewonden, gesneden en de uiteindes worden aan elkaar gelast. De mogelijkheid van overlappingen of gaten bij deze manier van werken is duidelijk. Ook in lat- erale richting kan de riem niet perfect geplaatst zijn. Binnen bepaalde productietoleranties zullen deze imperfecties voorkomen en dat heeft uiteraard zijn invloed op de symmetrie en uniformiteit van een band.

Niet-uniformiteiten van een band, zoals deze imperfecties in het algemeen worden genoemd, is het hoofd-onderwerp van dit rapport. Het doel van dit onderzoek is om basis inzicht te geven in de invloed van sommige van deze niet-uniformiteiten. Voornamelijk de krachtgeneratie karakteristieken van een niet-uniforme band en de interactie met een voertuig zijn onderzocht. Omdat dit een eerste aanzet is tot het modeleren van een niet-uniforme band zijn de geana- lyseerde niet-uniformiteiten beperkt tot 2D en is het gebruikte voertuigmodel voorzien van enkel één niet-uniforme band. Laterale krachtgeneratie en niet-uniformiteiten van een band en hun invloed op chassis trillingen zijn geen onderdeel van dit rapport.

Om het eect van een niet-uniforme band op chassis trillingen te onderzoeken, hoewel er een beperking tot 2D is in dit rapport, is een volledig voertuigmodel met gedetailleerde wiel- ophanging en compliantie ontwikkeld in cooperatie met een ander student.

Een niet-lineair, niet-uniforme bandmodel is ontwikkeld en gevalideerd aan de hand van on- derzoek gedaan door prof. H.B. Pacejka. Dit band model is gebruikt om krachtvariaties als gevolg van radiale runout, radiale stijfheidsvariatie en twee gemeten niet-uniformiteiten en hun eect op voertuigtrillingen te simuleren.

3 List of symbols

a position centre of gravity from front axle [m] b position centre of gravity from rear axle [m] c steering system compliance [Nm/rad] d steering system damping [Nms/rad]

dt tread thickness [m] l wheelbase [m] m vehicle mass [kg] r loaded tyre radius [m]

rc radial runout amplitude [m] re eective rolling radius [m] rf free rolling tyre radius [m] rp pinion radius [m] s x-position road [m] u longitudinal tyre deformation [m] w road height [m] z axle height [m]

C2 cornering stiness rear axle [N/rad] CoG centre of gravity

CF x longitudinal tyre stiness [N/m] CF z radial tyre stiness [N/m] CF za radial stiness variation amplitude [N/m] CF κ slip stiness [N/m] Fx horizontal or longitudinal force [N] Frack steering rack actuation force [N] Fz vertical or radial force [N] 2 Iw wheel inertia [kgm ] 2 Ixx vehicle body inertia around x-axis [kgm ] 2 Izz vehicle body inertia around z-axis [kgm ] 2 Jsw steering wheel inertia [kgm ] R turn radius [m] V vehicle speed [m/s]

VSx slip velocity [m/s]

4 β vehicle side slip angle [rad] δp pinion angle [rad] δsw steering wheel angle [rad] Ω wheel angular velocity [rad/s] φ tyre rotation angle [rad] ρ radial tyre deection [m] σκ relaxation length [-] θ phaseshift between radial runout and stiness variation [rad]

5 Chapter 1

Introduction

A tyre consists of several parts, see gure 1.1. During the production of a tyre all these separate parts have to be assembled into a single tyre. The tread, for example, is in fact a at strip of rubber and is wound around the other parts during assembly. Most of the tread-pattern is added mainly to the tread during the vulcanising of the so-called green tyre, which is the name used for the nal assembly of all tyre components. For the belts mentioned in gure 1.1 the same procedure is used. The belts are wound around the already assembled parts and the belt ends are welded together. It is not hard to imagine the possibility of overlapping or gaps when the belts are wound around the other components. It is also possible that the belts are not exactly positioned in the centre of a tyre in lateral direction. Within certain production tolerances all these things will happen to all the single components used in a tyre and this will o course have its inuence on, for example, the force generation of a tyre. This also makes that every tyre is unique. When the tyre is out of production tolerances it will be scrapped.

Figure 1.1: Example of a tyre.[1]

Tyre non-uniformities, as these imperfections are usually referred to, is the main subject of this thesis. To a tyre manufacturer it is of interest to know if their demands in relation to non-uniformities are realistic or if they may be changed in order to scrap less tyres. This is in fact the motivation behind this thesis, but the demands and conclusions hereabout are beyond the scope of this report.

6 The goal of this research is to provide some basic insight into the inuence of some of the known non-uniformities. This is a rst step in the validation of the demands mentioned before and is achieved by answering the following:

• Most customer complaints are in the speed range 80 to 120 km/h while production-end measurements are made at 6 km/h. What is the inuence of a specic non-uniformity on the longitudinal and radial force generation characteristic as a function of the rolling speed and is there a dierence between the specic types of non-uniformity?

• During production-end measurements two types of radial force characteristics are mea- sured. What is the dierence in inuence of both measured types on chassis vibrations?

• What is the vehicle's dynamic interaction with a non-uniform tyre? The non-uniformities used will be restricted to the 2D in-plane case and the vehicle model used will be equipped with a single tyre having non-uniformities. Lateral force generation and lateral non-uniformities of a tyre and their inuence on chassis vibrations is not part of this research and may be the subject of future research.

In order to evaluate what sorts of tyre non-uniformities exist, the research already done on this subject and the inuence of tyre non-uniformities on chassis vibrations, a literature survey is performed at rst, the results can be found in chapter 2.

To simulate the eect of these non-uniformities on chassis vibrations, although only 2D in- plane non-uniformities are analysed in this report, a full vehicle model with detailed suspension layout and compliance has been modeled in cooperation with a fellow student. A complete description on this full vehicle model, its development and validation can be found in chapter 3.

In chapter 4, a transient tyre model will be developed including non-uniformities, based upon research performed by prof. H.B. Pacejka. This tyre model is used to simulate force varia- tions generated by radial runout, radial stiness variation and two types of measured non- uniformities.

Analyses with the tyre model including non-uniformities can be found in chapter 5. A xed axle height model, which is in fact the same setup as is used during measurements at the end of the production of a tyre, a quarter vehicle model and the full vehicle model are used in the simulations discussed in this chapter.

At the end of the report, in chapter 6, conclusions and recommendations in relation to this research can be found.

The research has been performed in cooperation with Apollo Vredestein b.v.

7 Chapter 2

Literature survey

The manufacturing of a tyre involves four major steps which are shown in gure 2.1.

Figure 2.1: Lay-out of tyre manufacturing process.[2]

To obtain the compounds needed for the rubber parts in a tyre, ingrediënts like basic rubbers, oils, carbon black, pigments, antioxidants, accelerators and other additives are mixed in giant blenders, named after its inventor, Banbury. The compounds are then usually extruded in order to prepare, for example, the sidewalls and treads of a tyre.

To produce the beads, which are the parts of a tyre used to t the tyre correctly to the rim, spools of steel wire are driven through a small extruder to cover the wires with rubber com- pound. The rubber covered wires are then wound around a drum with a diameter depending on the tyre size desired. The actual beads are now ready for the assembly of the tyre.

To produce the plies a special sort of weaving, called calendering, is used. Textile and steel wires are calendered and covered with rubber to obtain the plies. The belts are calendered sheets consisting of two layers of rubber with in between steel cords oriented radially, hence the naming of a radial tyre.

8 The next step in the tyre manufacturing process is the assembly of all parts at a tyre building machine in order to produce the so-called green tyre. An example of such an assembly process can be divided in three phases, the carcass building, the treads and belts package building and the coupling of both. The carcass building starts with setting the bead wires at the applicators, as can be seen in gure 2.2. The positioning of the bead joint is of importance to avoid tyre imbalance.

Figure 2.2: First step in carcass building.[2]

The next step is adding the inner liner, this rubber part makes sure that no air can leak from the wheel. The inner liner is set at the drum and rotated around the drum one turn. The joint is cut with a hot knife in gure 2.3. The inner liner is set end to end so that the diagonal cutting seam is at the top and the ends are joined manually.

Figure 2.3: The addition of the inner liner.[2]

9 Next the body ply has to be mounted. A cord ply is wound around the drum once and is cut, by hand, with a join of two to ve overlapping wires, see gure 2.4.

Figure 2.4: The mounting of the body plies.[2]

Now the beads have to be mounted. This is done automatically and to ensure that the beads stay on place the edges of the carcass are turned up over the beads and stitched tightly as can be seen in gure 2.5

Figure 2.5: Joining the carcass and the beads.[2]

To nish this phase, the sidewalls have to be added. Here again the seam is fastened manually, see gure 2.6.

10 Figure 2.6: Finishing the rst phase by adding the sidewalls.[2]

The second phase in the production of the green tyre starts with winding a steel belt around a drum and joining the ends together as can be seen in gure 2.7.

Figure 2.7: First step in building the tread package.[2]

Next nylon ply is added by winding it around the drum, on top of the steel belts, one or two times. The ends are cut by hand again in the example of gure 2.8.

11 Figure 2.8: Adding the nylon plies.[2]

To nish the second phase, the tread is wound around the drum and the ends are jound automatically as shown in gure 2.9.

Figure 2.9: Adding the tread and nishing the second phase.[2]

12 To produce the green tyre both the carcass and the tread package are joined automatically, which is shown in gure 2.10.

Figure 2.10: The production of the green tyre.[2]

Some chemicals might be added to the green tyre before it will be vulcanised under high pres- sure in a mold. Most of the tread surface and the markings on the sidewall are also pressed into the tyre during this process.

The production and assembly of a tyre thus involves many steps and each step involves cer- tain tolerances. It might be clear that, at the end of the production process, a tyre is made which is within or out of certain tolerances set by the tyre manufacturer, or in the case of an OEM-tyre (Original Equipment Manufacturer): the tolerances set by the car manufacturer.

To verify whether a tyre is within tolerances at the end of the production process, a so-called non-uniformity test rig is used, see gure 2.11. Sample tyres are randomly picked out from the production line and measured, or each tyre that comes out of production is measured, depending on the strategy of the tyre manufacturer or the tyre type. This non-uniformity test-rig is usually a low-speed device (6 km/h), although high-speed devices also exist. The test-procedure starts with the mounting of the tyre on a test-rim and setting the tyre at its appropriate ination pressure. The tyre is rotated unloaded while a sensor measures the largest distance of the thread to the center of the rim in radial direction. Next the tyre is pressed against a test drum at a xed spindle height, guaranteeing a xed radial preload and is rotated again while keeping the axle height xed. The forces generated by the tyre in radial and lateral direction are measured [3]. Two criteria have to be met in order to pass these tests and the tyre being allowed to be sold on the market: the height variation may not be larger than a certain value and the force variations peak-to-peak values may not be larger than a certain value. The peak-to-peak value is the absolute dierence between the maximum and the minimum of a signal and will often be used in this thesis as not all signals are perfect sine-waves and amplitudes will therefore not suce.

13 This measurement only identies the force variation in radial and lateral direction and the runout in radial direction though. Moment and tangential force variations are not checked. In fact, all these variations in a tyre are called non-uniformities.

Tyre non-uniformities are usually divided into 3 categories [10]:

• Dimensional non-uniformities

• Force variations

• Unbalance

Figure 2.11: Example of a tyre non-uniformity test rig.[3]

Dimensional non-uniformities can be radial and lateral runout, caused by for example mis- placed tread rings and overlapping layers.

Force variations can occur in the radial, longitudinal and lateral directions. Well known examples of lateral direction force variations are conicity and ply-steer. Dimensional non- uniformities and unbalance will o course also generate force variations when the tyre is rotated, so it might be better to think of stiness variations next to runouts and unbalance. Moments generated by a tyre with non-uniformities are not discussed in the analysed litera- ture and will therefore not be in this report.

Tyre unbalance is usually of less interest than the other tyre non-uniformities in relation to vehicle vibrations as this can be (partly) counteracted by balancing the wheel [10]. The counteraction of these weights cannot be guaranteed for the complete speed range in which the wheel rotates though, because 2 factors prevent perfect balancing of mass unbalance at the tread ring [20]:

14 • The presence of damping, which causes the mass unbalance force vector to align or misalign with the counterbalance force vector as a function of speed.

• The presence of the elastic foundation between the tread ring and the rim, which makes forces on the bearings created by mass unbalance a function of the oscillatory behavior of the tread ring, while the forces from the counterbalance on the rim are not dependent on the oscillatory response. The inuence of speed and balancing can be seen in gure 2.12. It has to be noted though that the axis system used in the referred article deviates from the axis system generally used in this report, i.e. the x-axis is in longitudinal direction while the y-axis points in radial direction.

Figure 2.12: Peak-to-peak force diagrams for drum speeds of 80 and 120 km/h. [20]

15 Not only force variations created by unbalance are speed dependent, but the force variations created by runout are also very speed dependent as can be seen in gure 2.13. In both gure 2.12, for the balanced case, and gure 2.13 it can be seen that the force variations in radial direction Fy and R1H, which is the rst harmonic in radial direction, are not speed dependent. While the force variations in longitudinal direction Fx and T1H, which is the rst harmonic in tangential direction, are clearly speed dependent.

An other thing that can be seen is that at lower speeds the force variations in radial direction dominate, while for higher speeds the force variations in longitudinal direction dominate.

Figure 2.13: Experimentally measured data for a tyre with radial runout.[14]

The inuence of the speed on the response to radial stiness variations is signicantly less than for radial runout as is shown in gure 2.14.

16 Figure 2.14: F-Tire simulated data for a tyre with radial stiness variation.[14]

In [21] and [25] it is shown that other factors than the rotation speed and amount of non- uniformity also have a clear inuence on the force variations created by non-uniformities in a tyre. These factors are for example the tyre ination pressure and the position of concen- trations of material in the sidewall, which is in fact the position in the sidewall of a mass unbalance.

In order to understand the inuence of tyre non-uniformities on force variations and the pos- sible eects on vehicle vibrations, several studies have been performed in the past. The force variations measured on a non-uniformity test rig are measured as a function of the rotation angle of the tyre, see gure 2.15.

Figure 2.15: Experimentally measured force variation with rst harmonic approximation.[4]

17 The force variation signal is usually divided into several harmonics, of which the rst harmonic has the biggest inuence on the total force variation characteristic [13], [15], [16]. This is also illustrated by gure 2.16 where the peak-to-peak rst harmonic variations (y-axis) part of the complete peak-to-peak force variation (x-axis) can be seen.

Figure 2.16: The correlative relation between the complete force variation and its rst harmonic.[13]

Although usually only the rst harmonic is used in simulations, the inuence of the other harmonics and the phase shift between the several harmonics, has been studied by Gillespie [15] in the beginning of the 1980s. To have an idea of ride deterioration on a heavy truck several combinations of harmonics with dierent phases and amplitudes have been used as an input for a road-simulator setup on which a heavy truck has been placed. The simulation mimics for driving at 55 mph on a smooth road. Experienced ride engineers have then rated the ride deterioration caused by the non-uniformities in relation to cab shake, steering wheel and the seat vertical, lateral and longitudinal direction vibrations. Input was only in radial direction, so no statements can be made on the response to non-uniformity inputs in other directions. The main conclusions are that the sensitivity (ride degradation rate) is highest for the 2nd and 3th harmonic and that the 1st harmonic causes a rating-loss of 0.2 to 1.4 ride points (on a 0-10 point scale).

Demic [13] has been simulating the response of a multibody vehicle model to radial and lateral force variations and compared these against ISO comfort curves to determine the maximum allowable force variations that guarantee a comfortable ride. He concludes the following maximum allowable force variations:

• Peak-to-peak radial force variation -> 180N

• Peak-to-peak rst harmonic radial force variation -> 138N

18 • Peak-to-peak lateral force variation -> 190N

• Peak-to-peak rst harmonic lateral force variation -> 142N Neureder [23] has been investigating steering wheel rotational vibrations, also called nibble, at the Ford Research Centre in Aachen. He mentions that there are four main causes for steering wheel nibble, apart from road inputs:

• wheel imbalance: Imbalance at the outer side of the rim produces higher levels of nibble than imbalance at the inner side of the rim.

• tyre non-uniformity: Only tangential force variation has a signicant impact on nibble, while radial and lateral force variations have a negligible eect.

• wheel runout: Only radial wheel runout has an inuence and the eect is negligible.

• brake torque variation: Is the most critical disturbance factor for steering wheel nibble. His conclusion is that brake torque variations are a known critical item and much work is al- ready done in this eld. But the second most important inuence for steering wheel nibble is the tangential force variation, caused by tyre non-uniformities. He also mentions that further research is necessary to reduce tangential force variations.

Not only the inuences of non-uniformities on vehicle behavior have been studied, but several studies on tyre behavior have also been performed in the past to understand the background of force variations.

FEM tyre modeling is used for investigations on the inuence of variations in geometry and material on the radial force variations of tyres by Meijuan [25]. Comparisons between the FEM models and measurements on similar non-uniform tyres show that FEM can be used to predict the force variation characteristics, but also that for analysis on a dierent type of tyre design the model has to be adapted to the dierent design.

The rigid ring tyre model is a tyre model commonly used to investigate tyre behavior. This is possible for frequencies up to 100 Hz because above 100 Hz the inuence of deformation in the tread ring is no longer negligible [12]. Stutts [20] uses the rigid ring model to study the fore-aft (i.e. tangential, longitudinal) force variations as these are created by unbalance and radial runout. The rigid ring model is a model in which a rigid ring is connected to the rim trough radial and longitudinal springs and dampers and thus representing the simplest dynamics of a tyre. He concludes from the simulations that at higher speeds the horizontal forces are higher than the vertical forces and that complete balancing is not possible, for rea- sons already mentioned before. Dor [14] also uses a rigid ring model and compares it with F-Tire, for which the simulations represent experiments very well. His rigid ring model has some shortcomings though as it does not imply relaxation length. Therefore he also uses the transient tyre model, developed by Pacejka [24]. The transient tyre model accounts for footprint deformation and longitudinal force generation through the implementation of longitudinal relaxation length. This means that the model also needs a certain amount of distance to build up longitudinal force, just as a real tyre. The result of Dors model comparison is that the transient tyre model matches

19 experimental and F-Tire data very well, see gure 2.17.

Figure 2.17: Run-out tyre model comparison. [14]

Dor [14] has also done some research on the inuence of tyre non-uniformities on responses of a quarter vehicle model and on steering wheel accelerations. His ndings are that the front suspension resonance amplies the radial force variation and that the tangential force variations drive the steering wheel accelerations. The peak steering wheel acceleration is sig- nicantly increased by a larger tyre/wheel system inertia.

More detailed information on the transient tyre model can be found in chapter 4 as it will be used as a basis for further investigations that can be found in this report.

20 Chapter 3

Full vehicle multibody model development

3.1 Introduction

In the scope of this master's thesis only the inuence of non-uniformities in 2D have been studied, but further studies on the subject of tyre non-uniformity will certainly involve 3D analyses. Therefore a vehicle model with a detailed suspension system has been modeled in co- operation with Michel Groenendijk (DCT 2009.130). This vehicle model adds some additional value in comparison to a quarter vehicle, since it also gives the opportunity to get a rst idea of the inuence of tyre non-uniformities on the steering system and the inuence of a detailed suspension with compliance on the chassis vibrations due to (2D) tyre non-uniformities. More information on this subject can be found in chapter 5.

In gure 3.1 one can see a picture of the vehicle modeled. The vehicle is a BMW 5 series type e39 build in the years 1995 to 2004, detailed information on the models parameters and some pictures can be found in appendix A.

Figure 3.1: Vehicle on which the model is based, BMW 5 series (e39), [5]

21 An overview of the simulation model The model is divided in dierent multiple blocks; chassis, brake system, drive line with cruise control and front and rear axle, gure 3.2. The front and rear axle are described in detail in section 3.2 and 3.3. In this section the other blocks are explained briey.

The chassis consists of a mass with inertia based on rules from [17]. The chassis is connected to the ground by six degrees of freedom, which also provide the signals of the vehicle body movement. The brake system is implemented as a gain function where the input is multiplied with a brake moment. By multiplying this brake moment with a factor the brake balance between front and rear is achieved. This moment is applied between the hub and wheel. In the same manner the drive line is modeled, but somewhat more complex. The throttle signal is multiplied with the engine power, which is calculated by the average of the wheel velocities of the two rear wheels, and is limited by a revolution limiter. This is multiplied by the nal gear and results in the driving torque. The throttle signal originates from the cruise control block. Here a PID controller controls the actual velocity with the reference velocity of the model. The cruise control can be disabled and a custom throttle signal can be used to drive the vehicle. The tyre model MF-Tyre/MF-Swift version 6.1.0. is used and the tyre model parameters represent a Michelin 215x55R16.

Figure 3.2: Vehicle model with the dierent blocks, SimMechanics

22 3.2 Front axle

The front axle of the model is of the McPherson type, see gure 3.3 for a basic lay-out of this type of suspension.

Figure 3.3: Basic design of a McPherson suspension, [6]

This type of suspension exists of a lower wishbone (control arm), a spring-damper strut and a hub to which the wheel is connected. An important part of the front axle is the steering system which is also connected to the hub.

Steering system The steering system used in the BMW 5 series is of the rack and pinion type, of which an example can be seen in gure 3.4.

Figure 3.4: Rack and pinion steering system, including the exible parts [7]

23 The steering wheel is connected by a exible steering shaft and exible coupling to the pinion. The pinion is connected to the rack. When the steering wheel is rotated the pinion will rotate and translates the rack, the rack is connected to the wheel hub and will rotate the wheels. The steering system is modeled on the basis of the following momentum equilibrium, [19]:

¨ ˙ ˙ Jsw · δsw + d(δsw − δp) + c(δsw − δp) = Frack · rp (3.1)

In this equation steering compliance is introduced with the stiness c, d is the steering damp- ing and Jsw is the steering wheel inertia, Frack is the force with which the rack is actuated and rp is the pinion radius. δsw is the steerwheel angle and δp the angle of the pinion. Steering compliance originates from exible parts in the steering system, as shown in gure 3.4. These exible parts are added to the steering system to lter out vibrations from the road/tyres to the steering wheel, but also results in loss of steering angle. Part of the input to the steering system, whether steering wheel or road/tyre input, will result in deformation of these exible parts. The eect of implementing compliance can be seen in gure 3.5 where the response of the system to a step-input is shown.

Figure 3.5: Steering response of a step-input at 50 km/h

The connections of the steering links to the hubs are placed in such a way that Ackerman steering is introduced. A line is taken from the center of the rear axle to the contact point of the front tyres. The connection of the steering linkage to the hub has to be on this line to achieve Ackerman steering. Ackerman is used to get a dierence in steer angle between the outer wheel and inner wheel during cornering. Complete Ackerman steering is usually not feasible, this has to do with packaging space in wheel bays. In gure 3.6 one can see the amount of Ackerman steering in the model. This is not complete Ackerman steering but around 50 percent, the gure is comparable with literature, [26]

24 Figure 3.6: Ackerman principle

Suspension model The geometry of the McPherson suspension model is based on literature [8] and pictures of the vehicle, see appendix A. The basic lay-out of the front axle can be seen in gure 3.7.

Figure 3.7: Setup of the front axle, left front

25 For NVH (Noise, Vibration and Harshness) reasons a suspension is equipped with rubber elements so-called bushings. In the model a bushing is introduced at the front of the lower control arm, see gure 3.7. The rear connection of the lower control arm to the chassis is connected by a universal joint. The lower control arm is connected to the hub with a spher- ical joint, which is also the case for the connection of the steering linkage to the hub and steering rack. Not shown in gure 3.7 is the spring/damper-strut, which is modeled with a prismatic joint. The spring is modeled as a constant stiness and the damper with a look-up table, where the damping force is dependent on the velocity of the joint movement, see g- ure 3.8. The data used for the damping originates from measurements. The roll stabilizer is modeled as a stiness multiplied with the dierence in the movement of the left and right strut.

Figure 3.8: Front axle damper characteristic, [18]

Kinematics and Compliance The geometry of a suspension is important for vehicle handling, steering and ride comfort. Kinematics and compliance (K&C) determine how the suspension geometry changes due to wheel motions (kinematics) and forces (compliance). The front and rear axle are validated with K&C tests. Dierent sources are used to verify the K&C of the front and rear axle, [26, 27]. K&C tests are performed on a test-rig where the vehicle body is attached to a table and the wheels are placed on pads. The movement of the axle is measured at the wheel center, the forces on the axle are deployed by moving the wheel pads (MTS) or the table (Anthony Best Dynamics). The simulations to verify the front and rear axle are based on the MTS-method. The description of the following tests [28] are used as a guideline:

• Bounce: The table moves the vehicle through a sinusoidal vertical motion. During this motion the wheel pads are controlled to maintain zero force and moments in the horizontal plane, to ensure that the measured eects are purely kinematic. The bounce test is performed with locked brakes. During simulation the vehicle body is welded to the ground and the vertical force is applied in the tyre/road contact patch.

26 • Roll: The vehicle is put through a roll motion, controlled in such a way that the total load (vehicle force) remains constant. This is to reproduce the kinematics of a vehicle during cornering. The simulation is performed in the same way as the bounce test, but now the left and right wheel have an opposing motion.

• Lateral compliance: The pads perform a lateral motion to generate lateral forces. Equal forces on the left and right side are applied to determine the lateral stiness in the sus- pension and steering system. This test also delivers a lateral compliance steer gradient, which is derived from the measured toe angle. In the simulations, instead of moving the pads, a lateral force is applied at the tyre contact patch. When the forces on one axis are both in opposite direction the steering system stiness will not be included in the results. When the forces are in the same direction, and the steering wheel is kept xed, the steering system stiness is included. During the simulations the forces are both in opposite direction.

• Longitudinal compliance: Similar to the lateral compliance test but, with a longitudinal force applied, results are the longitudinal suspension compliance and compliance steer.

• Turning compliance: In the contact patch a moment around the z-axis is applied and the wheel angle is measured. In the simulation both moments are in the same direction and the steering wheel is locked, this ensures that the steering system stiness is also taken in with the results.

Some aspects which have been taken into account while modeling the suspension system, [28]:

• Toe-angle, in principle one wants to keep the toe angle as small as possible, to prevent tyre wear and high lateral forces during jounce. Small toe-change is useful to induce understeer behaviour. On the front axle toe-out during bounce is aimed for.

• Camber change inuences the understeer behaviour, for increased lateral grip in corner- ing, negative camber during jounce is needed.

• Longitudinal, the wheel will move rearward during jounce and forward during rebound. When driving over a bump one wants the wheel to move backward, this improves impact harshness (ride). The forces on the suspension are lower compared to the simulation where the wheel moves forward.

• Lateral wheel displacement, this characteristic equals half the track change. Track change causes a rolling tyre to slip which causes lateral forces, these forces result in increased rolling resistance and wear, therefore it is desirable to have minimal track change. Lateral displacement will also cause a change of the position of the rollcentre.

• In relation to longitudinal compliance, toe-in during braking and toe-out during driving is desired. For the rear axle almost no compliance during braking and driving is desired.

• Lateral compliance, toe-out for the front axle and toe-in for the rear axle is desired when the wheel is moved inward. This reduces the side slip angle, and generates more understeer. Lateral stiness needs to be as high as possible to meet vehicle requirements for steering and handling as it ensures a direct respons.

27 • The vehicle will dive/squat during braking and driving, so toe changes occur through jounce, compliance and the static toe-angle.

The model is tuned by hand, i.e. no automatic optimization process has been used. First a suspension model is build on the basis of the pictures of the vehicle. When the kinemat- ics where comparable with the gures from [27, 26], the model is modied to include the compliance. During the tuning process the following assumptions are used:

• The kinematic toe angle is mainly dependent on the placement of the steering linkage. Making the steering linkage longer/shorter results in a less/more circular characteristic. The imaginary center of the circular characteristic is determined by the height of the connections of the steering linkage (hub and rack) in relation to the placement of the under wishbones, [22]. To achieve the desired toe-changes through compliance, the bushing and steering system are placed in front of the center of the front axle. This might not be as in the BMW lay-out but priority was given to correct characteristics.

• Camber can be tuned by adjusting the length and placement of the lower control arm (wishbones).

• Lateral displacement is coupled with camber. The more circular the characteristic (shorter wishbones) the more lateral displacement.

• Longitudinal displacement is mainly determined by the dierence in height between the front and rear connection of the lower control arm to the chassis. Since a backward movement during jounce is desired, the front connection of the lower control arm to the chassis is placed a little higher than the rear connection.

• One leg of the lower control arm was at rst modeled as a spring to introduce compliance, later on this is replaced by a rod and bushing conguration as this gives better results during the K&C tests. The bushing gives the opportunity to dene stiness and damping in all six DOF, the damping has an inuence on the hysteresis in the K&C characteristics. In reality the whole suspension system is mounted in rubber elements, this would mean additional stiness elements in the model and this will increase simulation time which is not preferable. Therefore only one bushing is modeled.

• Compliance depends on the stiness used in the bushing and on the layout of the sus- pension. Since only one bushing is used in the model, the angle of the rod of the lower control arm connected to this bushing, determines the ratio between longitudinal and lateral compliance. The position of the connection of the steering linkage to the hub in relation to the bushing position largely inuences the direction of the toe-changes that occur during the compliance simulations.

28 Results

The following results are obtained with the nal model. In gure 3.9 the results for the bounce and roll test are shown. The toe-angle is kept within limits, also the other results give com- parable results with data from literature, [26, 28].

Figure 3.9: Characteristics of the front axle (static settings not included)

In gure 3.10 the results of the compliance tests are depicted. The results of the K&C tests are summarized in table 3.1. When this data is compared with the literature, the values are comparable with car ten of reference [27].

Table 3.1: Kinematics and compliance, characteristics of the front axle Bounce stiness 24 [N/mm] Roll stiness 44 [N/mm] Lateral stiness 1.40 [kN/mm] Toe-angle during lateral force 0.31 [deg/kN] Longitudinal stiness 0.2 [kN/mm] Toe-angle during longitudinal force 0.625e [deg/kN]

Steering stiness (Mz) 3 [deg/kNm]

29 Figure 3.10: Suspension compliance characteristics, front axle

3.3 Rear axle

The rear axle of the BMW 5 series is an integral multi-link suspension. In gure 3.11 one can see an example of the integral link suspension.

Figure 3.11: Integral link suspension, BMW 5 series (e39), [8]

30 The integral suspension consists of a lower arm (wishbone) where one link is connected directly to the hub and the other is connected to the hub via an integral link. Two rods connect the hub with the chassis, the leading link and rear link. The spring-damper strut is mounted between these two upper links. In gure 3.12 a schematic view of the integral link suspension is shown. An interesting aspect of this axle is the integral link, which is responsible for the torsional sti support of braking torques [9].

Figure 3.12: Model of an integral link suspension, [9]

Modeling

Geometric data from the rear-axle is not available, therefore, pictures of the vehicle (appendix A) and papers of the rear axle are used to estimate the dimensions of the rear axle, [8, 9, 29]. First a rectangular lay-out is chosen, with this suspension a bounce test is performed to obtain the kinematics. Since these kinematics were not good, with each iteration, a single rod is changed to obtain a good result. The targets for the kinematics come from literature [26, 27, 28]. After the kinematics simulations were satisfying, the suspension is modied to include the compliance of the axle. In gure 3.13 one can see a sketch of the rear axle to make clear where the stiness elements are positioned. A bushing is used on the front rod of the lower wishbone and stiness is added to degrees of freedom of the chassis-connection of the rear link (x-, y-, z-direction) and leading link (only y-direction).

Like with the front axle the spring is modeled as a constant stiness and for the damper a look-up table is used, gure 3.14. The assumptions used during tuning are:

• Longitudinal displacement during bounce is dependent on the angle of the lower wish- bone, the angle determines if the wheel will move rearward during bounce or forward, the length of the wishbone determines the curvature of the movement.

• The y- and z-position of the leading link on both hub and chassis side have an inuence on the camber movement, the camber movement is coupled with the lateral displacement, therefore a compromise between the camber and lateral displacement motion has to be made. As with the longitudinal displacement the length of the leading link determines the curvature of the characteristics.

31 Figure 3.13: Sketch of the rear axle without integral link

Figure 3.14: Rear axle damper characteristic, [18]

• Toe-angle during bounce is not depending on a single rod, the whole layout determines the toe movement. The toe-angle is coupled to the longitudinal displacement, damper placement and notably the x-position of all the rods, since these determine toe-out or toe-in during bounce. The toe angle also depends on the stiness of the dierent rods.

• Without the use of bushings or stinesses the whole rear suspension is innitely sti, rst only one bushing was included but since the suspension exists of three sti rods, an additional stiness is modeled in the form of extra degrees of freedom of the upper links.

32 • Lateral and longitudinal compliance is tuned with the stiness of the bushing and the stiness applied to the extra degrees of freedom, an other point which has inuence on the stiness is the layout of the suspension, increasing the angle of the rods more outwards (x-position on the chassis) gives a stier structure, also the angles of all the rods have an inuence on the stiness.

• The tuning process is not as simple as written above, because next to the inuences of all the independent rods on their characteristics, they also inuence the other character- istics, it is not that easy that one can say ve links determine ve characteristics, they all have an inuence on each other.

Results

Similar to the front axle also the rear axle is simulated on the K&C test rig. The kinematics of the rear axle are obtained by performing a vertical bounce test. In gure 3.15 one can see the results of this test, in the same gure the roll of the axle is depicted.

Figure 3.15: Characteristics rear axle (static settings not included)

The compliance of the rear axle is validated by performing tests where a longitudinal force, a lateral force and a turning moment are applied in the tyre road contact patch. In gure 3.16 the results of the compliance test are visible. In table 3.2 the values of the kinematic and compliance tests are given. As with the front axle the characteristics of the rear axle are comparable with the data from literature, [28, 26, 27].

33 Figure 3.16: Suspension compliance characteristics, rear axle

Table 3.2: Kinematics and compliance, characteristics of the rear axle Bounce 26 [N/mm] Roll 31 [N/mm] Lateral stiness 0.82 [kN/mm] Toe-angle during lateral force 0.048 [deg/kN] Longitudinal stiness 0.35 [kN/mm] Toe-angle during longitudinal force 0.34 [deg/kN]

Steering stiness (Mz) 2.7 [deg/kNm]

3.4 Full vehicle model validation

In gure 3.17 the virtual reality representation of the complete model can be seen. To validate the model, driving tests are performed. These tests are described in ISO standards. The tests are simulated with the model and the results are compared with measurements on the real vehicle which are documented in [18].

Steady state circular test (ISO 4138) During this test the vehicle follows a circular path with radius of 100 m with constant velocity. The test is repeated at dierent speeds giving a speed-dependent steering characteristic. With increasing velocity the lateral acceleration will increase, till approximately 8 m/s2. In gure 3.18 one can see the nal result of the simulation, compared with the measurements. As one can see, the side slip angle is somewhat higher than the measurement and the model is slightly more understeered in comparison to the real vehicle. Overall the model is in good agreement

34 Figure 3.17: Virtual Reality representation (wheels are not depicted) with the real vehicle.

Figure 3.18: Steady state circular test, measurement versus simulation

Some remarks that can be made in relation to the simulation of the steady state circular test are: The steering wheel angle starting point (low lateral accelerations) is determined by the steering ratio, the curvature is dependent on the steering and suspension compliance and a stier front suspension makes the vehicle less understeered. The side slip angle β is mainly dependent on the rear axle and for a large part on the lateral compliance toe-angle (rear axle). The roll angle can be inuenced by the position of the center of gravity. Changing the anti roll bar has an inuence on both steering behaviour and roll angle of the vehicle body.

35 Pseudo random steer test (ISO 7401) In this test a random steering input is generated by the test driver. The goal of this test is to excite the vehicle and measure the response over the entire frequency range of interest (0 to 4 Hz), so its transfer functions can be calculated from input and response data. The lateral acceleration during the test remains below 4 m/s2, the boundary below which vehicle behaviour can be regarded as linear. The forward velocity of the vehicle is kept constant at 100 km/h. In gure 3.19 the results of the pseudo random steer test are shown, the simulation is comparable with the measurement.

As with the steady state test, the steering compliance determines the graphs of the steering wheel angle versus lateral acceleration and roll angle, at the lower frequencies. By changing the inertia of the vehicle on the basis of rules from [11], the simulation results will change. In particular the place of the peaks and dips can be tuned in this manner, the whole graph can be shifted horizontally over the x-axis by changing the inertia around the z-axis. The peak of the roll angle versus lateral acceleration can be moved by changing the inertia in x-direction. The peaks/dips themselves can be tuned by increasing/decreasing the damping of the bushings which are included in both the front and rear suspension .

36 Figure 3.19: Random steer test, unloaded

37 Loaded vehicle model The vehicle tests have been executed for both the unloaded and loaded condition. For the loaded condition the mass is increased, the center of gravity moved backwards and the in- ertia is adapted, see appendix A. In gure 3.20 one can see the results of the steady state circular test with the loaded vehicle. Steering angle and roll angle are comparable with the measurement, only the side slip angle is too high. The side slip angle was already high with the unloaded model. 2 From the bicycle model theory it is known that the side slip angle is given by β = − b + amV R C2lR with a the position of the centre of gravity (CoG) from the front axle, b the position of the CoG from the rear axle, l the wheelbase, m the vehicle mass, R the radius of the turn and C2 the cornering stiness at the rear. Since the starting point of the slip angle graph is in agreement with the measurement, but the higher the lateral acceleration the higher the devi- ation between the measurements and the simulation becomes, this has to do with the second term of the equation mentioned above. Only the cornering stiness is not xed or linked to the rst term of the equation, thus a slight increase of the rear cornering stiness might make the simulation more accurate in relation to the measured slip angle. This will have its eect on all results though, which might imply the need to tune other model parameters as well. As the loaded situation is not used in this thesis, the ne-tuning of the model is recommended for other studies with the loaded vehicle model.

Figure 3.20: Steady state circular test, loaded situation

In gure 3.21 one can see the results of the random steer test with the loaded model and vehicle. The height of the sensor was not documented therefore this is tuned to give the peaks the right height, this results in a higher sensor position than for the unloaded situation. The inertia is adapted to get a better result from the simulation. Izz is increased to move the transfer functions of yaw and lateral acceleration to steering wheel angle to the left, next to that lateral acceleration peaks increase and yaw peaks decrease. Ixx is changed (also increased)

38 to get the gure of roll angle to lateral acceleration good. The height of the center of gravity has an eect on the roll angle. The simulations show comparable results, only in the graph of yaw velocity to steering wheel angle, the simulation is too high. This is in relation to the higher side slip angle during steady state cornering.

Additional tests The previous tests are performed to validate and tune the model, extra tests are performed to see if the model is representative for the real vehicle during dierent manoeuvres. More information on this can be found in appendix B.

Concluding remarks The simulations with the vehicle model for the unloaded situation show that the model is an acceptable representation of the real vehicle.

The simulations with the vehicle model for the loaded situation are less good in agreement with the measurements. The vehicle slip angle for the steady state test and the yaw velocity for the random steer test dier more from the measurements than is the case with the un- loaded vehicle model. Since the loaded situation is not used in this thesis it is recommended to ne-tune the loaded situation for uses in other research.

To simplify the modeling and tuning of a detailed vehicle model it is advisable to have all the (exact) data from the vehicle or have the real vehicle present. This way parameters can be measured instead of making assumptions.

39 Figure 3.21: Random steer test, loaded situation

40 Chapter 4

Modeling tyre non-uniformities

In this chapter a tyre model, based upon Pacejka's Transient Tyre Model [24] also mentioned in chapter 2, will be explained and analysed. The tyre non-uniformities modeled are:

• Radial runout

• Radial stiness variation

• Measured force variation typied as `peak`

• Measured force variation typied as `harmonic` The measured force variations will not be discussed in this chapter but will be used in the following chapter.

4.1 Full non-linear model

Pacejka's Transient Tyre Model is in fact a 2D representation of a tyre with the possibility to implement non-uniformity parameters. The relaxation length is part of this model. The basic ingredients of the model can be seen in gure 4.1.

Figure 4.1: Basic ingredients of Pacejka's Transient Tyre Model. [24]

41 To keep things simple the road prole ( and dw ) and rolling resistance will be −w = 0 − ds = 0 omitted, the axle height z is xed in this chapter for the cases in which a tyre non-uniformity is studied and the forward speed V is constant. Furthermore the tread thickness variation d˜t used by Pacejka in [24] will not be used in this thesis.

Tyre with radial runout

In gure 4.2 two instances of tyre rotation with a certain radial runout rc are sketched.

Figure 4.2: Tyre with radial runout at and π . φ = 0 φ = 2

The xed axle height makes that the loaded radius r is constant. Assuming a radial runout rc and an average free tyre radius rf0 the expression for the free tyre radius rf reads:

rf = rf0 + rcsin(φ) (4.1) The tyre deection (or compression) ρ is the dierence between the free tyre radius and loaded radius:

ρ = rf − r (4.2)

Assuming an uniform radial tyre stiness CF z the vertical force Fz can be calculated:

Fz = CF zρ (4.3)

The eective rolling radius re is dened as the distance between the rim center and the virtual slip point S, which is the center of the rotation of the wheel body at free rolling. The eective rolling radius re is also a function of the tyre deformation ρ and will therefore also change as a function of this radial runout.

42 Figure 4.3: Reduction of eective rolling radius re with respect to tyre free radius rf due to tyre radial deection ρ. The tread depth dt is not used in this thesis and the situation at nominal load Fz0 = 3000N has been indicated by circles. [24]

As can be seen in gure 4.3 the changes in the eective rolling radius are generally smaller in magnitude compared to the tyre deection ρ, in particular at higher loads. The eective rolling radius can be calculated from above gure if ρ and rf are known.

As a result of the variation in eective rolling radius the slip velocity VSx will vary also, inducing speed dependency:

VSx = V − Ωre (4.4) The longitudinal tyre deformation is calculated using the following dierential equation: 1 u˙ + V u = −Vsx (4.5) σκ

The longitudinal force Fx can be calculated as: u Fx = CF κ (4.6) σκ The derivation of (4.5) is explained next.

From the single contact point tyre model shown in gure 4.4 we can derive the time rate of change of the longitudinal deection u with VSx the speed of S attached to the rim and 0 VS0x the speed of S attached to the contact patch:

u˙ = −(VSx − VS0x) (4.7)

0 The formula for VSx is already mentioned, but in order to know the speed of S attached to the contact patch VS0x the transient tangential tyre behaviour has to be analysed.

43 Figure 4.4: Mechanical model of transient tangential tyre behaviour. [24]

The transient tangential tyre behaviour can be modeled as a spring-damper unit in series. The damper represents the sliding contact of the tyre tread elements, whereas the spring represents the stiness of the tyre sidewall. Using the force equilibrium in x-direction for this model, a function for VS0x can be derived. Starting with the damper:

−VS0x Fx = CF κ (4.8) |Vx| substituting the spring force Fx = CF xu gives:

CF x −VS0x = |Vx| u (4.9) CF κ

Using this equation for VS0x in (4.7) results in:

CF x u˙ = −(VSx + |Vx| u) (4.10) CF κ

CF κ introducing the relaxation length σκ = (4.5) is obtained. CF x Finally the dynamics of the tyre/wheel have to be considered:

IW Ω˙ = −rFx − Fzrccos(φ) (4.11) for which has a phase shift of π with respect to the loaded radius that is sensed Fzrccos(φ) 2 r at the contact center. I.e. the tyre deformation is largest at π while the moment due φ = 2 to the radial runout is largest at φ = 0, hence the change of sine to cosine in relation to the radial runout mentioned in (4.1). See also gure 4.2.

44 A tyre with a non-uniform radial stiness Next to radial runout, radial stiness variation is also a point of interest in this study on tyre non-uniformities. In gure 4.5 a sketch for a tyre with radial stiness non-uniformity can be seen.

Figure 4.5: Tyre with radial stiness variation

The axle height is again xed, thus the loaded radius r is constant. No radial runout is present, i.e. combinations of radial runout and stiness variation will be discussed in the following chapter, thus the free rolling radius rf is also constant resulting in a constant tyre deformation ρ.

The radial stiness CF z is not uniform however and the radial stiness can be divided into a nominal stiness and a stiness variation as function of the tyre rotation with θ the phase shift between radial runout and stiness variation used in the following chapter, thus in this chapter θ = 0:

CF z = CF z0 + CF zasin(φ + θ) (4.12)

Its inuence on the radial force Fz is clear, but since the radial tyre deformation ρ does not change, this non-uniform radial stiness will have no eect on changes of the eective rolling radius re.

As re will be constant, the longitudinal tyre deformation u will not change, resulting in the longitudinal force Fx not changing as a function of rotation speed Ω due to a non-uniform radial stiness.

45 4.2 Study with a nonlinear tyre model in the time domain

Summarizing, the following equations are used for the complete nonlinear tyre model: rf = rf0 + rcsin(φ) ρ = rf − r VSx = V − Ωre u˙ = −V − 1 V u Sx σκ F = C u x F κ σκ Fz = (CF z0 + CF zasin(φ + θ))ρ IwΩ˙ = −rFx − rccos(φ)Fz where re is derived from the implementation of a lookup-table for which gure 4.6 is a graph- ical representation. This gure is an approximation of gure 4.3 and depends on the type of tyre used for the analysis, i.e. in the case of gure 4.6 a new (dt = 7mm) bias-ply tyre.

Figure 4.6: re as a function of ρ

These equations have been implemented into Matlab/Simulink, see gure 4.7, with following additions:

• re is coupled to ρ. See gure 4.6. This makes it easy to change to other tyre character- istics (as e.g. used for the tyre in the vehicle model).

• the switch-blocks that can be seen in gure 4.7 are used to have no force generation in the case when the tyre is o ground.

46 • a reset-integrator block has been used to ensure that the tyres rotation angle φ is limited between 0 and 2π. This is needed for the measured force variations mentioned next.

• measured force variations (`peak` and `harmonic`) have been added to investigate the inuence of the form of the force variation when it is not a result of a perfect sine-wave. More information about this can be found in chapter 5.

This results in a nonlinear non-uniform tyre model that can be used to investigate the inu- ence of tyre non-uniformities on vehicle behaviour, see chapter 5.

3 IwdotOmega 2 V V

-K-

1/sigma_k u

V 1 u -K- s VSx u_dot u 1 Integrator1 Cfk/sigma_k Fx re 0 r_f-r_e -1

1 r rho 2 r Fz 0 rf0

3 1 sin rc omega rf omega s Integrator

-K-

Peak

-K-

Harmonic

sin cfza Cfz

-C- theta Cfz0

cos rc

Figure 4.7: Simulink scheme of the nonlinear tyre model.

The tyre parameters used in the nonlinear model are the same as for the bias-ply tyre men- tioned in [24] to have a good comparison between both models in the following section, see also appendix B.

To have an impression on the inuence of both radial runout and radial stiness variations simulations in the speed range 0 to 150 km/h have been performed. The results will be shown next, but a more thorough analysis will be performed in section 4.3. The xed axle height setup used for these simulations is shown in gure 4.8, where Fx, Fz and IwΩ˙ result from the tyre model. While V , Ω and r are sensed at the rim and are used as input to the tyre model. The speed V is also used as input to the complete model, while r is xed or variated with z in the case of axle height variations used in section 4.4.

47 Figure 4.8: Schematic of the simulation model with xed axle height position.

Simulated response of the horizontal and vertical forces to radial runout For this simulation the free rolling radius will be varied with a sine-wave with an amplitude of 2.5 mm and a frequency which is coupled to the rotation frequency of the wheel. The axle height is xed for these simulations. The results for the peak-to-peak force variations can be seen in gure 4.9.

Figure 4.9: Longitudinal and radial force variations responses to radial runout with amplitude of 2.5 mm

48 It is clear that the speed has a big inuence on the longitudinal force variations in case of radial runout. There is no speed dependency in the response of the radial force variations.

Simulated response of the horizontal and vertical force to radial stiness variation The inuence of a radial stiness variation on the force generation of a tyre is also studied.

Since no specic information is known on the stiness variation that occurs in production tyres, the stiness is varied such that the variations in Fz are equal to those for the specied radial runout. The results of the simulation can be seen in gures 4.10.

The force variation in longitudinal direction is signicantly less than (even 0) in the case of radial runout and is not inuenced by the velocity. This is in agreement with the informa- tion found in literature, [24], [14].

Figure 4.10: Longitudinal and radial force variations responses to a radial stiness variation.

4.3 Linearisation and transformation to the frequency domain

To analyse the inuence of some of the inputs on especially the frequency responses of the longitudinal force Fx, the model will be linearised and analysed in the frequency domain. As Laplace variable the usual s ∈ C is replaced by p ∈ C here to make a clear distinction between the road position s and the Laplace variable.

49 To keep the analytical study linear only small variations, indicated with a tilde, around a steady state value, indicated with 0, will be analysed by substitution of the following coordi- nates into the equations for the nonlinear tyre model: re = re0 +r ˜e rf = rf0 +r ˜f = rf0 +r ˜c ρ = ρ0 +ρ ˜ Ω = Ω0 + Ω˜ V = V0 + V˜

Starting with the radial force:

Fz = (Cfz0 + C˜fz)(ρ0 +ρ ˜) = CF z0ρ0 + CF z0ρ˜ + C˜fzρ0 + C˜fzρ˜ (4.13) and only being interested in the variations:

F˜z = CF z0ρ˜ + C˜fzρ0 (4.14) with

ρ0 +ρ ˜ = rf − r = rf0 +r ˜c − r0 + z (4.15) where z is the variation in axle height and r0 the nominal loaded tyre radius.

The linearization for the horizontal force Fx starts with the linearization of the function for VSx:

VSx = (V0 + V˜ ) − (re0 +r ˜e)(Ω0 + Ω)˜ = V0 + V˜ − re0Ω0 − re0Ω˜ − r˜eΩ0 − r˜eΩ˜ (4.16) again retaining the variations and V0 = re0Ω0:

VSx = V˜ − re0Ω˜ − r˜eΩ0 (4.17) This can be used in (4.5) to calculate u which is needed in the equation for the horizontal force variations: u F˜x = CF κ (4.18) σκ

The variations of the eective rolling radius r˜e can be approached with the following linear relation used from [24]:

r˜e =r ˜c − ηρ˜ (4.19) with ρ˜ the variation of the radial tyre deformation only due to radial runout for a xed axle height and η the slope dened as in gure 4.3.

To be able to analyse some of the responses of this system to dierent inputs, the Laplace transform of the horizontal force is taken. The radial force is not so interesting as from the previous section it can be concluded that the speed (i.e. in fact frequency) will not have inuence on the response of the radial force.

50 The derivation of F˜x starts with the derivation of Ω˜. Ω˜ used in (4.17) is derived from (4.11) with (4.6) substituted:

¨ r0 u Fz0 π 1 Ω˙ = φ = − CF κ − r˜c(t + ) (4.20) Iw σκ Iw 2 Ω0 Taking the Laplace transformation results in

r u F pπ 2 0 z0 2Ω p φ = − CF κ − r˜ce 0 (4.21) Iw σκ Iw and the function for pφ r u F pπ 0 z0 2Ω pφ = − CF κ − r˜ce 0 (4.22) Iwp σκ Iwp where the -term is due to the time-delay of π . e 2 (4.5) and (4.17) are Laplace transformed and (4.22) substituted which gives:

1 r u F pπ 0 z0 2Ω pu + V u = −px˜ + re0(− CF κ − r˜ce 0 ) +r ˜eΩ0 (4.23) σκ Iwp σκ Iwp which can be rewritten as the longitudinal tyre deformation u and with x˜ the variations in the x-position, see gure 4.1:

pπ Fz0 2Ω −px˜ − re0 r˜ce 0 +r ˜eΩ0 u = Iwp (4.24) p + 1 V + r r0 CF κ σκ e0 Iwp σκ

(4.24) substituted in (4.18) nally results in the function for F˜x:

pπ 2 2Ω ˜ CF κ −Iwp x˜ − re0Fz0r˜ce 0 + Iwpr˜eΩ0 (4.25) Fx = u = CF κ 2 σκ σκIwp + IwpV + re0r0CF κ 4.4 Analytical study in the frequency domain

Frequency response of the horizontal force to axle height variations

Function (4.25) will rst be used to analyse the frequency response of the horizontal force Fx to axle height variations z. This can be done by substituting iω for p and omitting radial runout r˜c and the variations in the x-position of the wheel x˜. Implementation of the resulting function for the eective rolling radius variations r˜e = −ηz gives the function for the frequency response to axle height variations:

˜ Fx −ηΩ0Iwiω (4.26) = CF κ 2 z −σκIwω + VIwiω + re0r0CF κ This frequency response is compared with the amplitude response of the nonlinear model and with Pacejka's linear transient tyre model. The plots can be seen in gures 4.11 and 4.12.

For the limit cases ω → 0 and ω → ∞ the amplitude of the frequency response tends to zero.

The amplitude responses reach a maximum at around 55 Hz, which is slightly less than for

51 Pacejka's model at 57 Hz. This can also be seen in the phase plots.

Pacejka's model takes into account a certain amount of rolling resistance which is modeled as a function of the variation in tyre compression [24], this makes that the plots for Pacejka's model are slightly dierent from the linearised and nonlinear model plots.

Figure 4.11: Frequency response of F˜x to axle height variations at 10 m/s.

52 Figure 4.12: Frequency response of F˜x to axle height variations at 40 m/s.

Frequency response of the horizontal force to radial runout

The frequency response function of Fx to radial runout r˜c is derived from (4.25) by omitting the axle height variations z and the variations in the x-position of the wheel x˜. Using the resulting function for the eective rolling radius r˜e =r ˜c − ηr˜c and replacing p with iΩ gives: iΩπ ˜ 2Ω Fx −re0Fz0e 0 + (1 − η)Ω0IwiΩ (4.27) = CF κ 2 r˜c −σκIwΩ + VIwiΩ + re0r0CF κ The comparison between the nonlinear, the linearised and Pacejka's linear model can be seen in gure 4.13.

For the limit case Ω → 0 (4.27) reduces to: F˜ −F x = z0 (4.28) r˜c r0 This clearly diers from 0 as is the case for the frequency response to axle height variations.

The minimum in the amplitude response is typical for the response to radial runout as it is related to the two terms in the numerator of (4.27). The minimum in the amplitude re- sponse occurs when:

iΩπ 2Ω −re0Fz0e 0 = (1 − η)Ω0IwiΩ (4.29)

53 Figure 4.13: Frequency response of F˜x to radial runout

Frequency response of the horizontal force to radial stiness variation The frequency response of the horizontal force to radial stiness variations is also studied. In Pacejka's book [24] this is given as an exercise.

From (4.25) can be concluded that a radial stiness variation C˜F z has no inuence on the horizontal force and only has inuence on the vertical force Fz, for which (4.14) clearly shows a C˜F z-term. The vertical force variations are not inuenced by speed however.

From Dor [14] is already known that the stiness variation is hardly inuenced by rota- tion speed of the wheel, see also chapter 2. A plot of the frequency response will therefore not provide extra information.

54 4.5 Concluding remarks

A nonlinear tyre model is developed which can be used to analyse the eect of radial runout and radial stiness variations on the radial and longitudinal force generation of a tyre.

The nonlinear tyre model is linearised and compared to Pacejka's transient tyre model in order to have some validation as no own measurements have been performed.

The longitudinal force variations of a tyre in the speed range 80 to 120 km/h, in the case of radial runout, are signicantly higher than at lower speeds. The longitudinal force varia- tions in the case of a stiness variation are negligible and are not inuenced by rotation speed of the wheel.

The radial force variations of a tyre due to tyre non-uniformities are not inuenced by the rotation speed of the wheel.

55 Chapter 5

Simulation study on tyre non-uniformities

Three dierent Matlab/Simulink models are used to study the inuence of dierent types of non-uniformity on vehicle vibrations. Next to the use of the complete vehicle model, discussed in chapter 3, a quarter vehicle model, with spring-damper-unit, and a xed axle height quar- ter vehicle model are used. All three models are equipped with one non-uniform tyre with parameters of the radial tyre used in the normal vehicle model, see also appendix B. This diers from the bias-ply tyre parameters used in the previous chapter.

Figure 5.1: Scheme for the xed axle height model.

The xed axle height quarter vehicle model is already discussed in the previous chapter as it is used to develop the nonlinear non-uniform tyre model, see gure 5.1. A spring-damper-unit has been added to this model, while the axle has not been kept xed anymore (not welded to the ground) to have the full quarter vehicle model, see gure 5.2. The top body is able to move freely in z-direction and is pushed in x-direction with a constant velocity.

56 Figure 5.2: Scheme for the quarter vehicle model.

For the full vehicle model only the left front wheel has been replaced by the non-uniform tyre model. This ensures that the model stays on track, since the non-uniform tyre model cannot generate lateral forces. Using the non-uniform tyre model on the front axle makes it is also possible to simulate steering system vibrations, as can be seen in gure 5.3.

Figure 5.3: Setup of the front axle, left front

57 The main dierences between the vehicle's left front suspension and the full quarter vehicle model are as follows:

• The full quarter vehicle model's hub has only one degree of freedom (DOF), which is the translation in z-direction, while the vehicle model's left front hub is attached to the chassis by means of a wishbone connected to the chassis through bushings. The vehicle model's suspension is also modeled with camber, caster, etc. This makes that the vehicle model's hub is not restricted to z-direction translation.

• The vehicle model's left front hub is connected to a steering linkage, which connects the left non-uniform tyre with the right uniform tyre through a steering system.

• The vehicle model's left front hub is also connected to an anti-roll spring which in fact makes the suspension stier when only one wheel is actuated in z-direction.

This chapter exists of several sections of which each section discusses one specic type of non- uniformity. Simulation results for all three models mentioned above are compared in order to get an idea on the inuence of the type of model on the force generation characteristics. The gures show amplitude values of the mentioned signal and the tyre parameters used in these simulations are from the radial tyre used in the complete vehicle model, see appendix B. To have a fair comparison between the longitudinal and vertical forces and the dierent types of tyre non-unformity, the amounts of non-uniformity are tuned in such a way that the vertical force variations for the xed axle model (no speed dependency) are 180 N, which is an acceptable value according to Demic's paper [13]. The forces shown are measured at the revolute joint connecting the wheel to the hub. The speed range from which most customer feedback is received is between 80 to 120 km/h.

5.1 Radial runout

The amplitude of the radial runout used is 0.9 mm which is signicantly less than the runout used in the previous chapter.

As discussed in chapter 4, and as expected, the radial forces do not vary as a function of speed for the xed axle height model, see gure 5.4. Thus speed dependency for the quarter vehicle and full vehicle model is related to the dynamics of the suspension.

Therefore two known eigenfrequencies for a quarter vehicle can be related to the characteristics in gures 5.4:

• At approximately 1 Hz, so-called bounce occurs, which is the resonance of the sprung mass. This means that with the used model-parameters around 7 km/h bounce occurs.

• At approximately 10 Hz, so-called wheel hop occurs, which is the resonance of the unsprung mass. With the used model-parameters, around 70 km/h wheel hop occurs.

As for the full vehicle model, the suspension is not restricted to 2D, the vehicle's body is cou- pled to four wheels and compliances have been modeled too, the eigenfrequencies (and their inuence) might dier from the quarter vehicle model. As can been seen in gure 5.4, where the

58 eigenfrequency (speed) at which wheel hop occurs is higher than for the quarter vehicle model.

Figure 5.4: Longitudinal and radial force variations due to a radial runout.

As well the radial as the longitudinal force variations are overall highest with the quarter vehicle model. The amplication due to eigenfrequencies is in accordance with the ndings of Dor [14], who also reported that the suspension resonances amplify the force variations, see also chapter 2.

Figure 5.5: Absolute accelerations as a result of a tyre with radial runout for the full vehicle model.

59 In gure 5.5 one can see the body accelerations and y-acceleration dierence between the steer- ing rack and body as simulated with the full vehicle model. The accelerations in z-direction are highest as might be expected since the forces in z-direction also clearly dominate the force variations. Although the tyre model does not have the possibility to create forces in y-direction, chassis accelerations in y-direction occur and slightly dominate the x-direction. This is due to the vertical force variations being transmitted to the chassis through a trian- gular suspension layout, i.e. the lower wishbone.

The eect and size of these accelerations and their inuence on the comfort related behaviour of a car is not part of this thesis but will only be used as an extra comparison possibility between the dierent types of tyre non-uniformity. It is highly recommended to have some research performed on this subject to have an idea of the inuence on comfort degradation due to tyre non-uniformities.

5.2 Radial stiness variation

The radial stiness variation used has an amplitude of 10721 N/m.

As seen in chapter 4, the inuence of speed on, and the size of longitudinal force varia- tions as a result of a radial stiness variation is signicantly less than is the case for radial runout. The radial force variations are highest for the full vehicle model. Compare gures 5.4 and 5.6. No longitudinal force variations occur with the xed axle model as expected from the theory in the previous chapter, thus the longitudinal force variations for the models with suspension need to be a result of the radial force variations and the extra degrees of freedom of the hub for these models. Therefore they are also lower than for the tyre with radial runout.

The inuence of the suspension on this response is very clear since the full vehicle radial force variations now clearly dominate the quarter vehicle model's force variations. Just as with the responses to radial runout, bounce and wheel hop are visible in the quarter vehicle and full vehicle model's responses shown in gure 5.6. The radial force variations are larger in the case of radial stiness variation than for radial runout.

The resulting chassis accelerations in z-direction are also slightly higher than for the radial runout case. The inuence of the radial force variations on the longitudinal force variations is also visible in the x- and y-direction accelerations. These are more like the radial force variation graph than is the case for radial runout where these accelerations have the form of the longitudinal force variations graph.

60 Figure 5.6: Longitudinal and radial force variations due to a radial stiness variation.

Figure 5.7: Absolute accelerations as a result of a tyre with radial stiness variation for the full vehicle model.

61 5.3 Measured tyre non-uniformity with a peak-characteristic

In this subsection the results obtained through simulations of a measured non-uniformity characteristic, named peak, will be shown. Since the non-uniformity of a tyre is usually measured as a radial force variation, a translation had to be made to be able to implement the measured characteristic into the non-uniform tyre model. From previous sections we can conclude that especially radial runout has the biggest inuence on force variations in relation to speed. Therefore the assumption is made that the measured force variation is largely caused by radial runout, and the measured force variation has been translated to a radial runout characteristic by dividing the measured force variation by a nominal radial tyre stiness of 200 N/mm resulting in the radial runout characteristic that can be seen in gure 5.8. An extra factor has been used in the model to create the vertical force variation of 180 N as used for a fair comparison mentioned before.

Figure 5.8: Radial runout as a function of the tyre rotation angle, derived from a measured force variation

The peak-type runout mainly has an inuence on the longitudinal force variations at lower speeds, as for higher speeds the force variation does not increase very much.

The radial force variations are also largely inuenced by the peak-type runout at lower speeds. The peak at bounce speed is almost as high as the peak at wheel hop speed, especially for the quarter vehicle model, see gure 5.9. It is remarkable that the radial force variations for the full vehicle model are noticeably smaller than for the quarter vehicle model. This is also the case in gure 5.4 but with a smaller dierence.

62 Figure 5.9: Longitudinal and radial force variations.

Figure 5.10: Absolute accelerations as a result of a tyre with radial runout typied as peak for the full vehicle model.

As a result of the higher force variations at lower speeds, the chassis accelerations, especially in z-direction, are up to about 50 km/h higher than for a perfectly harmonic radial runout or stiness variation. At higher speeds the accelerations due to perfect sine-wave signals domi- nate again.

Real measured runout and stiness variations would be a great asset to the non-uniform tyre model and the simulations performed with this model. The measured radial force variations

63 are now assumed to be 100% due to radial runout, but might also be partly or completely due to a radial stiness variation. Therefore some focus on this matter might be very useful in follow-up research.

5.4 Measured tyre non-uniformity with a harmonic-characteristic

The inuence of a second type of measured force variations, named harmonic, is also studied. The measured force variation has been translated to a radial runout characteristic in the same manner as for the peak-type force variation, see gure 5.11. Again a factor is used to tune to the desired 180 N.

Figure 5.11: Radial runout as a function of the tyre rotation angle, derived from a measured force variation

In comparison to the radial runout with a perfect sine-wave form as used in section 5.1, the responses are only slightly dierent.

The longitudinal force variations shown in gure 5.12 mainly dier from gure 5.4 in the lower speed range up to 50 km/h. The dip in the responses is not visible with the measured characteristic and therefore the build-up of the force variations starts earlier than for the per- fect sine-wave form radial runout.

The radial force variations are also slightly less at higher speeds than for the radial runout case of section 5.1., see gure 5.12. This minimal dierence is also visible in the accelerations gure 5.13.

64 Figure 5.12: Longitudinal and radial force variations.

Figure 5.13: Absolute accelerations as a result of a tyre with radial runout typied as harmonic for the full vehicle model.

5.5 Run-out and stiness variation combined

It is especially interesting for a tyre manufacturer to know if it is possible to compensate one type of non-uniformity with another type of non-uniformity. Combining non-uniformities might be a solution to lower the total force variations and thereby reducing the number of scrapped tyres. A problem posed here is that tyre manufacturers measure force variations,

65 while in the simulations several type of non-uniformities, each creating force variations, are used.

Nevertheless, this section shows the inuence of a combination of radial runout and radial stiness variation on force variations, for four dierent phaseshifts in steps of π . The non- 2 uniformities used are those used in the two rst subsections of this chapter. Only the full vehicle model has been used for the following simulations.

Figure 5.14: Longitudinal and radial force variations.

Figure 5.15: Absolute accelerations if both non-uniformities are in anti-phase for the full vehicle model.

66 The force variations are lowest when the non-uniformities are in anti-phase. This is the case when the radial runout is maximal while the stiness variation is lowest. The radial force variations for a phaseshift of π are signicantly lower than the radial force variations for only 2 runout or stiness variation. The other combinations show a higher amount of force variations.

Since the anti-phase combination gives the lowest force variations, the accelerations for this case are shown in gure 5.15 to be able to get an idea on the inuence on vibrations of this specic combination in relation to the accelerations due to non-combined non-uniformities. As expected the chassis accelerations in z-direction are signicantly lower than for the individual non-uniformities.

5.6 Concluding remarks

The goal of this chapter is to show the vehicles dynamic interaction with a non-uniform tyre. Therefore three dierent models are used and simulations with single, and combinations of, tyre non-uniformities are discussed. The main conclusions drawn from this are:

• The type of model has a signicant inuence on the generated force variations and resulting chassis vibrations. Suspension and vehicle resonances clearly have an inuence on the force variations in the complete simulated speed range, for which in the special range of interest (80 to 120 km/h) the force variations are amplied in relation to the xed axle model.

• A radial stiness variation shows the largest radial force variations in comparison with a radial runout. The longitudinal force variations for a radial stiness variation are smaller in comparison with a radial runout since they are a result of the radial force variation and not of the non-uniformity itself, see also the previous chapter.

• The comparison between the two types of measured non-uniformities show signicantly larger radial and longitudinal force variations in the speed range up to 50 km/h for the peak-type non-uniformity. Above 50 km/h the force variations are higher with the harmonic-type.

• A combination of radial runout and radial stiness variation for which they are in anti-phase reduce the radial force variations dramatically in relation to the single non- uniformities or the other simulated combinations of both non-uniformities. The longi- tudinal force variations are not signicantly inuenced by a combination of both tyre non-uniformities.

67 Chapter 6

Conclusions and recommendations

6.1 Conclusions

A full vehicle model with a detailed suspension has been developed to have a good repre- sentation of a real vehicle and to be able to simulate the eect of tyre non-uniformities on a car:

• The results from the kinematic and compliance simulations performed with the vehicle model show that the model is able to behave within realistic ranges.

• From the steady state simulation one can conclude that the model is slightly more understeered at higher lateral accelerations and the side slip angle beta is slightly higher than the real vehicle for the unloaded simulation, for the loaded steady state simulation the dierence is larger.

• The results from the random steer simulation are acceptable.

• Additional simulations show a similar behaviour between the model and the vehicle.

The rotation speed of a tyre has a large inuence on the force variations generated by tyre non-uniformities as can be concluded from the literature survey in chapter 2 and the simula- tion results of chapter 4 and 5. The longitudinal force variations are larger at the speed range 80 to 120 km/h than at the speed of about 6 km/h. This is the speed at which radial force variations are measured at the end of the tyre production. From the speed range 80 to 120 km/h most customer complaints are received.

A nonlinear tyre model has been developed that enables to simulate radial runout, radial stiness variations and measured non-uniformities and their eect on the force generation of a tyre. In the scope of this report, the tyre model has been restricted to in-plane (2D).

From simulations performed with the nonlinear non-uniform tyre model on a xed axle height model, a quarter vehicle model and the full vehicle model, the following can be concluded:

• The radial force variations generated through a non-uniform tyre are amplied by the suspension's eigenfrequencies.

68 • The measured tyre non-uniformity typied as peak, shows larger force variations (ra- dial and longitudinal) at speeds up to 50 km/h than the measured tyre non-uniformity typied as harmonic.

• A combination of radial runout and radial stiness variation for which they are both in anti-phase causes the lowest force variations of the four simulated combinations.

• The anti-phase combination of radial runout and radial stiness variation causes lower force variations than if only radial runout or radial stiness variation is simulated. At the speed range 80 to 120 km/h the maximal radial force variation for this specic combination is in the order of 50 N. For only radial runout this is in the order of 180 N and for only stiness variation 225 N.

6.2 Recommendations

• A general recommendation is, when one is making a full vehicle model with this level of detail in the suspension system, it is advisable to have all the exact data from the vehicle or have the real vehicle present. This way parameters can be measured instead of making assumptions. For this vehicle model the tuning started with kinematics and after that the compliance was tuned, it is advisable to rst collect all the data and then tune both kinematics and compliance at the same time, because all parameters in some way inuence each other.

• Additional tests are only performed with the unloaded vehicle model, since the model is validated and also the additional tests show that the model is similar to the real vehicle, however, one can perform these extra simulations with the loaded model.

• Power steering can be added to the steering system model when this is desired. Retuning of the steering system parameters will be necessary in this case.

• The nonlinear tyre model needs to be extended to three dimensions to study the inuence of lateral non-uniformities such as for example ply-steer.

• The nonlinear tyre model is validated against Pacejka's transient tyre model. To have a good validation for the nonlinear tyre model however, measurements in the complete simulated speed range are needed for which both the longitudinal as well as the radial force variations are measured. The tyre parameters, used as input to the tyre model for this validation, also need to be measured.

• More research is needed on what the causes are for a certain measured force variation as in this thesis the measured radial force variations are assumed as being caused by radial runout for 100%.

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71 Appendix A

Vehicle model parameters

Parameters

In this section all the parameters of the BMW full vehicle model are given. In table A.1 the vehicle is described in table A.2 and A.3 the parameters of the model are given for the unloaded situation, mass, inertia etc.. In table A.4 the parameters of the loaded model are given. In the tables A.5, A.6 the dimensions of the model are described, next in table A.7 the stiness parameters of the bushings and other exible parts are given.

The origin is dened at road level in the centre of the front axle and the axis system can be seen gure A.1.

Figure A.1: Scheme for the xed axle height model.

Coordinates for the left side of the model are dened, the right side is exactly the same only the y-axis is multiplied with minus one.

72 Table A.1: Global vehicle parameters Vehicle parameters BMW, 1996, 523i Gross vehicle weight 1495 [kg] Max. vehicle weight 1955 [kg] L x W x H 4.775 x 1.800 x 1.435 [m] Wheelbase 2.830 [m] Track front/rear 1.516 / 1.530 [m] Engine 2.5 liter, 6 Cylinder, 125 [kW]

Table A.2: Parameters of vehicle, unloaded parameter value vehicle wheelbase 2.830 [m] average vehicle track width 1.5230 [m] vehicle height 1.435 [m] weight front left 409.5 [kg] weight front right 419.0 [kg] weight rear left 406.5 [kg] weight rear right 424.5 [kg] total vehicle weight 1659.5 [kg] unsprung mass front left 45 [kg] unsprung mass front right 45 [kg] unsprung mass rear left 40 [kg] unsprung mass rear right 40 [kg] vehicle body mass 1489.5 [kg] vehicle body inertia about world x-axis 538.896 [kgm2] vehicle body inertia about world y-axis 1594.9 [kgm2] vehicle body inertia about world z-axis 1754.4 [kgm2] vehicle body centre of gravity coordinates [ -1.427 0 0.502] [m] rim mass properties [20 1 1.5] [mass(kg) Ixx(kgm2) Iyy(kgm2)] initial tyre loaded radius (kerb height) 0.3 [m] Tyre width 0.215 [m] tyre mass 9.3 [kg] maximum brake moment (sum of all wheels) 6000 [Nm] brake moment distribution 0.75 [%] steer ratio 18.3 [-] Engine Max Power 125 [kW] at 5500 [rev/min] Max Torque 245 [Nm] at 3950 [rev/min] 1st gear ratio engine-wheels 4.20*3.15 top gear ratio engine-wheels 1*3.15

73 Table A.3: Parameters of vehicle, unloaded (continued) parameter value front axle centre of front axle [0 0 0.3] [m] front axle track width 1.516 [m] king pin inclination 13.55 [deg] caster angle 6.46 [deg] camber angle 0.21 [deg] toe angle 0.08 [deg] scrub radius 3.4 [mm] mechanical trail 33.96 [mm] wheel center oset 70 [mm] roll center height ±0.1 [m] rear axle centre of rear axle [-2.83 0 0.3] [m] rear axle track width 1.530 [m] camber angle -2.17 [deg] toe angle 0.27 [deg]

Table A.4: Parameters of vehicle for loaded condition parameter value weight front left 422 [kg] weight front right 407.5 [kg] weight rear left 560.5 [kg] weight rear right 564 [kg] total vehicle weight 1954 [kg] vehicle body mass 1784 [kg] vehicle body inertia about world x-axis 794.4 [kgm2] vehicle body inertia about world y-axis 3500 [kgm2] vehicle body inertia about world z-axis 2800 [kgm2] vehicle body centre of gravity coordinates [ -1.65 0 0.58] [m]

74 Table A.5: Parameters of the McPherson front suspension

Function(name) Connection method coordinates [m] translated from Mass [kg] Inertia [kgm2] Chassis body CoG:[-1.427 0 0.502] World 1489.5 [538.9 0 0 0 1594.9 0 0 0 1754.4] Weld Cs1:[0 0 0] Adjoining Custom joint Cs2:[0 0 0] CoG Weld CS3:[0 0 0] Adjoining Virtual Reality CS5:[0 0 0.3] World chassis body CoG:[0 -0.02 0.8] World 51.2 eye(3)*1e-6 Weld Cs1:[0 0 0.8] World Spherical Cs2:[ -0.0566 0.5675 0.8]World Bushing Cs3:[0.09 0.358 0.305]World Prismatic Cs4:[0.13 0 0.3]World Universal Cs5:[-0.078 -0.358 0.3]World Bushing Cs6:[0.09 -0.358 0.305]World Spherical Cs7:[-0.0566 -0.5675 0.8]World Universal Cs8:[-0.078 0.358 0.3]World Damper CoG:[-0.0283 0.6277 0.55] World 1e-6 eye(3)*1e-6 Spherical Cs1:[0 0 0] Adjoining Prismatic Cs2:[0 0 0] Adjoining Lower wishbone CoG:[0.045 0.523 0.3025]World 1e-6 eye(3)*1e-6 Spherical Cs1:[0 0 0] Adjoining Bushing Cs2:[0 0 0]Adjoining Weld Cs3:[0 0 0] Cs1 Body CoG:[-0.039 0.523 0.3]World 1e-6 eye(3)*1e-6 Weld Cs1:[0 0 0]Adjoining Universal Cs2:[0 0 0]Adjoining Hub CoG:[0 0.758 0.3] 1e-6 eye(3)*1e-6 Spherical-Spherical Cs1:[0.13 -0.0352 -0.0035] Custom joint Cs2:[0 0 0] Prismatic Cs3:[0 -0.07 0] Spherical Cs4:[ 0 -0.07 0] Steering system Rack prismatic CoG:[0 0 0] Adj 2 [0.0344 0 0 0 0.0344 0 0 0 0.0344] Spherical-Spherical Cs2:[0 -0.375 0]CoG Spherical-Spherical Cs3:[0 0.375 0]CoG

75 Table A.6: Parameters of the multilink rear suspension

Function(name) Connection method coordinates [m] Mass [kg] Inertia [kgm2] subframe CoG:[-2.83 0 0.3] World 21.4 eye(3) Universal Cs1:[ 0.15 0.4 0.01 ]CoG Weld Cs2: [-2.83 0 0.8] World Spherical Cs3: [-0.06 0.615 0.5] CoG Custom joint Cs4: [0.15 0.18 0.25] CoG Custom joint Cs5: [ -0.05 0.35 0.1]CoG Bushing Cs6: [-0.1 0.4 0] CoG Spherical Cs7: [ -0.06 -0.615 0.5 ]CoG Custom joint Cs8: [0.15 -0.18 0.25] CoG Custom joint cs9: [-0.05 -0.35 0.1] CoG Bushing Cs10:[ -0.1 -0.4 0] CoG Universal Cs11:[ 0.15 -0.4 0.01] CoG Body CoG:[-2.71 0.7825 0.355] World 1e-6 eye(3)*1e-6 Spherical CS1:[0 0 0] Adjoining Spherical Cs2:[0 0 0] Adjoining Lower wishbone CoG:[-2.83 0.51 0.25] World 5 eye(3)*1e-6 Spherical Cs1:[ 0 0.4 0] Cs3 Universal Cs3:[0 0 0] Adjoining Bushing Cs4:[0.1 0 0] Adjoining Spherical Cs5: [0 0 0] Adjoining Rear link CoG:[-2.88 0.5575 0.4] World 1e-6 eye(3)*1e-6 Spherical Cs1:[0 0 0] Adjoining Custom joint Cs2:[0 0 0] Adjoining Integral link CoG:[-2.725 0.4975 0.5] World 1e-6 eye(3)*1e-6 Spherical Cs1:[0 0 0] Adjoining Custom joint Cs2:[0 0 0] Adjoining Damper1 CoG:[ -2.86 0.615 0.55] World 1e-6 eye(3)*1e-6 Revolute Cs1:[0 0 0] Adjoining Prismatic Cs2:[0 0 0] CoG Damper2 CoG:[ -2.86 0.615 0.55] World 1e-6 eye(3)*1e-6 Prismatic Cs1:[0 0 0] Adjoining Spherical Cs2:[0 0 0] Adjoining Hub CoG: [-2.83 0.765 0.3 ] World 1e-6 eye(3)*1e-6 Custom joint Cs1:[ 0 0 0 ] CoG Spherical Cs2: [ 0.09 0 0.1] CoG Revolute Cs3: [ 0 -0.15 0 ] CoG Spherical Cs4: [ 0.06 0.05 0.15 ] CoG Spherical Cs5: [ -0.05 0 0.1] CoG Spherical Cs6: [-0.1 0 -0.1] CoG

76 Table A.7: Stiness parameters of the model, compliance spring stiness front 28500 [N/m] spring stiness rear 21500 [N/m] roll stiness front 28500 [N/m] roll stiness rear 5000 [N/m] Steering system Stiness 2.9367e+006 [Nm/rad] Inertia 0.0344 [kgm2] Damping 2.9367e+004 [Nms/rad] Sr.ratio 0.0068 [-] Connection direction stiness [N/m] damping [Ns/m] Bushing, front axle x 576000 5760 y 576000 28800 z 5760000 57600 rear axle Bushing, lower wishbone x 800000 8000 y 2000000 60000 z 1000 10 Rearlink x 10000 100 y 50000 500 z 100 1 Leading link y 1100000 11000

77 Pictures of the vehicle

Photo's of the front and rear axle of a BMW type e60 for which the steering system diers from the e39 model.

Figure A.2: Photo of the front axle, left side front of the vehicle, (e60)

78 Figure A.3: Photo of the rear axle, taken from behind, (e60)

Figure A.4: Photo of the rear axle, taken from behind, (e60)

79 Appendix B

Additional vehicle model tests

The following tests are performed with the unloaded model:

• Lane change (ISO/TR 3888 ): Lane change is a discrete, closed loop, handling test, closed loop in the sense that the driver has to follow a prescribed path. The simulation is open loop since it is performed with cruise control on and the steering angle of the measurement is used as input.

• J-turn (ISO 7401): with the step input, the overshoot in yaw and lateral acceleration, response time and build up speed of the signals, the response of the vehicle is ana- lyzed. The simulation is performed with cruise control on and the steering angle of the measurement is used as input.

• Straight line braking (ISO 6597): with this test the braking behaviour is tested. The distance needed to come to a standstill from an average speed of 100 km/h is an im- portant parameter in this test. For the simulation the brake input is tuned to t the measurement.

• Brake in turn (ISO 7975): in this test the vehicle is driven at a constant lateral accel- eration and then braked with a constant brake-pedal pressure. This is a complex test and uses almost all relevant parameters of the vehicle: mass, geometrical parameters, suspension system, steering system and brake system etc. Important are the conicting aims between stability and attainable deceleration during the braking. Like the straight line braking test the brake input for the simulation is tuned in comparison with the measurement.

• Power o (ISO 7975): This test is similar to the brake in turn test, in this case instead of braking at a constant lateral acceleration of 4 m/s2 the throttle pedal is released on a circular path with radius of 100 m. Similarly to the brake in turn test, this test evaluates vehicle stability and predictability in response to the change in longitudinal tire forces. For the simulation instead of a brake input a negative throttle signal is used.

• Cross wind behaviour (ISO 12021-1): wind speed 18 m/s, length of wind zone 28.3 m, importance of this test is the stability of the vehicle during external disturbances. During the simulation a force is generated on the center of gravity of the chassis. This is dierent to the measurement where rst the front of the vehicle is inuenced by the windforce and in the end the rear of the vehicle.

80 Figures B.1, B.2, B.3, B.4, B.5 and table B.1 show the results of the extra tests. As one can see the simulation results are comparable with the measurements. Both lane-change and J-turn simulations are comparable with the measurements, although the model has a slightly faster response compared to the measurement. With straight line braking the model comes close to the vehicle only the deceleration is lower, though the braking distance is not very dif- ferent as shown in table B.1. For the brake in turn test the model shows a similar behaviour compared to the vehicle test. This is somewhat improved by changing the brake balance more to the rear, and decrease the brake input. The relaxation eect can be seen when the vehicle comes to a hold, there is an resonance at the end where without the relaxation behaviour the signal comes direct to a hold. With the Power o test the deceleration shows an eect that is dierent from the measurement, the lower starting point is because the model is cornering which causes the vehicle to slow down, the acceleration itself is comparable, both about 1 m/s2. Crosswind behaviour is not completely simulated like the real test, in the real test the vehicle drives across wind generators. During the simulation the vehicle is pushed sideways with a lateral force, the main dierence is that during the measurement rst the front, than the middle of the vehicle is inuenced by the wind force and in the end the rear part, this can cause a dierent yaw behaviour (later start compared to lateral acceleration) from the vehicle when compared with the simulation.

Table B.1: Straight line braking parameter measurement simulation braking distance 43.6 [m] 44.4[m] braking time 3.1 [s] 3.2 [s] average deceleration 9.36 [m/s2] ≈ 8.4 [m/s2]

81 Figure B.1: Lane change 82 Figure B.2: J-turn 83 Figure B.3: Brake in a turn 84 Figure B.4: Power o 85 Figure B.5: Crosswind behaviour 86 Appendix C

Tyre model parameters

Table C.1: Bias-ply tyre parameters as used by Pacejka Parameter Value

Fz0 3000 [N] CF κ 45000 [N/m] CF x 1e6 [N/m] CF z 190000 [N/m] Fr0 60 [N] Ar 0.02 [-] rf0 0.325 [m] r0 0.3 [m] η 0.4 [-] σkappa CF κ/CF x ρ0 Fz0/CF z CF za 30082 [N/m] rc 2.5e-3 [m] re0 0.325 [m] 2 Iw 0.8 [kgm ] in case of z-variations 2 Iw 1.1 [kgm ] in case of non-uniformities

87 Table C.2: Radial tyre parameters as used by Pacejka Parameter Value

Fz0 3000 [N] Cfκ 80000 [N/m] Cfx 565000 [N/m] Cfz 133000 [N/m] Fr0 25 [N] Ar 0.0083 [-] rf0 0.325 [m] r0 0.3 [m] η 0.6 [-] σkappa Cfκ/Cfx ρ0 Fz0/Cfz

Table C.3: Radial tyre parameters as used in the full vehicle model Parameter Value

Fz0 4000 [N] Cfκ 80000 [N/m] Cfx 565000 [N/m] Cfz 200000 [N/m] Fr0 25 [N] Ar 0.0083 [-] rf0 0.326 [m] r0 0.3136 [m] η not used, see gure C.1 σkappa Cfκ/Cfx ρ0 Fz0/Cfz CF za 30082 [N/m] rc 2.5e-3 [m]

88 Figure C.1: re as a function of ρ for the bias-ply tyre

89