View of a Ford Ranger Automatic Transaxle (Therangerstation.Com)

Analysis and Experimental Comparison of Models of a New Form of Continuously

Variable Transmission

A dissertation presented to

the faculty of

the Russ College of Engineering and Technology of Ohio University

In partial fulfillment

of the requirements for the degree

Doctor of Philosophy

Timothy J. Cyders

December 2012

This dissertation titled

Analysis and Experimental Comparison of Models of a New Form of Continuously

Variable Transmission

TIMOTHY J. CYDERS

has been approved for

the Department of Mechanical Engineering

and the Russ College of Engineering and Technology by

Robert L. Williams II

Professor of Mechanical Engineering

Dennis Irwin

Dean, Russ College of Engineering and Technology 3

ABSTRACT

CYDERS, TIMOTHY J., Ph.D., December 2012, Mechanical Engineering

Analysis and Experimental Comparison of Models of a New Form of Continuously

Variable Transmission

Director of Dissertation: Robert L. Williams II

Efficient, high-performance continuously variable transmission (CVT) technology is currently being pursued by many engineering companies as a way to combat manufacturing cost, improve the feasibility of systems such as electric cars and small- scale wind power generation, and improve efficiency of many mechanical systems such as industrial pumps and fans. Successful, wide-scale implementation of efficient CVT technology has the capacity to reduce global energy consumption, diversify feasible energy sources and improve production and operating cost of many mechanical systems, but most contemporary designs have worse efficiency, cost or performance than the systems they are meant to replace. A new, unique mechanism called the Beale CVT has the capacity to overcome all these obstacles, but presents difficulties in design based on its dynamic characteristics, which have not previously been modeled or verified through experiment. This work successfully developed an accurate dynamic model for the elements making up the Beale CVT to be used in future design of the mechanism, which was then verified by experiment on a physical transmission prototype.

DEDICATION

For Jess

Not those monkeys. Other monkeys.

ACKNOWLEDGEMENTS

Many thanks are due to my committee members, who helped with moral and technical support along the way, as well as driving me to learn more than I would have otherwise. Thank you all for your time, your patience, your guidance and your friendship.

I would also like to thank my many mentors who have helped me along the way with good advice, kind words and open doors. William Beale, the inventor of this concept, has been instrumental in my education and development not only with respect to this wonderful invention, but also as an engineer and inventor myself. Without his ingenuity, none of this work would exist.

The past few years have been an arduous journey, and the support I received from my friends and family members are the only reason I was able to weather it. My parents worked hard to instill in me an understanding of the sacrifices required and rewards reaped from the undertaking of pursuit of knowledge, and that such a pursuit is a never- ending quest replete with intrinsic rewards. Without them, I never would have made it.

My brothers have been there every step of the way as well with material and moral support; it was more valuable than they likely realize.

I would like to thank my grandfather for being a constant source of joy and pride, and for making the sacrifices he did to make this a possibility for our family.

Most of all, I wish to thank my beautiful wife for who she is, and the care she has taken of me for these past three years. Without her, this would all be meaningless, a chasing after the wind. I love her dearly. 6

TABLE OF CONTENTS

Page

Abstract ...... 3

Dedication ...... 4

Acknowledgements ...... 5

List of Tables ...... 9

List of Figures ...... 10

List of Abbreviations and Symbols...... 15

1. Introduction ...... 17

1.1 Background ...... 22

1.2 Functional Description of Beale CVT ...... 24

2. Literature Search ...... 27

2.1 Other Existing Transmission Designs ...... 27

2.1.1 Toroidal ...... 27

2.1.2 Cone Drive ...... 29

2.1.3 Hydrostatic ...... 31

2.1.4 Epicyclic ...... 32

2.1.5 Cam-Based IVT ...... 35 7

2.2 Modeling of One-Way Elements ...... 38

2.3 Relevant Software ...... 43

2.3.1 OpenModelica/Dymola ...... 44

2.3.2 MSC ADAMS ...... 46

2.3.3 Autodesk Simulation Mechanical ...... 47

3. Dissertation Goals ...... 50

3.1 Problem Statement ...... 50

3.2 Scope of Work ...... 50

3.3 Dissertation Objectives ...... 51

3.4 Experimental Criteria ...... 52

4. Introductory Pseudo-static Model ...... 55

4.1 Four-bar Mechanism Analysis ...... 55

4.2 Spring Design ...... 68

4.3 Complete Pseudo-static Model ...... 71

5. Dynamic Models ...... 76

5.1 Component-Based Model ...... 80

5.2 Finite-Element Model ...... 86

6. Experimental Design and Setup ...... 89

6.1 Parameter Measurement ...... 95 8

6.2 Instrumentation ...... 108

6.3 Experimental Procedure ...... 111

6.4 Data Analysis ...... 113

7. Experimental Results and Discussion ...... 115

7.1 Physical Phenomena ...... 116

7.2 Comparison of Model Results ...... 124

8. Conclusions ...... 136

8.1 Project Results ...... 136

8.2 Future Work ...... 138

References ...... 142

Appendix: Simulation Code ...... 149

FreeMat Four-bar Mechanism Model ...... 149

Data Processing Algorithm ...... 151

Modelica Sprag Unit Model ...... 154 9

LIST OF TABLES

Page

Table 1: Link lengths for four-bar crank-rocker mechanism ...... 57

Table 2: DC motor characterstics for testbed drive motor ...... 93 10

LIST OF FIGURES

Page

Figure 1: Sketch of a CVT concept by Leonardo DaVinci, c. 1490 (Anon) ...... 17

Figure 2: Exploded view of a Ford Ranger automatic transaxle (therangerstation.com) . 18

Figure 3: Beale CVT/IVT Schematic (Beale, 2006)...... 20

Figure 4: Schematic of four-bar mechanism system from Cyders and Williams (2010). 23

Figure 5: Schematic example of crank-rocker mechanism motion ...... 25

Figure 6: Example of a toroidal CVT (NSK, Inc. 2011) ...... 28

Figure 7: Cutaway view of toroidal CVT design (Dodge 1939) ...... 29

Figure 8: Evans friction cone-ring drive (Ukexpat 2009) ...... 30

Figure 9: Cone-ring drive as detailed in US Patent 3,257,837 (Davin 1966) ...... 30

Figure 10: Hondamatic Hydraulic CVT with adjustable swashplate (Anon 2000) ...... 31

Figure 11: 3-d model of Torvec hydraulic IVT (Torvec, Inc. 2010) ...... 32

Figure 12: Example of an epicyclic gearset (Longhurst 2011) ...... 33

Figure 13: Type I and Type II power flow, as depicted by Mantriota (2002) ...... 34

Figure 14: Epicyclic CVT invented by Paul Pires (Brown 1992) ...... 35

Figure 15: Epicyclic gear-based IVT analyzed by Benitez (2004) ...... 35

Figure 16: Cam driven analogue for epicyclic IVT (Lahr and Hong 2009) ...... 36

Figure 17: Designed implementation of cam-based IVT (Lahr and Hong 2009) ...... 37

Figure 18: Adjustable CVT using a one-way clutch (Zero-Max, Inc. 2011a) ...... 37

Figure 19: Manually-adjustable variable speed drives (Zero-Max, Inc. 2011b) ...... 38

Figure 20: Typical sprag clutch construction (Miura, Numata and Le Calve 2005) ...... 39 11

Figure 21: Schematic of idealized sprag clutch model ...... 40

Figure 22: 2-DOF model of automotive belt-pulley system (Zhu 2006) ...... 42

Figure 23: Example of a block diagram in OpenModelica's OM Edit ...... 45

Figure 24: Bouncing ball example simulation in OpenModelica ...... 46

Figure 25: MSC Adams simulation example (STU Bratislava 2009) ...... 47

Figure 26: Autodesk Mechanical Event Simulation example...... 49

Figure 27: Prototype of the Beale CVT ...... 56

Figure 28: Vector loop diagram of physical system ...... 57

Figure 29: Output angle 4 vs. input angle 2 ...... 58

Figure 30: Transmission angle vs. 2 ...... 59

Figure 31: Measured speed ratio vs. torque demand (Cyders and Williams 2010)...... 60

Figure 32: Free-body diagram of input to crank-rocker mechanism ...... 61

Figure 33: Instantaneous speed ratio vs. 2 ...... 62

Figure 34: FTI vs. input angle for crank-rocker mechanism (Lin and Chang 2002) ...... 64

Figure 35: Crank rocker mechanical advantage (black) with friction load (red) ...... 66

Figure 36: Crank rocker mechanical advantage with joint friction limitations ...... 68

Figure 37: Nonlinear spring deflection curve ...... 69

Figure 38: Range of effected motion for different torque levels ...... 70

Figure 39: Different transmission behavior resulting from different spring limits ...... 70

Figure 40: Simplified crank rocker triangles at singularity crossings ...... 73

Figure 41: Calculated (red) and measured (black) speed ratios vs. torque ...... 74

Figure 42: Error in estimation of singularity position assuming constant r4 ...... 75 12

Figure 43: Single unit setup of BCVT prototype for simplified model validation ...... 77

Figure 44: Basic free-body diagram of sprag unit ...... 78

Figure 45: Acausal model of RLC circuit from Fritzson (2004) ...... 82

Figure 46: Causal model of the same circuit in the previous Figure (Fritzson 2004) ...... 82

Figure 47: Modelica model of single sprag unit and output shaft ...... 84

Figure 48: Rotational spring model with flexible link r4 ...... 84

Figure 49: 2-D Finite element model of spring behavior in contact with backstop ...... 87

Figure 50: 3D MES calculating full forward dynamics ...... 88

Figure 51: Example testbed layout with conceptual instrumentation layout ...... 90

Figure 52: 3D model of final testbed design with instrumentation ...... 91

Figure 53: BCVT prototype, as built for experimentation ...... 91

Figure 54: Motor control and power wiring layout ...... 93

Figure 55: Measured (blue) and calculated (green) variables on the final testbed design 95

Figure 56: Composite beam spring with E-glass, carbon fiber and epoxy layers...... 96

Figure 57: Experimental setup for determination of composite spring elastic modulus .. 97

Figure 58: Spring force-deflection data for calculation of Young’s Modulus ...... 98

Figure 59: Experimental setup for determination of spring damping characteristics ...... 99

Figure 60: Displacement vs. time for determination of damping ratio,  ...... 100

Figure 61: Strain gauge mounted over hole on connecting link ...... 102

Figure 62: Meshed FEA model showing strain gauge location in magenta ...... 102

Figure 63: FEA model showing strain concentrations (red maximum, blue minimum) 103

Figure 64: Measured and predicted calibration data for connecting link strain gauge ... 104 13

Figure 65: Strain gauge used to measure torque output ...... 105

Figure 66: FEA model of strain concentrations on brake caliper bracket ...... 106

Figure 67: Experimental setup for torque strain gauge calibration...... 107

Figure 68: Measured and predicted calibration data for torque strain gauge ...... 107

Figure 69: Encoder mounted on input crank, measuring 2 ...... 109

Figure 70: Geared encoder arrangement for direct measurement of 4 ...... 110

Figure 71: 2 vs. 4 at 60 RPM, with predicted (red) and measured (black) values ...... 112

Figure 72: Cable and locking screw used to set brake torque...... 113

Figure 73: MATLAB data analysis output for single cycle selection ...... 114

Figure 74: Measured motor speed at 350 RPM, 2.5 N-m torque ...... 115

Figure 75: Measured speed ratio at various input speeds and torque levels ...... 117

Figure 76: Measured out, 4 and Tout at 350 RPM, 2.5 N-m ...... 119

Figure 77: Measured link force at 250 RPM, demonstrating low signal-to-noise ratio . 120

Figure 78: Sprag engagement and output angle at 200 (red) and 450 (black) RPM ...... 123

Figure 79: Calculated sprag velocities at low speed with measured output velocity ..... 124

Figure 80: Measured (black) and calculated (blue) out with measured 4 at 450 RPM 126

Figure 81: Measured (solid) and calculated (dashed) out at 200 and 450 RPM ...... 127

Figure 82: Measured (solid) and predicted (hollow) speed ratio vs. input speed ...... 128

Figure 83: Measured and predicted speed ratio, with emphasized "jumps" ...... 130

Figure 84: Simulated (blue) and measured (black/gray) brake torque at 2.5 N-m ...... 131

Figure 85: Calculated (blue) and measured (gray) link force, 350 RPM at 5 N-m ...... 132

Figure 86: Measured motor speed (black/gray) and simulated motor torque (blue) ...... 133 14

Figure 87: Hand-cut backstop from hand-cranked BCVT prototype ...... 134

Figure 88: 2-D MES results for force vs. deflection on hand-cut spring ...... 135

Figure 89: 2-D MES results for BVCT prototype spring and backstop ...... 135

Figure 90: Alternative form of transmission as a direct hydraulic pump ...... 139

LIST OF ABBREVIATIONS AND SYMBOLS

BCVT Beale Continuously Variable Transmission

CAD Computer Aided Design

CVT Continuously Variable Transmission

DAE Differential Algebraic Equation

FEA Finite Element Analysis

FTI Force Transmissivity Index

IVT Infinitely Variable Transmission

MES Mechanical Event Simulation

PWM Pulse Width Modulation

 Torque proportionality constant

F3 Connecting link force

spring Prescribed spring displacement k Spring stiffness k24 Instantaneous speed ratio ksprag Discontinuous sprag clutch spring stiffness ktorsion Torsional spring stiffness

 Transmission angle

k Coefficient of kinetic friction

T2 Transmission input torque

T4 Transmission output torque 16

i Inner bearing race angular velocity

out Output angle uncertainty

2 Crank angle uncertainty

o Outer bearing race angular velocity

R Speed ratio uncertainty

1. INTRODUCTION

Between every mechanical power source and its accompanying output mechanism there exists some medium for the transfer of mechanical power. Whether by direct shaft, fixed gearset, belt-pulley, chain-sprocket or otherwise, any application of mechanical power requires the existence of some form of transmission. For many applications, the simplest applicable transmission system is usually the most cost effective and least maintenance-intensive solution, so the aforementioned four forms of transmission are by far the most prevalent. These conventional forms of transmission all suffer from a common problem: fixed speed ratios from input to output are, in many cases, undesirable due to the transient nature of most loads compared to the torque-speed characteristics of most power sources. This was evident as early as 1490, when Leonardo DaVinci sketched the first-known illustration of a continuously variable transmission (CVT) concept, shown in Figure 1.

Figure 1: Sketch of a CVT concept by Leonardo DaVinci, c. 1490 (Anon) 18

In spite of this, conventional transmission devices continue to dominate the mechanical world. Adaptations of simple conventional designs to allow shifting between fixed speed ratios have been in common use for over a century in systems such as automotive transmissions and bicycle derailleurs. While these designs are not necessarily optimal, they have performed well enough in terms of cost, manufacturability and sustainability to continue to be employed for the past century. These designs, however, are beginning to show their age. Figure 2 shows an exploded view of a common four- speed automatic transaxle from a common light truck. It is readily recognizable that this system is quite complex, heavy and expensive, the result of adapting a fixed speed ratio design to a transient load application. Systems of this scale are commonplace in the automotive and industrial worlds, as most sources of mechanical power cannot necessarily be operated at any desired speed or torque level.

Figure 2: Exploded view of a Ford Ranger automatic transaxle (therangerstation.com) 19

Continuously Variable Transmission designs are currently being pursued by engineering efforts in a variety of markets to solve efficiency, reliability and other optimization problems for many different mechanical systems. This class of transmission allows continuous variability in speed ratio, as opposed to discontinuous selection of discrete ratios. This is the case, for example, for a conventional automobile transaxle.

Studies have repeatedly shown that even CVT types that suffer from efficiency problems

(such as friction-based designs) can provide a net gain in system efficiency due to the ability to use the system’s prime mover at a more efficient operating point (Kluger and

Long 1999). Currently, CVT technologies are being considered for and used in applications ranging from industrial equipment to automobiles and wind turbines

(Andersen 2007).

While the idea of a CVT is not new, reliable, efficient CVT designs capable of providing a wide torque conversion range have proven elusive. Currently, Toyota, Honda,

BMW and other major companies are using cone-belt and hydraulic designs in their products with limited success. These systems generally suffer from an efficiency loss over conventional geared systems, and also suffer from weight and reliability issues.

Moreover, torque conversion in such systems is still usually limited to the geometric size of the transmission just as in a conventional gear design, and manufacture or use of such designs requires specialized, modern tooling and processes. Therefore, application of these new technologies is economically feasible and sustainable only in developed economies and infrastructures, severely limiting their usefulness throughout the developing communities of the world. 20

Another type of transmission generally known as an Infinitely Variable

Transmission, or IVT, has also existed for quite some time. These designs can use many different principles to vary their torque conversion, but universally feature the ability to scale the speed ratio to 1:0. Of the many companies currently pursuing CVT technologies, many are looking to this specific class of transmission to solve problems associated with current and voltage optimizations in electromechanical systems such as industrial drive motors and wind turbine generators. One such design by Beale (2006), shown in Figure 3, uses one-way or overrunning clutches to effect this end. It connects an input crank (88) by a cable or link (92) to a nonlinear spring (94), which is connected to an output shaft (102) by a one-way clutch or ratchet (101). This mechanism, as will be explained, amounts to an automatic, continuously variable transmission.

Figure 3: Beale CVT/IVT Schematic (Beale, 2006).

Specifically, the Beale design is mechanically simple, lightweight and capable of extreme torque conversion. This particular transmission arrangement has many advantages for both technologically advanced systems available in developed communities and new, appropriate technologies for the developing world. In the case of the latter, its capacity to be easily constructed from widely available parts as common as strips of sheet metal and bicycle freewheels makes it appealing for application in developing communities as an appropriate technology, as simplicity and local serviceability are primary needs in these areas (Cyders 2008a; Darrow 1975; Adas 2006;

Cyders and Kremer 2009). These needs are not met by either conventional geared designs or more advanced technologically-dependent CVT designs that have resulted in high cost and infeasible long-term machine operation without sophisticated maintenance capabilities. Application of such a transmission to a human-powered utility vehicle could enable the use of human power for applications not previously considered feasible, since torque and resulting power output, which are generally relegated to a narrow performance envelope with a human motor, can be optimized for virtually any given scenario (Cyders

2008b; Wilson 2007; Too and Landwer 2008). In well developed areas, the Beale CVT still finds itself at the front of the technological curve, having torque-speed characteristics that are desirable for a range of basic and advanced technologies alike.

As simple as the Beale design is, the one-way elements it uses to both vary its output speed ratio and couple and decouple individual drive elements present a difficulty for design and dynamic modeling, as the resulting coupling and decoupling of the differential equations describing the overall motion becomes non-trivial to simulate. The 22 reciprocal nature of the mechanism lends itself to intense vibration, due presumably to inertial effects, which has resulted in unacceptable dynamic behavior in testing up to this point in time. Previous efforts to describe such machine elements mathematically have proven somewhat successful in certain specific applications, but none have done an in- depth study of interactions of the coupling/decoupling behavior between multiple parallel or serial clutch assemblies (Vernay et al. 2001; Zhu 2006; Popp 2005; Hoffmann and

Gaul 2004). Very simplistic studies have previously been completed internally by Beale

Innovations, Inc., but have only been successful with static loads and little consideration of inertial effects. Application of transient loads in the dynamic model is a great need for a useful understanding of the transmission’s performance. For any machine simulation or design involving the use of this transmission, the output of the transmission given the input must be known, both statically and dynamically.

1.1 Background

The first incarnation of the Beale CVT took form as a bicycle transmission. The inventor, William Beale, used a series of elastic bands to connect the pedals of a bicycle to the output wheel (wherein the one-way clutch already resided in the form of a ratchet- pawl), resulting in a mechanism that acted essentially as a torque limiter. Beale’s hope at the time was to develop a bicycle that provided extra torque conversion for climbing the many winding roads over steep hills in the immediate area of Athens, Ohio, while eliminating the need to deal with a dérailleur. On further experimentation, different arrangements were built using cams, four-bar linkages, gears and cables (Beale 2006), 23 and the scope of possibilities for application of the mechanism grew much wider. Those designs that worked generally involved the same principles to be herein explored.

The Beale CVT is a unique combination of simple machine elements which results in an intrinsically automatic mechanical transmission capable of extreme torque multiplication. Most usually, the design uses a Grashof four-bar crank rocker (or any analogue thereto) with a flexible link and a one-way bearing, or sprag clutch, arranged as shown in Figure 4. Each machine element involved in the design dates back at least to the

15th century, but the combination of the three as shown represents a new idea altogether.

According to the classifications afforded by Barker (1985), the specific linkage arrangement under study by this work is a Type 2, or Grashof crank-rocker-rocker mechanism; that is, only one element in the four-bar chain is capable of making one complete rotation with respect to the others, and that link (the short link on the left in

Figure 4, known as the crank) is grounded. The short link is the input, while the rocker

(the long link on the right) is coupled to the output.

Figure 4: Schematic of four-bar mechanism system from Cyders and Williams (2010). 24

While even just a single such assembly of the Beale CVT has the desired mechanical effects on speed ratio and torque conversion, one of the defining aspects of this design is the ability to combine multiple assemblies onto the same output shaft. As early as the first incarnation of the transmission on a bicycle, each pedal had its own effective four-bar linkage tied through one-way elements to the same output shaft on the rear wheel. This effect allows for even more innovation, as the transmission can be used in many different ways with the same application: it can function passively or actively as a coordinated or uncoordinated hybrid drive, a differential, or simply a balanced-load

CVT. It has also been shown that active control of the behavior of the transmission is feasible, even with multiple sprag assemblies acting in concert (Cyders 2011).

1.2 Functional Description of Beale CVT

A discussion of the Beale CVT’s operational principles is greatly informed by an understanding of the behavior of the mechanism upon which it is usually constructed. For clarity, it is helpful to go through the motions of a basic four-bar mechanism analysis, which is done in detail in Chapter 4 for the pseudo-static analysis of the system. Some basic concepts from the pseudo- static case will be discussed here briefly to introduce the operational principles of the mechanism.

Effectively, the crank-rocker mechanism behaves as a torque multiplier. It is used in many different applications to multiply the torque or force output of an otherwise incapable power source at the expense of speed, such as in oil horsehead pumps. Crank- rocker mechanisms composed of rigid links cannot do any work internally, thus resulting in the conservation of energy (and thus, power) from input link (crank) to output link 25

(rocker). It is easily concluded that, for constant power, the torque increase of the system is equal to the inverse of the rotational speed decrease. The method by which the crank- rocker mechanism achieves this decrease in rotational speed is depicted graphically in

Figure 5, wherein one full rotation of the input results in less than one rotation of the output.

Figure 5: Schematic example of crank-rocker mechanism motion

Without much analysis, it is recognizable that when the crank is in position 2 shown in the previous figure, the rocker approaches a dwell. As the links align, the rocker’s angular velocity reaches zero, regardless of input velocity. At this point, the speed ratio approaches zero, and thus the instantaneous torque ratio is greatly multiplied.

Effectively, the Beale CVT selectively transfers motion in areas surrounding this point 26 according to the applied torque. This allows the transmission to greatly multiply torque when needed, and scale torque multiplication down in periods of otherwise low-torque demand. The end result is an automatic transmission capable of supplying as much torque conversion as may be necessary, depending on the design of the mechanism. 27

2. LITERATURE SEARCH

A good understanding of the significance of the Beale CVT necessitates a purview of other CVT transmission technologies, and how they compare to the Beale

CVT design. Of similar importance is a background in modeling approaches to one-way clutches that have been taken in the past which will be useful for developing successful models of the various elements of the Beale CVT, as well as relevant software packages for rendering simulation results from these and other models, and their capabilities.

2.1 Other Existing Transmission Designs

Many different forms of CVT are currently or have previously been under study in the mechanical engineering community. While few designs have been successfully transformed into commonly used consumer or OEM products, recent developments in analysis, design and manufacturing capabilities have opened up possibilities in the realm of CVT not previously thought practical. Still, the fundamental principles by which most

CVTs transmit power suffer from efficiency, cost and weight concerns (Andersen,

Dalling and Todd 2008). The following is a brief introduction to the different types of

CVTs prevalent in the body of work, with some real-world examples of each type.

2.1.1 Toroidal

Toroidal CVTs transmit power by contact between three elements – a roller and two discs. By varying the diameter at which contact between the roller and the two discs occurs, it is possible to continuously vary the speed ratio from input to output, as shown in Figure 6. This is one of the oldest, most applied designs, as evidenced by the many 28 commercial products based on it, as well as the relatively old patents detailing its operation. An example is shown in Figure 7. The concept, however, suffers from marginal efficiency stemming from its use of solid-on-solid friction or, more appropriately, fluid dynamics to transmit torque, and is limited in overall speed ratio achievable being a function of its size and, and therefore, its weight. It also requires very specific lubrication, as torque is transmitted primarily through the shear forces in the lubricant, and viscosity and related geometries in the design heavily affect performance. Commercial examples include the Nissan Extroid, the Fallbrook Technologies NuVinci, and the Torotrak CVT.

Figure 6: Example of a toroidal CVT (NSK, Inc. 2011)

Figure 7: Cutaway view of toroidal CVT design (Dodge 1939)

2.1.2 Cone Drive

Another very basic, friction-based CVT concept is a cone drive. Again, this design is older, but has found limited use in several different end applications, as shown in Figure 8 and Figure 9. Cone drives generally work by linking input to output again by friction or contact, usually through a compressed ring or belt. Varying the position of the element that transmits the torque varies the diameter at which it acts, thus shifting the speed ratio. This system suffers from geometric constraints on torque ratio, as well as the same design and efficiency considerations that apply to toroidal CVTs. An example of a commercial product based on this design can be found in the WARKO transmission currently under development (www.warko.it).

Figure 8: Evans friction cone-ring drive (Ukexpat 2009)

Figure 9: Cone-ring drive as detailed in US Patent 3,257,837 (Davin 1966)

2.1.3 Hydrostatic

A class of designs based on hydraulic flow is known as a hydrostatic CVT.

Usually, these designs operate by using a variable flow through a hydrostatic motor to achieve smooth transition from one speed ratio to the next. While there are many methods to do this mechanically, an oft-used design employs staged pumps on an adjustable swashplate, as shown in Figure 10.

Figure 10: Hondamatic Hydraulic CVT with adjustable swashplate (Anon 2000)

In this case, variation of the swashplate’s displacement varies the flow of hydraulic fluid into the pump, thereafter varying the effective speed ratio. These assemblies are often combined with an epicyclic gearset to further enable diverse application of mechanical power through the system. These systems generally suffer from poor efficiency in the hydraulic motor, and high weight and cost, especially when 32 combined with a mechanical gearset. They are, however, capable of transmitting torque at high levels, as a result of using a hydrostatic motor. Examples of commercially- available products are the Hondamatic CVT shown previously in Figure 10, and the

Torvec IVT shown in Figure 10, which can achieve a 1:0 speed ratio, at the expense of performance compared to other systems.

Figure 11: 3-d model of Torvec hydraulic IVT (Torvec, Inc. 2010)

2.1.4 Epicyclic

Epicyclic gears (also known as planetary gears) are sets of gears arranged in an orbital fashion, as shown in Figure 12. These gearsets are extremely flexible in arrangement, resulting in assemblies which can exhibit considerable variation in speed ratio, depending not on the number of gear teeth, but simply how the gears are assembled.

Another property of this type of geared transmission is its ability to act as a differential: 33 input power from any one of the sun, planet or ring gears can be split between the other two, allowing either output to be slowed all the way to zero speed.

Figure 12: Example of an epicyclic gearset (Longhurst 2011)

Effectively, this gives the epicyclic gearset the same abilities as many infinitely variable transmissions, except that power not put into one output gear must be put into the other. In other words, instead of trading speed while scaling up torque, the power split in an epicyclic gearset will speed up one output proportionally to the slowing down of the other. In this fashion, the power to the sped-up gear cannot be wasted if the system is to be efficient. Epicyclic gears are thus the basis of a vast majority of infinitely variable transmissions, but in such a role must include a mechanical power feedback loop. There are two principal arrangements used to achieve this behavior, known as Type I and Type

II power flow. These are depicted in Figure 13, from Mantriota (2002), where P.G. 34 denotes a planetary gearset, F.R. denotes a fixed ratio mechanism, and C.V.T. represents a continuously variable transmission.

Figure 13: Type I and Type II power flow, as depicted by Mantriota (2002)

In general, the coordination of either power flow arrangement incorporates a fixed-range CVT in the power feedback loop. Due to the generally inefficient nature of conventional CVT devices, this inclusion is one of several sources of inefficiency for the overall hybrid IVT system. The concept of basing an infinitely variable transmission on an epicyclic gear is not new; many different designs have been based on this paradigm

(such as one presented in 1992, as shown in Figure 14), but few have been manufactured as useful products, largely due to their efficiency limits, as well as their high weight and cost. Figure 15 shows an epicyclic-based IVT analyzed by Benitez (2004), which uses multiple epicyclic gears linked through a one-way clutch for power flow. While such a design gets rid of the inefficient CVT link in the system, the mechanical complexity, weight and cost of such a system are higher still than its CVT-equipped counterparts. 35

Figure 14: Epicyclic CVT invented by Paul Pires (Brown 1992)

Figure 15: Epicyclic gear-based IVT analyzed by Benitez (2004)

2.1.5 Cam-Based IVT

Several new designs of infinitely variable transmission which have been recently developed use cams to vary speed ratio. These designs are similar in that the 36 shifting of cam geometry changes the overall speed ratio of the mechanism. They differ widely, however, in detailed operation and design. For example, an infinitely variable transmission shown in Figure 16 and Figure 17, recently developed by Lahr (Lahr and

Hong 2009; Lahr 2009), effectively combines the concept of an epicyclic gearset with one-way clutches and cam adjustments into a resultant transmission capable of varying speed ratio down to 1:0. While mechanically complex, this design effectively makes more compact the concepts previously reviewed with the epicyclic gearset, with similar operating principles. It still suffers from cost and weight issues associated with the mechanical complexity of its basic design.

Figure 16: Cam driven analogue for epicyclic IVT (Lahr and Hong 2009)

Figure 17: Designed implementation of cam-based IVT (Lahr and Hong 2009)

A different cam-driven design is shown in Figure 18. This design is much closer in nature to the Beale CVT, employing a one-way clutch with, effectively, a six bar mechanism to adjust speed ratio. It differs, however, in that its speed ratio is fixed at a given point of adjustment, and that it is traditionally built as a cam-based structure.

Heavy, manually-adjusted models of this transmission such as the one shown in Figure

19 are marketed by Zero-Max, Inc. in Plymouth, Minnesota.

Figure 18: Adjustable CVT using a one-way clutch (Zero-Max, Inc. 2011a)

Figure 19: Manually-adjustable variable speed drives (Zero-Max, Inc. 2011b)

2.2 Modeling of One-Way Elements

One-way elements have taken different forms in simulation depending on the needs of the model, but despite widespread use in various machine designs, only a small collection of literature dealing with their simulation has been published. In general, the simplest effective model is desired, but depending on the modeling approach being taken, varying levels of sophistication have proven necessary for simulations to even function, let alone yield accurate results. A basic cutaway view demonstrating the structure of most sprag clutches is shown in Figure 20. In this view, part 102 is the inner race, or the outer surface of the output shaft. When the input (the outer race, part 101a) rotates clockwise more quickly than the output, (part 102), the irregularly shaped elements, called “sprags”,

(part 105) rotate slightly, jamming between the output shaft (part 102) and the input (part

101a). This locks the output shaft and the input together, coupling the motion. Once the rotational velocity of the input becomes less than the output, the sprags rotate back to their original position, and allow overrunning motion between the input and output. 39

Figure 20: Typical sprag clutch construction (Miura, Numata and Le Calve 2005)

Generally, sprag clutches work by a set of spring-loaded eccentric elements sliding over the surface of a round driveshaft in one direction, while rotation in the other rolls the elements on end, compressing them between an outer race and the hard surface of the driveshaft. This creates a locking behavior much like a security bar on a door, usually with very little backlash. Upon release of the load, the internal springs return the sprags to a loose position, allowing them again to cam away from the driveshaft and slide. The simplest, idealized model of such a one-way clutch couples and decouples the outer race from the driveshaft instantaneously, using a nonlinear spring with infinite stiffness under a velocity condition. This is illustrated in Figure 21 and Equation (1), where i is the rotational velocity of the shaft and inner element of the clutch, o is the rotational velocity of the outer race, and k is a discontinuous spring constant that is dependent on the relative state of the two velocities. 40

Figure 21: Schematic of idealized sprag clutch model

{ (1)

While this simple model could be easy to implement in numerical simulation in some respects, it causes instability at the points where the spring constant changes instantaneously, and has been shown to be a somewhat inaccurate representation of the actual behavior of the sprag clutch (Vernay et al. 2001). As such, variations on the idealized model have been used in simulation, such as those developed by Leamy and

Wasfy (2002), Zhu and Parker (2008) and Mockensturm and Balaji (2005). The model by

Leamy and Wasfy, for example, is based on a concept of a torque, T, proportional to the relative angular velocities of the shaft and outer sprag race, as shown in Equation (2).

{ (2)

In this model,  is a proportionality constant which can be used to adjust the stiffness of the model, while T is the coupled torque load. The higher the value of , the more closely the model approximates the ideal example previously discussed. As can be seen, there is still a piecewise structure with a theoretical decoupling of the load when the sprag disengages, but instead of an infinite-slope step at the point of decoupling, the function ramps up from the totally decoupled condition with finite slope . This model reportedly gave better results in combination with a finite element-based dynamic simulation than the ideal model, but has only been comparable to date with results from steady-state simulation conditions with relatively small perturbations.

According to measured experiment, power losses in the return or overrunning behavior of the clutch at medium (80 rpm+) to high (250 rpm+) speed have been measured to be quite small compared to those at low speed. This study was carried out as an analysis on the individual element level. Most models presented here outright ignore these losses – it is likely this resistance can be ignored in simulation or lumped as a constant damping or friction parameter without affecting the overall accuracy of the simulation (K. Liu and Bamba 1998).

Zhu (2006) proposed a similar model to Leamy and Wasfy, but with two degrees of freedom in the context of vehicle belt pulley vibrations, using a serially connected spring and damper. A schematic of this system is shown in Figure 22. Analysis of this model was completed using several different numerical methods, with good agreement 42 between results in the studied cases. This method has been shown to be accurate, but can be computationally intense, and more complicated to integrate into a solution framework than simpler models.

Figure 22: 2-DOF model of automotive belt-pulley system (Zhu 2006)

If the loading conditions are known and predictably periodic, coupled and decoupled behavior can be separated into piece-wise systems in the time domain using matching conditions for angular position and velocity at the period boundaries

(Mockensturm and Balaji 2005). Also, as a projection of the work of Kim et al. (2003), a 43

Fourier series can be expanded on a square wave, effectively creating a continuous approximation of an oscillating binary switch, with finite slope in between switch positions. This has solved problems associated with instability at switching points, but assumes known, constant periodicity and amplitude of the load, and is incapable of effectively including transient, dynamic conditions. A family of continuous functions approximating the sign function (a function which returns –1 in the case of a negative input and 1 in the case of a positive input) can also be used to smooth oscillating models of piecewise nature (Kim, Rook and Singh 2003). Various numerical approximations for such systems have also been built and tested, such as the work of Najafi and Benjelloun-

Dabaghi in their Modelica simulation of the piecewise dynamics of a spark ignition engine (2008).

2.3 Relevant Software

Several notable software packages exist which may prove useful for the analysis of the Beale CVT. Specifically, the ability to approach the dynamic analysis of this system in different ways with different tools will work both to validate the results of the analysis, as well as to determine which and whether such tools may be useful for design and evaluation going forward. While several of these environments employ more of a

“black box” solution to the dynamics problem, it is understood that these will not make up the entire body of work for this dissertation. Environments in which ground-up, fundamental models can be constructed such as MATLAB, FreeMat or C++ bear only brief mention here, but it is worth noting that such custom-programmed modules are 44 important for a primary understanding of the functionality of different modeling approaches.

2.3.1 OpenModelica/Dymola

OpenModelica is an open-source modeling and simulation environment available freely on the Internet (http://www.openmodelica.org/). It is based on a commercial simulation language known simply as Modelica, wherein components of systems are created and manipulated in an object-oriented fashion, similar to bond graph modeling.

This allows models of complex electrical, mechanical or other systems to be built by linking together models of the components the system is composed of. For example, a mass-spring-damper system can be quickly put together linking together pre-programmed

(or custom-programmed, if necessary) blocks of a mass, a spring and a damper, and allowing the background solver to generate numerical solutions to the resulting system equations. Such a block structure is shown in Figure 23. It has been extensively employed in the engineering community for modeling simple machines to sophisticated systems composed of multidisciplinary components (Simic et al. 2007; Otter, Schlegel and

Elmqvist 1997; Martin-Villalba et al. 2007; Hoelemann and Abel 2009; Kral, A. Haumer and F. Pirker 2007). This software is relatively powerful in its ability to model even complex systems with relative simplicity, but requires, in many cases, significant effort to ensure integrity of simulations as the systems modeled become more complex. The commercial package Dymola is very similar to OpenModelica, albeit with easier integration of more powerful solvers, additional minor features and better technical support. 45

Figure 23: Example of a block diagram in OpenModelica's OM Edit

Particularly notable is the ease of application of boundaries in the Modelica language (Najafi and Benjelloun-Dabaghi 2008). Figure 24, for example, shows one of the most basic introductory examples that come with the software, a model of a bouncing ball. The red line depicts the vertical displacement of the ball over time, whereas the blue line depicts the Boolean description of whether the ball is bouncing (“flying”) or not. The ability of the language to handle Boolean and discretized modeling alongside continuous models is easily implemented, and integral to the language’s design, as explained by

Tiller (2001): “Often, it is convenient or even necessary to simulate both continuous and discrete behavior at the same time. Modelica allows both forms of behavior to be described within the same system model, or even the same component model.”

Figure 24: Bouncing ball example simulation in OpenModelica

2.3.2 MSC ADAMS

The ADAMS package from MSC Software, Inc. (www.adams.com) is a multiphysics package capable of multibody static and dynamic analysis. This is one of the most widely-used packages in the multibody simulation world, generally due to its easy integration with MSC’s finite-element packages, which are also widely used. For example, Adams has been used successfully to simulate systems from small mechanisms to whole vehicles (Xin Zhang et al. 2010; Xiaogang, Weixia and Hang 2010; Pengpeng

Huang, Lei and Wang 2011; Ge et al. 2011; Capitani et al. 2006).The stated usefulness of the software lies not in analysis of single components or simple systems, but in the combination of highly dynamic situations with sophisticated mechanical systems (MSC

Software, Inc. 2011). Adams can, however, pass its results on to FEA software for more localized analysis of mechanical phenomena in single components. It has also been used 47 in connection with SimuLink models, and integrated into control systems (Xiaogang,

Weixia and Hang 2010). Of particular interest with this package is its demonstrated usefulness for nonlinear dynamics and mechanical elements, which will be essential for a successful simulation of the transmission. A screenshot example from the software is shown in Figure 25.

Figure 25: MSC Adams simulation example (STU Bratislava 2009)

2.3.3 Autodesk Simulation Mechanical

Autodesk Simulation Mechanical is a finite-element based software package integrated with the Autodesk Inventor Suite (http://usa.autodesk.com/). This package is largely based on an earlier product known as ALGOR, which was procured by Autodesk,

Inc. in January, 2009. The mechanical simulation suite is a finite-element package with multiphysics support deriving from a time-series application of the finite element method. 48

With this package, CAD models of mechanical systems can be constructed and/or imported, and motion, deformation, contact and other linear or nonlinear phenomena can be simulated as simultaneous interactions in the system over discrete periods of time.

This methodology lends itself well to the needs of this project, since the system to be modeled includes interactions of contact stress, large body deformation and nonlinear dynamics.

The Finite Element Method has advantages in this regard, as it has been shown to be even more accurate than analytical models in some cases, as discussed by Laniado-

Jacome et al. (2010). The method does suffer from being computationally expensive, so its use is best relegated to specific phenomena and small time domains, but in this area, it has been validated as a very effective and accurate tool (Werner 1998; Laniado-Jacome,

Meneses-Alonso and Diaz-Lopez 2010; Liao and Chan 2008). Simulation times for a several-second time domain can vary from under an hour to more than a day at a time, depending on the complexity of the model, or more importantly, its ability to converge quickly. Thus, this approach is limited in its ability to simulate the overall mechanism with an interdependent power source and one-way clutch, but is useful for simulating localized interactions of the other components in the mechanism. Figure 26 shows a screenshot from an introductory mechanical event simulation tutorial of a piston on a crank. 49

Figure 26: Autodesk Mechanical Event Simulation example

3. DISSERTATION GOALS

The Beale CVT has specific advantages over other mechanical systems if coupled with a primary mover such as an internal combustion engine or an electric motor. In the case of an electric motor, for example, a high speed motor could conceivably be coupled with the Beale CVT to produce a low-cost, lightweight solution for even high torque demand applications. To this end, however, there must be a good understanding of the dynamic interactions of the Beale CVT design, and the influence of higher order effects on its behavior. To date, no full dynamic simulation of the mechanism capable of predicting transient response has been successfully constructed, but such a study is greatly needed in order to design a dynamically-balanced system. The purpose this work aims to achieve is the development of a tested, usable dynamic simulation of the Beale

CVT.

3.1 Problem Statement

Without a model capable of describing the dynamic behavior of the Beale CVT mechanism in response to transient and steady-state loads which is shown to be accurate by comparison to experimental results, effective design of a high-speed version of the transmission cannot be completed.

3.2 Scope of Work

The work to be completed in this dissertation includes developing one or more dynamic models of the Beale CVT, designing and building a single, physical dynamic system and testing apparatus for experimental data gathering, and comparing the model 51 results to said data. Approaches to be taken include open-ended conventional analytic or numerical approaches, mechanical event finite element solutions and solutions provided by otherwise closed-source, proprietary software, as all such methods are of potential interest. Many different incarnations of the device exist, but the development of a model for this particular design should suffice for describing other embodiments of similar design. In order to maintain focus, the experimental design, models developed and discussion thereto will be generally limited to the form of the Beale CVT as presented in this document, with the understanding that there is great value in the ease of adaptability of models to other, perhaps radically different forms of the mechanism.

3.3 Dissertation Objectives

Given the background and problem presented, the objectives of the proposed dissertation are as follows:

1.) Complete experimental validation and refinement of pseudo-static analysis for the

Beale CVT, ensuring both that it is capable of describing global transmission behavior

(e.g.: torque vs. speed ratio), and that it addresses the real-world limitations of the system and deviations from the ideal model. This will also serve to inform the overall construction of subsequent dynamic simulations.

2.) Create and develop multiple dynamic models of the Beale CVT, and document results as applied to different loading scenarios.

3.) Design and construct a physical prototype of the Beale CVT and instrumentation specifically for the purpose of dynamic testing, and measure CVT performance under loads comparable to those used in simulation.

4.) Compare model results to experimental data for the Beale CVT and refine as necessary.

5.) Compare final models of the Beale CVT with each other and to collected real-world data, and discuss implications for the effectiveness of each model type.

3.4 Experimental Criteria

An important understanding of what is to be measured and how to measure it is necessary to ensure the feasibility of the experiment. For this particular project, the data to be measured are dependent on the need for the model; the central purpose of the project is to develop models specifically to provide necessary information for design criteria. Therefore, these criteria will dictate the variables to be measured, the methods of measurement and the acceptable level of error. The internal and external characteristics of particular interest for design of the Beale CVT are listed below with a brief description of their significance for the design process, and their associated acceptable level of error for this study.

1.) Link stress – Stresses in the links of the four-bar mechanism are crucial for design life calculation, material selection and ability to optimize cost, weight and inertial 53 characteristics. The model must be able to predict this within ± 5% for effective link design.

2.)Torque input – Variations in input torque demand can have significant effects on the efficiency and reliability of the prime mover powering whatever system the transmission is connected to. This load should also be estimable within ±5% based on the model, as input shafting and mounting must also undergo a mechanical design process involving stress calculation.

3.)Speed output – The ability to precisely predict the speed output of the mechanism for different loading conditions is extremely important when designing the transmission for speed-controlled applications such as automotive, robotic and wind applications. Based on model output, it is necessary to estimate speed output at a given load to within ±3% for effectiveness in these applications, especially with respect to automotive control.

4.)Torque output – An understanding of the torque/speed relationship of the transmission with respect to loading conditions and internal design changes is important for virtually any application of the transmission. In the case of this experiment, the load on the system will be defined, but the ability to predict output torque variation would be a useful additional feature of a successful simulation.

5.)Bearing stress – For any high speed application, the design life of the bearings constraining the transmission is crucial for an understanding of reliability, feasibility and cost. Dynamic radial loads and reciprocating speeds should be within ±5% for reliable bearing life prediction.

6.)Sprag engagement – Engagement periods of the sprag clutches is important for optimizing the design of the flexible link (beam spring). The error allowance for this characteristic is dependent on the spring behavior in the design, and figures directly into the acceptable error in speed output.

7.) Casing vibration – In many applications, noise and oscillating stresses are important to keep within acceptable levels. Resonance behavior, frequency response and dynamic balancing of the system are of particular interest. For the purposes in this study, amplitude of the frequency response of the casing should be predicted within ±5% for a useful estimation of casing mount stresses.

4. INTRODUCTORY PSEUDO-STATIC MODEL

While it cannot describe transient or second-order effects, a pseudo-static analysis, sometimes also referred to as a kinetostatic analysis, of the Beale CVT can inform the overall torque-speed ratio behavior of the system. For spring design, this aspect of the analysis is important, as the deflection response of the spring must result in the desired overall torque characteristics of the system. Generally, pseudo-static models describe the dynamics of a moving system while ignoring inertial effects. For rigid bodies, the kinematics are the same at high speed as they are at low speed, allowing a forward kinematics solution approach by time differentiation of the vector loop-closure

(position) equations. This analysis also serves as an effective explanation of the mechanical phenomena that characterize the Beale CVT’s performance.

The pseudo-static analysis will make the following assumptions: first, a condition of no backlash on the output shaft is assumed, i.e.: the output shaft can move in the positive direction only. This is in contrast to previous studies by Cyders and Williams

(2010), but serves to simplify the validation process for the analysis, and is more closely approximate to a real-world system using multiple sprag units. Secondly, for a pseudo- static analysis, the model is assumed to be moving infinitesimally slowly, such that the effects of inertia and any damping are ignored.

4.1 Four-bar Mechanism Analysis

This analysis is based on a crank-rocker mechanism, as the Beale CVT is traditionally built in this basic form. A physical example of such a system from the 56 previous work by Cyders and Williams (2010) is shown in Figure 27. The four-bar mechanism approximation of the physical system under discussion is shown in Figure 28.

The model is a standard four-bar mechanism with the exception of flexible link r3, which models an arbitrary spring behavior based on applied torque demand on the output shaft.

In the past, this spring was modeled as a transversely flexible link r4, as was discussed by

Cyders and Williams (2010). Both cases have been shown to accurately describe the output of the mechanism at low speed, but a treatment of r3 as an axially flexible link somewhat simplifies the explanation. For the purposes of this short discussion, the simplified model is sufficient to demonstrate how the Beale CVT functions.

Figure 27: Prototype of the Beale CVT

Figure 28: Vector loop diagram of physical system

Link lengths to be used in this analysis correspond to the actual lengths of the physical system previously detailed in Figure 27, such that comparisons can be made between the analysis and the real-world system. These lengths are detailed in Table 1.

Table 1: Link lengths for four-bar crank-rocker mechanism

Link r1 r2 r3 r4 Length 229 mm 32 mm 178 mm 178 mm

It can be quickly determined that this mechanism meets the Grashof condition detailed in Equation (3), where S and L are the shortest and longest link lengths, respectively, and P and Q are the other two link lengths, arbitrarily.

(3) 58

A cursory run through the rigid-body position analysis begins with the standard four- bar mechanism loop-closure equations (variables reference Figure 28):

[ ] [ ] (4)

Here, f1 and f2 are arbitrary function placeholders set to zero for solution of the system. Axes are arranged such that 1 is equal to zero. Usually, a transcendental solution is applied to solve these equations. In this case, however, the Newton-Raphson numerical method was used, such that later inclusion of the nonlinear spring element would require little modification. Code for the FreeMat script used to solve this system can be found in the Appendix. The position analysis produces several important graphs, shown in Figure

29 and Figure 30. Figure 29 shows output angle 4 vs. input angle 2, while Figure 30 shows the transmission angle of the mechanism vs. input angle 2.

Figure 29: Output angle 4 vs. input angle 2 59

The transmission angle  is defined by Balli and Chand (2002) as:

− 2 휇 ( ) (5) 2

Figure 30: Transmission angle vs. 2

The previous two plots reveal much about the kinematic nature of the mechanism.

It can be clearly seen in Figure 29 that the output rocker only moves through approximately a 20° arc of rotation per full revolution of the input crank. Since the mechanism is currently being treated as having rigid body links, energy is conserved, and thus the angular displacement reduction (and, therefore, angular speed reduction) from input to output results in a torque increase of inverse proportion. In an unloaded condition at low speed, the flexible link of the mechanism acts as a rigid body, and a speed ratio of 60 approximately 1:16 is observed. This is confirmed by experiment in Figure 31 which shows measured speed ratio behavior of the physical model previously shown in Figure

27 by Cyders and Williams (2010); at zero torque, two sprag throws yield an output of roughly 1:8 (0.125).

Figure 31: Measured speed ratio vs. torque demand (Cyders and Williams 2010).

Transmission angle is an important facet of the Beale CVT’s operation in this design because of the transverse nature of the beam spring, and its load vs. displacement behavior. Generally, transmission angle is used as a measure of the quality of a mechanism in that high or low angles result in large proportions of shaking force to transmitted force within the mechanism. An average transmission angle of 90° is 61 desirable, with as little deviation as possible. This generally results in the lower fluctuation in output torque, which is advantageous for many designs (Balli and Chand

2002), including the Beale CVT with its spring design as applied in this arrangement.

As the crank rotates, the torque conversion from input to output fluctuates. Since the transmission angle is of good quality (i.e.: link r3 is nearly normal to link r4 with little variation), most of this fluctuation results from the variation in the difference between 2 and 3. This result is quickly found by a static torque balance about the crank’s pivot point, as shown in Figure 32 and Equation (6), where F3 is the force resultant from the torque on r4.

Figure 32: Free-body diagram of input to crank-rocker mechanism

∑ ⃑⃑⃑ (6)

From this, the torque ratio, which varies over the rotation of r2, can be calculated by dividing the output torque T4 by the input torque T2. The inverse of this, shown in

Equation (7), is equal to the instantaneous speed ratio, or the derivative of 4 with respect to 2. A plot of this result is shown in Figure 33.

(7)

Figure 33: Instantaneous speed ratio vs. 2

This result is also confirmed in experiment, in two fashions: first, a fixed load can be exerted on link r3, and the variations in torque can be felt on the hand crank. Secondly, the average instantaneous speed ratio over one half rotation (since the sprags are only engaged when the speed ratio is positive) is equivalent to the overall speed ratio at zero 63 load. Of note in Figure 33 are the points where 2 is approximately 45° and 230°, where

2 and 3 are either equivalent or antiphase, and the instantaneous speed ratio passes zero.

As the transmission approaches these operating points, the instantaneous torque conversion drastically increases, as the mechanism approaches a singularity. It is this phenomenon that allows the Beale CVT to convert torque at such high levels. The one- way clutch element and flexible link work in concert to alter the kinematic properties of the otherwise rigid mechanism, forcing it only to transmit motion over specific ranges of

2. By selecting progressively smaller ranges about the singularity as torque demand is increased, the transmission can selectively raise the average speed ratio to extremely high levels.

This singularity is well known in rigid-body mechanics. The physical ramifications of this mathematical phenomenon are significant, and merit short discussion here. When links r2 and r3 are perfectly aligned, no amount of force in link r3 can result in rotational motion of link r2. This results in the forces in link r2 and r3 being transmitted directly to the grounded revolute joint at 2. This also affects the motion of the mechanism, in this case manifested in variations in input torque demand for a given output load.

In real-world design, this singularity is overcome by inertia, which carries the links through the singularities. A recent treatment by Lin and Chang (2002, 1472) described the phenomenon well: “At these two positions [where links r2 and r3 are aligned], the input torque vanishes. Thus the loading is balanced by the reaction forces at the fixed pivots, and the effective force is transmitted to the output link with minimum 64 input effort (most effectively).” A plot of their Force Transmissivity Index (FTI, a measure of mechanical advantage) for a crank rocker mechanism remarkably similar to the model under study here is shown in Figure 34. This method can be applied to other linkages, demonstrating that the fundamental concept upon which the infinitely variable nature of the Beale CVT is predicated can be found on any linkage component or mechanical analogue thereto which experiences a sign change in its velocity while that of its driven link has a finite value. In short, this means that the design of the Beale CVT is not limited to link r4 on the crank rocker, or even to a crank rocker mechanism at all; the concept could also be easily applied to a drag link mechanism or double rocker mechanism, or even to virtually any four-bar mechanism in the case that extreme torque conversion is not needed.

Figure 34: FTI vs. input angle for crank-rocker mechanism (Lin and Chang 2002)

This result does not imply that the mechanism can transmit infinite torque. At the singular point where torque conversion becomes infinite, motion is necessarily zero, resulting in a finite energy transfer (since the product of torque and speed is energy). In actuality, the mechanism deviates from the ideal model; there is backlash in each sprag unit, non-rigidity in each link, and friction in the joint bearings. Deviation from the ideal model in any such way limits the singular torque conversion to a finite value. While non- infinite, the levels of torque conversion this mechanism is capable of delivering remain extremely high, based on spring design and common machine component characteristics.

To demonstrate these limitations, it is useful to consider a load with a Coulomb friction nature as opposed to a constant, direct torque. The result of this change in load is shown in Figure 35, where the mechanical advantage of the crank over the rocker is shown in black, with a sign change shown in red depicting the behavior when the load always opposes the direction of the rocker motion (as in a Coulomb friction model).

Figure 35: Crank rocker mechanical advantage (black) with friction load (red)

Considering this case, a lumped friction parameter can be applied to the joints of the mechanism. For the purposes of this discussion, the model will be simplified to a linear frictional torque, increasing in terms of a resistance on both the crank (r2), and the rocker

(r4). An additional torque at the bearings linking r2 to r3 and r3 to r4 proportional to the force in link r3 by a lumped friction coefficient k and bearing radius rbrg is thus added to the model in Equation (6), resulting in that shown in Equation (8). This reduces the ratio of torque output supplied (T4) per torque input required (T2) to that shown in Equation

(9), limiting the upper bound to a finite value at the singularities where 2 and 3 are equal or antiphase due to the lumped friction coefficient k. 67

∑ 휇 (8)

휇

휇 (9)

Figure 36 depicts the effect of this friction load on the mechanical advantage of the mechanism at the singularity point. Merely the introduction of such a term limits the otherwise infinite behavior. As the friction coefficient increases, the maximum mechanical advantage at the singularity decreases. Moreover, any backlash in the one- way element will further limit this maximum value by requiring motion to effectively be transferred over a larger range of 2. Typical values for bearing friction k can range anywhere from 0.0015 for small deep groove ball bearings to 0.15 for lubricated plain bearings operating in an elastohydrodynamic condition (Norton 2000, 629), while bearing radii will normally be less than 100 mm. As will be shown in the discussion of the total pseudo-static model, effective changes of length in any of the links due to deflection will also shift the singularity point from the original value of 2, albeit minutely. This is because 3 and 4 in equation (9) are functions of the link lengths. 68

Figure 36: Crank rocker mechanical advantage with joint friction limitations

4.2 Spring Design

The elastic element in any incarnation of the Beale CVT ultimately determines the automatic response of the mechanism to loads. The efficiency and fatigue life of these springs are also important factors in the design process, having strong effects on the performance and cost of the mechanism. Nearly any component capable of storing and dispensing mechanical energy efficiently can be used for the desired effect: designs have been explored using gas springs, beam springs, coil springs and even magnetic springs.

The measured spring behavior of the hand-cranked prototype is shown in Figure 37. It can be observed that the spring backstop in the particular design under discussion here 69 results in a highly nonlinear spring, whose deflection is largely dependent on the backstop’s geometry, and the beam spring’s contact with it.

Figure 37: Nonlinear spring deflection curve

The particular way in which the spring behavior affects the transmission’s output is demonstrated in Figure 38. For increasing levels of torque, the spring deflects more, focusing the region in which motion is transmitted more tightly about an advantageous speed ratio (i.e.: a speed ratio closer to zero). In this way, the spring can be designed to approach such levels of torque conversion at different levels of torque demand by configuring the spring’s deflection behavior according to the desired overall transmission characteristics. Additionally, it is possible to design a spring that either forces a maximum torque conversion by physically fixing displacement from input to output, as 70 shown in Figure 39 Example 1, or disallows motion at levels over a certain torque demand level by simply deflecting beyond the point of motion transmission, as shown in

Figure 39 Example 2. Thus, the Beale CVT can be employed as a plain CVT, limiting torque conversion, or as a torque limiter comparable to a ball-detent clutch.

Figure 38: Range of effected motion for different torque levels

Figure 39: Different transmission behavior resulting from different spring limits 71

Spring efficiency is also an important characteristic. A spring design allowing a full rotation of the input link without motion of the output link above a given torque level should result in the output shaft locking, preventing any energy transfer out of the mechanism. In this case, bearing friction and spring hysteresis are the most significant sources of energy loss. If those losses are small, the energy in the system is traded between the crankshaft inertia and the spring deflections relatively efficiently. The end result of this is that the output can be locked, and for a crankshaft with sufficient inertia to facilitate easy crossing of the singularities, the torque demand on the input power source should quickly approach a minimum value. This is useful, for example, for an application such as an electric vehicle; the motor should be able to idle at full speed with a minimal torque draw while the vehicle is stopped, thereby reducing the current spike experienced during the pursuant acceleration.

4.3 Complete Pseudo-static Model

As previously shown in Figure 38, different torque demand levels transmit motion over different ranges, resulting in variation of average speed ratio. If the spring’s deflection behavior is known, the torque-speed ratio relationship can be derived by combining the spring model with the four-bar mechanism model to determine this average resulting speed ratio over the range of 2 in which motion is transmitted by the one-way clutch. This is done by using the rigid four-bar mechanism analysis to find the singularities where 2 is equal to 3. At these points, link r4 will be at the extrema of its rocking cycle, where the sprag will engage and disengage at low speed. Under static 72 assumptions, flexible link r4 will deflect the prescribed amount shown previously in

Figure 37, at which point motion will be transferred to the output shaft.

Figure 40 shows the two triangles formed by the rigid links at the singularity points. Link lengths r1 and r4 remain constant, while the third side of the triangle is either the extension of r3 by the full length of r2, or the reduction of r3 by r2. Values for 2, 3 and 4 at these two points can be found analytically using the law of cosines, as shown in

Equations (10) and (11). 4 remains fixed at the top of the rocker’s cycle (2,top) while the spring deflects a given transverse angle based on the torque demand. For simplicity, r4 is assumed not to appreciably shorten along its length.

Once the spring deflects enough to meet the torque demand on the output shaft, the sprag unit begins to move, and4 increases until 2 reaches the bottom of the cycle, where the sprag clutch disengages. The crank angle 2 at which the torque demand is met by the spring can be found from the original four-bar mechanism position solution, using the spring’s force-deflection curve to determine the effective angular displacement of 4.

Hereafter, this value for 2 will be referred to as the engagement angle. Average instantaneous speed ratio can be calculated over the range of 2 between the engagement angle and bottom of the cycle (2,bottom). This result is equal to the average speed ratio of the transmission at the given level of torque.

This method was validated by measuring average input/output ratios on the prototype shown in Figure 27 at various torque levels. The input angle 2 necessary to produce one 360° rotation of the output out was measured with a protractor connected to the input crank while turning the input crank at a constant speed of 60 RPM. This 73 validation is shown in Figure 41, with calculated values as red asterisks and measured values as black triangles. The error in the measured values was smaller than the points on the graph.

Figure 40: Simplified crank rocker triangles at singularity crossings

2 (10) 2

2 (11) 2

Figure 41: Calculated (red) and measured (black) speed ratios vs. torque

While this simple model gives good results, it can be seen that at high torque demand, the differences between the model and actual performance begin to grow. This is due to the use of a beam spring on r4 for the flexible element. As discussed by Barten

(1944), large deflections of thin beams result in an effective shortening of the beam along its original neutral axis. This effectively shortens r4 in the four-bar mechanism, changing the crank angle 2 at which the singularity occurs. Figure 42 demonstrates this difference, where the connecting link depicted in dashed gray shows the singularity point with rigid links, while the connecting link in red shows the actual position when accounting for the 75 shortening of r4. Nonetheless, this simple method provides a quick means for approximating the speed ratio behavior of the transmission at low speed and various levels of torque with relatively good accuracy. Corrections can be added to the model depending on the design of the flexible member if more accurate results are necessary.

Figure 42: Error in estimation of singularity position assuming constant r4 76

5. DYNAMIC MODELS

Several dynamic modeling approaches were taken to describe the Beale CVT in different fashions. A basic free-body diagram and accompanying differential equations were constructed, but the nature of the one-way clutch and its specific use for instantaneously coupling and decoupling the full-scale torque to the mechanism demanded a focus on numerical methods for fully-developed models, as they have proven necessary for such applications in the past. In general, the focus of both approaches taken was on the sprag unit subassembly of the mechanism, comprised of the flexible link, the sprag clutch, the associated offset mass, the output shaft and the brake. As detailed in

Chapter 6 of this document, the design of the transmission was concurrently developed with the models with this goal in mind.

In order to approximate the overall behavior of the BCVT mechanism with a model of the single sprag unit, several assumptions were made. First, models were initially developed specifically for a single-sprag design, as shown in Figure 43. This was done such that model results could be validated against the simplest case concerning interactions of multiple sprag clutches. Second, motor speed was assumed to be constant, and third, motion of links r2 and r3 was assumed not to deviate from the rigid body, pseudo-static case, even at high speed. 77

Figure 43: Single unit setup of BCVT prototype for simplified model validation

Figure 44 shows a basic free-body diagram of the sprag unit, where the loads due to the shaft and spring have been separated into wrenches about the center of the output shaft, and the small wrench due to the encoder is ignored. In this basic form, any model for the behavior of both the beam spring and sprag clutch can be applied. Since the motion of links r2 and r3 are effectively prescribed, there is a fixed displacement on one end of the spring, and a boundary condition on the other end, depending on the model chosen. In like fashion, the sprag clutch can be modeled by any one of the methods previously discussed, coupled to an output shaft with a prescribed torque load, be it simple inertia, or something more sophisticated, such as Coulomb friction. 78

Figure 44: Basic free-body diagram of sprag unit

If, for example, the beam were freely vibrating, the Euler-Bernoulli beam equation could be applied with a fixed displacement on one end determined by the solution of the loop-closure equations previously given in Equation (4), and boundary conditions determined by the mounting of the end of the beam fixed to the sprag housing.

Effectively, this is what the time-series finite element solution does, with the exception that it handles the nonlinearities associated with the beam’s contact with the backstop on the sprag unit. Another proposed solution, the Duffing Equation, could be used, if not for the discontinuous behavior of the sprag clutch. Several such analytical methods still require numerical solution, as well as broad assumptions about behavior of the sprag clutch, thus unfortunately limiting their usefulness. 79

Considering a simple rotational or torsion spring with stiffness ktorsion and prescribed displacement spring, and a sprag clutch with discontinuous stiffness ksprag, the equation of motion for the sprag unit would reduce to the form shown in Equations (12) and (13). For this discussion, damping will be ignored, and the center of gravity of the sprag unit will be assumed to be at its center of rotation. This differential equation would be otherwise simple to solve, except for the discontinuity in ksprag, which is dependent on the relative velocity between the sprag unit and the output shaft. To compound this problem, shaft would be dependent on the outside load applied to the shaft, which is also often discontinuous.

̈ ( ) ( ) (12)

(13) {

Rigidity of the sprag unit and output shaft result in direct transmission of the reaction forces to the bearings and into the casing the bearings are mounted to, while the torques induce rotational motion. Thus, bearing stress and casing vibration can both be calculated directly from reaction forces between the sprag unit and output shaft. In reality, the center of gravity of the sprag unit is offset from the center of the shaft, adding a centripetal acceleration term to its mass, and thereby an eccentricity force, which is also transmitted directly to the bearings. In any case, if the motion of the sprag unit and sprag clutch engagement can be accurately ascertained with known spring behavior, speed output, bearing stress, casing vibration, link stress and torque input can be calculated 80 from it. Thus, models constructed will attempt to simulate the dynamics of the sprag unit and output shaft.

A finite element model will be constructed to explore the behavior of the beam spring and backstop, as well as to examine the possibilities for multibody simulation and forward dynamics. A component-based model will also be constructed to simulate sprag clutch behavior, and interactions between rotational masses, friction loads, damping and other measurable phenomena of the real-world system. Between these two models and the previously discussed pseudo-static model, there should be enough information to accurately describe the Beale CVT in its current form.

5.1 Component-Based Model

A powerful and sophisticated approach to simulation for design called bond-graph modeling can be used to model complex systems of components more easily than conventional methods, especially when the components in the system operate in different domains, such as analog electrical components and a four-bar mechanism. Using this method, components are described by their respective differential equations in terms of energy exchange, which is expressed in a common variable depending on the domain or domains in which the component operates. Components can then be easily connected

(bonded, hence the name of the approach) in acausal form, allowing the construction of large systems of many components quickly and easily. An extensive discussion on the approach was written by Broenink (1999).

The computing language Modelica uses this approach for physical modeling of component-based systems. The reasons acausality is so vital to this approach are twofold: 81 first, it alleviates the need for the programmer to determine the order in which the equations must be solved. Secondly, it allows for components to be easily reused and adapted to other system models in which they may act in a different capacity, albeit with the same basic behavioral equations. In conventional languages such as C++ or

MATLAB, the programmer must determine which variables are causal, and perform algorithmic solutions according to a fixed solution process, addressing values to memory spaces in a step-by-step fashion until the desired result is calculated. This approach is strong in its execution speed, as algorithmic calculations are well-suited to modern microprocessors. It is weak, however, in that assumptions must be made for causality, and that establishment results in custom-tailored code in which representations of the same component in different situations can vastly differ, drastically increasing the difficulty of model reuse. Figure 45 and Figure 46, from Fritzson (2004) demonstrate this, as they show the exact same system, expressed in acausal and causal block diagrams, respectively; note that the two resistors must be defined with different expressions in the causal model.

Figure 45: Acausal model of RLC circuit from Fritzson (2004)

Figure 46: Causal model of the same circuit in the previous Figure (Fritzson 2004) 83

For the Beale CVT, there are other advantages to this modeling approach. Due to the nature of the one-way clutch, the system must be defined as a set of differential algebraic equations (DAE’s), resulting in a severely discontinuous system. The acausal approach and the Modelica language are specifically powerful in this regard, due to the ability to use a set of libraries originally written by Linda Petzold, called the Differential

Algebraic System Solver Libraries (DASSL). These libraries, written in FORTRAN, currently provide some of the most effective means for solving stiff DAE’s, and can be used in conjunction with most Modelica environments as the solver engine.

A Modelica model of the single-sprag setup is shown in Figure 47. Parameters assigned to the various components are based on measured values or calculated values from solid part models, and can be found in the Modelica code in Appendix A. For a first try, the beam spring was modeled as a rotational spring, with a constant torque per angular deflection. Experimental determination of the linear behavior of the spring, as will be discussed in Chapter 6, was mapped onto a torque/rotation behavior, as shown in

Figure 47. This assumption should be valid in the linear region of the spring deflection, but cases where spring stiffness changes from deflection against the backstop to deflection in the other direction, or deflections of enough magnitude to effectively shorten r4 could cause significant deviation from reality. For reference, in this model, 4 is the angular displacement of the sprag unit, while spring is the angular displacement of the clevis joint on the end of the spring. The actual center of gravity of the sprag unit is also denoted in the Figure.

Figure 47: Modelica model of single sprag unit and output shaft

Figure 48: Rotational spring model with flexible link r4 85

Modelica models of one-way clutches are generally based on a combination of

Coulomb friction clutches and pure freewheels, exerting a frictional torque when the freewheel is decoupled. The nature of the sprag clutches in the Beale CVT makes this friction negligibly small, such that it was set to zero in the model. Numerical jumps are handled by a combination of smoothing functions over small discontinuities, in-situ reinitiation of the numerical method at the point of interest, matching the previous results with the new initial values, and the backwards-differentiation approaches used by

Petzold’s DASSL.

Parasitic losses were lumped into a bearing friction term on the output shaft, such that they could be matched to measured performance at a zero-load condition. A grounded one-way clutch component was coupled to the output shaft, so as to prevent backward motion that complicated results in previous studies (Cyders and Williams

2010). The actual load on the transmission may vary, but a Coulomb friction load exhibited some benefits for the prototype design that made it an attractive option. A simple brake was modeled, with on-off sliding torque. Much like the one-way clutch,

Modelica’s built-in numerical libraries can handle this otherwise difficult model well. An input table for the prescribed motion from links r2 and r3 was set up to import comma separated values from kinematic position and velocity results at different input speeds.

Once verified, this model should be capable of concurrent application to single outputs, as the CVT operates in a hybrid-drive mode, or with single inputs going to multiple outputs, serving as a differential. If necessary, additional components can be added for 86 refining damping, nonlinear spring behavior and the like. Results from this model will be discussed in Chapter 7.

5.2 Finite-Element Model

Time-series finite element modeling also has capabilities of interest for the Beale

CVT, especially for the current design. The ability to tailor spring behavior based on part geometry allows the automatic nature of the transmission to be customized for a given design. As shown in the pseudo-static analysis, the spring behavior directly dictates the

Torque-speed ratio relationship of the transmission at low speed. Also, finite element packages are becoming powerful for multibody simulation and forward dynamics in some cases. These approaches tend to be computationally expensive in the extreme, but simple 2-D models can run relatively quickly for modeling motion of the springs.

Overall, the use of the model is still limited by its inability to effectively incorporate components such as the one-way clutch, so in this study, its application is limited to the unloaded case.

Such a model is shown in Figure 49. In this model, a beam spring is being bent over a backstop with frictionless contact, using a prescribed nodal displacement at the clevis joint as the input. Raw data for deflection and resultant force can be output to comma-separated values, and plotted externally with other data. The contact algorithms can take time to converge, as iterations must be run in order to determine which elements are in contact at each time step. Models must be well-conditioned for convergence to occur easily, otherwise simulation time can increase considerably. A model such as that 87 shown in Figure 49 generally takes between 10 and 15 minutes to run on a modern, 6- core desktop.

Figure 49: 2-D Finite element model of spring behavior in contact with backstop

One method for dealing with the one-way clutch mentioned in the literature was to use a shooting method of sorts, whereby initial conditions for an instantaneous grab and release of the clutch mechanism, or as put into a finite element simulation, a torque that turns on and off instantaneously. A cycle could be run with a guess as to the engagement cycle, and at disengagement, the motion of the system at that point used as initial conditions for the unloaded portion of the cycle. By varying the points at which the clutch engages, one could converge to a solution where the engagement cycle ended with approximate final conditions that would double as the appropriate initial conditions triggering the engagement at the same spot. This approach was attempted by hand with 88 the finite element data, but became far too intense, and convergence was not achieved. If the process can be further automated in the future, it may provide sufficient results to simplify the one-way clutch as applied by the finite element method.

Lastly, a full, 3-D kinematic chain was constructed from the solid model, and appropriate boundary conditions applied to constrain the four-bar as in real life. This model still cannot easily integrate the one-way clutch, but can provide extremely accurate results when simulation conditions are set correctly. Unfortunately, simulation time on a

6-core, 3 GHz PC for one revolution was well over 6 hours, and a 5 second time span at

350 RPM took nearly a week to simulate, so it was not used extensively for this project.

Such a model is shown in Figure 50

Figure 50: 3D MES calculating full forward dynamics 89

6. EXPERIMENTAL DESIGN AND SETUP

In order to measure the real kinematic and dynamic behavior of the Beale CVT, a prototype was designed, fabricated and revised to provide adequate data for comparison to modeling results. The proposed measurement results included the following:

 Link stress

 Torque input

 Speed output

 Torque output

 Bearing stress

 Sprag engagement

 Casing vibration

The design process focused on maintaining the capability to measure each of these goals either directly, or by proxy through other means based on validated assumptions. The basic conceptual design is shown in Figure 51. In this design concept, the transmission was to be mounted on a rigid base (1), and coupled to a DC motor (2) through a balanced crankshaft (3) for operation at controllable speeds from 0 to 1800

RPM and beyond. The load was to be measured by a torque sensor (4), which would be attached to the load coupling mechanism, shown in Figure 51 as a roller chain sprocket

(5). Accelerometers (6) were to be used to measure the motion of the transmission base and angular motion of the sprag units, while strain gauges (7) would be used to 90 approximate linkage forces, and encoders (8) would measure shaft speeds. Torque input could be easily calculated from either motor current draw or the force on the connecting link, while bearing stresses could be calculated using the combination of linkage forces, sprag motion and output torque. This concept was adapted into a final design, a 3-D model of which is shown in Figure 52, and a photograph of which is shown in Figure 53.

Figure 51: Example testbed layout with conceptual instrumentation layout

Figure 52: 3D model of final testbed design with instrumentation

Figure 53: BCVT prototype, as built for experimentation 92

The original concept was modified to include four sprag units, with the ability to easily remove or add units for testing with different numbers of the one-way clutches interacting at multiples of a 90° phase separation. When operating with four sprag units, this could confirm whether loads were symmetric, as translational forces on the output shaft would be effectively balanced in the direction normal to both the center of shaft and the center of the angle through which the sprag units would be rocking.

All structural elements of the transmission were designed to be extremely stiff, such that the only significant deflection in the system would be the carbon-fiber springs.

For additional structural rigidity, the drive motor was mounted directly onto the baseplate at both ends, and was coupled to the crankshaft using a pair of 60-tooth involute gears, giving a 1:1 drive ratio between the crankshaft and the motor. The crankshaft design, along with the relatively large steel gears provided a high level of inertia on the input, so as to test the conjectured behavior of the system in the event that the output shaft was locked during operation. Acceptable parallelism of the crankshaft and output shaft axes was ensured by locating the bearing blocks with dowel pins, and maintaining tight constraints on the blocks themselves.

The drive motor selected was a Leeson model CM31D17NZ26D 12-volt 250 Watt brushed DC motor coupled with a Leeson 175290 pulse-width modulation (PWM) controller. This combination was powered by a Hengfu 12 volt, 18 amp DC power supply, and controlled by passing a 0-10V DC voltage signal from a TekPower HY1803D adjustable DC voltage source to the control terminals on the PWM controller. This setup is diagrammed in Figure 54. Detailed motor characteristics are summarized in Table 2. 93

Figure 54: Motor control and power wiring layout

Table 2: DC motor characterstics for testbed drive motor

Stall Torque Internal No-Load Shaft Inertia No-Load [N-m] Resistance [W] Speed [RPM] [kg-m2] Current [A] 4.97 1.9 2045 1.774x10-4 2.1

All revolute joints in the final design were supported by bearings, including the sprag units, due to the non-bearing nature of the one-way clutches used. This reduced the mechanical resistance in the mechanism, but came short of eliminating it; PTFE washers used to maintain alignment of the elements still provided noticeable resistance to motion, albeit relatively low. This resistance was roughly estimated to be less than 2% of the full- 94 power load at the motor. The load was designed to be a BB7 bicycle disc brake from

SRAM Corporation, as it was capable of creating one-way, speed-independent torque demand with reasonably fine adjustment well beyond the range needed for this experiment for relatively low cost. Also, replacement parts were readily available, in the event that pads wore down too quickly, or if the brake disc were to warp. The output torque measurement then became possible from a static location on the mounting bracket for the brake caliper, avoiding the need for wireless data transmission from the moving shaft.

Figure 55 shows a sketch of the design, depicting the variables that were measured and/or analyzed. 2 is the crank shaft input angle, 3 is the angle of the connecting link, 4 is the angle of the rigid housing containing the one-way clutch, and

out is the angle of the output shaft. T2 is the torque on the input shaft, F3 is the longitudinal force in the connecting link and Tout is the torque on the output shaft. FB2 and

FB4 are the radial bearing forces on the input and output shafts, respectively. 95

Figure 55: Measured (blue) and calculated (green) variables on the final testbed design

6.1 Parameter Measurement

In order to ensure agreement between the models and real-world behavior, this experimental design involved several parameters which required direct measurement or validation. These parameters included material constants for the composite springs, and the actual reactions of the two strain gauges to loads. Efficiency of the springs was also measured to determine the efficacy of the beam spring design for features such as the idling input capability.

The beam springs were fabricated from Gordon brand unidirectional carbon laminations 38.1 mm wide and 0.51 mm thick, and Gordon brand Bo-Tuff “E” fiberglass laminations 38.1 mm wide and 0.76 mm thick, as detailed in Figure 56. Fibers were oriented along the longitudinal axis of the beam. Two alternating layers of each laminate 96 were bonded with approximately 1mm thick layers of Smooth-On EA-40 clear epoxy adhesive, then wrapped in polyethylene, clamped into a precision-machined aluminum form, and baked under a heat lamp in an insulated box at approximately 82° C for eight hours, monitored periodically with a NIST-certified Extech 42510 infrared thermometer.

Since the composite springs were fabricated from raw materials, no dependable value for their modulus of elasticity could be derived from listed values, so a test was conducted to provide an accurate estimate of the value. This was done by arranging the spring in the sprag assembly, then hanging weights from the clevis joint mounting hole, and measuring the resulting deflection with a dial indicator, as shown in Figure 57.

Figure 56: Composite beam spring with E-glass, carbon fiber and epoxy layers 97

Figure 57: Experimental setup for determination of composite spring elastic modulus

Using the canonical beam deflection equation (14) along with the known values for beam width, b (30.8 mm), beam thickness, h (1.90 mm) and beam length, L (141.8 mm), an average value of 22.2 GPa was found for the modulus of elasticity, E, using a least-squares linear fit to the data, shown in Error! Reference source not found.. This value is consistent with accepted values for similar composite structures.

(14)

Figure 58: Spring force-deflection data for calculation of Young’s Modulus

Also of interest was the damping behavior of the spring material. If the material’s hysteresis was large, models would need to account for this energy loss to correctly model the spring vibrations. If the hysteresis characteristics were sufficiently small, damping could possibly be ignored with little loss of accuracy. This value would also be important for estimations of transmission efficiency. To measure this value, an Omega

ACC-103 single-axis accelerometer with 10 mV/g calibrated output was bolted to the end of the cantilevered beam spring, the base of which was rigidly mounted to a lab table, as 99 shown in Figure 59. As the accelerometer casing was not isolated, nylon hardware was used to insulate the casing from electrical connection to the rest of the clevis assembly.

Figure 59: Experimental setup for determination of spring damping characteristics

This orientation kept the motion in a horizontal plane, which eliminated gravitational forces. The tip of the spring was deflected 57.1 mm, and was then released, allowing free harmonic vibration. This resulted in a typical underdamped acceleration response, recorded using a Personal Daq 3000 USB data acquisition unit at a recording rate of 100 kHz. The resulting acceleration data was adjusted and numerically integrated to give displacement data versus time, as shown in Figure 60. Since the original displacement was known, the maximum displacement for each cycle was used to fit a 100 curve of the form of Equation (15), where max was the maximum displacement of each cycle at time t, X was the initial displacement of the accelerometer (34.2 mm),  was the damping ratio, and n was the fundamental frequency of the beam system, calculated from the mass on the end of the spring and measured value of Young’s modulus to be approximately 12 rad/sec. For simplicity, vibration was assumed to be first mode only.

Figure 60: Displacement vs. time for determination of damping ratio, 

− (15)

101

The resulting damping ratio  was 0.0027, corresponding to a viscous damping coefficient, c, of 0.032 N-s/m, or to a Q-value of approximately 190. This value is significant because one of the defining features of the Beale CVT is the ability to idle the transmission at full speed by simply locking the output. In this situation, the energy required to keep the transmission spinning is equivalent to the losses in the bearings and springs. Properly designed bearings coupled with springs of such low hysteresis may result in a very efficient system, allowing the exclusion of a clutch mechanism in applications such as vehicles. This measured value suggests such a feature is feasible.

Also of interest were conversion factors for the two strain gauges used to measure the force in the connecting link, and torque on the output shaft. These gauges were 2- contact pre-wired strain gauges from Omega Engineering, part number KFG-10-120-C1-

11L1M2R, with a listed gauge factor of 2.08 ±1%. Each gauge was applied to its respective location using a cold-cure adhesive, Loctite 496, and connected to a Vishay A2 bridge completion module/signal conditioner in a quarter-bridge arrangement. There was not enough space in the design to use a more sophisticated gauge arrangement to cancel off-axis strains and temperature effects. In addition to this, the strain states in question were non-uniform over the area of the gauge. Therefore, calibrations were performed to validate expected gauge behavior in response to known loads.

The connecting link strain gauge was attached near the middle of the link, along the longitudinal axis of the link, as shown in Figure 61. To improve the signal-to-noise ratio of the strain gauge at low levels of force in the link, a hole was drilled through the middle of the link at the gauge site prior to application of the gauge to concentrate strain 102 in that area. This created a non-uniform strain, which would be averaged over the area of the gauge during measurement. A simple finite-element model was created to calculate the average strains due to loading over the strain gauge area depicted by the magenta points in Figure 62, to ensure linearity.

Figure 61: Strain gauge mounted over hole on connecting link

Figure 62: Meshed FEA model showing strain gauge location in magenta 103

Figure 63 shows the longitudinal strain concentrations in the model, where areas of red are the highest concentrations, and areas of blue the lowest. The average strains over the area in question were found to behave linearly with load well beyond the expected load range for the physical linkage. These strain values, , were converted to expected bridge voltages using the listed gauge factor, GF, an excitation voltage, Vex, of

10 V and a gain of 500 in Equation (16).

Figure 63: FEA model showing strain concentrations (red maximum, blue minimum)

(16) 2

104

The connecting link was suspended from a static position, and calibrated weights were hung from the bottom, while the amplified bridge voltage was measured with a

Hewlett-Packard 34401A multimeter. This data was plotted with the data from the FEA model, and a least-squares linear regression was applied, giving a conversion factor of

1.592 mV per Newton of force. The results are shown in Figure 64, with calculated values from the finite-element model in blue, actual measured values in black, and the best-fit line in red.

Figure 64: Measured and predicted calibration data for connecting link strain gauge

As can be seen in the above figure, the calculated and measured data are very close, validating the model. Bridge voltages can be easily converted directly to 105 connecting link forces using the linear conversion factor. The same approach was taken to calibrate the strain gauge on the torque output, shown in Figure 65. In this case, a torque load caused a bending moment of the beam-like portion of the brake mounting bracket, resulting in a very strong signal-to-noise ratio, but the angle of the bracket imposed a multi-axial state of stress on the area. Again, a finite-element model was constructed, as shown in Figure 66, and average longitudinal strain values were applied to Equation (16) to calculate resulting bridge voltages.

Figure 65: Strain gauge used to measure torque output

106

Figure 66: FEA model of strain concentrations on brake caliper bracket

To test the actual response of the torque strain gauge, an aluminum beam was bolted to the brake disc, and the brake was locked tightly enough to prevent motion. The bridge voltage was then zeroed, and weights were hung from the beam, while the amplified bridge voltage was measured with the same multimeter as in the previous test.

This setup is shown in Figure 67. These measurements were compared to the predicted values from the FEA model, and a linear regression was applied to the measured data, giving a conversion factor of 12.90 mV per Newton-meter of torque. This plot is shown in Figure 68. The values for the gauge bridge voltage were again highly linear throughout the measured data, and continued that behavior well beyond that region, according to the

FEA model.

107

Figure 67: Experimental setup for torque strain gauge calibration

Figure 68: Measured and predicted calibration data for torque strain gauge

108

6.2 Instrumentation

In addition to the strain gauges, encoders and an accelerometer were used to measure the motion of the system components. All the encoders used were US Digital H5 differential quadrature-output ball-bearing rotary encoders with index channels. These units had a maximum resolution of 0.088° in quadrature mode, producing a square wave of ±5V, with a maximum rise/fall time of 15 nanoseconds, and were powered with a constant 5V DC signal from an Elenco Precision XP-581 DC voltage source. The encoders were terminated with a 120 resistor between the differential contacts of each channel, as prescribed by the manufacturer for noise reduction. One encoder was coupled to the crankshaft with a close-fit hole and setscrew to directly measure 2, as shown in

Figure 69. Only one pulse channel was used from this encoder, since the measurement of

2 did not require any higher precision than 0.3°, and channel space was limited on the data collection platform. Instead, full quadrature measurement was allocated for the other two encoders, measuring the absolute position of the shaft and first sprag unit, respectively. The index signal from 2 was measured, however, through a digital counter channel on the data acquisition system.

109

Figure 69: Encoder mounted on input crank, measuring 2

The output shaft angle out was measured using the same US digital H5 encoder in quadrature mode, without indexing. This encoder was connected to the output shaft in the same fashion as shown previously in Figure 53, with a close-fit hole and setscrew. 4 was directly measured with a third H5 encoder, coupled to a sprag housing with two 96-tooth,

14.5° pressure angle involute gears made of 6/6 Nylon, one mounted with a setscrew to the shaft of the encoder, and the other bolted to the aluminum sprag housing. This arrangement is shown in Figure 70. Based on the part tolerances, the maximum backlash between the encoder and sprag housing was 0.017°, corresponding to an error in absolute angle of ±0.0085°, an order of magnitude smaller than the precision of the encoder itself.

110

Figure 70: Geared encoder arrangement for direct measurement of 4

All instrumentation was connected to the analog inputs of an IOTech Personal

Daq 3000 USB data acquisition system. This system was capable of collecting up to 8 differential-ended analog signals with 16-bit depth using measurement periods down to 1

s per differential channel in use, corresponding to a maximum measurement frequency of 111 kHz for this experimental setup (8 analog channels and 1 digital channel). The accompanying software, DaqView, was used to collect the raw data at the maximum data rate, with no averaging or manipulation of the data as recorded. All encoder signals were measured in the 10V range with a listed accuracy of (±0.031% of the reading + 0.0008

V), while the strain gauge signals were measured in the 500mV range with a listed 111 accuracy of (±0.031% of the reading + 0.2 mV). The precision of the system’s built-in clock was listed as less than ±10 nanoseconds, well below the data collection frequency.

6.3 Experimental Procedure

In order to correctly zero the absolute positions of 2 and 4, data was recorded over the course of several cycles at 60 RPM, such that inertial effects on the sprag unit could be ignored. This data was then compared to the kinematic simulation, and offsets were set to correctly zero the index pulses of each encoder. This procedure was necessary every time the encoders were disconnected from their respective shafts, as the location of the setscrew could not be accurately controlled. A graph depicting the motion indicated by the corrected signal alongside the motion predicted by the kinematic simulation is shown in Figure 71, showing very close agreement between the simulation and measured values. The slight bumps in the signal are due to a visually noticeable vibration induced in the beam spring by bearing drag. 112

Figure 71: 2 vs. 4 at 60 RPM, with predicted (red) and measured (black) values

The motor was set to varying speeds from 200 RPM to 600 RPM in increments of

50 RPM at several different levels of loading with the brake. As shown in Figure 72, the brake load was set with a cable and locking screw mounted to the baseplate. Before powering up the motor, the bridge voltage on the torque transducer was zeroed to account for the load due to the cable tension. The motor was adjusted to the desired speed using a

DT-2234C+ digital tachometer, to within ±20 RPM. Data was collected over the course of several full rotations of the output shaft at a rate of 111 kHz, and saved into a tab- delimited value format for batch processing and analysis. This resolution was necessary to successfully pick up the encoder signals at the higher speed ranges. Most tests were 113 conducted with only one sprag unit in place, so as to directly compare the results to the simulation results. One test run was done with all four sprag units in place, to verify the interactions between multiple sprag units on one shaft.

Figure 72: Cable and locking screw used to set brake torque

6.4 Data Analysis

Data was extracted from the comma separated value files into Microsoft Excel, and edited to eliminate the unneeded timestamps and textual data from the file. The resultant data was read into MATLAB using a batch processing script, dbp.m, which can be found in Appendix A. This script read the raw data, converted strain gauge signals to their respective load units, and calculated angles based on the raw encoder signals. A binary representation of the quadrature state of the encoder was used to determine direction and absolute position based on the encoder index using the script quadenc2i.m, which can also be found in Appendix A. The resulting motion of 4 vs. 2 was then 114 plotted to the screen to ensure the integrity of the data. A typical cycle was selected from the plot as shown in Figure 73, and all data from this single cycle was projected onto an evenly spaced axis of 500 points over the time of one revolution of 2. A separate script, numdiff.m, performed numerical differentiation to obtain velocity results from the encoder data, as direct finite difference calculation was far too noisy. The script used a polynomial fit approach as discussed by Hoffman (1992). This data was output to a comma separated value format, and imported into SciDavis for plotting and analysis.

Figure 73: MATLAB data analysis output for single cycle selection

115

7. EXPERIMENTAL RESULTS AND DISCUSSION

As tests were run, many different overall phenomena were observed. First and foremost, the conjectured idling behavior whereby the springs allow the crank to complete a full rotation without necessarily causing motion in the output shaft was correct. With the maximum current on the motor severely limited so as to cause speed variations over the crank cycle, locking the brake instantly caused the motor to increase in speed up to the maximum dictated by the voltage, where inertia carried the velocity nearly constant through the singularities. Also, the assumption in the models that motor speed was constant was accurate to within ±3%. Slight speed variations were apparent, but measured values showed them to be very small, as shown for example in Figure 74.

The solid black line was differentiated using a polynomial fit routine.

Figure 74: Measured motor speed at 350 RPM, 2.5 N-m torque 116

7.1 Physical Phenomena

Summarized experimental results for overall speed ratio at various torque levels and input speeds are shown in Figure 75. Blue triangles represent points with zero braking load, while black circles, red crosses and orange diamonds represent approximate braking loads of 2.5 N-m, 5 N-m and 7 N-m, respectively. Some input speeds were unattainable at lower torque levels because high spikes of 4 caused the sprag unit to contact the transmission base. Torque loads were estimated from the average sliding torques measured in each loading case. The highest measurement uncertainty in speed ratio was ± 2.9x10-4 based on measurement outputs and encoder resolution. Accounting for variation in output over all cycles measured for a dataset increased this uncertainty to

-3 ±1.8x10 . Speed ratio uncertainty, R, was estimated using uncertainties in out and 2 using Equation Error! Reference source not found..

( ) ( ) (17)

117

Figure 75: Measured speed ratio at various input speeds and torque levels

In the unloaded case (blue triangles), the overall speed ratio of the transmission is greatly affected by input speed. This is because the inertia dominates the motion of the system, causing especially large displacements in the positive direction while the beam spring is displaced in the negative direction (away from the backstop). The addition of even a slight torque load limits this phenomenon, as can be seen in the other loading cases shown. Thus, as load is increased, the strength of input speed 2 on the overall speed ratio is reduced, as the effect of the inertia on the output motion is more restricted.

At high-enough torque levels (orange), the inertia begins to work against the throw of the sprag unit, actually reducing the speed output at higher levels of input speed. 118

An example of typical measured motion over a single cycle is shown in Figure 76.

This is from the second load case (black) previously shown in Figure 75, at an input speed of 350 RPM. Because of the input speed and inertia, 4 is negative beyond the point at which the input (shown previously in red in Figure 71) goes positive. Once the sprag unit begins to move in the positive direction at 2 = 90°, the sprag clutch locks, and the output shaft motion matches that of the sprag unit until the clutch is released at 2 =

225°. During this motion, output torque increases to a roughly constant value of sliding friction, and then decreases as the load is released. Torque signals were noisy, as shown in gray, so a post-process low-pass FFT filter was applied with a cutoff frequency twenty times higher than the rotational speed of the input shaft to clean up the signal. As shown, the filtered results remained representative of the original signals, and were consistent within 10% over the speed range for each torque setting. The noise shown is the worst case in the dataset for the torque measurements, having the lowest signal change due to the low torque; other cases have the same level of noise regardless of speed, but signal changes increased linearly corresponding to the increase in applied torque. This worst case had a signal-to-noise ratio of 6.7 dB.

119

Figure 76: Measured out, 4 and Tout at 350 RPM, 2.5 N-m 120

Signals for link force were unfortunately so noisy that useful information could not be extracted for most of the load cases. An example of the raw data from the 250

RPM, 5 N-m torque load case is shown in Figure 77, which has a signal-to-noise ratio of

-6.3 dB. Noise levels were far higher than they were during the calibration process, suggesting that the rapid movement of the strain gauge wire and proximity to the motor may have had significant contributions to measurement noise. While some data points may serve to roughly validate model results, most of the measurements lack substantial significance.

Figure 77: Measured link force at 250 RPM, demonstrating low signal-to-noise ratio

121

Figure 78 shows output angle and relative motion between the output shaft and sprag unit at two different speeds with zero torque load. The top and middle plots show measured behavior at 450 RPM and 200 RPM, respectively, with no brake load. Dotted lines represent the angular velocity of the sprag unit, while solid lines represent the angular velocity of the output shaft. When the sprag clutch is engaged, the two angular velocities are equal. These regions are bounded by triangular points on the plots.

Variations in the coupled velocities are largely due to artifacts from numerical differentiation.

In the 450 RPM case (depicted in black), the inertia of the shaft carries a positive velocity from one cycle of the sprag unit to the next. As a result, the clutch engagement is shortened, releasing for the remainder of the cycle at the sprag unit’s peak velocity, where it begins to decelerate. The output shaft continues with a positive velocity that decreases until the next engagement due to parasitic losses such as bearing friction. The initial engagement of the clutch is delayed in the regular cycle of 2, as the sprag unit velocity must increase to the output shaft’s positive velocity before engagement occurs. Clutch engagement, as expected, occurs in the portion of the cycle during which the second derivative of the output angle out is positive.

The 200 RPM case, depicted in red, shows different behavior in that the inertia doesn’t have as strong of an effect. There is still overshoot at the top of the first inflection point in the angular velocity of the sprag unit, 4, but it is small enough that there is a second engagement during a single cycle of 2. The inertia retains less velocity than the

450 RPM case, and displaces a smaller angle, as expected. The output shaft returns to 122 zero angular velocity by the end of each cycle, resulting in an initial clutch engagement at the point where the sprag unit’s angular velocity becomes positive.

123

Figure 78: Sprag engagement and output angle at 200 (red) and 450 (black) RPM

124

7.2 Comparison of Model Results

A brief test was run at 60 RPM with four sprag units operating at a 90° phase angle to verify properties of sprag engagement. Figure 79 shows the angular velocity of the output shaft in dashed black along with the angular velocities of four interacting sprag units in solid red. Since the four sprag units could not be simultaneously instrumented, calculated values for the sprag unit velocities were used instead. As shown previously in

Figure 71, the actual movement of each sprag unit is close to that calculated with the kinematic model.

Figure 79: Calculated sprag velocities at low speed with measured output velocity

125

Again, artifacts from numerical differentiation affect the measured values for out.

Nonetheless, the output shaft velocity closely follows that of whichever sprag unit has the most positive angular velocity at a given point in time. Once a given sprag unit has decreased in angular velocity below that of another, it is decoupled, and has no effect on the output thereafter until the sprag engages again. This experimentally confirms the previous work by Cyders and Williams (2010). For effectively rigid designs, or flexible designs at low enough speed, this result demonstrates that the pseudo-static model matches the measured output.

Figure 80 shows the same measured angular velocities as the 450 RPM case of

Figure 78, with results from the Modelica simulation. Again, the red dot-dashed line is the measured sprag unit angular velocity 4, the solid black line is the measured output shaft angular velocity out, and the dashed blue line is the simulation result from

Modelica. Initial results from the unloaded case were compared to the Modelica output to tune the system characteristics in the model to the apparent behavior of the experimental prototype.

The behavior of the output shaft due to the high velocity and inertia was taken advantage of, as differences in the slope of the simulated result with the measured result in the non-engaged portion of the cycle were matched by iterative calculation of bearing drag characteristics. Operating points were added at different speeds in the Modelica model to improve the accuracy of the results. At just three points, the model matched the output as shown. In this case, it can be seen that the simulation results show a higher output shaft velocity than was measured before the engagement, and a lower velocity 126 after the engagement period. This behavior oscillated around the measured result, with the following cycle having an initial simulated velocity lower than measured, and a final velocity higher than what was measured. In any case, the results are close.

Figure 80: Measured (black) and calculated (blue) out with measured 4 at 450 RPM

The output shaft angle results from the Modelica simulation are shown alongside measured results in Figure 81. The output angle for the 450 RPM case detailed in the previous figure is shown in the same colors and linetypes. As expected from the behavior of the angular velocity, the output angle out as simulated starts off the cycle with a higher slope, resulting in an overshoot of the angle out, with the error between the two decreasing over the cycle to 2.9%.

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Figure 81: Measured (solid) and calculated (dashed) out at 200 and 450 RPM

The output angle for the 200 RPM case is also shown in Figure 81, with measured values in solid red, and simulation results in dashed orange. As can be seen, the model initially overshoots the real case during the first clutch engagement in the cycle, undershoots until the second clutch engagement, then overshoots again with 5.4% error in the final result. Notably, the model accurately predicts the multiple clutch engagements, as can be seen by comparing the curve deflections due to a positive second derivative of

out. Further tuning of bearing resistance and other model parameters would likely further improve the modeling results. 128

After initially adjusting model parameters to match the free case, the model was run to simulate all the experimentally-measured load and speed cases. Model results for speed ratio at the different torque loads and speeds are shown in Figure 82, where solid points represent measured values, and hollow or stick (in the case of the red cross) points represent simulation results. Again, error in the measured results is smaller than the solid points themselves, not accounting for variability in output from one cycle to the next.

Figure 82: Measured (solid) and predicted (hollow) speed ratio vs. input speed

As can be seen in the above figure, simulated results have good agreement with measured values. For the 2.5 N-m and 5 N-m cases, results are very close, due to the 129 relatively small, controlled motion of the sprag unit, and the applicability of linearity to the beam spring. This linearity breaks down for the 7 N-m load case, resulting in differences between the simulation and reality. At high loads, the larger deflections of the spring not only take the force-deflection relationship into a nonlinear region, but also effectively shorten link r4 in the mechanism, shifting the characteristic singularity upon which the torque conversion is based. Developing a nonlinear spring component would improve model performance in these regions, as would an acausal multibody simulation capable of describing changes in the kinematics due to the large deflections involved. The simulation also accurately predicts the “jumping” behavior where the speed ratio at a given level of torque suddenly increases with increasing input speed, as seen at 400 RPM for the unloaded case, 450 RPM for the 2.5 N-m case and 500 RPM for the 5 N-m case.

These points are emphasized in Figure 83 with magenta arrows. Many of the results are accurate to less than ±3%, within the specified requirements. 130

Figure 83: Measured and predicted speed ratio, with emphasized "jumps"

Figure 84 shows measured brake torque in solid gray, with a filtered signal in solid black, and simulated brake sliding torque overlaid in dashed blue for the 2.5 N-m case at 350 RPM input speed. This variable was effectively an independent variable in the simulation in terms of the magnitude of the sliding torque, but the timing is dependent on the results from the model. As expected from the rest of the results, sliding occurs relatively accurately with the measured motion of the model. The measured results are similar for most of the cases, with a flat peak during the bulk of the sliding friction, and steep ramps from and to zero torque during the period of clutch disengagement. 131

Unfortunately, dynamic signals for link force were so noisy as to be essentially useless for real measurement. Nonetheless, the signal can be compared with the simulated value to show agreement of scale. Simulation results were obtained by converting the input torque to a linear force using the kinematic approximation for angle and acceleration calculated from the rigid-body motion, and known component masses.

Figure 85 shows measured force signal in solid gray with adjusted Modelica simulation results in dashed blue, which agree reasonably well.

Figure 84: Simulated (blue) and measured (black/gray) brake torque at 2.5 N-m 132

Figure 85: Calculated (blue) and measured (gray) link force, 350 RPM at 5 N-m

Simulated link forces were also resolved to motor torques using Equation (6) with position values based on the kinematic analysis. Motor torque results from the 350 RPM case at 2.5 N-m of torque are plotted in dashed blue along with measured signals as differentiated by direct finite difference (solid gray) and polynomial fit (solid black) in

Figure 86. As expected, major inflections in the motor speed roughly line up with points where the motor torque goes to zero, while areas of highest rate of change of speed follow peaks in the torque demand. 133

Figure 86: Measured motor speed (black/gray) and simulated motor torque (blue)

Testing of finite element models was also performed. Initially, the 2D Mechanical

Event Simulation approach was tested by photographing a hand-cut backstop, and digitally extracting the points from it to construct a model in a CAD environment. This photograph is shown in Figure 87. Once the geometry was imported into the finite element model, an estimate of Young’s Modulus for the beam spring was made by hanging weights on it in a cantilevered arrangement, and calculating the modulus as previously done in Chapter 6.

With this information, the MES model was constructed, and measurements for spring deflection were compared to real measurements taken with calibrated weights. 134

Results are shown in Figure 88, where black points are measured values and red crosses are simulation results. The simulation accurately predicted a sudden increase in stiffness due to a ridge in the backstop difficult to see with the naked eye. The same approach was applied to the backstop as designed for the prototype in this study, with measured (black) and simulated (red) data plotted together in Figure 89. Results again have good agreement with measured values, showing a smooth, yet sudden increase in stiffness as deflection approaches the crossover of the crank.

Figure 87: Hand-cut backstop from hand-cranked BCVT prototype

135

Figure 88: 2-D MES results for force vs. deflection on hand-cut spring

Figure 89: 2-D MES results for BVCT prototype spring and backstop 136

8. CONCLUSIONS

The Beale CVT presents some important developments in the transmission of mechanical power for many different applications. As future designs take shape, successful modeling approaches focused on assisting the design process and improving understanding of the mechanism’s behavior must be developed and validated by experiment. As a wider body of knowledge about the transmission is constructed, models can be improved in their sophistication, and designs can be further improved and brought to market.

8.1 Project Results

The Beale CVT prototype as designed was largely a success. While link force measurements were too noisy for useful data, model results very closely matched other measured phenomena. Many of the operational characteristics at high speed performed as expected, including the unique ability of the transmission to idle with a stopped output shaft in an efficient manner. The balanced implementation of the mechanism minimized vibrations, and allowed for smooth output with sufficient sprag units and inertia.

Kinematic and pseudo-static modeling demonstrated the fundamental operational principle upon which the Beale CVT is based, revealing that a whole family of mechanisms exists which can be adapted into the design. The basic model also provided a quick, simple approximation for the limitations on the transmission’s ability to convert torque. Without specific, intense modeling, this approach can be used for very fast estimation of design parameters for the transmission in its current form, providing a 137 useful design tool. Moreover, measured data showed that under moderate loads, deviations in speed ratio from the pseudo-static case were relatively small, suggesting that the simple pseudo-static model results may be useful beyond low-speed cases.

The component-based Modelica model accurately predicted motion of the sprag unit and output shaft within the desired parameters at a plurality of points, even with rough assumptions. Desired output variables were consolidated based on mechanics, and the relevant variables necessary to calculate the sought-after results were simulated with sufficient accuracy according to the initial project goals over a moderate range of operating points. This suggests that input load predictions were also accurate, but could not be confirmed based on the state of the data measured in experiment. Behavior of the sprag clutch in the model was also close to that measured in experiment, even to the extent of accurate prediction of multiple engagements per cycle of the input crank.

The finite element model very accurately predicted behavior of the beam spring in contact with the sprag unit backstop, providing an avenue for further development of and integration with the Modelica model. The method was extremely computationally expensive with simulation times extending to multiple days, but worked well for modeling specific phenomena when needed. The approach did not present a convenient avenue for integration of the one-way clutch mechanism, but other software packages may provide such utility going forward.

In summary, the project was a success. Several modeling approaches were taken, providing different types of information about the Beale CVT, with good accuracy in each domain. The complete pseudo-static model provided a quick and accurate 138 description of the transmission at low speed, and advanced understanding of the singularity where links r2 and r3 align. The finite element model provided a method of accurate modeling of the beam spring’s nonlinear contact with the sprag unit backstop, and changes in kinematic link lengths. The Modelica model provided an accurate means for modeling the transmission dynamically, with the many mechanical components involved in the motion, and all three models were validated by experiment with the physical prototype.

8.2 Future Work

While this dissertation has answered some of the questions about the Beale CVT, it has opened many more new questions to be answered by continued work. A detailed study on the classification of mechanisms with the Beale CVT’s characteristic singularity will improve the understanding of possible areas of applicability of the concept. Notably, the transmission can take many forms, with energy storage elements not limited to beam springs, and one-way elements not limited even to the mechanical domain. Any mechanism in which a driving element is in motion while another, linked element experiences a direction change will possess the singularity upon which this mechanism is based; any one-way flow element (e.g.: sprag clutches, check valves, diodes, etc.) coupled with an energy storage component (e.g.: springs, pressure vessels, capacitors, etc.) can be employed to achieve the continuously and infinitely variable nature of the mechanism developed in this work. As a result, it is possible to create a variable speed pump, for example, using a slider-crank mechanism with a flexible member, a piston and a one-way valve, as shown in Figure 90. In such a case, the flow produced by each stroke 139 of the piston would vary with the pressure against which it would be working, resulting in a direct, efficient conversion of rotational mechanical power to continuously variable hydraulic power. Similarly, the piston and check valve could be directly replaced with a linear alternator and diode, resulting in a direct transduction of rotational power to electrical power, again with continuous variability. The Modelica approach to modeling makes crossing these domains, whether from rotational to linear, or to hydraulic/pneumatic through a check valve or even to electrical through a linear alternator and diode, quite easy. Designs taking advantage of this property of the mechanism may provide elegant solutions for otherwise overlooked problems.

Figure 90: Alternative form of transmission as a direct hydraulic pump

The property of the transmission allowing idling at full input speed will also be important for future study, especially for integration with transient drive systems for 140 vehicles, as well as hybrid and differential drives. Specifically, this phenomenon demands an interest in material damping and fatigue characteristics for different spring elements, whether based on beams, coils, gas pressure, or even electromagnetic arrangements. A good understanding of hysteresis in the energy storage element of the mechanism is important for an understanding of not only efficiency of the mechanism in this and other operational modes, but also damage and effective life of individual components.

Further improvement and sophistication of the Modelica model to handle spring nonlinearities and kinematic relationships should yield even more consistently-accurate results over an expanded range of parameters. Development of acausal components to describe the kinematics of the mechanism would allow connection of power sources directly to the model, resulting in a better understanding of the reaction of the prime mover to the load through the transmission. Design revisions to the prototype platform to fundamentally change the method used for measurement of this variable should serve to further confirm modeled behavior, and better represent mechanical phenomena.

Further explorations of interactions between multiple sprag clutches are also important; any design of the transmission for real-world implementation will include multiple sprag units to provide smooth power output, and thus their interactions should be well understood. The component model allows easy adaptation of single models to multiple interacting models, provided that numerical problems can continue to be avoided. While forward dynamics for closed kinematic chains can be difficult to solve, multibody packages are becoming available in acausal form for Modelica-based simulation environments, and pose important possibilities for future modeling of different 141 incarnations of the Beale CVT, especially with prospects for integration with the models of the rest of the components.

Integration of the transmission with real-world systems such as pumps, fans and motors for further testing is also an important step to bringing this technology to fruition.

Experimental work with electric motor drives and internal combustion engines can demonstrate the efficacy of the mechanism, and generate additional interest in the design.

The ramifications of this mechanism’s characteristics are far-reaching, and using it in real-world applications will be an important step to its adoption by industry.

142

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APPENDIX: SIMULATION CODE

This Appendix contains the various scripts, programs and codes used to perform the analyses detailed in this dissertation. Most FreeMat codes were used for post- processing data, while Modelica codes were used for simulation.

FreeMat Four-bar Mechanism Model

% fourbar.m % author: TJ Cyders % originated: 4/27/2011 % last update: 9/8/2011 % purpose: uses Newton-Raphson to solve four-bar equations r1 = 229; r2 = 32; r3 = 178; r4 = 178; % Set link lengths dth = 2*pi/360; % resolution of input angle (interval width for theta2) th2 = [0:dth:2*pi]; % initialize theta 2 as one full rotation theta3 = zeros(size(th2)); % initialize theta 3 and theta 4 theta4 = zeros(size(th2)); n = length(th2); % number of iterations for loop for JIT compiler th3 = 0.8; th4 = 1.5; % initial guesses at unknowns for i = 1:n

deltath = [10;10]; % while loop starter

while abs(max(deltath)) > 1e-5 % epsilon

%% --- Loop Closure Equations --- %% f1 = r1 + r4*cos(th4) - r2*cos(th2(i)) - r3*cos(th3); f2 = r4*sin(th4) - r2*sin(th2(i)) - r3*sin(th3);

%% --- Loop Closure Derivatives --- %% df1th3 = r3*sin(th3); df2th3 = -r3*cos(th3); df1th4 = -r4*sin(th4); df2th4 = r4*cos(th4);

%% --- Newton Raphson Setup --- %% F = [f1; f2]; J = [df1th3 df1th4;df2th3 df2th4];

deltath = J\F; % Gaussian Elimination step th3 = th3 - deltath(1); % update values for theta 3 and theta 4 th4 = th4 - deltath(2); end

150

%% --- Update Stored Values --- %% theta3(i) = th3; theta4(i) = th4; end

151

Data Processing Algorithm

% script dbp.m Dissertation Batch Processing % imports tab-delimited data from BCVT test rig setup as detailed % in dissertation, and outputs angular data clear; clc; clf; tic; data = dlmread('testdata.TXT'); % be sure to name output file daqv.TXT t = data(:,1); th2 = dualenci(t, data(:,2),data(:,end))-211.19; % Input shaft angle [deg], with index correction th4 = quadenc2i(t, data(:,6), data(:,7), data(:,8)) +126.966; % Sprag angle [deg], with index correction thout = quadenc2(t, data(:,4), data(:,5)); % Ouput shaft angle [deg] f_link = data(:,3)/1.592; % Link load [N] t_out = data(:,9)/12.9; % Output torque [N-m] th4t = interp1(t,th4,linspace(0,max(t),1000)'); th2t = interp1(t,th2,linspace(0,max(t),1000)'); plot(th2t, th4t); uaxis = axis; hold on toc cyclenum = input('Which *full* cycle would you like to output?\n'); countercounter = 1; indrr = []; for i = 2:length(data(:,end)); if data(i,end) ~= data(i-1,end); indrr(countercounter) = i; countercounter = countercounter + 1; end end plot([360*(cyclenum-1), 360*(cyclenum-1)], [uaxis(3) uaxis(4)], '--k'); plot([360*(cyclenum), 360*(cyclenum)], [uaxis(3) uaxis(4)], '--k'); ind = find(th2 > (cyclenum-1)*360 & th2 < cyclenum*360); data = [t th2 th4 thout f_link t_out]; data = data(ind(1):ind(end),:); datai = interp1(data(:,1),data,linspace(data(1,1),data(end,1),500)); datai(:,1) = datai(:,1)-datai(1,1); % Reset time index to start at zero datai(:,2) = datai(:,2)-datai(1,2); % Reset th2 to start at zero csvwrite('testdata.csv',datai);

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% DUALENCi DUALENCi Dual-Mode Encoder Signal Transduction with Index % % Usage % % This routine uses the following syntax: % % [t, theta] = dualenci(t, A, ind) % % where t is the time vector associated with voltage signal A and counter % signal ind. This function transforms the signals into angle data % correlated with the original time vector. Output angle is in degrees, and % is *not* direction sensitive. % % % NOTE: THIS FUNCTION CURRENTLY REQUIRES zerocross.m AND sign.m, which % can be found in the script library here: % % http://oak.cats.ohiou.edu/~tc285202/scriptlibrary.html % % Copyright (c) 2010 Timothy Cyders % Licensed under the GPL

function ret = dualenci(t, A, ind)

resolution = 1024; % Number of cycles per revolution of the encoder n = length(t); theta = zeros(n,1); % Preallocate theta vector dth = 360/(resolution);

%% --- Find Zero Crossings for Both Signals --- %%% Apzc = find(zerocross(A)); % Finds element number of each zero crossing in the array Anzc = find(zerocrossn(A)); % A negative zero crossings

tApzc = t(Apzc); tAnzc = t(Anzc);

%% --- Assume Positive Direction --- %% index = sort(vertcat(Apzc, Anzc)); tindex = t(index); n2 = length(index); thetar = [1:n2]*dth; theta = interp1(tindex, thetar, t)'; theta(index(end):end) = theta(index(end)-1); theta(1:index(1)) = 0;

ind = ind - ind(1); % Set counter to start at zero indfind = find(ind>0); % Find first element to cross index theta = theta - theta(indfind(1)); % Zero angle to index ret = theta';

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% QUADENC2i QUADENC2i Quadrature Encoder Signal Transduction With Index % % Usage % % This routine uses the following syntax: % % [t, theta] = quadenc2i(t, A, B) % % where t is the time vector associated with voltage signals A and B. % This function transforms the signals into angle data correlated with % the original time vector. Output angle is in degrees, and is direction % sensitive. % % % NOTE: THIS FUNCTION CURRENTLY REQUIRES sign.m, which % can be found in the script library here: % % http://oak.cats.ohiou.edu/~tc285202/scriptlibrary.html % % Copyright (c) 2010 Timothy Cyders % Licensed under the GPL

function ret = quadenc2i(t, A, B, ind)

resolution = 1024; % Number of ticks per revolution of the encoder n = length(t); theta = zeros(n,1); % Preallocate theta vector dth = 360/(resolution*4);

%% --- Find Element Number of First Index Trip --- %% ind = find(ind<-2); % Change to I>2 for positive index pulse

%% --- Convert Signals to Binary --- %% sigA = (sign(A)==1); sigB = (sign(B)==1); bincomb = bin2dec(num2str([sigA sigB]))+1; % Converts binary state of pulse signals to indexed position, 1-4

refarray = [0 -1 1 0; 1 0 0 -1; -1 0 0 1; 0 1 -1 0]; % Additive reference array, (oldval, newval) = dir

for i = 2:n theta(i) = theta(i-1) + refarray(bincomb(i-1),bincomb(i))*dth; end

theta = theta - theta(ind(1)); % Index angle to zero at index trip

ret = theta;

154 function ret = numdiff(x,y,numpoints,order)

for i = (1:length(x)-numpoints) xtemp = x(i:i+numpoints); % Get x vector for polynomial fit subdomain ytemp = y(i:i+numpoints); % Get y vector for polynomial fit subdomain ptemp = polyfit(xtemp,ytemp,order); % Fit polynomial to subdomain xtemp, ytemp dptemp = polyder(ptemp); % Take derivative of polynomial approximation yout(i) = polyval(dptemp, mean(xtemp)); % Output value of derivative at middle of subdomain xout(i) = mean(xtemp); % Output x value at middle of subdomain end ret = [xout' yout']; end

Modelica Sprag Unit Model

model BCVTdiss3 Modelica.Mechanics.Rotational.Components.Inertia SpragUnit(J=0.004371 ); Modelica.Mechanics.Rotational.Components.Spring BeamSpring(c=24.07); Modelica.Mechanics.Rotational.Components.Inertia EncoderShaft(J=0.000 0058467); Modelica.Mechanics.Rotational.Components.ElastoBacklash GearBacklash( c=500, d=50, b=0.00087266462599716); Modelica.Mechanics.Rotational.Components.Inertia OutputShaft(J=0.0002 2129); Modelica.Mechanics.Rotational.Components.Brake brake(mue_pos=[0,0.5]) ; Modelica.Mechanics.Rotational.Components.OneWayClutch SpragClutch; Modelica.Mechanics.Rotational.Components.Inertia BrakeWithHub(J=0.001 3916); Modelica.Blocks.Sources.Constant SpragFriction(k=0); Modelica.Blocks.Sources.Constant BrakeNormalForce(k=5); Modelica.Mechanics.Rotational.Sources.Position position; Modelica.Blocks.Sources.TimeTable timeTable(table=[]); Modelica.Mechanics.Rotational.Components.BearingFriction bearingFrict ion(tau_pos=[ 0,0.09; 8,0.1787; 10,0.1787]); Modelica.Mechanics.Rotational.Components.OneWayClutch oneWayClutch; Modelica.Mechanics.Rotational.Components.Fixed fixed; Modelica.Blocks.Sources.Constant SpragFriction1( k=0); equation connect(BeamSpring.flange_b, SpragUnit.flange_a); connect(SpragUnit.flange_b, SpragClutch.flange_a); 155

connect(SpragClutch.flange_b, OutputShaft.flange_a); connect(OutputShaft.flange_b, BrakeWithHub.flange_a); connect(GearBacklash.flange_b, SpragUnit.flange_b); connect(SpragFriction.y, SpragClutch.f_normalized); connect(EncoderShaft.flange_b, GearBacklash.flange_a); connect(position.flange, BeamSpring.flange_a); connect(timeTable.y, position.phi_ref); connect(BrakeWithHub.flange_b, bearingFriction.flange_a); connect(bearingFriction.flange_b, brake.flange_a); connect(BrakeNormalForce.y, brake.f_normalized); connect(fixed.flange, oneWayClutch.flange_a); connect(SpragFriction1.y, oneWayClutch.f_normalized); connect(oneWayClutch.flange_b, OutputShaft.flange_a); end BCVTdiss3; partial package Modelica.Icons.Package "Icon for standard packages" end Package; model Modelica.Mechanics.Rotational.Components.Inertia "1D-rotational component with inertia" import SI = Modelica.SIunits; Rotational.Interfaces.Flange_a flange_a "Left flange of shaft"; Rotational.Interfaces.Flange_b flange_b "Right flange of shaft"; parameter SI.Inertia J(min=0, start=1) "Moment of inertia"; parameter StateSelect stateSelect=StateSelect.default "Priority to use phi and w as states"; SI.Angle phi(stateSelect=stateSelect) "Absolute rotation angle of com ponent"; SI.AngularVelocity w(stateSelect=stateSelect) "Absolute angular velocity of component (= der(phi))"; SI.AngularAcceleration a "Absolute angular acceleration of component (= der(w))"; equation phi = flange_a.phi; phi = flange_b.phi; w = der(phi); a = der(w); J*a = flange_a.tau + flange_b.tau; end Inertia; partial package Modelica.Icons.InterfacesPackage "Icon for packages containing interfaces" //extends Modelica.Icons.Package; end InterfacesPackage; connector Modelica.Mechanics.Rotational.Interfaces.Flange_a "1-dim. rotational flange of a shaft (filled square icon)" SI.Angle phi "Absolute rotation angle of flange"; flow SI.Torque tau "Cut torque in the flange"; end Flange_a; type Modelica.SIunits.Angle = Real ( final quantity="Angle", 156

final unit="rad", displayUnit="deg"); type Modelica.SIunits.Torque = Real (final quantity="Torque", final unit="N.m"); connector Modelica.Mechanics.Rotational.Interfaces.Flange_b "1-dim. rotational flange of a shaft (non-filled square icon)" SI.Angle phi "Absolute rotation angle of flange"; flow SI.Torque tau "Cut torque in the flange"; end Flange_b; type Modelica.SIunits.Inertia = MomentOfInertia; type Modelica.SIunits.MomentOfInertia = Real (final quantity="MomentOfInertia", final un it= "kg.m2"); type Modelica.SIunits.AngularVelocity = Real ( final quantity="AngularVelocity", final unit="rad/s"); type Modelica.SIunits.AngularAcceleration = Real (final quantity="AngularAcceleration", final unit= "rad/s2"); model Modelica.Mechanics.Rotational.Components.Spring "Linear 1D rotational spring" extends Modelica.Mechanics.Rotational.Interfaces.PartialCompliant; parameter SI.RotationalSpringConstant c(final min=0, start=1.0e5) "Spring constant"; parameter SI.Angle phi_rel0=0 "Unstretched spring angle"; equation tau = c*(phi_rel - phi_rel0); end Spring; type Modelica.SIunits.RotationalSpringConstant = Real(final quantity="RotationalSpringCons tant", final unit="N.m/rad"); partial model Modelica.Mechanics.Rotational.Interfaces.PartialCompliant

"Partial model for the compliant connection of two rotational 1- dim. shaft flanges" Modelica.SIunits.Angle phi_rel(start=0) "Relative rotation angle (= flange_b.phi - flange_a.phi)"; Modelica.SIunits.Torque tau "Torque between flanges (= flange_b.tau)" ; Flange_a flange_a "Left flange of compliant 1- dim. rotational component"; 157

Flange_b flange_b "Right flange of compliant 1- dim. rotational component"; equation phi_rel = flange_b.phi - flange_a.phi; flange_b.tau = tau; flange_a.tau = -tau; end PartialCompliant; model Modelica.Mechanics.Rotational.Components.ElastoBacklash "Backlash connected in series to linear spring and damper (backlash i s modeled with elasticity)" import SI = Modelica.SIunits; parameter SI.RotationalSpringConstant c(final min=Modelica.Constants. small, start=1.0e5) "Spring constant (c > 0 required)"; parameter SI.RotationalDampingConstant d(final min=0, start = 0) "Damping constant"; parameter SI.Angle b(final min=0) = 0 "Total backlash"; parameter SI.Angle phi_rel0=0 "Unstretched spring angle"; extends Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelati veStates; extends Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryCondition alHeatPortWithoutT; protected final parameter SI.Angle bMax = b/2 "Backlash in range bMin <= phi_rel - phi_rel0 <= bMax"; final parameter SI.Angle bMin = -bMax "Backlash in range bMin <= phi_rel - phi_rel0 <= bMax"; SI.Torque tau_c; SI.Torque tau_d; SI.Angle phi_diff = phi_rel - phi_rel0; // A minimum backlash is defined in order to avoid an infinite // number of state events if backlash b is set to zero. constant SI.Angle bEps = 1e-10 "Minimum backlash"; equation if initial() then /* During initialization the characteristic is modified, in order that it is a strict monoton rising function. Otherwise, initializati on might result in a singular system when the characteristic has to be inverted. The characteristic is modified in the range 1.5*bMin <= phi_rel - phi_rel0 <= 1.5 bMax, so that in this range a linear characteristic is present that a pproaches the original function continuously at its limits, e.g., original: tau(1.5*bMax) = c*(phi_diff - bMax) = c*(0.5*bMax) initial : tau(1.5*bMax) = (c/3)*phi_diff = (c/3)*(3/2)*bMax = (c/2)*bMax */ 158

tau_c = if phi_diff > 1.5*bMax then c*(phi_diff - bMax) else if phi_diff < 1.5*bMin then c*(phi_diff - bMin) else (c/3)*phi_diff; tau_d = d*w_rel; tau = tau_c + tau_d; lossPower = tau_d*w_rel; else /* if abs(b) <= bEps then tau_c = c*phi_diff; tau_d = d*w_rel; tau = tau_c + tau_d; elseif phi_diff > bMax then tau_c = c*(phi_diff - bMax); tau_d = d*w_rel; tau = smooth(0, noEvent(if tau_c + tau_d <= 0 then 0 else tau _c + min(tau_c,tau_d))); elseif phi_diff < bMin then tau_c = c*(phi_diff - bMin); tau_d = d*w_rel; tau = smooth(0, noEvent(if tau_c + tau_d >= 0 then 0 else tau _c + max(tau_c,tau_d))); else tau_c = 0; tau_d = 0; tau = 0; end if; This is written in the form below, in order that parameter "b" is not evaluated during translation (i.e., in the above form it cannot be changed anymore after translation). */ tau_c = if abs(b) <= bEps then c*phi_diff else if phi_diff > bMax then c*(phi_diff - bMax) else if phi_diff < bMin then c*(phi_diff - bMin) else 0; tau_d = d*w_rel; tau = if abs(b) <= bEps then tau_c + tau_d else if phi_diff > bMax then smooth(0, noEvent(if tau_c + tau_d <= 0 then 0 else t au_c + min(tau_c,tau_d))) else if phi_diff < bMin then smooth(0, noEvent(if tau_c + tau_d >= 0 then 0 else t au_c + max(tau_c,tau_d))) else 0; lossPower = if abs(b) <= bEps then tau_d*w_rel else if phi_diff > bMax then smooth(0, noEvent(if tau_c + tau_d <= 0 then 0 e lse min(tau_c,tau_d)*w_rel)) else if phi_diff < bMin then smooth(0, noEvent(if tau_c + tau_d >= 0 then 0 e lse max(tau_c,tau_d)*w_rel)) else 0; end if; end ElastoBacklash; type Modelica.SIunits.RotationalDampingConstant = 159

Real(final quantity="RotationalDampingCo nstant", final unit="N.m.s/rad"); partial model Modelica.Mechanics.Rotational.Interfaces.PartialCompliant WithRelativeStates "Partial model for the compliant connection of two rotational 1- dim. shaft flanges where the relative angle and speed are used as prefe rred states" Modelica.SIunits.Angle phi_rel(start=0, stateSelect=stateSelect, nomi nal=phi_nominal) "Relative rotation angle (= flange_b.phi - flange_a.phi)"; Modelica.SIunits.AngularVelocity w_rel(start=0, stateSelect=stateSele ct) "Relative angular velocity (= der(phi_rel))"; Modelica.SIunits.AngularAcceleration a_rel(start=0) "Relative angular acceleration (= der(w_rel))"; Modelica.SIunits.Torque tau "Torque between flanges (= flange_b.tau)" ; Flange_a flange_a "Left flange of compliant 1- dim. rotational component"; Flange_b flange_b "Right flange of compliant 1- dim. rotational component"; parameter SI.Angle phi_nominal(displayUnit="rad")=1e-4 "Nominal value of phi_rel (used for scaling)"; parameter StateSelect stateSelect=StateSelect.prefer "Priority to use phi_rel and w_rel as states"; equation phi_rel = flange_b.phi - flange_a.phi; w_rel = der(phi_rel); a_rel = der(w_rel); flange_b.tau = tau; flange_a.tau = -tau; end PartialCompliantWithRelativeStates; partial model Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditional HeatPortWithoutT "Partial model to include a conditional HeatPort in order to dissipat e losses, used for textual modeling, i.e., for elementary models" parameter Boolean useHeatPort = false "=true, if heatPort is enabled" ; Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a heatPort( final Q_flow=-lossPower) if useHeatPort "Optional port to which dissipated losses are transported in form o f heat"; Modelica.SIunits.Power lossPower "Loss power leaving component via heatPort (> 0, if heat is flowing out of component)"; end PartialElementaryConditionalHeatPortWithoutT; connector Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a "Thermal port for 1-dim. heat transfer (filled rectangular icon)" extends HeatPort; end HeatPort_a; 160

partial connector Modelica.Thermal.HeatTransfer.Interfaces.HeatPort "Thermal port for 1-dim. heat transfer" Modelica.SIunits.Temperature T "Port temperature"; flow Modelica.SIunits.HeatFlowRate Q_flow "Heat flow rate (positive if flowing from outside into the componen t)"; end HeatPort; type Modelica.SIunits.Temperature = ThermodynamicTemperature; type Modelica.SIunits.ThermodynamicTemperature = Real ( final quantity="ThermodynamicTemperature", final unit="K", min = 0, start = 288.15, displayUnit="degC") "Absolute temperature (use type TemperatureDifference for relative te mperatures)"; type Modelica.SIunits.HeatFlowRate = Real (final quantity="Power", final unit="W"); type Modelica.SIunits.Power = Real (final quantity="Power", final unit="W"); model Modelica.Mechanics.Rotational.Components.Brake "Brake based on Coulomb friction " extends Modelica.Mechanics.Rotational.Interfaces.PartialElementaryTwoFlange sAndSupport2; parameter Real mue_pos[:, 2]=[0, 0.5] "[w,mue] positive sliding friction coefficient (w_rel>=0)"; parameter Real peak(final min=1) = 1 "peak*mue_pos[1,2] = maximum value of mue for w_rel==0"; parameter Real cgeo(final min=0) = 1 "Geometry constant containing friction distribution assumption"; parameter SI.Force fn_max(final min=0, start=1) "Maximum normal force "; extends Rotational.Interfaces.PartialFriction; extends Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryCondition alHeatPortWithoutT; SI.Angle phi "Angle between shaft flanges (flange_a, flange_b) and su pport"; SI.Torque tau "Brake friction torqu"; SI.AngularVelocity w "Absolute angular velocity of flange_a and flang e_b"; SI.AngularAcceleration a "Absolute angular acceleration of flange_a and flange_b"; Real mue0 "Friction coefficient for w=0 and forward sliding"; SI.Force fn "Normal force (=fn_max*f_normalized)"; 161

// Constant auxiliary variable Modelica.Blocks.Interfaces.RealInput f_normalized "Normalized force signal 0..1 (normal force = fn_max*f_normalized; brake is active if > 0)"; equation mue0 = Modelica.Math.tempInterpol1(0, mue_pos, 2); phi = flange_a.phi - phi_support; flange_b.phi = flange_a.phi; // Angular velocity and angular acceleration of flanges flange_a and flange_b w = der(phi); a = der(w); w_relfric = w; a_relfric = a; // Friction torque, normal force and friction torque for w_rel=0 flange_a.tau + flange_b.tau - tau = 0; fn = fn_max*f_normalized; tau0 = mue0*cgeo*fn; tau0_max = peak*tau0; free = fn <= 0; // Friction torque tau = if locked then sa*unitTorque else if free then 0 else cgeo*fn*(if startForward then Modelica.Math.tempInterpo l1( w, mue_pos, 2) else if startBackward then - Modelica.Math.tempInterpol1(-w, mue_pos, 2) else if pre(mode) == Forward then Modelica.Math.tempInterpo l1( w, mue_pos, 2) else - Modelica.Math.tempInterpol1(-w, mue_pos, 2)); lossPower = tau*w_relfric; end Brake; type Modelica.SIunits.Force = Real (final quantity="Force", final unit="N"); connector Modelica.Blocks.Interfaces.RealInput = input Real "'input Real' as connector"; function Modelica.Math.tempInterpol1 "Temporary function for linear interpolation (will be removed)" input Real u "input value (first column of table)"; input Real table[:, :] "table to be interpolated"; input Integer icol "column of table to be interpolated"; output Real y "interpolated input value (icol column of table)"; protected Integer i; Integer n "number of rows of table"; Real u1; Real u2; Real y1; Real y2; 162 algorithm n := size(table, 1); if n <= 1 then y := table[1, icol]; else // Search interval if u <= table[1, 1] then i := 1; else i := 2; // Supports duplicate table[i, 1] values // in the interior to allow discontinuities. // Interior means that // if table[i, 1] = table[i+1, 1] we require i>1 and i+1= table[i, 1] loop i := i + 1; end while; i := i - 1; end if; // Get interpolation data u1 := table[i, 1]; u2 := table[i + 1, 1]; y1 := table[i, icol]; y2 := table[i + 1, icol]; assert(u2 > u1, "Table index must be increasing"); // Interpolate y := y1 + (y2 - y1)*(u - u1)/(u2 - u1); end if; end tempInterpol1; partial model Modelica.Mechanics.Rotational.Interfaces.PartialElementaryTwoFlangesA ndSupport2 "Partial model for a component with two rotational 1- dim. shaft flanges and a support used for textual modeling, i.e., for e lementary models" parameter Boolean useSupport=false "= true, if support flange enabled, otherwise implicitly grounded"; Flange_a flange_a "Flange of left shaft"; Flange_b flange_b "Flange of right shaft"; Support support(phi = phi_support, tau=-flange_a.tau- flange_b.tau) if useSupport "Support/housing of component"; protected Modelica.SIunits.Angle phi_support "Absolute angle of support flange" ; equation if not useSupport then phi_support = 0; end if; end PartialElementaryTwoFlangesAndSupport2; connector Modelica.Mechanics.Rotational.Interfaces.Support "Support/housing of a 1-dim. rotational shaft" 163

SI.Angle phi "Absolute rotation angle of the support/housing"; flow SI.Torque tau "Reaction torque in the support/housing"; end Support; partial model Modelica.Mechanics.Rotational.Interfaces.PartialFriction "Partial model of Coulomb friction elements" // parameter SI.AngularVelocity w_small=1 "Relative angular velocity near to zero (see model info text)"; parameter SI.AngularVelocity w_small=1.0e10 "Relative angular velocity near to zero if jumps due to a reinit(.. ) of the velocity can occur (set to low value only if such impulses can occur)"; // Equations to define the following variables have to be defined in su bclasses SI.AngularVelocity w_relfric "Relative angular velocity between frictional surfaces"; SI.AngularAcceleration a_relfric "Relative angular acceleration between frictional surfaces"; //SI.Torque tau "Friction torque (positive, if directed in opposite dir ection of w_rel)"; SI.Torque tau0 "Friction torque for w=0 and forward sliding"; SI.Torque tau0_max "Maximum friction torque for w=0 and locked"; Boolean free "true, if frictional element is not active"; // Equations to define the following variables are given in this class Real sa(final unit="1") "Path parameter of friction characteristic tau = f(a_relfric)"; Boolean startForward(start=false, fixed=true) "true, if w_rel=0 and start of forward sliding"; Boolean startBackward(start=false, fixed=true) "true, if w_rel=0 and start of backward sliding"; Boolean locked(start=false) "true, if w_rel=0 and not sliding"; constant Integer Unknown=3 "Value of mode is not known"; constant Integer Free=2 "Element is not active"; constant Integer Forward=1 "w_rel > 0 (forward sliding)"; constant Integer Stuck=0 "w_rel = 0 (forward sliding, locked or backward sliding)"; constant Integer Backward=-1 "w_rel < 0 (backward sliding)"; Integer mode( final min=Backward, final max=Unknown, start=Unknown, fixed=true); protected constant SI.AngularAcceleration unitAngularAcceleration = 1; constant SI.Torque unitTorque = 1; equation /* Friction characteristic locked is introduced to help the Modelica translator determining the different structural configurations, if for each configuration special code shall be generated) */ startForward = pre(mode) == Stuck and (sa > tau0_max/unitTorque or pr e(startForward) and sa > tau0/unitTorque) or pre(mode) == Backward and w_relfric > w_small or 164

initial() and (w_relfric > 0); startBackward = pre(mode) == Stuck and (sa < - tau0_max/unitTorque or pre( startBackward) and sa < - tau0/unitTorque) or pre(mode) == Forward and w_relfric < -w_small or initial() and (w_relfric < 0); locked = not free and not (pre(mode) == Forward or startForward or pr e( mode) == Backward or startBackward); a_relfric/unitAngularAcceleration = if locked then 0 el se if free then sa e lse if startForward then sa - tau0_max/unitTorque else if startBackward then sa + tau0_max/unitTorque else if pre(mode) == Forward then sa - tau0_max/unitTorque else sa + tau0_max/unitTorque; /* Friction torque has to be defined in a subclass. Example for a clutc h: tau = if locked then sa else if free then 0 else cgeo*fn*(if startForward then Math.tempInterpol1( w_r elfric, mue_pos, 2) else if startBackward then -Math.tempInterpol1(- w_relfric, mue_pos, 2) else if pre(mode) == Forward then Math.tempInterpol1( w_r elfric, mue_pos, 2) else -Math.tempInterpol1(- w_relfric, mue_pos, 2)); */ // finite state machine to determine configuration mode = if free then Free else (if (pre(mode) == Forward or pre(mode) == Free or startForward) a nd w_relfric > 0 then Forward else if (pre(mode) == Backward or pre(mode) == Free or startBackward) a nd w_relfric < 0 then Backward else Stuck); end PartialFriction; model Modelica.Mechanics.Rotational.Components.OneWayClutch "Series connection of freewheel and clutch" extends Modelica.Mechanics.Rotational.Icons.Clutch; extends Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelati veStates; parameter Real mue_pos[:, 2]=[0, 0.5] "[w,mue] positive sliding friction coefficient (w_rel>=0)"; parameter Real peak(final min=1) = 1 165

"peak*mue_pos[1,2] = maximum value of mue for w_rel==0"; parameter Real cgeo(final min=0) = 1 "Geometry constant containing friction distribution assumption"; parameter SI.Force fn_max(final min=0, start=1) "Maximum normal force "; parameter SI.AngularVelocity w_small=1e10 "Relative angular velocity near to zero if jumps due to a reinit(.. ) of the velocity can occur (set to low value only if such impulses can occur)"; extends Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryCondition alHeatPortWithoutT; Real u "Normalized force input signal (0..1)"; SI.Force fn "Normal force (fn=fn_max*inPort.signal)"; Boolean startForward(start=false) "true, if w_rel=0 and start of forward sliding or w_rel > w_small"; Boolean locked(start=false) "true, if w_rel=0 and not sliding"; Boolean stuck(start=false) "w_rel=0 (locked or start forward sliding) "; protected SI.Torque tau0 "Friction torque for w=0 and sliding"; SI.Torque tau0_max "Maximum friction torque for w=0 and locked"; Real mue0 "Friction coefficient for w=0 and sliding"; Boolean free "true, if frictional element is not active"; Real sa(final unit="1") "Path parameter of tau = f(a_rel) Friction characteristic"; constant Real eps0=1.0e-4 "Relative hysteresis epsilon"; SI.Torque tau0_max_low "Lowest value for tau0_max"; parameter Real peak2=max([peak, 1 + eps0]); constant SI.AngularAcceleration unitAngularAcceleration = 1; constant SI.Torque unitTorque = 1; public Modelica.Blocks.Interfaces.RealInput f_normalized "Normalized force signal 0..1 (normal force = fn_max*f_normalized; clutch is engaged if > 0)"; equation // Constant auxiliary variable mue0 = Modelica.Math.tempInterpol1(0, mue_pos, 2); tau0_max_low = eps0*mue0*cgeo*fn_max; // Normal force and friction torque for w_rel=0 u = f_normalized; free = u <= 0; fn = if free then 0 else fn_max*u; tau0 = mue0*cgeo*fn; tau0_max = if free then tau0_max_low else peak2*tau0; /* Friction characteristic (locked is introduced to help the Modelica translator determinin g the different structural configurations, if for each configurati on special code shall be generated) */ 166

startForward = pre(stuck) and (sa > tau0_max/unitTorque or pre(startF orward) and sa > tau0/unitTorque or w_rel > w_small) or initial() and (w_rel > 0); locked = pre(stuck) and not startForward; // acceleration and friction torque a_rel = unitAngularAcceleration* (if locked then 0 else sa - tau0/unitTorque); tau = if locked then sa*unitTorque else (if free then 0 else cgeo*fn* Modelica.Math.tempInterpol1(w_rel, mue_pos, 2)); // Determine configuration stuck = locked or w_rel <= 0; lossPower = if stuck then 0 else tau*w_rel; end OneWayClutch; model Modelica.Mechanics.Rotational.Icons.Clutch "Icon of a clutch" end Clutch; partial package Modelica.Icons.SourcesPackage "Icon for packages containing sources" //extends Modelica.Icons.Package; end SourcesPackage; block Modelica.Blocks.Sources.Constant "Generate constant signal of type Real" parameter Real k(start=1) "Constant output value"; extends Interfaces.SO; equation y = k; end Constant; partial block Modelica.Blocks.Interfaces.SO "Single Output continuous control block" extends BlockIcon; RealOutput y "Connector of Real output signal"; end SO; connector Modelica.Blocks.Interfaces.RealOutput = output Real "'output Real' as connector"; partial block Modelica.Blocks.Interfaces.BlockIcon "Basic graphical layout of input/output block" end BlockIcon; model Modelica.Mechanics.Rotational.Sources.Position "Forced movement of a flange according to a reference angle signal" import SI = Modelica.SIunits; extends Modelica.Mechanics.Rotational.Interfaces.PartialElementaryOneFlange AndSupport2; parameter Boolean exact=false "true/false exact treatment/filtering the input signal"; parameter SI.Frequency f_crit=50 "if exact=false, critical frequency of filter to filter input signa l"; 167

SI.Angle phi(stateSelect=if exact then StateSelect.default else State Select.prefer) "Rotation angle of flange with respect to support"; SI.AngularVelocity w(start=0,stateSelect=if exact then StateSelect.de fault else StateSelect.prefer) "If exact=false, Angular velocity of flange with respect to support else dummy"; SI.AngularAcceleration a(start=0) "If exact=false, Angular acceleration of flange with respect to sup port else dummy"; Modelica.Blocks.Interfaces.RealInput phi_ref(final quantity="Angle", final unit="rad", displayUnit="deg") "Reference angle of flange with respect to support as input signal" ; protected parameter Modelica.SIunits.AngularFrequency w_crit=2*Modelica.Constan ts.pi*f_crit "Critical frequency"; constant Real af=1.3617 "s coefficient of Bessel filter"; constant Real bf=0.6180 "s*s coefficient of Bessel filter"; initial equation if not exact then phi = phi_ref; end if; equation phi = flange.phi - phi_support; if exact then phi = phi_ref; w = 0; a = 0; else // Filter: a = phi_ref*s^2/(1 + (af/w_crit)*s + (bf/w_crit^2)*s^2) w = der(phi); a = der(w); a = ((phi_ref - phi)*w_crit - af*w)*(w_crit/bf); end if; end Position; type Modelica.SIunits.Frequency = Real (final quantity="Frequency", final unit="Hz"); type Modelica.SIunits.AngularFrequency = Real (final quantity="AngularFrequency", final unit= "rad/s"); partial model Modelica.Mechanics.Rotational.Interfaces.PartialElementaryOneFlangeAn dSupport2 "Partial model for a component with one rotational 1- dim. shaft flange and a support used for textual modeling, i.e., for el ementary models" parameter Boolean useSupport=false "= true, if support flange enabled, otherwise implicitly grounded"; 168

Flange_b flange "Flange of shaft"; Support support(phi = phi_support, tau = -flange.tau) if useSupport "Support/housing of component"; protected Modelica.SIunits.Angle phi_support "Absolute angle of support flange" ; equation if not useSupport then phi_support = 0; end if; end PartialElementaryOneFlangeAndSupport2; block Modelica.Blocks.Sources.TimeTable "Generate a (possibly discontinuous) signal by linear interpolation i n a table" parameter Real table[:, 2] "Table matrix (time = first column; e.g., table=[0, 0; 1, 1; 2, 4]) "; parameter Real offset=0 "Offset of output signal"; parameter SIunits.Time startTime=0 "Output = offset for time < startT ime"; extends Interfaces.SO; protected Real a "Interpolation coefficients a of actual interval (y=a*x+b)"; Real b "Interpolation coefficients b of actual interval (y=a*x+b)"; Integer last(start=1) "Last used lower grid index"; SIunits.Time nextEvent(start=0, fixed=true) "Next event instant"; function getInterpolationCoefficients "Determine interpolation coefficients and next time event" input Real table[:, 2] "Table for interpolation"; input Real offset "y-offset"; input Real startTime "time-offset"; input Real t "Actual time instant"; input Integer last "Last used lower grid index"; input Real TimeEps "Relative epsilon to check for identical time in stants"; output Real a "Interpolation coefficients a (y=a*x + b)"; output Real b "Interpolation coefficients b (y=a*x + b)"; output Real nextEvent "Next event instant"; output Integer next "New lower grid index"; protected Integer columns=2 "Column to be interpolated"; Integer ncol=2 "Number of columns to be interpolated"; Integer nrow=size(table, 1) "Number of table rows"; Integer next0; Real tp; Real dt; algorithm next := last; nextEvent := t - TimeEps*abs(t); // in case there are no more time events tp := t + TimeEps*abs(t) - startTime; if tp < 0.0 then // First event not yet reached 169

nextEvent := startTime; a := 0; b := offset; elseif nrow < 2 then // Special action if table has only one row a := 0; b := offset + table[1, columns]; else // Find next time event instant. Note, that two consecutive tim e instants // in the table may be identical due to a discontinuous point. while next < nrow and tp >= table[next, 1] loop next := next + 1; end while; // Define next time event, if last table entry not reached if next < nrow then nextEvent := startTime + table[next, 1]; end if; // Determine interpolation coefficients next0 := next - 1; dt := table[next, 1] - table[next0, 1]; if dt <= TimeEps*abs(table[next, 1]) then // Interpolation interval is not big enough, use "next" value a := 0; b := offset + table[next, columns]; else a := (table[next, columns] - table[next0, columns])/dt; b := offset + table[next0, columns] - a*table[next0, 1]; end if; end if; // Take into account startTime "a*(time - startTime) + b" b := b - a*startTime; end getInterpolationCoefficients; algorithm when {time >= pre(nextEvent),initial()} then (a,b,nextEvent,last) := getInterpolationCoefficients(table, offset, startTime, time, last, 100*Modelica.Constants.eps); end when; equation y = a*time + b; end TimeTable; type Modelica.SIunits.Time = Real (final quantity="Time", final unit="s"); model Modelica.Mechanics.Rotational.Components.BearingFriction "Coulomb friction in bearings " extends Modelica.Mechanics.Rotational.Interfaces.PartialElementaryTwoFlange sAndSupport2; parameter Real tau_pos[:, 2]=[0, 1] "[w,tau] Positive sliding friction characteristic (w>=0)"; parameter Real peak(final min=1) = 1 "peak*tau_pos[1,2] = Maximum friction torque for w==0"; 170

extends Rotational.Interfaces.PartialFriction; extends Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryCondition alHeatPortWithoutT; SI.Angle phi "Angle between shaft flanges (flange_a, flange_b) and su pport"; SI.Torque tau "Friction torque"; SI.AngularVelocity w "Absolute angular velocity of flange_a and flang e_b"; SI.AngularAcceleration a "Absolute angular acceleration of flange_a and flange_b"; equation // Constant auxiliary variables tau0 = Modelica.Math.tempInterpol1(0, tau_pos, 2); tau0_max = peak*tau0; free = false; phi = flange_a.phi - phi_support; flange_b.phi = flange_a.phi; // Angular velocity and angular acceleration of flanges w = der(phi); a = der(w); w_relfric = w; a_relfric = a; // Friction torque flange_a.tau + flange_b.tau - tau = 0; // Friction torque tau = if locked then sa*unitTorque else (if startForward then Modelica.Math.tempInterpol1( w, ta u_pos, 2) else if startBackward then -Modelica.Math.tempInterpol1(- w, tau_pos, 2) else if pre(mode) == Forward then Modelica.Math.tempInterpol1( w, ta u_pos, 2) else -Modelica.Math.tempInterpol1(- w, tau_pos, 2)); lossPower = tau*w_relfric; end BearingFriction; model Modelica.Mechanics.Rotational.Components.Fixed "Flange fixed in housing at a given angle" parameter SI.Angle phi0=0 "Fixed offset angle of housing"; Interfaces.Flange_b flange "(right) flange fixed in housing"; equation flange.phi = phi0; end Fixed;

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