Coupled Belt-Pulley Mechanics in Serpentine

COUPLED BELT-PULLEY MECHANICS IN SERPENTINE

BELT DRIVES

DISSERTATION

Presented in Partial Fulfillment of the Requirements of the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Lingyuan Kong, B.S., M.S. *****

The Ohio State University 2003

Dissertation Committee:

Prof. Robert G. Parker, Advisor Approved by

Prof. Stephen E. Bechtel

Prof. Chia-Hsiang Menq ______Advisor Prof. Rajendra Singh Department of Mechanical Engineering

ABSTRACT

Belt vibration and slip are primary concerns in the design of serpentine belt drives.

Belt-pulley coupling is essential for the analysis. This work investigates issues to advance the understanding of belt-pulley mechanics.

Closed-form eigensolution approximations for an axially moving beam with small bending stiffness are given. This model is the first order approximation for the transverse vibration of each span in a serpentine belt drive. Perturbation techniques for algebraic equations and the phase closure principle are used. The eigensolutions are interpreted in terms of propagating waves.

For a complete serpentine belt drive, a hybrid continuous-discrete model is built.

Incorporation of belt bending stiffness introduces linear belt-pulley coupling. This model can explain the transverse span vibrations caused by crankshaft pulley fluctuations at low engine idle speeds where other coupling mechanisms do not. For the steady state analysis, a novel transformation of the governing equations to a standard ODE form for general-purpose BVP solvers leads to numerically exact steady solutions. A closed-form singular perturbation solution is developed for the small bending stiffness case. A coupling indicator based on the steady state is defined to quantify the undesirable belt- pulley coupling. A spatial discretization is developed to find the free vibration ii eigensolutions. In contrast to prior formulations, this discretization is numerically robust and free of missing/false natural frequency concerns. New dynamic properties induced by bending stiffness are characterized. Dynamic response calculations using the discretized model follow naturally. The effects of major design variables are investigated. This provides knowledge to help optimize structural design, especially to reduce large belt transverse vibration.

Finally, to better predict the belt-pulley contact interactions applicable to serpentine belt drives an improved model is established for the steady state mechanics. Bending stiffness is considered while other factors in the literature such as belt-pulley friction and belt inertia are retained. An iterative solution based on general-purpose BVP solvers is presented to determine the belt deflections and the distributions of speed, tension, and friction along the belt as well as the belt-pulley contact points and adhesion/slip zones on the pulleys. Key design criteria like maximum transmissible moment and power efficiency are examined.

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Dedicated to my wife, Hong Chi

ACKNOWLEDGMENTS

I wish to express my sincere thanks to Dr. Robert G. Parker for his assistance and guidance throughout the research project upon which this dissertation is based.

Especially, I am grateful for his help in continuously correcting and modifying my writing, from which I benefited greatly. I also thank Dr. Stephen E. Bechtel, Dr. Chia-

Hsiang Menq, and Dr. Rajendra Singh, who served as members of my dissertation committee. I would like to acknowledge the generous financial support given to this project by Mark IV/Dayco Corporation. Furthermore, I thank my colleagues for their suggestions regarding my dissertation. Finally I thank my wife who constantly supported me throughout this endeavor.

VITA

March, 1972…..……………….Born Guizhou, China

1993……………………………B.S., Shanghai Jiaotong University

1999……………………………M.S., Tsinghua University

1999 – present …………………Graduate Research Assistant, The Ohio State University

PUBLICATIONS

1. L. Kong and R. G. Parker, 2003, “Equilibrium and Belt-Pulley Vibration Coupling in Serpentine Belt Drives,” ASME Journal of Applied Mechanics, 70(5), pp.739-750.

FIELDS OF STUDY

Major Field: Mechanical Engineering

TABLE OF CONTENTS

Abstract ...... ii Dedication ...... iv Acknowledgments...... v Vita...... vi List of Figures ...... ix List of Tables...... xiii Chapters

CHAPTER 1 INTRODUCTION ...... 1 1.1 Motivation and Objectives ...... 1 1.2 Literature Review...... 6 1.2.1 Vibration of axially moving materials ...... 7 1.2.2 Serpentine belt drives...... 8 1.2.3 Belt-pulley steady state contact mechanics...... 12 1.3 Scope of Investigation...... 13 CHAPTER 2 APPROXIMATE EIGENSOLUTIONS OF AXIALLY MOVING BEAMS WITH SMALL BENDING STIFFNESS...... 17 2.1 Introduction ...... 18 2.2 Model Equations ...... 22 2.3 Application of the Phase Closure Principle...... 26 2.4 Other Boundary Conditions ...... 32 CHAPTER 3 MODELING AND STEADY STATE ANALYSIS OF SERPENTINE BELT DRIVES WITH BENDING STIFFNESS...... 36 3.1 Introduction ...... 37 3.2 System Model...... 40

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3.3 Numerical Solution ...... 49 3.4 Numerical Results and Discussion...... 52 3.5 Approximate Closed-Form Solution ...... 64 CHAPTER 4 DYNAMIC ANALYSIS OF SERPENTINE BELT DRIVES WITH BENDING STIFFNESS...... 79 4.1 Introduction ...... 80 4.2 Linearization of Equations of Motion ...... 83 4.3 Extended Operator Formulation...... 88 4.4 Galerkin Discretization ...... 94 4.5 Results and Discussion...... 96 CHAPTER 5 STEADY STATE MECHANICS OF BELT-PULLEY SYSTEMS ...... 113 5.1 Introduction ...... 114 5.2 Nonlinear Equations of a Moving Curved Beam...... 116 5.3 BVP Solver Based Method for Problem with Unknown Boundaries ...... 119 5.4 Steady State Analysis of Belt Pulley Drives: an Iteration Method ...... 122 5.4.1 Regular Moment Transmission Problem ...... 125 5.4.2 Maximum Transmissible Moment Problem...... 131 5.5 Results and Discussion...... 134 5.5.1 Example of regular moment transmission problem ...... 136 5.5.2 Example of maximum transmissible moment problem ...... 140 CHAPTER 6 SUMMARY AND FUTURE WORK...... 145 6.1 Summary ...... 145 6.2 Future Work ...... 150 REFERENCE……………… …………………………………………………………..158

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LIST OF FIGURES

Figure Page

Figure 1.1 Installed serpentine belt drive...... 3 Figure 1.2 Sketch of a seven-pulley serpentine belt drive...... 4 Figure 2.1. Waves in a finite moving beam with small bending stiffness ...... 26 =− =2 −222 − Figure 2.2. Phase angle of D RrrrrA ( 3 231)/( ) from equation (2.21),

=−εω 2 =−−+εω2 22 ω D1 2/1vv, Dvv2 {[/(1)][/(1)]}, =− −2 +εω2222 − − ω + Dvvv3 (1 ) {[ /(1 )] [ /(1 v )] } ...... 28 Figure 2.3. Comparison of fundamental natural frequency for a simply supported beam.

−−−, perturbation; , exact...... 30 Figure 3.1. A prototypical three-pulley serpentine belt drive system...... 38 Figure 3.2. Detail of tensioner region and pulley 2, defining alignment angles...... 43 Figure 3.3. Steady state deflections of spans 1 and 3 for varying belt bending stiffness. = = γ = = = = s 0, ks 4 , 400 , P1 P2 P3 1...... 54 Figure 3.4. Steady state deflections of spans 1 and 3 for varying span tensions. ε = 0.05 , = = γ = s 0, ks 4 , 400 ...... 55 ε = = Figure 3.5. Steady state deflections of spans 1 and 3 for varying speed. 0.05 , ks 4 , γ = = = = 400 , P1 P2 P3 1...... 58 Figure 3.6. Steady state deflections of spans 1 and 3 for varying tensioner spring ε = = γ = = = = stiffness. 0.05 , s 0 , 400 , P1 P2 P3 1...... 59 Figure 3.7. Steady state deflections of spans 1 and 3 for varying longitudinal belt ε ======stiffness. 0.05 , s 0 , ks 4 , P1 P2 P3 1...... 60

Figure 3.8. System coupling indicator Γ for varying belt bending stiffness. s = 0 , = γ = = = = k s 4, 400 , P1 P2 P3 1...... 61 Figure 3.9. System coupling indicator Γ for varying span tensions. ε = 0.015 , s = 0 , = γ = k s 4, 400 ...... 62 Γ ε = = Figure 3.10. System coupling indicator for varying speed. 0.015 , k s 4, γ = = = = η = 400 , P1 P2 P3 1, 0.78 corresponds to tensioner orientation β = o β = o η = 1 135.79 , 2 178.74 in Table 3.2; while 0 corresponds to tensioner β = o β = o orientation: 1 68.53 , 2 111.47 ...... 72 = γ = Figure 3.11. Variation of tension with different tensioner orientation. k s 4, 400 , = = = β = o β = o η = β = o P1 P2 P3 1, a) 1 135.79 , 2 178.74 , 0.78 , b) 1 68.53 , β = o η = 2 111.47 , 0 ...... 73 Figure 3.12. Steady state deflections (dimensionless) of the first and third spans. ε = = = γ = = = = 0.01, s 0.6 , ks 4 , 400 , P1 P2 P3 1...... 74 Figure 3.13. Steady state deflections (dimensionless) of the first and third spans.ε = 0.01, = = γ = = = = s 0.9 , ks 4 , 400 , P1 P2 0.9395 , P3 1.5395...... 75 Figure 4.1. A prototypical three-pulley serpentine belt drive...... 83 Figure 4.2. Rotationally dominant mode (ε = 0 ) for increasing belt bending stiffness. The dimensionless natural frequency for ε = 0 is ω = 4.1205 . a) ε = 0.01, b) ε = 0.04 , ε = ε = = = γ = = = = β = o c) 0.07 , d) 0.1. s 0, ks 4 , 400 , P1 P2 P3 1, 1 135.79 , β = o 2 178.74 ...... 98 Figure 4.3. Span 2 transversely dominant mode (ε = 0 ) for increasing belt bending stiffness. The dimensionless natural frequency for ε = 0 is ω = 3.0951. a) ε = 0.01, ε = ε = ε = = = γ = = = = b) 0.04 , c) 0.07 , d) 0.1. s 0, ks 4 , 400 , P1 P2 P3 1, β = o β = o 1 135.79 , 2 178.74 ...... 99 Figure 4.4. Span 3 transversely dominant mode (ε = 0 ) for increasing belt bending stiffness. The dimensionless natural frequency for ε = 0 is ω =1.9968. a) ε = 0.01,

ε = ε = ε = = = γ = = = = b) 0.04 , c) 0.07 , d) 0.1. s 0, ks 4 , 400 , P1 P2 P3 1, β = o β = o 1 135.79 , 2 178.74 ...... 100 = = Figure 4.5. Natural frequency spectrum for varying belt bending stiffness. s 0, ks 4 , γ = = = = β = o β = o 400 , P1 P2 P3 1, 1 135.79 , 2 178.74 ...... 102 Figure 4.6. Natural frequency spectrum for varying belt bending stiffness. −−−, fix

= = steady state; , fix bending stiffness value in (4.39)-(4.45). s 0 , ks 4 ,

γ = = = = β = o β = o 400 , P1 P2 P3 1, 1 135.79 , 2 178.74 ...... 105

Figure 4.7. Natural frequency spectrum for varying belt speed. , ε = 0.1; −−−,

ε = = γ = = = = β = o β = o 0.01. ks 4 , 400 , P1 P2 P3 1, 1 135.79 , 2 178.74 ...... 106

η = β = o Figure 4.8. Natural frequency spectrum for varying belt speed. , 0 ( 1 68.53 ,

β = o −−− η = β = o β = o ε = = 2 111.47 ); , 0.78 ( 1 135.79 , 2 178.74 ). 0.04 , ks 4 , γ = = = = 400 , P1 P2 P3 1...... 107

Figure 4.9. Natural frequency spectrum for varying tensioner effectiveness η . ,

ε = −−− ε = = = γ = = = = 0.1; , 0.01. s 0 , ks 4 , 400 , P1 P2 P3 1...... 108 Figure 4.10. Fifth vibration mode for varying tensioner effectiveness η . a) η = 0

β = o β = o η = β = o β = o ( 1 68.53 , 2 111.47 ), b) 0.5 ( 1 95.53 , 2 138.47 ), c) η = β = o β = o ε = = = γ = 0.78 ( 1 135.79 , 2 178.74 ). 0.1, s 0, ks 4 , 400 , = = = P1 P2 P3 1...... 112 Figure 5.1 Free body diagram of a moving curved beam...... 117 Figure 5.2. Single span boundary value problem with unknown boundaries...... 120 Figure 5.3. Two-pulley belt drive with inclusion of belt bending stiffness...... 127 Figure 5.4. Steady solutions for the system properties specified in Table 5.1. a) EI = 0.0015, b) EI = 0.015, c) EI = 0.05 N ⋅m2 ...... 135

Figure 5.5. Variations of tension and speed in the tight and slack spans for the belt-pulley drive in Table 5.1...... 138 Figure 5.6. Deflections of the free spans for two different belt-pulley models. (a) and (b) correspond to the current model (symbols denote span endpoints); (c) and (d) correspond to the fixed boundary model in chapter 3 [11]. The system is specified in Table 5.1...... 139 Figure 5.7. Steady solutions for the system properties specified in Table 5.3. Full slip occurs on the driver pulley. a) EI = 0.0015, b) EI = 0.015, c) EI = 0.05 N ⋅m2 . .. 142 = Figure 5.8. Comparison of maximum transmitted moment M maxMTR 2 _ max / ini 2 , power η efficiency , Tt _ midspan/T ini , and Ts _ midspan/T ini between the string and beam models for the belt drive in Table 5.3...... 144 Figure 6.1. Model considering the radial stiffness of the belt...... 150 Figure 6.2. Steady motion of an extensible belt considering bending stiffness...... 154 Figure 6.3. a) Self-sustained system, b) friction force characteristics, c) limit cycle. .... 155 Figure 6.4. Two-dimensional friction between belt and grooved pulley...... 157

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LIST OF TABLES

Table Page

Table 2.1. Comparison of approximate roots of (2.4) from (2.15) with numerically exact roots...... 24 Table 2.2. Reflection coefficients for different end supports...... 33 Table 3.1. List of symbols defining serpentine belt drives...... 42 Table 3.2. Physical properties of the prototypical system...... 53 Table 3.3 Comparison of approximate analytical and numerical solutions. Case 1 parameters are those used in Figure 3.12, and Case 2 parameters are those used in Figure 3.13...... 70 Table 4.1. Physical properties of the example system, from which nominal dimensionless parameters are calculated...... 97 Table 5.1. Physical properties of the belt drive with two identical pulleys...... 136 Table 5.2. Numerical results for the belt drive specified in Table 5.1...... 137 Table 5.3. Physical properties of the belt drive with two different pulleys...... 137

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CHAPTER 1

INTRODUCTION

1.1 Motivation and Objectives

Serpentine belt drives with a long, flat, multi-ribbed belt are widely used in the automobile industry for passenger vehicles and heavy duty trucks (Figure 1.1). With a single belt, the engine power is delivered from the crankshaft to all of the individual accessories (air conditioner, alternator, power steering, water pump, etc.). To maintain proper belt tension as accessory loads and engine speed vary, a spring-loaded tensioner is introduced (Figure 1.2). There are many advantages of the serpentine belt drives over the

V-belt drives including compactness, longer life, automatic tension loss compensation, ease of assembly, and so on [1-3].

Crankshaft torque pulsations, resulting from cylinder combustion in the engine, and dynamic accessory torques excite rotational vibration of the pulleys. These pulley vibrations can then be further transmitted to the belt spans. This belt-pulley coupling has strong influence on the entire system’s dynamic behavior. Under certain circumstances, 1 the coupling is so strong that some belt spans experience large transverse vibrations although no direct transverse forces are applied to these belt spans. Belt-pulley coupling is undesirable because it allows crankshaft excitation, which drives pulley rotation directly, to indirectly drive transverse belt vibrations that cause noise, fatigue failure, and belt slip. A more complete understanding of belt-pulley vibration coupling will allow engineers to design for reduced span vibrations.

To accurately predict the entire system’s dynamic behavior, both the discrete motions of the pulleys and the continuous motions of the belt need to be considered in the model. This results in a coupled continuous-discrete system. Former models concentrate mainly on the pulley rotational motions while treating the belt spans only as springs linking the pulleys [2,4-7]. These discrete models can not predict the system behavior without losing fidelity, especially when coupling between the belt spans and pulleys is strong. Furthermore, they provide no prediction of the troublesome belt span response.

Thus, a comprehensive hybrid model is preferable, in which both the pulley rotational motions and the transverse span motions are incorporated.

Because the transport speed of the belt is high for high engine speeds, this system has the gyroscopic characteristics of an axially moving medium [8-10]. Many industrial applications fall into this class, like band saws, tape drives, paper-handling machinery, textile processing, high-speed printers, and copiers. Due to the mathematical complexities, hybrid continuous-discrete gyroscopic systems have been rarely studied in the literature. Novel mathematical formulations are needed to present the governing equations, and new analytical and numerical methods are needed to solve the problems

2 arising from these kinds of systems (such as finding the steady state mechanics at a specified speed, solving the corresponding eigenproblems, sensitivity analysis, dynamic response, and so on). Thus, developing mathematical methods for hybrid continuous- discrete systems is an objective of this research.

Figure 1.1 Installed serpentine belt drive. 3

J5 Power Steering J4 θ θ 5 4 J J 7 Idler 3 w (x ,t) θ θ i-1 i-1 7 3 Alternator J θ Tensioner t t β 1 ζ w (x ,t) ζ1 2 i i K Water Pump t β θ 2 6

J Air 6 Conditioner Crankshaft θ 2 J θ 1 1 J2

Figure 1.2 Sketch of a seven-pulley serpentine belt drive.

For practicing accessory drive designers in the automotive/heavy vehicle industries, the natural frequencies and dynamic response are of utmost interest. Predicting the system’s dynamic behavior in the design stage is important because changes are difficult to accomplish once prototypes are built. This requires a comprehensive hybrid continuous-discrete, gyroscopic model of serpentine belt drive vibration. Thus, a practically important engineering need creates the opportunity for analytical scientific research.

Reducing belt-pulley coupling is one of the main concerns in the design stage.

Undesirable transverse span vibrations reduce belt life, produce noise, and accelerate belt fatigue. Large belt vibrations can even cause the belt to jump a pulley groove, dramatically increasing belt stress that ultimately snaps the belt and fails the system. The mechanisms causing large transverse vibrations in some belt spans of the serpentine system are not clear. Better understanding of those coupling mechanisms can deepen our knowledge of this system and provide useful information to designers of serpentine systems. Correspondingly, development of a mathematical model that can capture the belt-pulley coupling is one of the main objectives of this research. This dynamic model should explain the large belt transverse vibration, which occurs at engine firing frequency, as observed by our research sponsor. Further analytical investigation of design parameters on this belt-pulley coupling mechanism follow naturally.

Another objective of this research is to advance the investigation of steady state contact mechanics for general belt-pulley drives, where serpentine belt drives are one application. This analysis is somewhat different in spirit from the above investigation of

5 belt-pulley dynamical coupling phenomenon. This analysis focuses closely on the detailed steady state mechanics of the belt, the pulley, and their interactions, such as the distribution of friction and belt tension on the belt-pulley contact zones. For this analytical purpose, fewer modeling assumptions are made, producing a more general and complicated model. For example, belt speed and tension are no longer assumed to be uniform throughout the belt, the belt has slip and stick arcs on the pulleys, and the belt- pulley contact points are not known a priori. Only steady state analysis on multi fixed pulley systems is pursued in this research. This steady state belt-pulley mechanics analysis is useful in practice to better predict the stress distribution of the belt-pulley drives, which is one of the dominant factors used to calculate belt fatigue and predict belt durability. It can also better predict some important design performance criteria such as maximum transmissible moment and power efficiency. The analysis along this path can be extended to more complicated situations like free vibration analysis or nonlinear response prediction of entire serpentine belt drives, which is planned for future work.

1.2 Literature Review

In this section, a brief review of the related work is given. The review is divided into three parts. Part 1 focuses on the discussion of the general class of axially moving materials. The belts in the free spans of serpentine belt drives belong to this class. Part 2 discusses papers that address the entire serpentine belt drive. Part 3 reviews work concentrating on the steady state belt-pulley mechanics.

1.2.1 Vibration of axially moving materials

The moving belt of the serpentine belt system belongs to the class of axially moving materials. Extensive research has been done for these materials. The topics studied include steady state, free vibration, dynamic response, stability, and so on [8-15]. Most research in this field concerns the transverse vibration of axially moving materials and treats the continua as moving strings or beams with the boundary conditions of the materials as simple supports. This modeling excludes pulley rotations at the ends of the belt span. This results in purely continuous systems and no interaction between the span and the pulleys is considered. See papers [10-12] for comprehensive reviews for these axially moving materials.

For axially moving (gyroscopic) systems, the natural frequencies are velocity dependent and the eigenfunctions are complex [13], which means that the different components of a system do not have the same phase in the vibration modes (that is, they do not pass through the steady state position at the same time even in single mode response). The property results from the transport speed of the material, which makes the system gyroscopic. The axial velocity introduces two convective acceleration terms that are not present in the equivalent stationary system: one is the centripetal acceleration term and the other is the Coriolis acceleration term. The canonical form of a gyroscopic

+ + = system is MU tt GU t KU F where M and K are self-adjoint operators and G is skew-self-adjoint [16-18].

One prominent characteristic of this study is the incorporation of the belt bending stiffness into the modeling of the entire serpentine belt drive. The belts in serpentine belt drives have small bending stiffness relative to their tensions. The belts can be modeled as either a traveling string or a traveling, tensioned, Euler-Bernoulli beam with small dimensionless bending stiffness. One contribution of this research is to develop a uniformly valid perturbation method for closed-form approximate eigensolutions of moving, tensioned beams with vanishing bending stiffness. Thus, the works in the literature investigating the transition behavior from the axially moving string case to the beam case is highlighted here. Pellicano and Zirilli [19] study the nonlinear vibrations of large amplitudes for axially moving beams with small bending stiffness. Boundary layers due to the consideration of small bending stiffness appear at the end boundaries, which need the special treatment of singular perturbation techniques (like the method of matched asymptotic expansions). Oz et al. [20] and Ozkaya and Pakdemirli [21] examine the transition from axially moving string to beam for an axially accelerating material. O’Malley [22] obtains the perturbation solution for the eigenvalue problem of a stationary beam for clamped boundaries.

1.2.2 Serpentine belt drives

For serpentine belt drives, considerable research has been done with models that only consider pulley rotational motions. The pulleys are linked by the belt spans as springs, and no belt transverse vibrations are considered. This results in a completely discrete model. This model is relatively simple, non-gyroscopic, and has been studied extensively. Many factors have been incorporated to describe the system dynamic

8 behavior such as bearing damping, Coulomb friction of the tensioner arm, and so on [2,4-

6]. The modeling assumptions exclude the span transverse vibrations, which may be large and interact strongly with the pulleys, so this is only an approximation for the whole serpentine belt system.

The above two models represent two distinct types of vibration found in serpentine belt systems: 1) transverse vibration of individual belt spans, 2) rotational vibration of pulleys with the belts acting as linking springs. In contrast to the above models, there are much fewer papers in the literature that address coupled vibration of the belt spans and the pulley rotations. Models that consider the interactions between the continuous (belt span) and discrete (pulley) components are more complicated and less developed. The study of these hybrid models appeared later than the aforementioned two kinds of models. Initially, only the simpler band saw system was investigated. The band saw consists of two belt spans and two pulleys. Mote and Wu [23] first noticed the coupling phenomena between the belt span vibration and the pulley rotations from experiments.

Wang and Mote [24] established an analytical model to obtain a linear belt-pulley coupling mechanism due to the finite bending stiffness of the metal belt. (Although a lot of research has been done related to band saw systems, most of it treats the pulleys as simple supports for the spans, which results in a purely continuous, single span system decoupled from the pulleys [10,11,13]). Hwang and Perkins [25-27] focused mainly on the response of axially moving beam-like elements at translation speeds that exceed the classical “critical speed stability limit.” Further, they applied their theory to high-speed band/wheel systems and studied the response of the entire hybrid system. Leamy and

Wasfy [28] developed a dynamic finite element model to study the response of a simple two pulley belt-drive system by modeling the belt as truss elements while treating the pulleys as rotating circular constraints.

Because the serpentine belt system has more spans and pulleys plus the spring- loaded tensioner, modeling the entire system is more complicated than the two pulley system. Ulsoy, et al. [29] considered the possibility of parametric instability and presented a mechanism which may cause large transverse vibration in some spans due to tension fluctuations under certain operating conditions. Note this model incorporates the pulley-tensioner coupling. In [14,15], this behavior is considered further for the case of parametric instability in a single span. The boundaries of the belt spans are specified, thus excluding the interaction between the belt span vibrations and pulley rotations. In addition to giving a comprehensive review of available models for analyzing the free and forced vibrations of the power transmission belt, Abrate [12] first mentioned the challenging research direction of modeling the entire serpentine belt drives with the consideration of geometric nonlinearity due to the introduction of the tensioner device.

Beikmann, et al. [30,31] treated the belt as a moving string and studied a prototypical three-pulley model that has all essential components of a real serpentine belt drive. He captured a linear coupling mechanism between the tensioner rotation and the transverse vibrations of the two spans adjacent to the tensioner. This coupling results from tensioner rotation moving the boundary points of the two adjacent spans. Zhang and Zu [32] and

Zhang, et al. [33] further refined this linear model by incorporating belt damping and gave a complex modal analysis of the hybrid model for the serpentine belt drive system.

Parker [34] developed a spatial discretization of this model extended to n pulleys.

Beikmann, et al. [35] demonstrate a nonlinear coupling mechanism between all spans and pulleys due to finite belt stretching. He studied the two-to-one internal resonance using a numerical method. For the same nonlinear model, Zhang, et al. [36] investigated a one- to-one internal resonance combined with primary external resonance using the multiple scales method.

In practice, transverse vibrations of the belt generally appear on those spans not adjacent to the tensioner and not satisfying the conditions of parametric instability. None of the aforementioned models can predict this behavior. This suggests that belt span transverse vibrations can be excited through as yet unidentified belt-pulley coupling mechanisms. Finding the mechanisms accounting for the belt-pulley coupling, investigating their roles under different operating conditions, and predicting the system’s dynamic behaviors are the main concerns of this research.

After the models are established, solving them analytically or numerically also presents challenges due to the complication of the hybrid continuous-discrete modeling.

Up to now, mathematical methods and tools dealing with hybrid systems are rare in the literature, and only a few papers in the literature relate to such systems. The methods presented in these papers are quite useful. Parker and Mote [37,38] developed an extended operator form to study the vibration of disk-spindle systems where the conventional operator form is expanded to act on an extended variable that includes multiple unknown functions and discrete unknown variables. With proper definition of the inner product, the system is cast as a canonical gyroscopic one and powerful

11 analytical methods can be applied naturally (e.g., Galerkin discretization, perturbation, etc.) For serpentine drives, it is preferable to study the system in an extended operator formulation since the system involves both discrete (pulleys) and continuous (belt spans) components. Beikmann [31] presented the eigenvalue problem in the extended operator form and proved that these operators satisfy the symmetric/skew-symmetric properties for a conservative gyroscopic system. In his free vibration analysis, however, this extended operator form is not used (instead, a modification of Holzer’s method was used). In this research, the extended operator form and its associated properties are used to the full extent.

1.2.3 Belt-pulley steady state contact mechanics

Serpentine belt drives belong to the broad class of belt-pulley drives, which have been widely used to transmit power for hundreds of years. The power is transmitted from the driver pulley to the driven pulleys through friction between the belt and the pulleys.

The belt-pulley contact mechanics are important in industrial applications as they impact belt tension, belt life, power transmission efficiency, maximum transmissible moment, and noise. Considerable research has been done in this field. The earliest work can be traced back to Leonard Euler’s study of a belt wrapped around a fixed pulley or capstan

[39]. Fawcett [40] gives a comprehensive review of belt-pulley contact mechanics up to

1981. Two different theories have been used to describe the belt behavior. One is known as creep theory, which assumes that the belt is elastically extensible, friction is developed due to the relative slip motion between the belt and pulley, and a Coulomb law describes the belt-pulley friction. Another model is the shear theory, which addresses shearing

12 deformation of the belt and assumes that the belt is inextensible. The shear theory is developed recently in [41,42]. Alciatore and Traver [43] give a comparison between these two different theories. In this research the creep theory is adopted with the refinement of incorporating belt bending stiffness.

Creep theory has been widely used to calculate and design belt-pulley drives in industry. Johnson [44] gives a review of the classic creep theory. Recently, by considering inertial effects, Bechtel et al. [45] update the classic creep theory to include belt inertia and present a complete solution for a two-pulley belt drive. Independently,

Rubin [46] investigates the effects of the same inertia terms and presents a method to find solutions for general multi-pulley systems. Although the derivations in [45,46] seem different, the analysis and main conclusions are essentially the same. The main contribution of these two papers is that they include belt inertia terms and determine the relative errors of prior creep theories that neglect these terms. Belt bending stiffness is ignored in both studies.

1.3 Scope of Investigation

The main scope of this project is to advance the knowledge and understanding of serpentine belt drive mechanics. A prominent characteristic of this research is the incorporation of bending stiffness into the modeling of the axially moving belt. The background information and literature review is introduced in chapter 1. Chapter 2 analytically explores the transition behavior of modal properties from an axially moving string to a tensioned, axially moving, Euler-Bernoulli beam with small bending stiffness.

An approximate perturbation method is developed. Closed-form, uniformly valid 13 approximate eigensolutions are derived for combinations of different boundary conditions. Chapters 3 and 4 investigate the steady state and dynamic behavior of the entire serpentine belt drive when the belt bending stiffness is considered. This part of the analysis is based on the assumption of fixed belt-pulley contact boundaries. Belt-pulley mechanics on the contact arc are not considered in detail. Chapter 3 first establishes the model. A computational method based on boundary value problem solvers is developed to obtain the numerically exact solution of the nonlinear steady state equations. An approximate analytical solution of closed-form is also obtained for the case of small bending stiffness. Based on these solutions, the effects of design variables on the steady state deflections and span-pulley coupling are investigated. Chapter 4 then studies the linear vibration about the obtained nontrivial steady state. A robust spatial discretization is presented to solve the corresponding eigenproblem. New dynamic characteristics of the system induced by belt bending stiffness are discussed. Belt-pulley dynamic coupling is investigated thoroughly through evolution of the vibration modes. Chapter 5 focuses on the investigation of steady state contact mechanics for belt drives of fixed pulley systems.

More complicated factors such as belt inertia, elastic extension, and Coulomb friction are retained, and belt bending stiffness is also included. A numerical iteration method based on general purpose BVP solvers is developed. This overcomes the obstacles induced by inclusion of the bending stiffness (e.g. non-uniform distribution of the tension and speed in the free belt spans, unknown belt-pulley contact points a priori, etc).

The benefits of this research include:

1. A uniformly valid approximate perturbation technique is developed for the eigenvalue

problem of axially moving beams with small bending stiffness. The closed-form

approximate eigensolutions for different boundary conditions can be used as a basis

to analyze nonlinear and parametric excited moving media problems. The knowledge

applies to most of the belts in belt-pulley drives, that is those with small bending

stiffness relative to tension.

2. The developed model for the entire serpentine belt drives is suitable for the

explanation and understanding of the belt-pulley coupling phenomena observed in

experiment. Belt-pulley coupling induced by the consideration of bending stiffness is

thoroughly investigated and can be used in troubleshooting large belt vibration caused

by this coupling mechanism.

3. Knowledge of the natural frequencies and vibration modes is essential for almost all

dynamic analyses. This study characterizes the special properties that are applicable

for serpentine belt drives with bending stiffness, which is a fundamental advance for

serpentine belt vibration research.

4. The research of steady state contact mechanics between the belt and pulley reveals

how the belt bending stiffness alters the distribution of belt tension, speed, and

friction in the belt-pulley drives. Further some important design criteria like

maximum transimmible moment and power efficiency are also analytically obtained.

5. Effective methods, both theoretical and numerical, are developed to address the

steady state, eigenvalue, and dynamic response problems arising from the above

investigation. Some of these methods can be generalized to other coupled continuous-

discrete systems.

6. A comprehensive simulation tool for the dynamics of serpentine drives based on this

analysis has been developed for the analysis of practical serpentine belt drives.

Applied in practice by our research sponsor, it provides practical design guidance to

designers of serpentine drive systems.

CHAPTER 2

APPROXIMATE EIGENSOLUTIONS OF AXIALLY

MOVING BEAMS WITH SMALL BENDING STIFFNESS

For the simplest case, without considering the dynamic interaction between the pulley and belt, each free span of a serpentine belt drive can be modeled as an axially moving string or a tensioned, axially moving beam with small bending stiffness. In this chapter, a perturbation method based on the phase closure principle is developed to find the closed-form approximate eigensolutions of an axially moving beam with vanishing bending stiffness. This method is suited for different combinations of boundary conditions and uniformly valid approximate eigenfunctions are obtained. Clear physical interpretation based on wave propagation and reflection is available from this perturbation method.

2.1 Introduction

Before studying the belt interaction with the pulleys, it is beneficial to first understand the belt transition behavior from the known axially moving string case to the beam case. In the simplest sense, the transverse vibration of each span of serpentine belt drives is typically modeled as either a traveling string or a traveling, tensioned, Euler-

Bernoulli beam. This chapter focuses on the transition of modal properties between these two cases for an axially moving material.

Closed-form solutions for the natural frequencies and vibration modes are available for the string model [8,47]. For the axially moving beam model, due to the beam’s dispersive property, only numerical solutions are available [13,48]. Because most axially moving media have small bending stiffness relative to their tension, they can be modeled as an axially moving beam with small dimensionless bending stiffness. The transition of modal properties from the axially moving string case to the beam case is desirable from both practical and theoretical viewpoints. Finding closed-form approximate solutions of the eigenvalue problem for such transitional systems is the main objective of this chapter.

Before highlighting the works in this field, we first review work related to the phase closure principle, which is one of the main tools used in this chapter.

The vibration of elastic structures can be described in terms of waves propagating and attenuating in structures. The phase closure principle [49] states that if the phase change for propagating (or evanescent) waves is an integer multiple of 2π as they return to their start point after traveling forward and back along a finite structure, then the frequency at which the waves travel is a natural frequency and the corresponding

18 vibration mode is the superposition of the component waves. In the field of acoustics, solids, and fluids, wave propagation and attenuation in waveguides and wave reflection/transmission at a boundary point has been studied extensively. The phase closure principle links the knowledge of wave motion in these fields to computation of natural frequencies and modes of finite structures. Mead [50] applies the method to find the eigensolutions of stationary beams. Exact frequency equations are established that differ from the conventional ones but have identical roots. These frequency equations have clear physical interpretation and deepen understanding of the beam vibration modes.

Mace [51] develops a numerical matrix method based on wave propagation, reflection, and transmission at a point support (or a change of cross-section or material property) to calculate the natural frequencies and modes for beams. Tan and his coworkers [52,53] extend this method to some complex beam structures, like those consisting of several different uniform segments.

For axially moving continua it is well known that the vibration modes can be viewed as the superposition of pairs of opposite–going propagating waves. The phase speeds in the opposite directions are different due to the convective effect of the medium’s axial speed. Lengoc and McCallion [54] study the relation between wave propagation and natural frequency, but their work is limited to non-dispersive system like taut strings. Lee and Mote [55] investigate the energy transfer due to the interaction between the translating continua and its boundary supports. The phase closure principle is used to obtain the natural frequencies of an axially moving string. Chakraborty and

Mallik [56] study the free vibration of a traveling beam simply supported at both ends.

The frequency equation is derived based on the phase closure principle. This work applies only to beams with finite bending stiffness and zero tension; transition behavior from a taut string to a tensioned beam is not investigated.

More commonly, researchers investigate axially moving continua mathematically without consideration of the physical wave propagation. They study transition behaviors for moving beams with vanishing bending stiffness by using perturbation techniques directly on the differential equations. Because the main concern of this chapter is the modal properties, only those aspects of related works are reviewed here. Pellicano and

Zirilli [19] study axially moving beams with simple supports at both ends. While not specifically addressed, their natural frequencies can be extracted from the results. These natural frequencies depend only on the displacement boundary condition of each end, suggesting that the remaining two beam boundary conditions do not affect the natural frequencies. Oz et al. [20] and Ozkaya and Pakdemirli [21] examine the transition from axially moving string to beam for an axially accelerating material. By letting the accelerating terms vanish, the free vibration solutions for constant belt speed follow. In

[20], multiple scales perturbation is applied to find the approximate natural frequencies.

Problems are apparent because no boundary conditions are considered in the derivation, indicating that different boundary conditions yield the same natural frequencies. The problem considered in [21] is similar to that in [20] and similar techniques are used. The improvement is that the spatial boundary layer terms arising from small bending stiffness are considered. Two sets of boundary conditions are considered. The solutions in [21] incorrectly imply that the approximate natural frequencies for these two sets of boundary

20 conditions are the same. Further, for clamped boundaries the zero speed solution fails to give the approximate solution given by O’Malley [22]. In contrast, the present analysis gives different natural frequencies for these two kinds of boundary conditions, and the zero speed results converge to the exact solution (simply supported) and that given by

O’Malley (fixed-fixed), although the adopted methods differ. O’Malley’s work [22] treats stationary beams with two clamped ends. When we extended this method to axially moving beams, the procedure became cumbersome and no explicit solutions were obtained.

In this chapter, a different perturbation method is developed to find closed-form, approximate eigensolutions of axially moving beams with small bending stiffness. Wave propagation considerations lead to an algebraic equation with a small dimensionless bending stiffness parameter. Taking advantage of the simplicity of the propagation and attenuation properties of the waves, which are determined by the roots of an algebraic equation, the phase closure principle is used to find the natural frequencies. The complex vibration modes are obtained naturally from the superposition of all component waves in the beam. Approximate eigensolutions for different boundary conditions are presented.

The perturbation solutions are confirmed by comparison with numerically exact ones. For the special cases mentioned above where the exact or approximate solutions are available, the derived approximate solutions agree with them.

Instead of considering spatial and temporal variations for the governing partial differential equation (like in [19,21]), this approach focuses on perturbation of algebraic equations. No boundary layers or secular terms need to be considered explicitly in the

21 derivation. Although the method is simple, no completeness of the solutions is sacrificed.

For example, the evanescent wave components (if not zero) automatically generate boundary layer terms for those beams where small bending stiffness creates edge effects at the boundaries. Unlike prior perturbations where assumed mode spatial expansions are only suited for certain boundary conditions, this method handles different boundary conditions with a consistent treatment.

2.2 Model Equations

The dynamic equation for an axially moving beam is

+−−+=<<2 mwmcwPmcwEIwtt2() xt xx xxxx 0,0 xL (2.1) where m is the belt mass per unit length, w(x,t) is the transverse displacement, c is the belt transport speed, P is the tension, and EI is the bending stiffness. The following non- dimensional variables are introduced,

x wPEIm xwttˆˆ===,,ˆ ,ε 2 = , vc ˆ = (2.2) L LmLPL22P

Substitution of (2.2) into (2.1) leads to the dimensionless equation (after dropping the hat)

+−−+22ε =<< wtt2vw xt(1 v ) w xx w xxxx 0, 0 x 1 (2.3)

Assuming we= i()rx−ω t , where r is the wavenumber and ω is the wave propagation frequency, equation (2.3) yields

εωω24rvrvr+−(1)2 2 2 + − 2 =0 (2.4)

Note that the small parameter ε <<1 multiplies the highest power of r. The roots of such a polynomial equation have two possible forms [57].

In the first form, the roots of (2.4) are expressed using the straightforward expansion

=+εε +2 + rx01 x x 2... (2.5)

Substitution of (2.5) into (2.4) leads to the ε 0 order result

ω x = (2.6) 0 v ±1 and the ε 1 order result

−+=2 ω = 2(1vx )0 2 v 0 or x1 0 (2.7)

= ε 2 The first equation in (2.7) is discarded because it contradicts (2.6), so x1 0 . The order equation gives

ωω3 −=1  4 , when x0 −x4 2 (1)v ++v 1 ==0  x2  (2.8) 22(1)vvxω +−2 ωω3 − 0  1 = 4 , when x0  2(1−−vv ) 1

Equations (2.5)-(2.8) provide two of the four roots of (2.4).

The remaining two roots are expressed as the singular expansion

y rx=++..., λ >0 (2.9) ε λ 0

Substitution of (2.9) into (2.4) gives

yyy εωω2422( ++xvxvx...) +− (1 )( ++ ...) + 2 ( ++ ...) − 2 =0 (2.10) εεελλλ000

The dominant terms in (2.10) are y4 / ε 4λ−2 and (1)/− vy2 22ε λ . Balancing these leads to

4λλλ−=22⇒ =1 (2.11)

Equation (2.10) then becomes

εε−−24+− 22 + 1 3 + − 2 + ω += [yvyxyvxyvy(1 ) ] [400 2(1 ) 2 ] ... 0 (2.12) with the solutions

y =±iv1 − 2 (2.13)

vω 2xv [(1−−−22 ) 2(1 v )] + 2 vω = 0 ⇒ x= (2.14) 001− v2

Case Results r1 r2 r3 r4

Exact 0.7222 -6.4335 2.8556+60.2421 i 2.8556-60.2421 i ε = 0.01 v = 0.80 ω =1.30 Approx. 0.7222 -6.4313 2.8889+59.9999 i 2.8889-59.9999 i

Exact 0.6482 -1.1501 0.2510+19.2243 i 0.2510-19.2243 i ε = 0.05 v = 0.28 ω = 0.83 Approx. 0.6482 -1.1501 0.2522 +19.2000 i 0.2522 -19.2000 i

Exact 1.2949 -1.7342 0.2197+10.0070 i 0.2197-10.0070 i ε = 0.10 v = 0.15 ω =1.50 Approx. 1.2947 -1.7324 0.2302 +9.8869 i 0.2302 -9.8869 i

Table 2.1. Comparison of approximate roots of (2.4) from (2.15) with numerically exact roots. 24

In summary, the wave dispersion equation (2.4) has the four roots

ωω1133 ωω rOrO=−εε2 +( 323)( =−+ εε +) 1212(1)++vv44 12(1) −− vv (2.15) vvωω11−−22 vv riOriO=+ +()εε =− + () 3411−−vv22εε

Table 2.1 compares the approximate and numerically exact roots for three cases.

Consistent with (2.15), r1 and r2 are best approximated by perturbation. Physically, the real parts of the roots represent the phase change between two points unit distance apart, and the imaginary parts represent the variation of the wave amplitudes for two points unit

distance apart. Specifically, r1 represents the wave propagating in the positive direction,

r2 the wave propagating in the negative direction, r3 the evanescent wave attenuating in

the positive direction, and r4 the evanescent wave attenuating in the negative direction.

The beam motion is the superposition of the four components

− ω =+++ir12 x ir xir3 x ir 4 x it w(,)xt [ ce12 ce ce 3 ce 4 ] e (2.16)

where cc14~ are complex coefficients.

ε << For small bending stiffness 1, the imaginary parts of the evanescent waves r3

and r4 become very large. Consequently the r3 component can exist only close to the

= = boundary x 0 , and the r4 component exists only close to the boundary x 1 (Figure

2.1). They can be viewed as part of the reflected waves as the propagating waves ( r1 and

= r2 ) travel forward and back along the beam between the boundary points A( x 0 ) and

B( x =1) (Figure 2.1). The phase closure principle can now be applied to the propagating waves to find the eigensolutions for different boundary conditions. 25

r3 r r1 v 1

A x=0 Beam with small ε = EI/PL2 x=1 B

r2 r2

Figure 2.1. Waves in a finite moving beam with small bending stiffness

2.3 Application of the Phase Closure Principle

To apply the phase closure principle to the propagating waves ( r1 and r2 ), one needs to find four different phase changes: first, the propagating wave leaves from

= boundary A and arrives at boundary B with a phase change Re (rr11) (because the span is normalized to unit length); second, it reflects at boundary B with a phase change

φ ()RB ; third, it travels from boundary B to boundary A with another phase change -

=− Re ()rr22 (minus sign due to leftward propagation); finally, it reflects at boundary A

φ with the fourth phase change ()RA and returns to the start point boundary A.

Mathematically, the phase closure principle requires

+−+=φφπ =±± rRrR12( BA)()2,0,1,2, nn ... (2.17)

Consider the case of a simply supported beam with boundary conditions

== == wt(0,)(0,)0,(1,)(1,) wxx t wt w xx t 0 (2.18)

= At x 0 (point A in Figure 2.1), there is no r4 evanescent component as noted above, and

(2.16) becomes

− ω =++ir12 x ir x ir3 x it w(,)xtA [ ce12 ce ce 3 ] e (2.19)

Substitution into the boundary conditions at x = 0 yields

11c 1 c 1 rr22− 1 =− c ⇒ 1 = 23c (2.20) 22 22  22− 22− 2 rr13 c 3  r2 c3 rr31rr12

For the two propagating waves, the relative phase due to the reflection at the left boundary is given by the phase of

c rr22− {−− (1v2 ) +εω22222 [vv /(1 − )] − εω [ /(1 − vivv )] } + 2 εω / 1 − 2 R ==−1 32 =− (2.21) A 22− 2 22222 2 crr231{−− (1v ) +εω [vv /(1 − )] − εω [ /(1 + vivv )] } + 2 εω / 1 −

One can prove mathematically that the phase angle of RA is

φ =+πε3 (ROA )() (2.22) as shown graphically in Figure 2.2, where π is from the leading minus sign preceding

2 −−222 (r32rrr)/( 31) in (2.21).

2 Im D2= O( ε )

D1= O( ε) Denominator of D = -RA Ο(ε3)

Numerator of D = -RA = D3 O(1) Re

=− =2 −222 − Figure 2.2. Phase angle of D RrrrrA ( 3 231)/( ) from equation (2.21), =−εω 2 =−−+εω2 22 ω D1 2/1vv, Dvv2 {[/(1)][/(1)]}, =− −2 +εω2222 − − ω + Dvvv3 (1 ) {[ /(1 )] [ /(1 v )] } .

= At x 1(point B), there is no r3 evanescent component in (2.16). In seeking the relative phase between propagating waves at B, it is notationally convenient to introduce

ξ =− = irk x 1 and express the coefficients in (2.16) using bkkce . This gives

ξξξ − ω =++ir12 ir ir3 it w(,)xtB [ be12 be be 4 ] e (2.23)

Substitution of (2.23) into the x =1 boundary conditions yields

11b 1 b 1 rr22− 2 =− b ⇒ 2 = 14b (2.24) 22 21  22− 22− 1 rrb24 4 r 1 b4 rr42rr21

For the two propagating waves, the relative phase due to the reflection at the right boundary B is given by the phase of

b rr2 − 2 R ==2 14 (2.25) B 2 − 2 b1 rr42

Similar to the handling of RA , the phase angle of RB is

φ =+πε3 (ROB )() (2.26)

Substitution of (2.15), (2.22), and (2.26) into (2.17) leads to

21ω 1 1 −++==±±εω23[ ] ... 2nn π , 0, 1, 2,... (2.27) 12(1)(1)−+−vvv244

This is an algebraic equation (for ω ) with the small parameter ε multiplying the highest power. Application of the previously discussed algebraic perturbation technique leads to three different roots. Only the root from the straightforward expansion form is retained.

The two roots from the singular expansion form are discarded because they yield complex roots, and physically the natural frequency ω must be real for subcritical

ωω=+ εωεω +2 + speeds. Substitution of 01 2... into (2.27) leads to

ωπ=−+222242 επ +++ = n nv[1 nvv ( 6 1) / 2 ...], n 1,2,3,... (2.28)

Figure 2.3 compares the fundamental ( n =1) natural frequencies obtained from

(2.28) with numerically exact solutions for different belt speed v and different bending stiffness. The approximation results are best for small bending stiffness and low axial belt speeds. This is because for such cases the four roots in (2.15) have the best perturbation approximation. For large bending stiffness or high speed, more terms need to be incorporated in the perturbation approximation.

3.5 v=0 3 v=0.3 2.5

2 v=0.6 1.5 perturbation

1 v=0.9 exact 0.5 Fundamental natural frequency 0 0 0.02 0.04 0.06 0.08 0.1 0.12 Bending stiffness, ε

Figure 2.3. Comparison of fundamental natural frequency for a simply supported beam. −−−, perturbation; , exact.

When v = 0 , (2.28) becomes

1 ωπ=+nnn[1 επ22 ( ) ] += ..., 1,2,3,... (2.29) n 2

The exact eigensolution for the special case v = 0 is

ωπεπ=+22 4 = π = exact (nnwxnxn)(),()sin(),1,2,3,... (2.30)

Expansion of the eigenvalue in (2.30) for small ε yields (2.29).

Computation of the eigenfunctions requires additional consideration of the evanescent waves at the boundaries A and B. From (2.20) and (2.24),

c rr22−+−−[/(1)][/(1)]ωω v 2 v 2 R ==3 12 = (2.31) A 22− 2 222 2 2 crr231{[vωεωωε /(1−−−−vv )] (1 ) / [ /(1 ++ vivv )] } 2 /( 1 −)

brr22−−+−−[ωω /(1 v )] 2 [ /(1 v )] 2 R ==−412 = (2.32) B 22− 2 222 2 2 brr142{[vωεωωε /(1−−−−−−vv )] (1 ) / [ /(1 vivv )] } 2 /( 1 −)

The eigenfunction can be written as

ξ =+irx1 irx24 ++ir3 x ir w()xcececebe12 3 4 (2.33)

Normalization of (2.33) by dividing it by c2 , application of (2.21), (2.31), (2.32), and use

ξ =− = ir1 of x 1 and b11ce give

− =+++ir12 x ir xir3 x ir 14 ir( x 1) wx() ReAAAB e Re Re Re (2.34)

≠ When v 0 , there are boundary layer terms from the evanescent r3 and r4 terms. But

= =− == when v 0 , RA 1 and RRAB0 , leading to the eigenfunctions

w()xee=−irx1 + ir2 x (2.35)

Substitution of (2.15), (2.29), and v = 0 into (2.35) gives the eigenfunction approximation

wx( )sin(),1,2,3== nπ x n ... (2.36) in agreement with the exact solution in (2.30). The eigenfunctions have no boundary layer terms for v = 0 while they do for v ≠ 0 .

2.4 Other Boundary Conditions

The above perturbation method for the eigenvalue problem can be applied to other

boundary conditions. Table 2.2 lists the reflection coefficients ( RA , RRRBA,,B ) for two different end supports. These can be used to determine the eigensolutions for combinations of such boundary conditions. For example, for the fixed-simple boundary

== == =− − conditions wt(0, ) wx (0, t ) 0 , wt(1, ) wxx (1, t ) 0 , we have RA (rr2 331)/( rr) ,

R =−(rr)/( rr −) , R =−(rr2 222)/( rr −) , and R =−(rr2 222)/( rr −) . The A 1 231 B 1 442 B 2 142 presented method yields the approximate eigenvalues

ωπ=−+2 1 ε + = n nv(1 )[1 ...], n 1,2,3... (2.37) 1− v2

The corresponding eigenfunctions still have the form (2.34) but with revised coefficients

RA , RA , RB . The roots rr14∼ in (2.15) have the same functional form but differ between boundary condition cases because of changes in the expression for ω (e.g. (2.27) and

(2.37)).

Boundary Type A ( x = 0 ) B ( x =1)

r 2 − r 2 r 2 − r 2 R =− 3 2 R =− 4 1 A 2 − 2 B 2 − 2 r3 r1 r4 r2

r 2 − r 2 r 2 − r 2 R = 1 2 R =− 1 2 Simple support A 2 − 2 B 2 − 2 r3 r1 r4 r2

r − r r − r R =− 3 2 R =− 4 1 A − B − r3 r1 r4 r2

r − r r − r R = 1 2 R =− 1 2 Fixed support A − B − r3 r1 r4 r2

Table 2.2. Reflection coefficients for different end supports.

Another example is for the fixed-fixed boundary conditions wt(0, )== wt (1, ) 0 ,

== wtwtxx(0, ) (1, ) 0 . The approximate eigenvalues are

ωπ=−+2 2 ε + = n nv(1 )[1 ...], n 1,2,3... (2.38) 1− v2

The approximate eigenfunction is the superposition of the four component waves. Again,

ω (2.34) holds for this case, in which is different for rr14∼ in (2.15) according to (2.38),

and the coefficients RA , RA , and RB are available in the second row of Table 2.2.

For the case of fixed-fixed supports, letting v = 0 in (2.38) gives

ωπ=++= ε n nn(1 2 ) ..., 1,2,3... (2.39) with corresponding eigenfuncitons

sin(nxπ ) −+−εε wx( )=+−−++−+=επ [ (1 2 x )cos( nx ) exnx/1(1)/2 ( 1) e ] ε (...), n 1,2,3...(2.40) n nπ

For this zero speed case, by using a different perturbation method (directly assuming the eigenfunction as the combination of outer solution and boundary layer inner solutions),

O’Malley [22] obtains the approximate eigensolutions

ωπεπ22=+ 2 += n (nnn) 4 ( ) ..., 1,2,3... (2.41)

sin(nxππ ) sin( nx ) wx()=+επ [ −− (12)cos( x nx ) n nnππ (2.42) ++−ee−−xnx/(1)/2εε( 1) ] +ε (...), n = 1,2,3...

These eigenvalues are the same as (2.39) for the leading terms. There are two differences between (2.40) and (2.42). First, in (2.42), the coefficient before the boundary layer term at x =1 is (−1)n while in (2.40) the coefficient is (−1) n+1 . Second, there is no term in

(2.40) corresponding to the sin(nππxn ) /( ) term inside the bracket of (2.42). For the first difference, it can be easily checked that (−1)n in (2.42) is a typographical error that should be (−1) n+1 . The second difference is due to different normalization used in (2.42) and (2.40); multiplication of (2.40) by (1+ ε ) yields the same leading terms as (2.42).

Thus, the eigenfunctions (2.40) and (2.42) agree with each other. For such boundary conditions, there are always boundary layer terms at the ends of the beam. O’Malley’s method is only for stationary beams and does not extend simply to problems of axially moving beams.

CHAPTER 3

MODELING AND STEADY STATE ANALYSIS OF

SERPENTINE BELT DRIVES WITH BENDING STIFFNESS

In this chapter, a serpentine belt system model incorporating the belt bending stiffness is established. The finite belt bending stiffness causes nontrivial transverse span steady state, in contrast to string models with straight span steady state. Nontrivial span steady state cause linear span-pulley coupling, and the degree of coupling is determined by the steady state curvatures. A computational method based on boundary value problem solvers is developed to obtain the numerically exact solution of the nonlinear steady state equations. An approximate analytical solution of closed-form is also obtained for the case of small bending stiffness. Based on these solutions, the effects of design variables on the steady state deflections and span-pulley coupling are investigated.

3.1 Introduction

A prominent characteristic of serpentine belt drives is the introduction of a spring- loaded tensioner assembly, which greatly improves the system dynamic performance by automatically compensating for tension changes as accessories are activated or deactivated. Figure 3.1 shows a prototypical model consisting of three pulleys and a tensioner, which is first studied by Beikmann et al. [3,30,31]. By treating the belt as a continuum string, Beikmann et al. [3,30,31] captures linear coupling between the tensioner rotation and the transverse vibration of the two spans adjacent to the tensioner.

This coupling results from tensioner rotation moving the boundary points of the two adjacent spans. For spans away from the tensioner bounded by fixed pulleys, the belt is modeled as an axially moving string whose boundary points have no transverse deflections, thereby decoupling the span vibrations from pulley rotations. Parker [34] used similar modeling to analyze vibration of a general n-pulley system. Beikmann’s experiments [30,31], however, show a degree of linear coupling between pulley rotations and the span transverse vibrations between fixed pulleys, which can be excited despite the fact that the motions of the two end pulleys are only along the axis of the span treated as a string. Observations on current automobiles also demonstrate strong, apparently linear coupling where pulley rotations excite undesirable transverse vibration of the adjacent spans.

K ( r r (1) t M u w S 1 1

θ 2 ( J M (2) 2 E

θ 1 (

M u ( (1) 3 2 (2) M M J M E J 1 3 w S 1 θ 3 2 w

3 ( (3) c ( M u M (3) S 3 E

Figure 3.1. A prototypical three-pulley serpentine belt drive system.

In this chapter, general equations of motion that incorporate belt bending stiffness are formulated using Hamilton’s principle for a three-pulley system. The equations show that vibration coupling exists between spans away from the tensioner and the pulley rotations. The mechanism of the coupling depends upon the steady state curvature of the

Γ = Γ spans. Correspondingly, a coupling indicator ∑ i determined by the steady state is

Γ defined to measure the magnitude of vibration coupling of the whole system, where i denotes the coupling indicator for individual spans.

The main work of this chapter is to determine the span steady state deflections from the set of nonlinear equations. The steady state equations are differential-integral equations coupled with several algebraic equations. A key step is the application of ordinary differential equation (ODE) conversion techniques to reformulate the governing equations into a standard boundary value problem (BVP) form that can be readily accepted by general-purpose BVP solver codes, [58]. Taking advantage of the reliability and high performance of modern BVP solver codes, almost exact numerical results are found with little programming effort. For the practically important case of small dimensionless bending stiffness, singular perturbation techniques are used to derive an approximate closed-form solution.

Based on these numerical and analytical solutions, the effects of major design variables on the steady state deflections and coupling indicators are investigated. The analytical solution explicitly reveals the qualitative and quantitative impact of design variables on the span steady state and magnitude of the vibration coupling. A closed-form approximation for the coupling indicator captures this information in a simple expression.

The steady state solution presented here is essential for subsequent material in chapter 4 on the dynamic analysis of coupled belt-pulley serpentine drive vibration [59].

3.2 System Model

The prototypical system of Figure 3.1 was first used by Beikmann et al. [3,30,31] to study the vibration of serpentine belt systems. It contains the essential components in automotive serpentine drives: a driving pulley (pulley 1), a driven pulley (pulley 3), a belt, and a spring-loaded tensioner assembly. The tensioner assembly consists of a tensioner arm spring-loaded at its pivot with an idler pulley (pulley 2) in contact with the

= belt. The accessory driving torques M 1 (t ) and M 3 (t ) are present, but M 2 (t) 0 for the idler pulley. (See Table 3.1 for definitions of the symbols.)

θ = The dynamic motions are the pulley rotations i (t),i 1,2, 3, the tensioner arm

θ rotation t (t) , and the transverse (wi (xi ,t ) ) and longitudinal (ui (xi ,t ) ) displacements of each belt span. The spans are modeled as continua translating with constant speed c. Each span is subjected to constant moments at its ends arising from the bending of the belt around the pulleys. No microslip or gross slip is considered at the belt-pulley interface, which is taken to be a single contact point that does not vary with belt motion [23,24].

All motions are measured relative to a reference state. The reference state corresponds to the steady state for a stationary belt with no bending stiffness, that is, the steady state for the system with the belt modeled as a string. Steady accessory torques are present in the reference state, so different spans may have different reference state tensions. Beikmann et al. [30] presented a method to calculate the reference steady state.

Measuring displacements from this reference state clearly shows the effect of belt bending stiffness.

Consider the reference steady state when all accessory torques are zero. In this case,

all span tensions are Po , which is used in the subsequent nondimensionalization procedure.

The equations of motions are derived from Hamilton’s principle. The kinetic energy

T is

33 c li T =+++111JJmwcwucucd()θθ22 [()( −+−− 2 )] 2x (3.1) 222∑∑i itt∫ 0 itixitix,, ,, i ii==11ri

= + 2 where J t J arm m2 rt , J arm is the rotational inertia of the tensioner arm, m2 is the

θ mass of the tensioner pulley, and 2 is the absolute rotation angle of the tensioner idler pulley (as opposed to rotation relative to the tensioner arm). The potential energy V is

3 li = 1 θ +θ 2 + EA Pi + + 1 2 2 + 1 2 V kr ( t tr ) [ ( ui,x wi,x ) EIwi,xx ]dxi (3.2) 2 ∑∫0 2 EA 2 2 i=1

θ where tr is the rotation angle of the tensioner arm in the reference state (Figure 3.2).

The virtual work is

333 δδ=+−()ii () δ δθ WMwtMwltM∑∑∑S ix,,(0, ) E ix ( i , ) i i (3.3) iii===111

J i rotation inertia of pulley i

M i applied static torque on accessory i

Pi belt tension of span i in the reference state

Po static tension in reference state without accessory torques c steady state belt speed

kr tensioner spring stiffness

li length of belt span i m belt mass per unit length

ri radius of pulley i

ui longitudinal displacement of span i

wi transverse displacement of span i θ i rotation of pulley i θ tr tensioner arm rotation in the reference state θ t rotation of the tensioner ζ ζ β β 1, 2 , 1, 2 alignment angles of tensioner (see Figure 3.2) (i) (i) M S , M E end moments of span i

Table 3.1. List of symbols defining serpentine belt drives.

θtr parallel to r ζ spans 1,2 t 1 β 1 θ β 2 ζ 2 J 2 β ζ 2 1 =θ tr− 1 β ζ 2 =θ tr− 2 u u w 2 1 1 w 2

Figure 3.2. Detail of tensioner region and pulley 2, defining alignment angles.

(i) (i) where M S is the end moment on the ith span start point and M E is the end moment on the ith span end point. Beam theory applied to the belt at the pulley contact point requires (i) = (i) = M S EI / rS M E EI / rE (3.4)

where rS and rE are the radii of the pulleys that bound a span, [23,24].

The kinematic constraints are obtained from Figures 3.1 and 3.2

= = θ β w1 (0,t) 0 w1 (l1 ,t) rt t cos 1 (3.5)

=−θθθ =− − β ut1(0, ) r11 ultr 1 ( 1 , ) 2 2 rtt sin 1 (3.6)

= θ β = w2 (0,t) rt t cos 2 w2 (l2 ,t) 0 (3.7)

=−θθ −β =− θ u2 (0,trr )22tt sin 2 ultr 2 ( 2 , ) 33 (3.8)

= = w3 (0,t) 0 w3 (l3 ,t) 0 (3.9)

=−θθ =− u3 (0,tr )33 ultr 3 ( 3 , ) 11 (3.10)

β = θ − ζ where 1,2 tr 1,2 are the orientation angles of the tensioner arm relative to the two spans adjacent to the tensioner. For the details of the tensioner alignment, see Figure 3.2.

The positive direction of wi is always to the inside of the belt loop, the positive direction

θ θ of ui is counterclockwise, and the positive direction of belt travel c is clockwise; tr ,t

θ are positive counterclockwise; i is positive if its rotation is in the direction of belt

ζ travel; i is the angle from due east to the outside of the ith span; the pulleys and spans are numbered sequentially in the counterclockwise direction (Figures 3.1 and 3.2). The

β definition of the tensioner orientation using 1,2 differs from prior research, [3,30-32]; this new definition seems easier to understand, although the two approaches are equivalent.

Application of Hamilton’s principle yields the equations of motion. The field equations for the spans are

m( w−+2){[()]}0,1,2 cw c22 w − EA u ++1 w P w + EIw == i ,3 (3.11) i,,, tt i xt i xx i , x2 ix, i i ,,, x x i xxxx

mu( −+2)[()]0,1,2 cu cu22 − EAu ++==1 w P i ,3 (3.12) itt,,, ixt ixx ix ,2 ix, i , x

For practical serpentine drives, the longitudinal stiffness EA is much greater than the transverse stiffness from belt tension and bending. As a consequence, longitudinal waves propagate much more rapidly than transverse waves, and one may adopt the quasi-static assumption, [60]. Under this assumption, the inertia terms are neglected in the field

+ 1 2 equations for longitudinal motion, and the dynamic tension EA(ui,x 2 wi,x ) becomes uniform throughout the span

~ EA li P (t) = EA(u + 1 w2 ) = [u (l ,t) − u (0,t) + 1 w2 (x ,t)dx ],i = 1,2,3 (3.13) i i,x 2 i,x i i i ∫0 2 i,x i i li

The transverse vibration equations become

−+2 −++ == mw( i,,, tt2)[()]0,1,2 cw i xt cw i xx P i Pw i i ,,, x x EIw i xxxx i ,3 (3.14)

= = EI = θ β = − EI w1 (0,t) 0 EI w1,xx (0,t) w1 (l1,t) rt t cos 1 EI w1,xx (l1 ,t) (3.15) r1 r2

= θ β = − EI = = EI w2 (0,t) rt t cos 2 EI w2,xx (0,t) w2 (l2 ,t) 0 EI w2,xx (l2 ,t) (3.16) r2 r3

= = EI = = EI w3 (0,t) 0 EI w3,xx (0,t) w3 (l3 ,t) 0 EI w3,xx (l3 ,t) (3.17) r3 r1

The equations for the tensioner and pulleys are

Jkθθ ++[()( mcwlPmcPwlEIwlr +−+22 )()()]cos( +ββ + mcPr − )sin tt rt1,1 t 1 1 1,1 x 1,1 xxx t 1 1 t 1 (3.18) −+−++22ββ −−= [mcw2,t (0) ( P 2 mc P 2 ) w 2, x (0) EIw 2, xxx (0)] r t cos 2 ( mc P 2 ) r t sin 2 0

θ +−= JPrPr11 11 31 0 (3.19)

θ −+ = JPrPr22 12 22 0 (3.20)

θ −+= JPrPr33 23 33 0 (3.21) where

l ~ = EA θ − θ − θ β + 1 1 2 P1 (r2 2 r1 1 rt t sin 1 w1,x dx) (3.22) l1 ∫0 2

l ~ = EA θ − θ + θ β + 2 1 2 P2 (r3 3 r2 2 rt t sin 2 w2,x dx) (3.23) l2 ∫0 2

l ~ = EA θ − θ + 3 1 2 P3 (r1 1 r3 3 w3,x dx) (3.24) l3 ∫0 2

−+ββθ += Note the reference steady state equations Pr1122ttrtrsin Pr sin k 0 ,

−+=−+ = −++= Pr11 Pr 31 M 1 0 , Pr12 Pr 22 0, and Pr23 Pr 33 M 3 0 have been used to simplify the equations (3.18)-(3.21).

The following nondimensional variables are defined

~ x w l + l + l P P P xˆ = i wˆ = i l = 1 2 3 ˆ = o Pˆ = i Tˆ = i i i t t 2 i i li li 3 ml Po Po

ε 2 = EI = m = kr γ = EA 2 s c ks (3.25) Pol Po Port Po

where Po is the initial tension of the string model at rest with no accessory torques, as mentioned previously. l is the characteristic length taken as the average span length.

Eliminating time derivative terms and dropping the hat on dimensionless variables yields the nondimensional steady state equations

ε 2 ( l )2 w − [P − s2 + T ]w = 0 0 < x < 1 i = 1,2, 3 (3.26) li i,xxxx i i i,xx

r l = = l1 = t β θ = − 1 w1 (0) 0 w1,xx (0) w1 (1) cos 1 t w1,xx (1) (3.27) r1 l1 r2 46

l = rt β θ = − l2 = = 2 w2 (0) cos 2 t w2,xx (0) w2 (1) 0 w2,xx (1) (3.28) l2 r2 r3

l = = 3 = = l3 w3 (0) 0 w3,xx (0) w3 (1) 0 w3,xx (1) (3.29) r3 r1

[−(P − s 2 + T )w (1) + ε 2 ( l ) 2 w (1)]cos β 1 1 1,x l1 1,xxx 1

−[−(P − s 2 + T )w (0) + ε 2 ( l ) 2 w (0)]cos β 2 2 2,x l2 2,xxx 2

+ − 2 β − − 2 β − θ = (T1 s )sin 1 (T2 s )sin 2 k s t 0 (3.30)

− = T1 T3 0 (3.31)

− + = T1 T2 0 (3.32)

− + = T2 T3 0 (3.33) with the dimensionless tensions

1 ==−Prr121γθθθβ + −rt + 1 2 T12111,( txsin wdx) (3.34) Plllo 111 ∫ 0 2

1 ==−Pr22γθθθβrr3 + +t + 1 2 T23222,( txsin wdx) (3.35) Plllo 222 ∫ 0 2

1 ==−Pr33γθθr1 + + 1 2 T3133,()wdx x (3.36) Pllo 33∫ 0 2

Equations (3.26)-(3.36) are a mixed differential-integral-algebraic system. The

θ θ unknowns are t , wi , and i . The design variables that control the steady state span

ε 2 deflections are the initial span tensions Pi , bending stiffness , speed s , tensioner

γ spring stiffness ks , longitudinal belt stiffness , and drive geometry.

From the pulley equations (3.31)-(3.33),

= = = T1 T2 T3 T (3.37)

From (3.34)-(3.36),

1 Tl11=− r 2θθθβ + r 1 −rt + l 1 1 2 γ 21txsin 1w 1,dx (3.38) ll ll l∫ 0 2

1 Tl22=−rr3 θθθβ + r 2 +t + l 2 1 2 γ 32txsin 2w 2,dx (3.39) ll l l l∫ 0 2

1 Tl33=−r1 θθ + r 3 + l 3 1 2 γ 13w 3,xdx (3.40) ll l l∫ 0 2

Addition of (3.38), (3.39), and (3.40) and substitution of the relation (3.37) yield

+ + 1 1 1 l1 l2 l3 T − l1 1 2 − l2 1 2 − l3 1 2 + rt β − β θ = ( ) γ w1,x dx w2,x dx w3,x dx (sin 1 sin 2 ) t 0 (3.41) l l ∫0 2 l ∫0 2 l ∫0 2 l

Defining T as the new unknown variable and substituting (3.37) into (3.26) and

(3.30) gives

ε 2 ( l )2 w − (P − s2 + T )w = 0 0 < x < 1 i = 1,2, 3 (3.42) li 1,xxxx i 1,xx

[−(P − s 2 + T )w (1) + ε 2 ( l ) 2 w (1)]cos β 1 1,x l1 1,xxx 1

−[−(P − s 2 + T )w (0) + ε 2 ( l ) 2 w (0)]cos β 2 2,x l2 2,xxx 2

+ − 2 β − − 2 β − θ = (T s )sin 1 (T s )sin 2 ks t 0 (3.43)

Equations (3.41)-(3.43) and boundary conditions (3.27)-(3.29) define a simplified nondimensional system that is equivalent to the original system (3.26)-(3.36). Compared

θ θ with the original steady state equations, the variables t and T replace t and

θ = i ,i 1,2,3 .

3.3 Numerical Solution

The above system consists of boundary value problem (BVP) equations (3.42) coupled with algebraic equations (3.41) and (3.43). The boundary conditions (3.27) and

(3.28) for the two spans adjacent to the tensioner w1 and w2 are nontrivial and coupled

θ with the tensioner rotation t . The algebraic equation (3.41) contains integral terms

involving the wi . Furthermore, all equations are nonlinear. The combination of these characteristics initially makes it seem difficult to formulate an accurate numerical solution. By applying ODE conversion techniques, however, the above system can be transformed into a standard form defined on the interval [0,1], [58]. This formulation can be accepted by general-purpose BVP solvers yielding convenient and highly accurate solutions with minimal programming.

The standard form required for most BVP solvers is

y' (x) = f (x, y(x)), a < x < b (3.44)

g( y(a), y(b)) = 0 (3.45) where f , y , and g are n-dimensional vectors and f and g may be nonlinear [58]. This standard form cannot contain integral terms or algebraic equations as are present in the current system.

To adapt the belt drive steady state equations to standard form, the following three conversion techniques are used:

• θ = θ = θ Define the constants T and t as functions T T (x) , t t (x) governed by

dT (x) dθ (x) = 0 t = 0 0 < x < 1 (3.46) dx dx 49

x • = 1 2 σ For the integral terms in the algebraic equation , define I i (x) wi,σ d , which ∫0 2

gives

dI (x) i = 1 w2 I (0) = 0 i = 1,2, 3 (3.47) dx 2 i,x i

1 1 2 Ii (1) is then equivalent to the original integral terms wi,x dx in (3.41). ∫0 2 • θ With T and t defined as functions of x as in (3.46) and the definition of Ii (x )

in (3.47), the algebraic equations (3.41) and (3.43) are treated as boundary

conditions as seen in equations (3.54) and (3.55) below. This conveniently draws

the discrete variable equations (3.41) and (3.43) into the continuum BVP

formulation.

This process yields the following differential equations

= θ = < < T,x 0, t,x 0 0 x 1 (3.48)

− 1 li 2 − 2 + = = 1 2 = < < wi,xxxx ε 2 ( l ) (Pi s T )wi,xx 0 I i,x 2 wi,x i 1,2,3, 0 x 1 (3.49) with boundary conditions

= = = I1 (0) 0 I 2 (0) 0 I 3 (0) 0 (3.50)

w (0) = 0 w (1) = rt cos β θ (1) w (0) = l1 w (1) = − l1 (3.51) 1 1 l1 1 t 1,xx r1 1,xx r2

w (0) = rt cos β θ (1) w (1) = 0 w (0) = − l2 w (1) = l2 (3.52) 2 l2 2 t 2 2,xx r2 2,xx r3

w (0) = 0 w (1) = 0w (0) = l3 w (1) = l3 (3.53) 3 3 3,xx r3 3,xx r1

+ + l1 l2 l3 1 − l1 − l2 − l3 + rt β − β θ = l γ T (1) l I1(1) l I 2 (1) l I 3 (1) l (sin 1 sin 2 ) t (1) 0 (3.54)

[−(P − s 2 + T (1))w (1) + ε 2 ( l ) 2 w (1)]cos β 1 1,x l1 1,xxx 1

−[−(P − s 2 + T (1))w (0) + ε 2 ( l ) 2 w (0)]cos β 2 2,x l2 2,xxx 2

+ − 2 β − − 2 β − θ = (T (1) s )sin 1 (T (1) s )sin 2 k s t (1) 0 (3.55)

Notice that the 17 boundary conditions (3.50)-(3.55) equal the total order of the eight differential equations (3.48)-(3.49). The equations (3.48)-(3.55) involving higher derivatives can be readily reduced to standard first order form (3.44)-(3.45) with the

= = = definitions y1 (x) w1 (x) , y2 (x) w1,x (x) , y1 '(x) y2 (x), and so on. Also note that the problem is cast entirely on the interval x ∈[0,1] even though the problem involves multiple spans of different lengths. This standard form is readily implemented in BVP solver software. Here, the solver BVP4C in the Matlab software is adopted, [61].

This approach for solving the steady state problem of coupled continuous-discrete systems has several advantages

• It is easy to implement in readily available professional software once the

problem is cast in standard form. This minimizes software development needs and

setup time.

• Second, with the high quality and robustness of general-purpose codes, the

numerical results can be excellent. For example, in this study the relative

tolerance RelTol=0.001 was used for the BVP4C calculations, which is a high

criterion for numerical computation, [61]. Because there is no spatial

discretization, the final results can be viewed as numerically exact.

• Finally, the method can be extended to other nonlinear, continuous-discrete BVP

systems. By applying similar conversion techniques, these systems can be

transformed into a standard BVP system. The algebraic equations associated with

the discrete variables (e.g., tensioner rotations) typically serve as boundary

conditions.

3.4 Numerical Results and Discussion

In this section, numerical steady state results are presented for a prototypical three- pulley serpentine belt system. The physical properties shown in Table 3.2 are drawn from

[3,30,31]. Steady state deflections are plotted in three-dimensional perspective for span 1 adjacent to the tensoner and span 3 away from the tensioner. Tensioner rotation are

= β θ evident from these plots using w1 (1) rt / l1 cos 1 t .

Figure 3.3 shows that nondimensional bending stiffness ε 2 strongly influences the steady state deflections. When bending stiffness increases, the steady state deflection increases markedly. When ε 2 is small, there are boundary layers in the steady state deflections. As ε 2 grows larger, the boundary layers become less pronounced and the steady state deflections become bigger and smoother. Approximating the bending stiffness of a poly-ribbed belt typical of vehicle applications by

EIm=−( 1)2.867 × 10−32 Nm ⋅ (where m is the number of ribs), reasonable values of ε fall in the range 0.01≤≤ε 0.12.

Pulley 1 Pulley 2 Pulley 3 Tensioner

Center (0.5525,0.0556m) (0.3477,0.05715m) (0,0) (0.2508,0.0635m)

Radius 0.0889 m 0.0452 m 0.02697m 0.097 m

Other Belt modulus: EA = 120000 N

= = = physical Span lengths: l1 0. 1548 m, l2 0. 3449 m, l3 0. 5518 m

= properties Tensioner spring stiffness: kr 116. 4 N-m/rad

θ = Tensioner rotation at reference state: tr 0. 1688 rad

= Calculate Belt tension at reference state:P0 300 N

d values β = β = Tensioner alignment angles: 1 135. 79 deg, 2 178.74 deg

Table 3.2. Physical properties of the prototypical system.

0.01

-0.01

-0.02 Span 1 deflection

-0.03 4000 3000 1 0.8 2000 0.6 1/ ε 2 1000 0.4 0.2 x1 0 0

-0.01 Span 3 deflection

-0.02 4000 3000 1 0.8 2000 1/ 0.6 ε 2 1000 0.4 0.2 x3 0 0

Figure 3.3. Steady state deflections of spans 1 and 3 for varying belt bending stiffness. = = γ = = = = s 0, ks 4 , 400 , P1 P2 P3 1. 54

0.01

-0.01 Span 1 deflection

-0.02

1 1 0.95 0.8 P 0.6 1 0.9 0.4 0.2 x1 0.85 0

-3 x 10 0

-4

-8

-12 Span 3 deflection

-16

2.5 1 2 0.8 P 0.6 3 1.5 0.4 0.2 x3 1 0

Figure 3.4. Steady state deflections of spans 1 and 3 for varying span tensions. ε = 0.05 , = = γ = s 0, ks 4 , 400 .

Figure 3.4 illustrates the influences of tensions Pi on the steady state deflections.

The unequal ranges of these tensions shown in Figure 3.4 result from calculating span tension variations as the steady accessory torques M ,M (M = − r3 M ) are varied 1 3 3 r1 1

≤ ≤ ⋅ across 0 M1 37.34 N m. Notice that the variation range of the tensions of the two

= spans adjacent to the tensioner P1 P2 is much smaller than that of the third span P3 .

This occurs because, when the accessory torques change, the tensioner assembly compensates for the tension loss or gain to stabilize the tensions of the two spans adjacent to it, [30]. The span steady state increase considerably with decreasing tension.

ε 2 Compared with and Pi , the speed s has smaller influence on the steady state deflections as seen in Figure 3.5. Small steady state changes with speed seem contradictory to physical intuition because increasing the speed means decreasing the effective tension. This phenomenon can be explained by the mechanism of the tensioner, whose main purpose is automatic tension loss compensation to keep variations of the

− 2 + tractive tension of the belt ( Pi s T ) small when the speed is changed, [30].

The tensioner spring stiffness ks has small influence on the steady state deflections over a wide range of variation (Figure 3.6). Although for span 1 the steady state

deflection appears to change significantly when ks increases, these changes are predominantly from rotation of the whole span (which is caused by tensioner rotation and

56 the span boundary condition w (1) = rt cos β θ ). Subsequent results using the coupling 1 l1 1 t indicator confirm this. Figure 3.7 shows that the longitudinal belt stiffness γ has very small influence on the steady state deflections.

Linearization of the general dynamic equations about the steady state configuration shows that the degree of pulley-span vibration coupling is determined mainly by the steady state curvature of each span. For example, the linearized equation for span 3 is

1 ll33222**−−+εγθθl −−++r1 r3 = ()wswPw3,tt 2 3, xt 3 3, xx () w 3, xxxx ( 1 3 wwdxw 3, x 3, x ) 3, xx 0(3.56) ll l333 ll∫ 0

1 = − 2 + * ****2=−γθr1 +r3 θ + 1 where P3 P3 s P3 , P3133,[ ()wdx x] , the asterisk denotes the ll33∫ 0 2

θ θ steady state configuration, and w3 , 1 , and 3 are small vibrations about steady state (for

θ θ this equation only). Looking at the 1 and 3 terms coupling the span and pulley

* vibrations, the steady state curvature w3,xx governs the magnitude of coupling. Returning

to the notation of wi (x) representing steady state deflections, one can define the steady

1 Γ = 2 state parameter i wi,xx dx as the coupling indicator for each span and the sum ∫0

3 3 1 Γ = Γ = 2 Γ i w dx as the coupling indicator for the whole system. Increasing ∑ ∑ ∫0 i, xx i=1 i=1 indicates increasing coupling between the rotations of the pulleys and transverse motions of the belt.

0.02

0.01

-0.01 Span 1 deflection

-0.02 0.8 0.6 1 0.8 0.4 s 0.6 0.4 0.2 1 0.2 x 0 0

-3 x 10 0

-5

-10

Span 3 deflection -15

0.8 0.6 1 0.8 0.4 s 0.6 0.2 0.4 0.2 x3 0 0

ε = = Figure 3.5. Steady state deflections of spans 1 and 3 for varying speed. 0.05 , ks 4 , γ = = = = 400 , P1 P2 P3 1. 58

0.01

-0.01 Span 1 deflection -0.02

60 50 1 40 0.8 30 0.6 k s 20 0.4 10 0.2 x1 0

0 -0.002

-0.006

-0.01

-0.014 Span 3 deflection

-0.018 60 50 1 40 0.8 k 30 0.6 s 20 0.4 10 0.2 x3 0

Figure 3.6. Steady state deflections of spans 1 and 3 for varying tensioner spring ε = = γ = = = = stiffness. 0.05 , s 0 , 400 , P1 P2 P3 1. 59

0.005

-0.01

Span 1 deflection -0.02

2000

1500 1 0.8 1000 0.6 γ 0.4 500 0.2 x1 0

-0.002

-0.006

-0.01 Span 3 deflection -0.014

2000

1500 1 0.8 1000 0.6 γ 0.4 500 0.2 x3 0

Figure 3.7. Steady state deflections of spans 1 and 3 for varying longitudinal belt ε ======stiffness. 0.05 , s 0 , ks 4 , P1 P2 P3 1.

ε 2 Figures 3.8 and 3.9 show that belt bending stiffness and span tensions Pi strongly affect coupling. Generally, large ε 2 and small tensions result in a more beam- like belt with relatively large steady state span deflections and strong pulley-span vibration coupling. In such cases, bending stiffness cannot be ignored.

Γ perturbation

10 exact

5 Coupling Indicator,

0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 ε

Figure 3.8. System coupling indicator Γ for varying belt bending stiffness. s = 0 , = γ = = = = ks 4, 400 , P1 P2 P3 1.

4.2

4.1

4 Γ

3.9

3.8 perturbation

3.7

Coupling Indicator, exact 3.6

3.5

3.4 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 P (While P varies from 1 to 0.8119) 3 1

Figure 3.9. System coupling indicator Γ for varying span tensions. ε = 0.015 , s = 0 , = γ = ks 4, 400 .

Belt speed s weakly affects coupling for properly designed systems, but the effect of speed can rise if the tensioner is not properly designed. These results are shown in

Figures 3.10 and 3.11 where η = 0.78 corresponds to a well designed system (as used for all other results in this chapter) and η = 0 is a poorly designed system. η is the tensioner effectiveness, and it indicates the ability of the tensioner to maintain constant tractive belt

∂T tension in response to changes in belt speed, [30]. Mathematically, η = at s = 0 , ∂(s 2 )

ε = 0 (an analytical approximation is derived subsequently). η is close to unity for well designed systems, while small η indicates poor tensioner design. Figure 3.11 shows the variation of tension for ranges of ε and s2 for the values η = 0.78 and η = 0 (the change

η β in is induced by adjusting the tensioner orientation angles 1,2 ). For the well-designed system η = 0.78 , the surface remains nearly planar and Ts≈η 2 over the entire region.

For η = 0 , the surface is clearly curved except for ε ≈ 0 . In this case, Ts≈η 2 is a poor approximation for ε > 0 ; η is no longer effective to describe the change of tension with respect to speed changes. Physically, the reason behind this phenomenon is that when tensioner effectiveness is small, the system cannot maintain constant tractive tension through tensioner rotation as speed increases. Increased speed then decreases the tractive tension and, because of the non-zero bending stiffness, enlarges the belt span deflections

(string model steady state always have zero span deflections and are unaffected by decreased tension). The increased span deflections in turn cause additional tension due to the increased span lengths. Thus, when bending stiffness is considered, η cannot correctly describe the relationship between tension and speed for poorly designed systems.

Γ γ The plots of versus tensioner spring stiffness ks and longitudinal belt stiffness are not shown because coupling indicator is a weak function of these quantities. When

ε = = = = = Γ 0.015, s 0, P1 P2 P3 1, the value of varies less than 3% over the ranges of

γ ks and corresponding to that in Figures 3.6 and 3.7.

3.5 Approximate Closed-Form Solution

When the belt bending stiffness is small, which means ε << 1 (this is the usual case for practical serpentine belt systems), an approximate solution can be obtained using singular perturbation. This closed-form solution shows explicit dependence of the steady state on system parameters. It also leads to a simple equation that reveals quantitative relations between the coupling indicator and the key design variables.

First, let us investigate the linearized equations of the steady state system (3.42)-

(3.44). When nonlinear terms are neglected, one has the linear system

ε 2 ( l )2 w − (P − s 2 )w = 0 i = 1,2, 3 0 < x < 1 (3.57) li i,xxxx i i,xx

[−(P − s 2 )w (1) + ε 2 ( l ) 2 w (1)]cos β 1 1,x l1 1,xxx 1

−[−(P − s 2 )w (0) + ε 2 ( l ) 2 w (0)]cos β 2 2,x l2 2,xxx 2

+ − 2 β − − 2 β − θ = (T s ) sin 1 (T s )sin 2 ks t 0 (3.58)

+ + l1 l2 l3 T + rt β − β θ = l γ l (sin 1 sin 2 ) t 0 (3.59)

Boundary conditions for the wi are still equations (3.27)-(3.29). While readily solvable, the linear system approximation is unsatisfactory as seen in Figures 3.12 and 3.13. Thus, the steady state problem is nonlinear in essence, and the linear system does not give an effective approximation.

Returning to the nonlinear model, the variables are represented as

T = T (0) +εT (1) +... (3.60)

θ =θ (0) +εθ(1) + t t t ... (3.61)

w = wc + ... (3.62) i i where wc is the leading composite combination of the inner and outer solution of the ith i span. Substitution of equations (3.60)-(3.62) into equation (3.42) yields equations for the spans.

First consider span 1, which has the equation

ε 2 ( l ) 2 w c − ( P − s 2 + T ( 0 ) + εT (1) )w c = 0, 0 < x < 1 (3.63) l1 1, xxxx 1 1, xx

The outer expansion has the form

O = + ε + ε 2 + w1 y0 (x) y1 (x) y2 (x) ... (3.64)

Substitution of (3.64) into (3.63) gives

= y0,xx 0 (3.65)

− 2 + (0) = − (1) (P1 s T )y1,xx T y0,xx (3.66)

(P − s 2 + T (0) )y = ( l ) 2 y − T (1) y (3.67) 1 2,xx l1 0,xxxx 1,xx

Solution of these problems in sequence yields

O = + + ε + + ε 2 + w1 D0 D1x (D2 D3 x) (D4 D5 x) (3.68)

where D0 ~ D5 are arbitrary constants. This outer expansion is not required to satisfy any boundary conditions. It must be matched with two boundary layer expansions, one valid near x = 0 and the other valid near x = 1.

To find the inner expansion near x = 0, one introduces the stretching transformation

x ξ = λ > 0 (3.69) ε λ

Denoting the inner expansion near x = 0 by the superscript i , (3.63) becomes

4 i 2 i − λ d w − λ d w ε 2 4 l 2 1 − − 2 + (0) + ε (1) ε 2 1 = ( ) (P1 s T T ) 0 (3.70) l1 dξ 4 dξ 2

ε → λ = i As 0 , the distinguished limit corresponds to 1. Then, w1 is governed by

d 4 wi d 2 wi d 2 wi l l 2 1 − − 2 + (0) + ε (1) 1 = i = 1 = ε 2 1 ( ) (P1 s T T ) 0 , w1 (0) 0 (0) ( ) (3.71) l1 ξ 4 ξ 2 ξ 2 d d d r1

i = ξ + ε ξ + ε 2 ξ With the inner expansion w1 W0 ( ) W1( ) W2 ( ) , (3.71) gives

d 4W d 2W l 2 0 − − 2 + (0) 0 = = = ( ) (P1 s T ) 0, W0 (0) 0 W0,ξξ (0) 0 (3.72) l1 dξ 4 dξ 2

d 4W d 2W d 2W l 2 1 − − 2 + (0) 1 = (1) 0 = = ( ) (P1 s T ) T , W1(0) 0 W1,ξξ (0) 0 (3.73) l1 dξ 4 dξ 2 dξ 2

d 4W d 2W d 2W l l 2 2 − − 2 + (0) 2 = (1) 1 = = 1 ( ) (P1 s T ) T , W2 (0) 0 W2,ξξ (0) (3.74) l1 ξ 4 ξ 2 ξ 2 d d d r1

The general solution of (3.72) is

−ξ ( l1 ) P −s2 +T (0) ξ ( l1 ) P −s2 +T (0) ξ = + ξ + l 1 + l 1 W0 ( ) a0 b0 c0e d0e (3.75)

ξ ξ The constant d0 must be zero; otherwise, W0 ( ) would grow exponentially with , making it unmatchable with the outer expansion. The boundary conditions in (3.72) lead

ξ = ξ to W0 ( ) b0 . Subsequent solution of (3.73) and (3.74) for W1 and W2 yields the inner expansion

l 2 (0) l −ξ ( 1 ) P −s +T i ξ = ξ + ε ξ + ε 2 ξ − l 2 1 1 − l 1 w1 ( ) b0 b1 [b2 ( l ) ( r ) − 2 + (0) (1 e )] (3.76) 1 1 P1 s T where exponential growth terms have been eliminated.

With use of (3.69) and λ = 1, the inner expansion of (3.68) is

O i = + ε ξ + + ε 2 ξ + (w1 ) D0 (D1 D2 ) (D3 D4 ) (3.77)

The outer expansion of (3.76) as ε → 0 is

i O = ξ + ε ξ + ε 2 ξ − l 2 l1 1 (w1 ) b0 b1 [b2 ( l ) ( r ) − 2 + (0) ] (3.78) 1 1 P1 s T

O i = i O The matching principle (w1 ) (w1 ) yields

======− l 2 l1 1 D0 0, b0 0 , b1 D1 , D3 b2 , D2 0, D4 ( l ) ( r ) − 2 + (0) (3.79) 1 1 P1 s T

For the boundary layer near x = 1, the stretching transformation is ζ = (1− x) / ε .

Denoting the inner expansion of (3.63) near x = 1 by the superscript I , the governing equation as ε → 0 is

d 4 wI d 2 wI l 2 1 − − 2 + (0) + ε (1) 1 = ( ) (P1 s T T ) 0 (3.80) l1 dζ 4 dζ 2

i I Application of similar procedures as for w1 to w1 yields

w I = rt cos β θ (0) + B ζ + ε (B ζ + rt cos β θ (1) ) 1 l1 1 t 0 1 l1 1 t

l (3.81) l −ζ 1 P −s2 +T (0) + ε 2 ζ + l 2 1 1 − l 1 [B2 ( l ) ( r ) − 2 + (0) (1 e )] 1 2 P1 s T

where B0 , B1 , and B2 are arbitrary constants.

The inner expansion near x = 1 of the outer solution (3.68) is

O I = + + ε − ζ + + + ε 2 − ζ + + (w1 ) D0 D1 [ D1 D2 D3 ] [ D3 D4 D5 ] (3.82)

O I = I O The matching condition (w1 ) (w1 ) leads to

r D + D = rt cos β θ (0) , − D = B , D + D = 3 cosβ θ (1) 0 1 l1 1 t 1 1 2 3 l1 1 t

− = = + = l 2 l1 1 D3 B2 , B0 0 , D4 D5 ( l ) ( r ) − 2 + (0) (3.83) 1 2 P1 s T

Solution of equations (3.83) and (3.79) gives

D = 0 , D = rt cos β θ (0) , D = 0 , D = rt cos β θ (1) 0 1 l1 1 t 2 3 l1 1 t

2 l r = − l 1 1 = l 2 1 l1 + l1 = = t β θ (0) D4 ( l ) ( r ) − 2 + (0) , D 5 ( l ) − 2 + ( 0 ) ( r r ) , b0 0 , b1 l cos 1 t 1 1 P1 s T 1 P1 s T 1 2 1

b = rt cos β θ (1) , B = 0 , B = − rt cos β θ (0) , B = − rt cos β θ (1) (3.84) 2 l1 1 t 0 1 l1 1 t 2 l1 1 t

The composite expansion for span 1 is

c = O + i + I − i O − I O w1 ~ w1 w1 w1 w1 (w1 ) (w1 ) x l 2 (0) x−1 l 2 (0) l l l l − ( 1 ) P −s +T l ( 1 ) P −s +T = ε 2 l 2 1 − 1 + 1 + 1 + 1 ε l 1 − 1 ε l 1 ( l ) − 2 + (0) [ r ( r r )x r e r e ] 1 P1 s T 1 1 2 1 2 (3.85) + rt cos β (θ (0) + εθ (1) )x l1 1 t t

Similar perturbation processes for w2 and w3 yield

x l 2 (0) x −1 l 2 (0) l l l l − ( 2 ) P −s +T l ( 2 ) P −s +T c = ε 2 l 2 1 2 + − 2 − 2 − 2 ε l 2 + 2 ε l 2 w2 ~ w2 ( l ) − 2 + (0) [ r ( r r )x r e r e ] 2 P2 s T 2 2 3 2 3 (3.86) − rt cos β (θ (0) + εθ (1) )(x −1) l2 2 t t

l 2 (0) − l 2 (0) l l l l − x ( 3 ) P −s +T l x 1( 3 ) P −s +T c = ε 2 l 2 1 − 3 + 3 − 3 + 3 ε l 3 + 3 ε l 3 w3 ~ w3 ( l ) − 2 + (0) [ r ( r r )x r e r e ](3.87) 3 P3 s T 3 3 1 3 1

Substitution of (3.60), (3.61), (3.85), and (3.86) into the tensioner arm equation

(3.43) gives the following conditions corresponding to the ε 0 and ε 1 terms, respectively,

− 2 + (0) β − β ( s T )(sin 1 sin 2 ) (3.88) − [k + rt cos2 β (P − s 2 + T (0) ) + rt cos2 β (P − s 2 + T (0) )]θ (0) = 0 s l1 1 1 l2 2 2 t

T (1) (sin β − sin β ) −[k + rt cos2 β (P − s 2 + T (0) ) + rt cos2 β (P − s 2 + T (0) )]θ (1) 1 2 s l1 1 1 l2 2 2 t

− ( rt cos 2 β + rt cos 2 β )T (1)θ (0) = 0 (3.89) l1 1 l2 2 t

Substitution of (3.60), (3.61), and (3.85)-(3.87) into (3.41) yields the ε 0 and ε 1 conditions

++ 11ll123 l(0)ll 1 rtt 2 2 r 2 (0) 2 r t (0) γ T −+[(cos)ββθββθ (cos)]( ) +−= (sin sin) 0 (3.90) llllll2 1212tt 12

++ 1 ll123 l(1)ll 1 rtt 2 2 r 2 (0) (1) r t (1) γ T −+[(cos)ββθθββθ (cos)] +−= (sin sin) 0 (3.91) lllll1212tt l 12 t

Exact Approximate θ θ T T Exact t Approximate t Case 1 0 .2828 0.2836 −0.0106 − 0.0113 Case 2 0 .6408 0.6417 −0.0247 − 0.0255

Table 3.3 Comparison of approximate analytical and numerical solutions. Case 1 parameters are those used in Figure 3.12, and Case 2 parameters are those used in Figure 3.13.

(0) θ (0) Generally, T and t must be solved numerically from the nonlinear algebraic

(0) θ (0) equations (3.88) and (3.90). After solving for T and t , higher order terms can be

(1) θ (1) calculated successively, like the T and t are obtained by solution of (3.89) and

(1)==θ (1) (3.91) (which give T t 0 ). These values complete the leading order composite span solutions of (3.85), (3.86), and (3.87). Representative agreement between analytical and numerical solutions are shown in Figures 3.8-3.10, 3.12-3.13, and listed in Table 3.3.

Note if the bending stiffness vanishes, then the analytical results reduce to those for the string model. For example, according to (3.87), the deflection in span 3 is zero throughout the span when ε is zero; also see (3.85) and (3.86). When the bending stiffness approaches zero, no matter how small it is, boundary layers always exist at the ends of each span, although their thickness and height are very small, as seen in Figures

3.3 and 3.12.

Substitution of (3.85)-(3.87) into the definitions of system and span coupling indicator gives

3 3 ε l l Γ = Γ 1 l i 2 + i 2 i ~ [( ) ( ) ] (3.92) ∑ ∑ 2 (0) i=1 i=1 2 − + l r r Pi s T i si ei

where rsi and rei are the radii of the two pulleys at the ends of the ith span. Analytical and numerical solutions agree well as shown in Figures 3.8, 3.9, and 3.10. For general serpentine belt systems having n spans, generalization of (3.92) gives

1 n ε l l l Γ ~ [( i ) 2 + ( i ) 2 ] (3.93) ∑ 2 (0) 2 i=1 − + l r r Pi s T i si ei

Note that from (3.93), when the bending stiffness is zero (ε = 0), then Γ = 0 . This corresponds to the string model where there is no steady state curvature.

7.5

6.5 Γ

5.5 perturbation η=0 exact 5 Coupling Indicator, 4.5 perturbation exact 4 η=0.78

3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 s

Γ ε = = Figure 3.10. System coupling indicator for varying speed. 0.015 , ks 4, γ = = = = η = β = o 400 , P1 P2 P3 1, 0.78 corresponds to tensioner orientation 1 135.79 , β = o η = 2 178.74 in Table 3.2; while 0 corresponds to tensioner orientation: β = o β = o 1 68.53 , 2 111.47 .

0.6

0.5

T 0.4

0.3

Tension, 0.2

0.1

0 0.8 0.6 0.015 0.4 0.01 s 2 0.2 0.005 ε 0 0

0.2

0.15 T

0.1 Tension, 0.05

0 0.8 0.6 0.015 0.4 0.01 s 2 0.2 0.005 ε 0 0

= γ = Figure 3.11. Variation of tension with different tensioner orientation. ks 4, 400 , = = = β = o β = o η = β = o P1 P2 P3 1, a) 1 135.79 , 2 178.74 , 0.78 , b) 1 68.53 , β = o η = 2 111.47 , 0 .

-3 7 x10

5 linear

4 exact perturbation 3

2 Span 1 deflection

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1

-4 x 10 2

-2

-4 perturbation -6 exact linear -8 Span 3 deflection

-10

-12

-14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x3

Figure 3.12. Steady state deflections (dimensionless) of the first and third spans. ε = = = γ = = = = 0.01, s 0.6 , ks 4 , 400 , P1 P2 P3 1.

-3 x 10 20

perturbation 10 linear exact

5 Span 1 deflection

-5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x1

-4 x 10 2

-2

-4 exact perturbation

-6

Span 3 deflection linear -8

-10

-12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x3

Figure 3.13. Steady state deflections (dimensionless) of the first and third spans.ε = 0.01, = = γ = = = = s 0.9 , ks 4 , 400 , P1 P2 0.9395 , P3 1.5395.

From (3.60) and (3.61), the leading order tension and tensioner rotation approximations

(0) θ (0) ε 2 (0) (T and t ) are independent of the belt bending stiffness , which means that T

θ (0) and t are determined by the string model of the serpentine belt drives. For string models, former researchers, [13,30] have indicated that

dT T (0) ≈ ηs 2 , η = at s 2 = 0 (3.94) d(s 2 ) where η is the tensioner effectiveness, as mentioned before. Equations (3.88) and (3.90) lead to

η = 1 1 2 β + 1 2 β + (3.95) l + l + l  P1 ( l cos 1 l cos 2 ) k s  1 2 3  1 2  + 1 γ (sin β − sin β ) 2  1 2  which corresponds to that defined in [30].

Further comparison with numerical solutions shows that the above approximation is good for properly designed system, but the approximation becomes poor if η is away from unity (Figure 3.10). In such cases, T (0) ≈ ηs 2 is no longer the dominant part of T due to the poor speed compensation ability of the tensioner, as mentioned previously; bending stiffness significantly impacts tension. For good approximation for η ≈ 0 , more terms need to be incorporated in (3.60) and (3.61) (with no change in (3.62) for approximation through ε 4 ). This leads to additional equations like (3.89) and (3.91).

Approximation through ε 4 gives excellent agreement with Figure 3.11b where η = 0 .

Considering well-designed systems, substitution of (3.94) into (3.93) leads to

1 n ε l l l Γ ~ [( i ) 2 + ( i ) 2 ] (3.96) ∑ 2 2 i=1 − −η l r r Pi (1 )s i si ei

The simple relation (3.96) confirms the coupling indicator’s dependence on bending stiffness, tensions and speed as shown by the numerical solution. The effect of speed is small due to the small coefficient 1−η preceding it; the influence of speed would rise as

η gets smaller. The effects of tensioner spring stiffness ks and longitudinal belt stiffness

γ are very small because their impact is felt only through the tensioner effectiveness η .

Equations (3.93) and (3.96) reveal that the span-pulley vibration coupling is determined primarily by three factors: the system geometry (especially the ratios of the pulley radii to the span lengths), the bending stiffness, and the span tensions. Large Γ is necessary for strong vibration coupling between the spans and pulleys; reduced Γ prevents pulley rotations from generating undesirable span vibration. By tuning the system design variables, the degree of coupling Γ can be adjusted. The most effective methods to reduce this coupling are: (1) decreasing the belt bending stiffness, (2) increasing the span tensions, which are determined by the initial tensioner torque and the accessory torques exerted on the pulleys, and (3) increasing the ratio between the pulley radii and the span lengths. Higher bearing loads negatively balance the benefits of higher span tensions. The increased bearing loads occur at all pulleys, even though the troublesome coupling is typically concentrated at a single span. Increasing a pulley radius

77 can be an effective, low cost solution to reduce a practical span vibration problem where pulley rotations drive large span vibrations. This solution is localized to address only the problem span vibration.

CHAPTER 4

DYNAMIC ANALYSIS OF SERPENTINE BELT DRIVES

WITH BENDING STIFFNESS

In this chapter, a method is developed to evaluate the natural frequencies and vibration modes of serpentine belt drives where the belt is modeled as a moving beam with bending stiffness. Inclusion of bending stiffness leads to belt-pulley coupling not captured in moving string models. New dynamic characteristics of the system induced by belt bending stiffness are investigated. The belt-pulley coupling is studied through the evolution of the vibration modes. When the belt-pulley coupling is strong, the dynamic behavior of the system is quite different from that of the string model where there is no such coupling. The effects of major design variables on the system are discussed. The spatial discretization can be used to solve other hybrid continuous/discrete eigenvalue problems.

4.1 Introduction

By incorporating the belt bending stiffness, chapter 3 presents a model for a simplified three-pulley serpentine drive (Figure 4.1). In this chapter, the linear free vibration about the nontrivial steady state obtained in chapter 3 is investigated. In contrast, Beikmann et al. [30,31] investigate the free vibration analysis based on the string assumption for the belt. Zhang et al. [32,33] build on this string model by adding damping and give a complex modal analysis of the serpentine belt drive system. Parker

[34] develops a spatial discretization of this model extended to n pulleys.

Each span is an Euler elastica [62] moving with constant speed. Modeling the belt as a moving beam shows that transverse vibration of every span is linearly coupled with the rotations of the two adjacent pulleys at its ends. The degree of span-pulley coupling depends on the steady state curvature of the span. Further, a coupling indicator is defined for each span to quantify coupling strength. In last chapter [63], the attention is focused on analytical and numerical methods to calculate the steady motion and the effects of design parameters on the steady state and coupling indicator.

Belt-pulley coupling from bending stiffness is more consistent with observed automotive serpentine drive vibration problems than prior models. In these cases, span vibrations most commonly occur at low (engine idle) speeds and at the engine firing frequency from pulsations to the crankshaft pulley. This precludes belt or pulley runout as root causes because they lead to span vibrations at frequencies other than the firing frequency. Parametric instability from tension or speed fluctuations [14,15,64] occurs in practice only at high engine speeds.

Based on the moving beam-pulley model in chapter 3 [63], this chapter investigates serpentine belt drive dynamic analysis. Computation of the natural frequencies and vibration modes is a central task. Serpentine belt drives belong to the class of hybrid continuous/discrete systems, and solving the eigenvalue problem for such a system is challenging. For the simpler case of a string model of the belt, three papers [31,32,65] develop numerical methods to solve the serpentine drive eigenvalue problem. The first two methods [31,32] fail when the spans are modeled as moving beams because both require the explicit solution form for axially moving continua while no such form exists for a traveling beam [8]. Both methods retain the continuum model and seek roots of a numerically ill-behaved characteristic equation. The numerical singularities (see [65]) can lead to missing or false roots and significant computational expense to try to avoid these errors. To address these issues, the third method [65] discretizes the two spans adjacent to the tensioner and uses Lagrange multipliers to enforce the geometric boundary conditions at the belt-tensioner interface. The method is presented for a general n-pulley drive.

One of the main developments of this chapter is a spatial discretization to solve the serpentine belt drive eigenvalue problem with a moving beam model. Compared with characteristic equation methods [31,32], the present approach can incorporate bending stiffness, is numerically economical with dramatic reduction in computation time, avoids numerical singularities, and does not require advance estimation of the natural frequency bandwidth. After a coordinate transformation and some mathematical modifications, the governing equations are rewritten into an extended operator form that retains the mathematical structure of a gyroscopic continuum. Galerkin discretization is readily

81 applied in this extended operator context. Although the reformulation initially seems complicated, the key ideas are such that the method is straightforward to implement when devising code. The concepts can be naturally extended to other hybrid continuous/discrete systems.

The three-pulley system in Figure 4.1 is used to demonstrate the method and results. The method can be extended to multi-pulley serpentine drive systems. A comprehensive, multi-pulley analysis code based on the method has been developed for use in the automotive industry.

The relationship between belt-pulley coupling and bending stiffness is investigated from the perspective of evolution of the vibration modes. When the bending stiffness is appreciable, all the continuous (span) and discrete (pulley) components interact strongly with each other, and the former classification of the modes for the string model (pulley rotationally dominant vibration modes and span transversely dominant vibration modes

[31,32,65]) does not apply. When the bending stiffness is small, the dynamic behavior converges to that of string models. Finally, the effects of key design parameters on the natural frequencies are investigated.

4.2 Linearization of Equations of Motion

Figure 4.1 depicts the prototypical three-pulley system used in chapter 1, which includes the primary components in automotive serpentine drives [31-33]. The spans are modeled as Euler-Bernoulli beams translating with constant speed c. Each span is subjected to constant moments at its ends arising from the bending of the continuous belt around the pulleys. Movement of the belt-pulley contact point due to belt vibration is neglected [23,24,66]. Detailed description of the model is given in last chapter [63].

Some essential equations and some key concepts are repeated here. Hamilton’s principle applied to the prototypical serpentine belt drive leads to the equations of motion

K ( r r (1) t M parallel to w S spans 1,2 u 1 J 1 2 θ 2 ( u M (2) 2 β E w β 1 2 2 l 1 θ 1

J l ( 3 2 ( (1) M (2) M 1 M M E J 3 θ 3 S 1 w

3 ( (3) u l ( M 3 3 c M (3) S E

Figure 4.1. A prototypical three-pulley serpentine belt drive.

−+2 −++ = = mw(2itt,,, cw ixt cw ixx )[()] P i Pw i ix ,,, x EIw ixxxx 0 i 1,2, 3 (4.1)

= = EI = θ β = − EI w1(0,t) 0 EIw1,xx (0,t) w1(l1,t) rt t cos 1 EIw1,xx (l1 ,t) (4.2) r1 r2

= θ β = − EI = = EI w2 (0,t) rt t cos 2 EIw2,xx (0,t) w2 (l2 ,t) 0 EIw2,xx (l2 ,t) (4.3) r2 r3

= = EI = = EI w3 (0,t) 0 EIw3,xx (0,t) w3 (l3,t) 0 EIw3,xx (l3 ,t) (4.4) r3 r1

θ +− = J1 11131Pr Pr M1 (4.5)

θ −+ = JPrPr22 12 22 0 (4.6)

θ −+= J3 32333Pr Pr M3 (4.7)

θθ ++ +−+22 +ββ + − Jktt rt[()( mcwlPmcPwlEIwlr1,1 t 1 1 )()()]cos( 1,1 x 1,1 xxx t 1 mcPr 1 )sin t 1

−+−++22ββ −−= [mcw2,txxxxtt(0) ( P 2 mc P 2 ) w 2, (0) EIw 2, (0)] r cos 2 ( mc P 2 ) r sin 2 0 (4.8)

l =−+−EA θθθβ +1 1 2 P1221111,( rrrttsin wd xx) (4.9) l1 ∫ 0 2

l =−+EA θθθβ + +2 1 2 P233222( rrrttsin wd 2, xx) (4.10) l2 ∫ 0 2

l =−+EA θθ +3 1 2 P311333,()rr wdx x (4.11) l3 ∫ 0 2

θ and Mti () are dynamic accessory torques. All dynamic motions (pulley rotations i (t),

θ tensioner arm rotation t ()t , transverse belt deflections (wi (xi ,t ) )) are measured relative to the reference state corresponding to a stationary, string model system subject to any

84 steady accessory torques [63]. Different spans may have different reference tensions Pi

β due to the accessory torques. 1,2 are the orientation angles of the tensioner arm relative to the two spans adjacent to the tensioner in the reference state (Figure 4.1).

The following non-dimensional variables are introduced

+ + x wi l l l P P 2 EI m xˆ = i wˆ = l = 1 2 3 ˆ = o Pˆ = i ε = s = c i i tt 2 i 2 li li 3 ml Po Pol Po

~ k EA J J M k = r γ = = i = t = i s mi 2 mt 2 M i (4.12) Port Po mri l mrt l Po ri

where Po is the uniform tension at zero speed with no accessory torques using a string model [63]. Substitution of (4.12) into (4.1)-(4.11) leads to the non-dimensional dynamic equations of motion for the prototypical serpentine belt system, from which the equations governing the steady motion can be obtained by equating time derivative terms to zero.

Methods to determine the steady motion and its properties are discussed in last chapter

[63].

Linearization for small motions about the steady state configuration yields the

θ θ following non-dimensional equations, where i (t), t ()t , wi (xi ,t) now represent small vibrations about the steady motion (not about the reference state as in (4.1)-(4.11)); steady motion quantities are denoted by asterisks. The hats on dimensionless variables have been dropped. The span vibration equations are

()()ll11222ws−−+ 2()() ll 11 w () l 1 Pwε ()()l l 1 w rltttt1,ttxtxxxxx lr 1, r 1 1, l1 r 1, x

1 −−+−γθθθβlr12 r 1 rt +** = ( )(21txx sin 1wwdxw 1,1, ) 0 (4.13) rlt 111 l l ∫ 0 1,xx

w (0,t) = 0 w (1,t) = rt cos β θ w (0,t) = 0 w (1,t) = 0 (4.14) 1 1 l1 1 t 1,xx 1,xx

()()ll22222ws−−+ 2()() ll 22 w () l 2 Pwε ()()l l 2 w rltttt2,tt lr 2, xt r 2 2, xx l2 r 2, xxxx

1 −−++γθθθβlr22rr3 t +** = ( )(32txxxx sin 2wwdxw 2,2,2, ) 0 (4.15) rlt 222 l l ∫ 0

w (0,t) = rt cos β θ w (1,t) = 0 w (0,t) = 0 w (1,t) = 0 (4.16) 2 l2 2 t 2 2,xx 2,xx

()()ll33222ws−−+ 2()() ll 33 w () l 3 Pwε ()()l l 3 w rltttt3,tt lr 3, xt r 3 3, xx l3 r 3, xxxx

1 −−++γθθlr33r1 ** = ( )(1 3wwdxw 3,xx 3, ) 3, xx0 (4.17) rlt 33 l ∫ 0

= = = = w3 (0,t) 0 w3 (1,t) 0 w3,xx (0,t) 0 w3,xx (1,t) 0 (4.18)

1 =−+2*****=−γθrr21 + θθ −rt β + 1 *2 P11Ps P 1, P12111,[ txsin (wd ) x] (4.19) lll111 ∫ 0 2

1 =−+2*****=−γθrr32 +r θ +t θ β + 1 *2 P22Ps P 2, P23222,[ txsin (wd ) x] (4.20) lll222 ∫ 0 2

1 =−+2*****2=−γθr1 +r3 θ + 1 P33Ps P 3, P3133,[ ()wdx x] (4.21) ll33∫ 0 2

The pulley and tensioner arm equations are

1 rrrr1121θγ+−+− θ θθβγrt + r 1* m1() 1,tt ()( 2 1 t sin) 1 () wwd 1, x 1, x x rrllltt111 r t∫ 0

1 −−+−γθθγrr11r3 r 1* = r 1 ( )(1 33,3, ) ( )w xxwdx ( ) M1 (4.22) rlttt33 l r∫ 0 r

1 rr22θγ+−++rr3 θ r 2 θt θβγ + r 2* m22,()tt ()( 3 2 t sin) 2 () wwd 2,2, x x x rrllltt222 r t∫ 0

1 −−+−γθθθβγrr22 r 1rt − r 2 * = ( )(21txx sin 1 ) ( ) wwdx 1,1, 0 (4.23) rltt111 l l r∫ 0

1 rr33θγ+−++r1 θθγ rr 33* m33,()tt ()( 1 3 ) () wwd 3,3, x x x rrllrtt33 t∫ 0

1 −−++γθθθβγrr33r2 rt − r 3* = r 3 ( )(3 2txx sin 2 ) ( )w 2,wdx 2, ( ) M3 (4.24) rlttt222 l l r∫ 0 r

1 θ +++−+−+lrr1 β β γβ21 θθθβrt ** mstt( ) cos11, w t (1) P 1 cos 11, w x (1) cos 1 ( 2 1 t sin 1 wwdxw 1,1, x x ) 1, x (1) llll111 ∫ 0

1 −−−−+++l2 β β γβθθθβrr3 r2 t ** swPw( )cos22,tx (0) 2 cos 22, (0) cos 2 ( 3 2 txxx sin 2 wwdxw 2,2, ) 1, (0) llll222 ∫ 0

− ε 2 ( l )2 cos β w (1) + ε 2 ( l )2 cos β w (0) l1 1 1,xxx l2 2 2,xxx

1 −−γ rr21θθθ + −rt ββ + * (21txx sin 1wwdx 1,1,1 )sin lll111 ∫ 0

1 +−γθθθβrr32 +r +t +* β + θ = ( 32txxstsin 2wwdx 2,2,2 )sin k 0 (4.25) lll222 ∫ 0

Equations (4.13)-(4.25) reveal that the transverse vibrations of all spans are coupled with the pulley rotational motions, in contrast to string models for the belt. Notice that equation (4.17) for span 3 shows that its transverse vibration is now coupled with the two adjacent pulleys’ rotational motions and the degree of the span-pulley coupling is

* determined by its steady state curvature w3,xx [63]. If there is no bending stiffness, this span (and any others between fixed center pulleys in an n pulley system) remains a

* = straight line with w3,xx 0 at steady state, meaning that its motion is completely decoupled from the rest of the system. This expanded coupling is the primary ramification of including belt bending stiffness.

4.3 Extended Operator Formulation

The above system can be expressed in the extended operator form [31,32,65]

MW + GW + KW = F (4.26) where the displacement vector, external force vector, and inner product are

= θ θ θ θ T W {w1,w2,w3, 1, 2, 3, t } (4.27)

F = {0,0,0,( r1 )M ,0,( r3 )M ,0}T (4.28) rt 1 rt 3

3 < >= 1 + 1 + 1 + θ σ +θ σ W,U w1u1dx w2u2 dx w3u3dx i i t t (4.29) ∫0 ∫0 ∫0 ∑ i=1 and the overbar means complex conjugate. The differential operators M and K are symmetric while G is skew-symmetric. Therefore, the above linear model constitutes a conservative gyroscopic system. The factors ( li ) in the span equations and ( ri ) in the rt rt pulley equations are necessary to preserve the symmetric/skew-symmetric properties of

M , G, and K .

Although the above model is linear, solution of the corresponding eigenvalue problem is difficult because of the belt-pulley coupling in the differential equations, belt- tensioner coupling in the boundary conditions (see (4.14) and (4.16)), multiple spans, and gyroscopic character. Even for the simpler case of modeling the spans without bending

88 stiffness, which eliminates the belt-pulley coupling, solution is difficult. In that case, the main obstacle lies in the inhomogeneous boundary conditions (4.14) and (4.16) that result from rotation of the tensioner arm moving the endpoints of the two adjacent spans.

Three distinct methods have been presented for the eigenvalue problem of the string model. The first one is by Beikmann et al. [31], who determine a boundary condition error function akin to a characteristic equation. The second approach is by Zhang and Zu

[32], who established a closed-form characteristic equation, from which the eigenvalues for the belt drive system are numerically computed. The above two methods, however, can not be used to attack the eigenvalue problem presented here because both methods require the explicit solution form for an axially moving string [47]. For the present model, no such explicit solution exists [8]. Furthermore, both of the two methods need a pre-specified bandwidth to search for roots of the characteristic equations. Singularities and numerical ill-behavedness of both characteristic equations complicate the root finding process [31,65], leading to time consuming calculations, especially when coded for general multi-pulley systems. Due to the numerical concerns, the accuracy and completeness of the calculated natural frequencies can not be guaranteed (that is, some computed natural frequencies may be false and true natural frequencies in the specified range may be missed).

In [65], a third method is developed to solve the string model eigenvalue problem.

There the spans adjacent to the tensioner (which are the only ones coupled to the pulleys) are expanded in a series of basis functions. The inhomogeneous boundary conditions at the belt-tensioner interface are treated as constraints and the Lagrange multiplier method

89 is applied to impose them. This method overcomes the drawbacks of the first two methods and could be extended to models with bending stiffness. A result of the

Lagrange multiplier approach is that the discretized matrices lose the symmetric/skew- symmetric properties of a conservative gyroscopic system, although this does not influence the accuracy of the results.

A different technique is developed in this work to solve the eigenvalue problem of the moving beam model. The key concepts are to reformulate the span deflections in terms of variables satisfying homogeneous boundary conditions, cast the equations into a structured symmetric/skew-symmetric extended operator form, and apply Galerkin discretiztion to this form. The reformulation is needed to transform the troublesome mixed continuum/discrete boundary conditions at the belt-tensioner interface, (4.14) and

(4.16), into homogeneous boundary conditions.

First the following coordinate transformations are applied

y = w − rt x cos β θ 1 1 l1 1 t

y = w + rt (x −1) cos β θ 2 2 l2 2 t

= y3 w3 (4.30)

The new unknown functions yi satisfy the trivial boundary conditions

= = = = = yi (0,t) 0 yi (1,t) 0 yi,xx (0,t) 0 yi,xx (1,t) 0, i 1,2,3 (4.31) instead of the mixed continuum/discrete boundary conditions in (4.14) and (4.16).

Substitution of (4.30) into (4.13)-(4.25) leads to a new set of equations. When directly 90 rewritten in the extended operator form (4.26), however, these new equations do not lead to the requisite symmetric/skew-symmetric properties of the M, G, and K operators. To recover these operator properties, more manipulations are needed.

In the following derivation, the wi in (4.13)-(4.25) have been replaced by the yi through equation (4.30). Multiplying equation (4.13) for the first span by rt x cos β and l1 1 integrating over the span yields

1 1 l1 2 β + l1 2 rt β 2 θ ( ) x cos 1 y1,tt dx ( ) ( ) (cos 1 x) dx t l ∫0 l l1 ∫0

11 −−ll11ββrt 2 θ 2()coss x11, yxt dx 2()()cos s x 1 dx t lll∫∫001

11 −+βε22l β P111,xydxxydcosxx ( ) cos 11, xxxx x ∫∫00l1

11 1 −−γθθθβrr21 + −rrtt +*** + βθ β = [ 21txxtxxxsin 1ywdx 1,1, ( )cos 11, wdxx ] cos 11, w dx 0 (4.32) lll111∫∫00 l 1 ∫ 0

Similar manipulations of equation (4.15) for the second span (first multiplying

−−r3 (x 1) cos β and then integrating) give l2 2

1 1 − l2 2 − β + l2 2 2 β rt − 2 θ ( ) (x 1)cos 2 y2,tt dx ( ) cos 2 ( )(x 1) dx t l ∫0 l ∫0 l2

11 +−ll22ββ −rt −2 θ 2()s (x 1)cos22, yxt dx 2()()( s x 1)cos 2 dx t lll∫∫002

11 +−βε −22l − β P222,22,(xydxxyd 1)cosxx ( ) ( 1)cos xxxx x ∫∫00l2

1 + γ (− r3 θ + r2 θ + rt θ sin β + y w* dx − l2 3 l2 2 l2 t 2 ∫0 2,x 2,x (4.33) 1 1 rt β θ * − β * = ( l )cos w dx) (x 1)cos w dx 0 ∫0 2 2 t 2,x ∫0 2 2,xx

Addition of (4.32), (4.33), and (4.25) leads to a new equation for the tensioner arm, which is not given here for the sake of brevity. This process is similar in spirit to pre- multiplying by the transpose of the transformation matrix to retain a symmetric form when transforming coordinates in symmetric, discrete equations of motion.

Equations (4.13)-(4.24) and this new tensioner equation can be expressed compactly in the following extended operator form

MY + GY + KY = F (4.34)

= θ θ θ θ T Y {y1, y2, y3, 1, 2, 3, t } (4.35)

ll1122+ l 1 βθ ()()yx11 ()cos t rlt l ll2222−− l 2 βθ ()()rlyx22 ()cos( l 1)t t ()()ll332 y rlt 3  = m ()r1 θ MY 11rt (4.36) r2 θ m22() rt r3 θ m33() rt ll11 1222ββ−− +θ ()x cos11 y dx () ( x 1)cos 2 y 2 dx M d t ll∫∫00 92

−−ll11 l 1 βθ 2()()sys1,xt 2()cos 1 lrt l −+ll22 l 2 βθ 2(syslr )()2,xt 2( l )cos 2 t −2(syll33 )() lrt 3,x =  GY 0 (4.37) 0  0 11 −−−ll12ββ s[()ll 2cosx11, yxx dx () 2( x 1)cos 2 y 2, dx] ∫∫00

= T KY {K1, K 2, K 3, K 4, K 5, K 6, K 7 } (4.38)

1 =−lll111 +εγ2 l 2** − K1 ()Py11,xx ()() y 1, xxxx () w 1,1, x y x dxw 1, xx rlrrttt1 ∫ 0 −+γ ()rr12ww**θ γ ()θ +γβ sin w *θ −γβ cos ww ** (1) θ (4.39) rrtt1,xxxxxxtxx 1 1, 2 1 1, 1 1 1, t

1 =−lll222 +εγ22l − * * K2 ()Py22,xx ()() y 2, xxxx () w 2,2, x y x dxw 2, xx rlrrttt2 ∫ 0 −+−γ ()r2 ww**θ γ ()r3 θ γβ sin w *θ −γβ cos ww ** (0) θ (4.40) rrtt2,xxxxxxtxx 2 2, 3 2 2, 2 2 2, t

1 =−lll333 +εγ22l − * * K3 ()Py33,xx ()() y 3, xxxx () w 3,3, x y x dxw 3, xx rlrrttt3 ∫ 0 +−γ ()r1 ww**θ γ ()r3 θ (4.41) rrtt3,1xx 3, xx 3

11 =−γγrr11** K41,1,3,3,()wydxxx () wyd x xx rrtt∫∫00 ++−γ ( rr11)( r 1 )θ γ ( rr 12 )( )θ −γββ ( r 1 )[sin − cosw* (1)]θ −γ ( r 1 )(r3 )θ (4.42) rltt13 l1 rl 12111 l 1t rlt 33

11 =−γγrr22** + K51,1,2,2,()wydxxx () wyd x xx rrtt∫∫00 − γ ( r2 )( r1 )θ + γ ( r2 )( r2 + r2 )θ − γ ( r2 )( r3 )θ rt l1 1 rt l1 l2 2 rt l2 3 +−γββ( rr22)[sin cosww** (1)]θ +γββ ( )[sin + cos (0)]θ (4.43) ll12111tt 2 22

11 =−γγrr33** + K62,2,3,3,()wydxxx () wyd xxx rrtt∫∫00 −−γ ( rr33)(rr12 )θ γ ( )( )θ −γββ ( r 3 )[sin + cosw* (0)]θ ++γ ( rrr 333 )( )θ (4.44) rltt3221 rl2222 lt rlt 23 l 3

1 1 = −γ β * + γ β * * K 7 sin 1 w1,x y1,x dx cos 1w1,x (1) w1,x y1,x dx ∫0 ∫0

1 1 + γ β * + γ β * * sin 2 w2,x y2,x dx cos 2 w2,x (0) w2,x y2,x dx ∫0 ∫0

−−γβγβrr11* θ [()sin1111 ()cosw (1)] ll11 −−γβγβγβrr22 − + r 2** − γ r 2 βθ [ ( )sin12 ( )sin ( )cos 11222ww (1) ( )cos (0)] ll12 l 1 l 2 rr r r +++++[()sinγβγβtt22 ()sinkP ()cos t 2 β P ()cos t 2 β ll12121122s l 1 l 2 + γ ( rt ) cos2 β (w* (1))2 + γ ( rt ) cos2 β (w* (0))2 l1 1 1 l2 2 2 − 2γ ( rt )sin β cos β w* (1) + 2γ ( rt )sin β cos β w* (0)]θ l1 1 1 1 l2 2 2 2 t + [γ ( r3 )sin β + γ ( r3 )cos β w* (0)]θ (4.45) l2 2 l2 2 2 t where Mm=+1 [()()cosrrttll1222ββ + ()()cos 22]. The external force vector F dt3 ll121 l l 2 remains the same as (4.27). After these manipulations, the system seems more complicated. The key advantage, however, is that the new operators M and K are symmetric and G is skew-symmetric with an inner product analogous to (4.29).

4.4 Galerkin Discretization

The mathematical structure of the extended operator form (4.34)-(4.45) and the trivial boundary conditions in (4.31) allow classical Galerkin discretization. The extended variable Y is expanded in a series of basis functions as

+ + + N1 N2 N3 4 = ψ Y ∑ ak (t) k (x) k =1 ,

ψ = {sin(kπx),0,0,0,0,0,0}T k = 1,2,...N , k 1

ψ = {0,sin(mπx),0,0,0,0,0}T k = N +1,..., N + N m = k − N , k 1 1 2 1

ψ = {0,0,sin(nπx),0,0,0,0}T ,k = N + N + 1,...N + N + N k 1 2 1 2 3 , = − + n k (N1 N 2 )

ψ = {0,0,0,1,0,0,0}T k = N + N + N +1, k 1 2 3

ψ = {0,0,0,0,1,0,0}T k = N + N + N + 2 , k 1 2 3

ψ = {0,0,0,0,0,1,0}T k = N + N + N + 3, k 1 2 3

ψ = T = + + + k {0,0,0,0,0,0,1} k N1 N 2 N 3 4 (4.46) where N is the number of basis functions for the ith span. The ψ are global i k comparison functions where each one describes a deflection of the entire system and satisfy all boundary conditions. They form a complete set. After substitution of (4.46) into (4.26) (with w → y ), the error (residual) is constrained to be orthogonal to the ψ k using the inner product (4.29). This gives the equations of motion and eigenvalue problem

[M ]A + [G]A +[K ]A = f (4.47)

-ω 2 [M ]ρ + iω[G]ρ + [K ]ρ = 0, A = ρ eiωt (4.48)

T ρ = {a ,a ,...a + + + } (4.49) 1 2 N1 N2 N3 4

= ψ ψ = ψ ψ = ψ ψ = + + + M ij M j, i Gij G j, i Kij K j, i , i, j 1...N1 N 2 N3 4 (4.50)

T f = { f , f ,... f + + + } f = F,ψ , i = 1...N + N + N + 4 (4.51) 1 2 N1 N2 N3 4 i i 1 2 3 where the inner product is an extended one similar to that in (4.29). The matrices [M],

[K], and [G] inherit the symmetry/skew-symmetry of the corresponding differential operators. These properties ensure that the eigenvalues are purely imaginary, as required for a conservative gyroscopic system.

The present method has several advantages over continuum characteristic equation approaches [31,32]: 1) It is easy to implement because of the simple basis functions and trivial boundary conditions. 2) It is efficient, accurate, and greatly reduces computational time. 3) It does not require a user-specified bandwidth to search for natural frequencies.

4) It is numerically robust and free of missing/false natural frequency concerns. 5)

Because the method uses Galerkin discretization, all properties of that approach are retained, including convergence of the eigenvalues from above. 6) Dynamic response analysis is trivial to implement using (4.47).

4.5 Results and Discussion

In this section, results are presented for a prototypical three-pulley system (Figure

4.1); the physical properties are shown in Table 4.1. Because the motion of the crankshaft

θ ( 1 ) is typically prescribed in practical applications, it is treated as a specified excitation

θ = source and 1 0 in the following free vibration analysis. Special attention is given to the interaction between span 3 and the rest of the system because this span is bounded by fixed pulleys, and its motion is decoupled from the rest of the system for vanishing bending stiffness.

Pulley radius r1 0.0889m Pulley center (,x1 y1 ) (0.5525,0.0556)m

Pulley radius r2 0.0452m Pulley center (,x2 y2 ) (0.3477,0.05715)m

Pulley radius r3 0.02697m Pulley center (,x3 y3 ) (0,0)

Tensioner arm rt 0.097m Pulley center (,xt yt ) (0.2508,0.0635)m

⋅ 2 Rotational inertia J1 0.07248 kgm Belt Modulus EA 120000 N

⋅ 2 Rotational inertia J2 0.000293kgm Initial tension P0 300 N

⋅ 2 Rotational inertia J3 0.000293kgm Belt mass density m 0.1029kg / m

⋅ 2 Rotational inertia Jt 0.001165kgm Tensioner stiffness kr 116.4 N-m/rad

β o Span length l1 0.1548 m Alignment angle 1 135.79

β o Span length l2 0.3449 m Alignment angle 2 178.74

θ Span length l3 0.5518 m Tensioner rotation tr 0.1688rad

Table 4.1. Physical properties of the example system, from which nominal dimensionless parameters are calculated.

a) X

b) X

c) X

d) X

Figure 4.2. Rotationally dominant mode (ε = 0 ) for increasing belt bending stiffness. The dimensionless natural frequency for ε = 0 is ω = 4.1205 . a) ε = 0.01, b) ε = 0.04 , c) ε = ε = = = γ = = = = β = o 0.07 , d) 0.1. s 0, ks 4 , 400 , P1 P2 P3 1, 1 135.79 , β = o 2 178.74 .

Figure 4.3. Span 2 transversely dominant mode (ε = 0 ) for increasing belt bending stiffness. The dimensionless natural frequency for ε = 0 is ω = 3.0951. a) ε = 0.01, b) ε = ε = ε = = = γ = = = = 0.04 , c) 0.07 , d) 0.1. s 0, ks 4 , 400 , P1 P2 P3 1, β = o β = o 1 135.79 , 2 178.74 .

Figure 4.4. Span 3 transversely dominant mode (ε = 0 ) for increasing belt bending stiffness. The dimensionless natural frequency for ε = 0 is ω =1.9968. a) ε = 0.01, b) ε = ε = ε = = = γ = = = = 0.04 , c) 0.07 , d) 0.1. s 0, ks 4 , 400 , P1 P2 P3 1, β = o β = o 1 135.79 , 2 178.74 .

100

Figures 4.2 and 4.3 show that when the belt bending stiffness is small (ε = 0.01), there are two distinct types of modes: pulley rotationally dominant vibration modes and span transversely dominant vibration modes. These mode types are similar to the results computed with the string model except that here span 3 has some small transverse deflection (as opposed to being straight in the string model [31]). These two figures also show that when bending stiffness increases, the magnitude of the span 3 deflection increases accordingly, and the relative magnitudes of the initially dominant components diminish. The coupling between the spans and pulleys becomes stronger for these modes.

Figure 4.4 describes a type of vibration mode not captured with the string model.

When ε is small ( ≈ 0.01), the dominant motion is the transverse motion of span 3 while all other components have small motions. But when ε is appreciable, say ε = 0.07 or

ε = 0.1, the modal amplitudes of other components are significant, indicating a strongly coupled mode. Note that as bending stiffness decreases, the span 3 deflection increases markedly relative to other deflections. In the limit as ε → 0 , span 3 is completely decoupled from the rest of the system, as in prior string model results [31].

A key point of Figures 4.2-4.4 is that bending stiffness induced modal coupling with spans connecting fixed center pulleys (such as span 3) provides an explanation for the observed vibration of these spans in vehicle applications. String models have no means to capture this known behavior except using parametric excitation models [14,15] that are not relevant at the idle/low speed regions where span vibrations are commonly observed.

101

ω 9

Dimensionless Natural Frequency, 2

1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ε Belt bending stiffness,

= = Figure 4.5. Natural frequency spectrum for varying belt bending stiffness. s 0, ks 4 , γ = = = = β = o β = o 400 , P1 P2 P3 1, 1 135.79 , 2 178.74 .

102

These figures also reveal another tendency for all types of modes: when the belt bending stiffness increases or the span tensions decrease (that is, as ε increases), the magnitude disparities between the dominant components and the other parts of the system decrease. In essence, the distinction between different types of modes becomes less pronounced. Eventually, this classification of vibration modes does not apply any more.

This can be seen from the cases of ε = 0.1 in Figures 4.2-4.4 where all the spans and pulleys have nearly the same order amplitude. In this case, all the pulley and span vibrations are strongly coupled, and the system’s dynamic operating condition response is quite different from that when belt bending stiffness is neglected. ε = 0.04 and ε = 0.07 fall into the transition region between the string and the beam models. As mentioned in last chapter, for practical serpentine belt drives, manufacturers approximate the bending stiffness of a poly-ribbed belt typical of vehicle applications by

EIm=−( 1)2.867 × 10−32 Nm ⋅ (where m is the number of ribs). Notice that ε depends on

ε 2 = EI the relative magnitude of bending stiffness and belt tension because 2 . For the Pol range of belts and span tensions in use, reasonable values of ε fall in the range

0.01≤≤ε 0.12.

Figure 4.5 shows the relationship between the natural frequencies and the bending stiffness. As bending stiffness increases, some natural frequencies decrease when the belt bending stiffness is small. This interesting phenomenon is inconsistent with the linear system requirement that natural frequencies increase with stiffness. The root cause is that although increased bending stiffness tends to increase the natural frequencies for a fixed

103 steady state, the increased bending stiffness changes the steady state, which also influences the natural frequencies, as shown in the dynamic equations (4.39)-(4.45). This change in the steady state causes some natural frequencies to decrease with increasing bending stiffness. The mechanism is that increased bending stiffness increases the steady state curvature, and correspondingly the coupling between the belt spans and pulleys also increases [63]. The increased belt-pulley coupling causes the strange phenomenon because for transversely dominant modes the increased belt-pulley coupling is similar to relaxing constraints at the boundaries of the dominant span. Numerical experiments confirm that if the steady state is fixed (let the steady state terms marked by asterisk in the equations (4.39)-(4.45) be assumed constant), then all the natural frequencies increase as the belt bending stiffness is increased (Figure 4.6), which is consistent with our physical intuition. If the changing belt bending stiffness only influences the steady state

ε 2 while the value of in (4.39)-(4.45) is held fixed, then Figure 4.6 shows that some natural frequencies decrease due to the increased belt-pulley interactions. Eventually for large enough ε , further increases in bending stiffness lead to monotonic increases in all natural frequencies (Figure 4.5).

104

ω 9

Dimensionless Natural Frequency, 2

1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 ε Belt bending stiffness,

Figure 4.6. Natural frequency spectrum for varying belt bending stiffness. −−−, fix = = γ = steady state; , fix bending stiffness value in (4.39)-(4.45). s 0 , ks 4 , 400 , = = = β = o β = o P1 P2 P3 1, 1 135.79 , 2 178.74 .

105

ω 10

8 transversely dominant mode

4 rotationally dominant mode 3

Dimensionless Natural Frequency, 2

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Speed, s

Figure 4.7. Natural frequency spectrum for varying belt speed. , ε = 0.1; −−−,

ε = = γ = = = = β = o β = o 0.01. ks 4 , 400 , P1 P2 P3 1, 1 135.79 , 2 178.74 .

106

ω 9

2 Dimensionless Natural Frequency,

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Speed, s

η = β = o Figure 4.8. Natural frequency spectrum for varying belt speed. , 0 ( 1 68.53 , β = o −−− η = β = o β = o ε = = 2 111.47 ); , 0.78 ( 1 135.79 , 2 178.74 ). 0.04 , ks 4 , γ = = = = 400 , P1 P2 P3 1.

107

14 ω

10 transversely dominant mode

4 rotationally dominant mode

Dimensionless Natural Frequency, 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Tensioner effectiveness,η

Figure 4.9. Natural frequency spectrum for varying tensioner effectiveness η . , ε = −−− ε = = = γ = = = = 0.1; , 0.01. s 0 , ks 4 , 400 , P1 P2 P3 1.

108

The relationship between the dimensionless natural frequencies and belt speed is shown in Figure 4.7. When the bending stiffness is small (ε = 0.01), the spectrum is similar to the string model. The frequencies of transversely dominant modes decrease quickly with speed, but speed has little influence on the natural frequencies of rotationally dominant modes. As the bending stiffness increases to ε = 0.1, the influence of speed on the natural frequencies is markedly smaller, and there is no clear distinction between rotationally dominant modes or transversely dominant modes. Notice that the natural frequencies do not decrease monotonically as they do for the single span moving string or beam systems.

Generally, only four or five basis functions per span are needed for all natural frequencies of practical importance to converge to within 3%. More terms are needed for increasing speed and higher natural frequencies, as discussed by Jha and Parker [67]. For the string model critical speed s = 1 and ω <11 in Figure 4.7, six terms per span are needed for the eight natural frequencies to convergence to within 3% with ε = 0.1; for

ε = 0.01, 16 terms per span are required for the 16 natural frequencies.

Through changing the orientation of the tensioner arm, the effects of the tensioner effectiveness η on the natural frequencies are described in Figures 4.8 and 4.9. η is an indicator of the ability of the tensioner to maintain constant tractive belt tension despite changes in belt speed or accessory torques [30]. For a general n pulley system with the tensioner pulley as pulley i, one has [63]

109

1 η = (4.52) n  1 1  l 2 β + 2 β + ∑ j  Pi ( cos 1 cos 2 ) k s  = l l j 1  i−1 i  + 1 γ β − β 2  (sin 1 sin 2 )     

For well designed systems, η is close to unity, while for poorly designed systems, η is away from unity. Figure 4.8 shows how the tensioner effectiveness η influences the relationship between natural frequencies and belt speed. For larger η = 0.78 , the decrease rate of the natural frequencies with belt speed is smaller than that for η = 0 because of the stronger ability of the tensioner to compensate for the tension loss induced by belt speed due to centrifugal action (η = 0 and η = 0.78 can be visualized in Figure 4.10).

Similar behavior occurs with the string model [31]. Further inspection of Figures 4.7 and

4.8 reveals that for the properly designed system with η = 0.78 , the decrease rate of the natural frequencies with speed in Figure 4.8 (ε = 0.04 ) is between the two cases ε = 0.01 and ε = 0.1 in Figure 4.7. This is because ε = 0.04 falls in the transition region between the string and beam models. Speed has its strongest effect for small ε .

Figure 4.9 shows the relationship between natural frequencies and the tensioner effectiveness η for different bending stiffness. The variation of η is caused solely by

β changing the tensioner orientation 1,2 . For small bending stiffness, only rotationally dominant modes are influenced significantly while transverse dominant modes are insensitive to the orientation of the tensioner. This agrees with conclusions from the string model [31,32]. As η increases, the natural frequencies of rotationally dominant modes increase. Physically, this is because larger η means the corresponding tensioner

110 orientation provides increased resistance torque from the belts and makes it more difficult for the tensioner to rotate around its pivot (Figure 4.10). This increases the effective rotational stiffness of the tensioner. Transverse dominant modes are insensitive to this effect.

For significant bending stiffness (or small tension), all natural frequencies are affected when the tensioner effectiveness η is changed (Figure 4.9). Furthermore, the dominant tendency of the natural frequencies is to decrease with η , in contrast with the small ε case. These differences result because of the expanded vibration mode coupling that leads to all spans and pulleys deflecting in a given mode rather than the division into rotational and transverse dominant modes. For these coupled modes with modal deflections distributed throughout the system, there are two competing effects as η increases. The first is described above for the small ε case and tends to increase the natural frequencies for modes with appreciable tensioner rotation. With higher bending stiffness, the opposing effect is from the resistance of the tensioner to deflections of the endpoints of the spans adjacent to the tensioner. For small η , the tensioner orientation is such that it strongly resists translational deflection of the tensioner pulley in the radial direction that span deflections want to move it because the rigid tensioner arm can not be compressed (Figure 4.10a). In contrast, for larger η the tensioner resistance to span endpoint deflection comes primarily from the compliant tensioner spring (Figure 4.10c).

Consequently, this effect causes the natural frequencies of coupled belt-pulley modes to decrease for increasing η . This second effect tends to dominate the first one noted above as seen by the decreasing natural frequencies in Figure 4.9. The exceptions are modes

111 dominated by tensioner rotation such as the lowest natural frequency in Figure 4.9. Still, the increase rate of this natural frequency is markedly smaller for ε = 0.1 than ε = 0.01 because of the additional coupling with the adjacent spans and the increased resistance to span endpoint deflection for small η .

X a)

b) X

Figure 4.10. Fifth vibration mode for varying tensioner effectiveness η . a) η = 0 β = o β = o η = β = o β = o η = ( 1 68.53 , 2 111.47 ), b) 0.5 ( 1 95.53 , 2 138.47 ), c) 0.78 β = o β = o ε = = = γ = = = = ( 1 135.79 , 2 178.74 ). 0.1, s 0, ks 4 , 400 , P1 P2 P3 1.

112

CHAPTER 5

STEADY STATE MECHANICS OF BELT-PULLEY

SYSTEMS

In this chapter, steady state analysis of a two-pulley belt drive is conducted where the belt is modeled as a moving Euler-Bernoulli beam with bending stiffness. Other factors in the classical creep theory, such as elastic extension and Coulomb friction with the pulley, are retained, and belt inertia is included. Inclusion of the bending stiffness leads to non-uniform distribution of the tension and speed in the free belt spans and alters the belt departure points from the pulley. Solutions for these quantities are obtained by a numerical iteration method that generalizes to n-pulley systems. The governing boundary value problem (BVP), which has undetermined boundaries due to the unknown belt- pulley contact points, is first converted to a standard fixed boundary form. This form is

113 readily solvable by general purpose BVP solvers. Bending stiffness reduces the wrap angles, improves the power efficiency, reduces the maximum transmissible moment, and increases the span tensions.

5.1 Introduction

In spirit, the steady state analysis conducted in this chapter is different from that in chapter 3. The analysis in chapter 3 is more like those done by Wang and Mote [24] or

Hwang and Perkins [68], where calculation of the steady state is mainly for subsequent linearized free vibration analysis when belt bending stiffness is considered.

In contrast, the steady state analysis in this work and those by Bechtel et al. and

Rubin [45,46] focuses on the distribution of tension, speed, and friction force along the belt, which addresses the problems like belt slip on the pulleys, power transmission efficiency, and maximum transmissible torque, etc. This is along the lines of traditional belt contact mechanics, which started with Euler [39] studying a belt wrapped around a fixed pulley or capstan. Some simplifying assumptions adopted in chapter 3 and [24,68] are abandoned. For example, the belt speed is no longer assumed to be uniform throughout the system, friction between the pulley and belt is not to be neglected, and the belt can adhere to or slide relative to the pulleys. The belt-pulley friction follows the classic Coulomb law. This leads to a non-conservative system where energy is dissipated due to friction. Another prominent difference between the analysis here and that in chapter 3 is that the boundaries of the free spans are not fixed at the belt-pulley contact points of the string model any more, instead they are not known a priori and need to be determined in the analysis.

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In both [45,46], the belt is treated as a string and belt bending stiffness is not considered. Bending stiffness introduces additional, non-uniform tension in the belt due to the induced curvatures in the free spans, leads to non-uniform speed along the belt free spans, alters the wrap angles, and influences performance criteria such as maximum transmissible moment and efficiency. When bending stiffness is appreciable compared to tension stiffness (especially for thick, low tension, short span belts), the effects of bending stiffness can be more significant than the inertia terms introduced in [45,46]. The main concern of this chapter is to investigate the influence of the bending stiffness on the steady motion while keeping belt inertia.

The belt is modeled as a moving Euler-Bernoulli beam. Even in the free spans of the belt, the distributions of tension and speed are no longer uniform and there is no explicit analytical solutions for the ordinary differential equations (ODE) governing the free belt spans, in contrast to the string models [45,46]. Furthermore, inclusion of the bending stiffness makes the contact points between the belt and pulleys (boundaries for the ODE) not known a priori. Consequently, the steady motion analysis is governed by a boundary value problem with unknown domain. This presents one of the main obstacles.

By suitable transformation using ordinary differential equation conversion techniques, however, this problem is formulated as a standard BVP with fixed boundaries. This form is accepted by general-purpose two-point BVP solvers. No spatial discretization (e.g.,

Galerkin, Ritz) is used; the final result can be viewed as numerically exact. Although the iteration method is presented for two-pulley belt drives, it can be readily extended to multi-pulley drives.

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This chapter is organized as follows. Section 2 presents the governing differential equations of the moving Euler-Bernoulli beam in the steady state. Section 3 introduces techniques to transform the unknown boundary BVP to fixed-boundary form. These techniques are presented for the case where a single belt span wraps around two pulleys.

This simple example is a fundamental building block for the overall belt-pulley system solution. Section 4 presents the steady state analysis for a two-pulley belt drive system.

Two different problems are considered. One finds the steady motion when the moments on the pulleys are specified. The second problem finds the maximum transmissible moment that can be exerted on the pulleys. Section 5 presents the numerical results of two examples corresponding to these two problems.

5.2 Nonlinear Equations of a Moving Curved Beam

Figure 5.1 shows the free body diagram of an extensible belt, which is modeled as a moving Euler-Bernoulli beam. Rotary inertia and shear deformation are ignored. An

Eulerian formulation is adopted for the control volume. The radius of curvature ρ()s is a function of the arclength coordinate. f ()s and ns()are contact forces per unit length exerted on the belt.

For steady motions, conservation of mass requires that

GmsVs==( ) ( ) constant (5.1) where ms() is the belt mass density per unit length. Euler-Bernoulli beam theory requires

M = EI κ (5.2) where EI is the bending stiffness and κ is the curvature of the beam. κ equals the varying rate of inclination of the beam

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κθ= d / ds (5.3) where the inclination angle θ is measured from the due east direction (Figure 5.2). The balance of angular momentum with respect to the center of mass of the control volume yields

dM−= Qds 0 , Q ==dM/ ds EI(/ dκ ds) (5.4)

ρ(s) ρ(s+ds)

dθ n(s)

T(s) M(s) f (s) M(s+ds) T(s+ds)

V(s) control volume V(s+ds) Q(s) Q(s+ds)

Figure 5.1 Free body diagram of a moving curved beam.

117

Balance of linear momentum projected in the tangential direction leads to

dTfdsGdVQd−= −θ (5.5)

Substitution of (5.2)-(5.4) into (5.5) leads to

()T −+GV′′ EIκκ =f (5.6) where ' denotes differentiation with respect to the arclength s . Balance of linear momentum projected in the normal direction yields

nds=−() T GV dθ − dQ (5.7)

Substitution of (5.4) leads to

()T −−=GVκκ EI ′′ n (5.8)

The above derivation assumes that dθ 1, cos(dθ / 2) ≈1, sin(ddθθ / 2)≈ / 2 , and products of infinitesimal quantities are negligible.

For the belt in the free spans, the contact forces f and n are zero, and equations

(5.6) and (5.8) become

(TGVEI−+)′′κκ =0 (TGV−−=)κκ EI′′ 0 (5.9)

This converges to the string model [45] when the bending stiffness is zero. Because the curvature is constant for the belt on the pulleys, equations (5.6) and (5.8) in those regions become

()T −=GV ′ f ()/T −=GV R n (5.10)

The governing equations (5.10) are the same as those for a string model [45,46].

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5.3 BVP Solver Based Method for Problem with Unknown Boundaries

Figure 5.2 presents a single belt span stretched between two fixed pulleys. This boundary value problem has unknown boundaries; the belt departure points from the pulleys are determined in the analysis. This case is a key step in the subsequent solution for general belt-pulley systems. Furthermore, it shows the techniques for converting the problem with unknown boundaries into a standard form [58] that can be solved by general purpose BVP solvers.

For this reduced problem, the belt speed is zero. This reduces (5.9) to

TEI′′+=κκ 0 TEIκκ−=′′ 0 (5.11)

Three boundary conditions are needed. For reasons to be evident later, the tension at the start point is assumed to be known

= TT(0) 0 (5.12)

The curvatures κ at the two boundaries are specified by the geometric relations (Figure

5.2) κ =− κ ==− (0) 1/ R1 , (s LR)1/2 (5.13) where L is the total arclength of the belt in the free span. What makes the problem unusual is that L is not known a priori. The compatibility requirement of L is that the curved belt should contact and be tangent with both pulleys.

119

)

) −θ()L s=L y s

θ(s) ) −θ()L θ(0) R θ(0) x R1 R2 L

Figure 5.2. Single span boundary value problem with unknown boundaries.

By applying ODE conversion techniques, the above system is transformed into a standard form defined on the interval (0,1). This formulation is readily accepted by most general-purpose BVP solvers, yielding accurate solutions with minimal programming. As mentioned in chapter 3, the standard form is

y '(xfxyx)(,(= )) a < x < b , g( y(a), y(b)) = 0 (5.14) where f , y , and g are n dimensional vectors, a and b are fixed boundaries, and f and g may be nonlinear. First, the following non-dimensional variables are introduced

120

s xy TL2 sxyˆˆˆ====,,,κκˆ Lp, ˆ = (5.15) L LL EI where xs() and ys() are the rectangular coordinates of belt particles (Figure 5.2).

Substitution of (5.15) into (5.11) yields

pˆˆˆ′′+=κκˆˆ0, κ ˆ ′′ −=ps κ ˆ 0 0 << 1 (5.16)

The boundary conditions (5.12) and (5.13) become = 2 κ =− κ =− pˆ(0)TL0 / EI , ˆ(0)LR / 1 , ˆ(1)L / R2 (5.17) The unknown constant L is defined as function L = L()sˆ , governed by L′ =<<0, 0sˆ 1 (5.18)

The geometric relations lead to

θκ′′==, xˆˆcos θ ,ys ′ = sin θ 0 << ˆ1 (5.19)

The corresponding boundary conditions are =− θ = θ Lx(0)ˆ (0) R1 sin (0) Ly(0)ˆ (0) R1 cos (0) (5.20) −+2 2 =2 −=−θ [L (1)xLLyRˆˆ (1) ] [ (1) (1)] 2 R2 sin (1)Lxˆ (1) L (5.21) where L is the distance between the centers of the two fixed pulleys (Figure 5.2).

Equations (5.20) and (5.21) ensure that the belt contacts, and is tangent to the bounding pulleys. Tangency is imposed by the angles on the pulleys to the contact points being equal to θ (0) and −θ (1) .

The seven boundary conditions (5.17), (5.20), and (5.21) equal the total order of the six differential equations (5.16), (5.18), and (5.19). Equation (5.16) involving higher derivatives can be reduced to standard first order form (5.14) with the

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= κ = κ′ ′ = definitions y1()s ˆ ()s , y2 ()s ˆ ()s , y12()sy ()s . The problem is cast entirely on the interval x ∈(0,1) even though the problem involves unknown boundaries. This standard form is readily accepted by general-purpose, two-point BVP solvers. Here, again the solver BVP4C in Matlab is adopted [61].

There are several advantages of the BVP solver based method: 1) Because there is no spatial discretization and because of the high quality and robustness of state-of-the-art

BVP solver codes, the final result can be viewed as numerically exact. 2) It is easy to implement. 3) It not only gives the tension and the curvature of the belt but also gives the explicit positions of the entire belt ( xs() and ys()) simultaneously.

Further, the method can be extended to other problems that do not seem amenable to a general-purpose BVP solver at first sight. In section 4, two additional required ODE conversion techniques first introduced in chapter 3 are presented. One shows how to incorporate an integral term in the BVP system. The other shows how to incorporate an algebraic equation.

5.4 Steady State Analysis of Belt Pulley Drives: an Iteration Method

In this part, the steady motion analysis is presented for a general two pulley belt drive with different pulley radii. Following [45], the specified parameters are: driver

pulley radius R1 , driven pulley radius R2 , center distance between the two fixed pulleys

ω L , belt longitudinal stiffness EA , constant rotation speed 1 of the driver pulley,

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µ µ coefficients of friction 1 and 2 on the two pulleys, static tension of the belt Tini of the reference state (where there is no moment exerted on the pulleys and the belt speed is zero), and belt mass flow rate G. Due to consideration of the bending stiffness, the

reference state tensions are no longer uniform along the belt; Tini is assumed to be the tension at the midpoints of the free spans. (Note that in the string model of [46], instead

ω = of specifying 1 and G, the undeformed belt mass per unit length m0 and c Gm/ 0 are the chosen parameters. In section 5.5.2, these definitions are shown to be equivalent.)

Two problems are considered: (1) If the moment M 2 on the driven pulley is specified, one finds the driving moment, distributions of tension, speed, and friction along the belt as well as the slip and adhesion angles on the two pulleys, and (2) The moments on the pulleys are not specified, and one first finds the maximum transmissible moment and then calculates the corresponding steady state mechanics; the implicit condition here is that one or both of the adhesion angles vanishes and no additional moment can be transmitted.

As derived above, the governing equations for the free spans are (5.9). In the contacting zones, the governing equations are (5.10). On the boundaries between the free spans and contacting zones, the tension, curvature, and speed must be continuous. To complete the problem, a constitutive law is needed. Following [69], a differential belt

element with undeformed length dso has deformed length

=+ dsTEAd(1 / ) so (5.22) and the mass density per unit length ms() becomes

123

mds m ms()==00o (5.23) dsTE1+ / A

Substitution of (5.23) into (5.1) leads to

m G ==msVs( )()0 V (s) (5.24) 1+T / EA

So, the constitutive law is

=−= T EA( V/1), Vref V ref G / m0 (5.25)

as used in [45]. Note that Vref is not known a priori; it is determined from the compatibility condition

(0) (0) (0) (0) (0) (0) (0) L +++++=LLLLLLββαα (5.26) TS 1212

(0) (0) (0) where L and L are the unstretched lengths for the free tight and slack spans, Lα and T S 1

(0) Lα are the unstretched lengths of the adhesion zones on the driver and driven pulleys, 2

(0) (0) Lβ and Lβ are the unstretched lengths of the slip zones on the driver and driven 1 2 pulleys, and L(0) is the unstretched length of the total belt when it is removed from the pulleys. L(0) can be calculated from the reference state. Physically, the compatibility equation (5.26) means that the unstretched length computed from the steady state should equal that computed from the reference state, guaranteeing that the same belt is considered [46].

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The key compatibility equation (5.26) is that used in [46] and prior belt drive analysis [30]. In [45], the compatibility equation is mistakenly omitted and this makes the

−= derivation inconsistent. In [45], equation (17) leads to TtsTM/ r . This contradicts

−−−= their equation (3), which requires that, after integration, ()()/TttssGV T GV M r .

5.4.1 Regular Moment Transmission Problem

In this problem, the moment M 2 exerted on the driven pulley is a specified value less than the maximum transmissible moment.

To find the driving moment and distributions of tension, speed, and friction along the belt loop, an iteration method is used. The iteration starts from the tight span, which has the following governing ODE

−+′′κκ = ()TGVEI11 11 0 (5.27)

−−=κκ′′ ()TGV1111 EI 0 (5.28) where the subscript 1 represents the tight span (subscript 2 represents the slack span).

First, the tension at the left boundary (contact point between the driver pulley and the tight span) is assigned an initial guess

= T1(0) T1_0 (5.29)

= At the same boundary of the tight free span ( s1 0 ), the speed of the belt is the same as the driver pulley

= ω VR1(0) 1 1 (5.30)

Substitution of (5.30) into the constitutive law (5.25) leads to

125

ω R V = 11 (5.31) ref + 1(T1_0 /EA )

Thus the relationship between the tension T1 and speed V1 can be completely determined

− from (5.25) (for the assumed T1_ 0 ), and T11GV can be treated as one unknown field variable. Equations (5.29) and (5.30) give the boundary condition for this unknown field variable. The other two boundary conditions for the governing equations (5.27) and

(5.28) are κ =− κ =− 11(0) 1/ R , 1(L1)1/R2 (5.32) where L1 is the total arclength of the tight span, which is to be determined. This single span problem with unknown boundary L1 is of the form discussed previously. By adopting the ODE conversion techniques presented earlier, the solution for the tight span can be found.

126

) s1=L1 y s1 ) V1(S1) Sliding θ 1(s) Tight Span −θ1()L1

− ) Adhesion θ 1(0) θ1()L1 β θ 1(0) Driver pulley Driven pulley 2 α1 x M2 α β 2 1 M1 Sliding L R2 Adhesion

s ) 2 s =L Slack Span 2 2 V2(S2)

Figure 5.3. Two-pulley belt drive with inclusion of belt bending stiffness.

For the belt in the belt-pulley contact zones, the governing equation (5.10) is the same as for the string models. All relations for these segments of the belt derived from prior string models [45,46] still hold. For the adhesion zones, the tension and speed are constant. But for the sliding zone on the driver pulley (see Figure 5.3), (5.10) leads to

1 TGV(0)− (0) β = ln 11 (5.33) 1 µ − 12TGV(0) 2 (0)

where T2 (0) and V2 (0) are the boundary values of tension and speed for the free slack span. The balance of angular momentum on the driver and driven pulley leads to

127

β R MR/ ==−−−11 fdsT[ (0) GV (0)] [ T (0) GV (0)] (5.34) 11∫0 1 1 1 2 2 =− − − M 2 / RTLGVLTLGVL2111122[() ()][( ) 22 ( )] (5.35)

Using (5.29)-(5.31), (5.34) and the constitutive law (5.25) give two equations for

T2 (0) and V2 (0) . This gives one boundary condition for the equations governing the free slack span

−+′′κκ = −−=κκ′′ (TGVEI22) 220 (TGV2222) EI 0 (5.36)

The other two boundary conditions for (5.36) are κ = κ = 21(0) 1/ R , 2 (L2 )1/R2 (5.37) where L2 is the total arclength of the slack span (to be determined). Again, this boundary value problem has the form discussed previously for the single span case and can be solved by a general purpose BVP solver.

Integration of (5.27) and the first of (5.36) yields

−−−=−+11κκ22 + [(()T22sGVsTsGVs 22 ()][(() 11 11 ()]22 EIs 2 () 2 EIs 1 () 1 D (5.38) = = = = where D is a constant. Taking the two choices s1 0 , s2 0 and then sL11, s22L in

(5.38), subtracting the two resulting equations, and using (5.32) and (5.37) gives −−−=−−− [T22() L GV 22 ()][() L T 11 L GV 11 ()][(0) L T 2 GV 2 (0)][((0) T 1 GV 1 (0))] (5.39)

= Comparison of (5.34), (5.35), and (5.39) yields M1 (RRM12/)2 for steady operation.

For the adhesion zones of the belt on pulleys, the adhesion angles are απβθ=− − + θ απβθ=− + − θ 1112(0) (0) 2 21122(L )(L ) (5.40)

128

Thus, the steady motion has been calculated for the assumed tension T1_ 0 . This includes the torque on the driver pulley M1 , the free span lengths ( L1 , L2 ), the deflected

belt shapes in the free spans ( x1()s1 , y1()s1 , x2 ()s2 , y2 ()s2 ), the tension and speed

distributions along the belt (T1()s1 , V1()s1 , T2 ()s2 , V2 ()s2 ), the belt-pulley contact points θ θ θ θ α α β β ( 1(0) , 1()L1 , 2 (0), 2 ()L2 ), and the adhesion and slip zones ( 1 , 2 , 1 , 2 ).

The calculated result for the assumed tension T1_ 0 is a possible steady state that can physically exist. But whether or not it is the same belt specified in the reference state, which has the unstretched length L(0) , depends on if the system satisfies the compatibility equation (5.26). To check the compatibility condition, one must find the unstretched

= length of the total belt for the assumed case (T1(0) T1_0 ).

For the adhesion and slip zones, the unstretched lengths are

α α (0) 11R (0) 22R Lα = Lα = (5.41) 1 + 2 + 1 T1(0) / EA 1 T22()/LEA

β R GV +−− (0)11 1 * ref R1 1 EA/[ T1111 (0) GV (0) M / R ] Ldsβ ==−(1 ) ln (5.42) 1 ∫0 ++−* µ 1()/T s EA EA111 1 EA /[(0)(0)] T GV GV +−− (0) ref R2 1 EA/[() T11 L GV 11 () L M 2 / R 2] Lβ =−(1 ) ln (5.43) 2 µ +− EA211111/[()()] EA T L GV L where s* is the local arclength for the slip zone of the belt. The unstretched length of the tight free span is

L (0) = 1 1 LT ds1 (5.44) ∫0 + 1()/Ts11 EA

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To calculate this term, direct integration method could be used because the distribution of the tension T1()s1 and the total arclength of the free tight span L1 have been calculated.

Instead we use the alternative method introduced in chapter 3 that integrates the integral term into the standard BVP form for the tight span with little additional effort. We define

= s 1 IT ()sds1 and add an additional ODE and boundary condition to the ∫0 + 1()/Ts11 EA corresponding BVP standard form for the free tight span

dI() s 1 T =<<,0 sL with I (0) = 0 (5.45) + 1 T ds1()/ T1 s EA

(0) Then IT ()L1 is equivalent to the desired integral term LT and is a natural product of the

BVP solution. Although the added ODE and boundary condition (5.45) are written in the dimensional form over the range (0, L1 ), (5.15) transforms them into the necessary form on (0,1) . Similar operations can be performed on the BVP for the slack free span to

(0) obtain the unstretched belt length LS .

The error between the unstretched length for the assumed T1_ 0 and the actual unstretched length L(0) (discussed below) is

(0) (0) (0) (0) (0) (0) (0) (0) L =()LLLLLL +++++ββαα − L (5.46) error T S 1212

(0) < Physically, if Lerror 0 the assumed T1_ 0 is larger than the true T1(0) and should be

(0) > reduced in the next iteration step; Lerror 0 implies T1_ 0 is smaller than the true T1(0) .

(0) Lerror is a monotonically decreasing function of the assumed tension T1_ 0 . This property

130 allows use of the bisection method in the iteration loop. This gives rapid convergence of the iteration to the true solution. In this study, the results converge such that

(0) (0) < LLerror / 0.1% .

The unstretched length of the total belt L(0) must be calculated from the reference

(0) state when the analysis follows those in [45,46] and Tini is specified (instead of L ). A similar iteration method as already presented can be applied to obtain L(0) with the only key difference that during the trial-and-error process, instead of checking the compatibility condition, one checks if the calculated tension at the midpoints of the free

spans equals the specified value Tini . Because in the reference state the tension of the free

spans is non-uniform due to the inclusion of bending stiffness, Tini is assumed to be the tension of the span midpoints (versus the entire span as in string models). In practical applications, the more appropriate problem formulation provides the unstretched belt

(0) (0) length L instead of Tini . In that case, the iteration to determine L is not necessary.

5.4.2 Maximum Transmissible Moment Problem

The above iteration method is valid for the case of specified pulley moments. How is this extended to calculate the maximum moment that can be transmitted? One obvious

solution is to first specify a small moment M 2 exerted on the driven pulley and calculate

131 the steady motion by using the above iteration method, then increase the specified

moment M 2 towards the unknown maximum transmissible moment M 2 _max until one of the adhesion angles reaches zero. This method is feasible, but it involves two iteration loops and is cumbersome. In the following analysis, a modification of the above iteration method is introduced that only involves one iteration loop.

As in the specified moment case, the tension T1(0) changes in each iteration loop.

The first step is also the same, namely calculating the steady mechanics for the tight free span using the single span BVP building block. After this step, instead of addressing the belt in the contacting zones and the belt in the free slack span successively, we address

both of them simultaneously to find M 2 _max (for the assumed T1_ 0 ) as follows.

= The unknown M 2 _max is defined as a field variable M 2 _maxMs 2_max() 2 governed by

dM() s 2_max 2 = 0 (5.47) ds2

forcing M 2 _max to be a constant. This ODE is added to the BVP (with unknown boundaries) for the slack free span. Correspondingly, one additional boundary condition

is needed to make the ODE system complete. Because M 2 _max is constant along the

= domain, the boundary value of M 2_max(0) at s2 0 equals the maximum transmissible

= moment. Substitution of (5.34) and MRRM1 (/)12 2_max into (5.33) yields

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1 M (sR= 0) / β =−ln[1 − 2_max 2 2 ] (5.48) 1 µ − 111TGV(0) (0)

Similarly, for the belt on the driven pulley,

1 M (0)/sR= β =−ln[1 − 2_max 2 2 ] (5.49) 2 µ − 21111TL( )() GVL

Without loss of generality, we assume that full slip appears first on the driver pulley

α = ( 1 0). Substitution of (5.48) into the first of (5.40) yields

1 Ms(0)/= R πθθ+−ln[12_max 2 2 ] −+= (0) (0) 0 (5.50) µ − 12 111TGV(0) (0)

θ where 1(0) has been calculated in the last step when solving the BVP for the free tight span. Equation (5.50) serves as the required additional boundary condition for the ODE

(5.47). The rest of the iteration process is the same as discussed previously. M 2 _max emerges naturally as part of the solution along with all other quantities giving the

= mechanics for MM2 2_max .

Since most general-purpose BVP solvers can not directly handle coupled

BVP/algebraic equations, this ODE conversion technique is useful for addressing such coupled systems.

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5.5 Results and Discussion

Two example two-pulley belt drives are examined. The first is for the regular moment transmission problem. Except for the bending stiffness, all specified data of this drive (Figure 5.4) are the same as for the string model example in [45]. The second example is for the maximum transmissible moment problem. Except for the bending stiffness, all specified data of this drive (Figure 5.7) are the same as in [46].

For a poly-ribbed belt typical of vehicle applications, the belt bending stiffness can be approximated by EIm=−( 1)2.867 × 10−32 Nm ⋅ (where m is the number of ribs and six ribs are common). For V-belts, the bending stiffness can be much larger. Three different belt bending stiffness values ( EINm=⋅0.0015,0.015,0.05 2 ) are considered in the examples.

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a) tight span

driver α pulley 1 β 2 R1 R2 β driven 1 α pulley 2

slack span

b) tight span

driver α 1 pulley β 2 R1 R2 β driven 1 pulley α2

slack span

c) tight span

driver

α1 pulley β R1 R2 2 β 1

driven α2 pulley

slack span

Figure 5.4. Steady solutions for the system properties specified in Table 5.1. a) EI = 0.0015, b) EI = 0.015, c) EI = 0.05 N ⋅m2 .

135

5.5.1 Example of regular moment transmission problem

The data for this system are specified in Table 5.1. The calculated results for

θ slip/adhesion zones, belt-pulley departure points 1,2 (0) , and span lengths are presented in Table 5.2 for three values of bending stiffness. The deflected belt shapes are shown in

Figure 5.4, where the strokes of the belt in the free spans are thickened. Inclusion of the bending stiffness decreases the wrap angles. For appreciable bending stiffness, the adhesion angles are reduced significantly. Notice that the belt transverse deflections are significantly increased for large bending stiffness. Figure 5.5 shows the variations of the tension and speed in the free spans. When EI = 0 , this converges to the string model where the tension and speed are uniform throughout the spans [45,46]. Large percentage increases in belt tension result with increased bending stiffness, and this impacts belt life.

Comparing line lengths in Figure 5.5 shows the increased span lengths for increased bending stiffness. The distribution of tension and speed on the pulley-belt contact zones are not given here because in the adhesion zones, the tension and speed are uniform, and in the sliding zones, the tension and speed are exponentially distributed, similar to the string models.

== ==π µµ== R12R 0.05m L R1 0.1571 m EAk= 25 N 120.6 = ω = =⋅ TNini 50 1 500rad / s M 2 2 N m G = 0.5kg / s

Table 5.1. Physical properties of the belt drive with two identical pulleys. 136

ββ= αα= θ −θ EI 1(0) 2 (0) L L 12 12 1 2 ⋅ 2 −θ θ L L ()N m (deg) (deg) ( 11(L )) ( 22(L )) (deg) (deg) 0 109.4 70.60 0 0 1 1 0.0015 100.2 64.67 5.645 9.503 1.063 1.106 0.0150 87.18 50.05 16.86 25.92 1.189 1.293 0.0500 90.26 15.91 29.82 44.00 1.339 1.511

Table 5.2. Numerical results for the belt drive specified in Table 5.1.

= = = µµ== Rm1 0.01 Rm2 0.05 EAk40 N 121 = ==ρ* Lm= 0.1 = TNini 800 m00Akg0.0056 / s ccr 374m / s

Table 5.3. Physical properties of the belt drive with two different pulleys.

137

90 25.012 EI=0.05 ) ) 88 EI=0.05 25.01 N

86 m/s 84 25.008 EI=0.015 82 25.006 80 25.004 78 EI=0.015 76 25.002 EI=0.0015 74 EI=0.0015 25 Tensions of tight span ( Speeds of tight span ( 72 EI=0 N.m2 EI=0 N.m2 70 24.998 -0.05 0 0.05 0.1 0.15 0.2 -0.05 0 0.05 0.1 0.15 0.2 x coordinate (free tight span) x coordinate (free tight span)

50 24.972

EI=0.05 ) ) 48 24.97 EI=0.05 N 46 m/s 24.968 44 EI=0.015 42 24.966 40 24.964 38 EI=0.015 36 24.962 EI=0.0015 34 EI=0.0015 24.96

Speeds of slack span ( . 2 Tensions of slack span ( 32 EI=0 N.m2 EI=0 N m 30 24.958 -0.05 0 0.05 0.1 0.15 0.2 -0.05 0 0.05 0.1 0.15 0.2 x coordinate (free slack span) x coordinate (free slack span)

Figure 5.5. Variations of tension and speed in the tight and slack spans for the belt-pulley drive in Table 5.1.

138

0.064 0.064 ) ) m m (a) (c)

0.06 0.06 EI=0.05

EI=0.05 0.055 0.055 EI=0.015 EI=0.015 EI=0.0015 EI=0.0015 0.05 0.05 EI=0 N.m2 EI=0 N.m2

0.045 0.045 Deflections of free tight span ( Deflections of free tight span ( 0.042 0.042 -0.05 0 0.05 0.1 0.15 0.2 -0.05 0 0.05 0.1 0.15 0.2 x of tight span (moving boundaries) x of tight span (fixed boundaries)

-0.035 ) ) -0.035 m (b) m (d) -0.04 -0.04

-0.045 -0.045 EI=0 N.m2 EI=0 N.m2 -0.05 -0.05 EI=0.0015 EI=0.0015 -0.055 -0.055 EI=0.015

-0.06 -0.06 EI=0.015 EI=0.05 -0.065 -0.065 EI=0.05 -0.07 -0.07 Deflections of free slack span ( -0.075 Deflections of free slack span ( -0.075 -0.05 0 0.05 0.1 0.15 0.2 -0.05 0 0.05 0.1 0.15 0.2 x of slack span (moving boundaries) x of slack span (fixed boundaries)

Figure 5.6. Deflections of the free spans for two different belt-pulley models. (a) and (b) correspond to the current model (symbols denote span endpoints); (c) and (d) correspond to the fixed boundary model in chapter 3 [63]. The system is specified in Table 5.1.

139

Figures 5.6a,b give the details of the free span deflections. The boundaries of the free spans change as the bending stiffness changes. For the concern of bending stiffness in belt-pulley systems, this theory is different from that used in the chapters 3 and 4

[23,24,59,63], which assumes that the boundaries of the free spans are fixed at the belt- pulley contact points of the string model, the speed is uniform throughout the system, the tensions are uniform throughout the free spans, and at the boundaries the beam displacement satisfies EIw=± EI / r , where r is the radius of the pulley. Figures ,xx BC

5.6c,d give the deflections of the free spans derived from the fixed boundary analysis in chapter 3 [63]. Comparison of results from these two theories shows that when the bending stiffness is small, the differences between these two theories are negligible; but as the bending stiffness increases, the differences are pronounced.

5.5.2 Example of maximum transmissible moment problem

Following [46], the data are specified in Table 5.3. ccr is the critical speed for the string model where the belt expands such that the maximum transmissible moment

ω vanishes. Note that in the string model of [46], instead of specifying 1 and G , m0 and

=≤≤ c (or Ccc/ cr ,0 C1) are specified. This does not change the definition of the

=+ problem. Because in [46], equation (16)4 is dii1 TE/ A and equation (23)2 is

ω = i cdi / Ri ,

=−ω TEARciii( / 1) (5.51)

Comparison of (5.25), (5.30), and (5.51) leads to

= c Vref (5.52)

140

= Equations (5.25) and (5.52) yield G m0c . For any assumed T1_ 0 , c and m0 can be

ω obtained from G and 1 , or vice versa. For example, suppose c and m0 are specified,

VT T = ω =+=+ref 1_0c 1_0 then G m0c and, using (5.31), 1 (1)(1) . R11EA R EA

Figure 5.7 depicts the steady states of the belt drive transmitting maximum moment

= α = = at cccr /2; full slip occurs on the driver pulley, 1 0. When EI 0 , the results are for the string model. Especially for pulleys of small radius, the wrap angle is reduced significantly with increasing bending stiffness as shown by the belt-pulley departure angles listed on the figure. Because generally it is the small pulley that first reaches the full slip state, and this determines the maximum transmissible moment, inclusion of the bending stiffness can greatly reduce the maximum transmissible moment, as shown in

Figure 5.8.

141

tight span

ο driven ο 29.45 22.37 driver pulley R1 pulley ο 38.83ο 20.32

R2 slack span

tight span

ο 43.98ο driven 19.13 pulley R1 driver pulley ο 60.05 ο 15.18 R2 slack span

c) tight span

12.96ο 68.58ο driven pulley R1 driver pulley ο 77.92 10.48ο

R2 slack span

Figure 5.7. Steady solutions for the system properties specified in Table 5.3. Full slip occurs on the driver pulley. a) EI = 0.0015, b) EI = 0.015, c) EI = 0.05 N ⋅m2 . 142

= In Figure 5.8, four non-dimensional variables are plotted with respect to C cc/ cr

= at different bending stiffness values: M maxMTR 2_ max/( ini 2 ) , Tt _ midspan/T ini , Ts _ midspan/T ini ,

η and , where Tt _ midspan and Ts _ midspan are tensions at the midpoints of the free tight and slack spans, respectively, and the power efficiency η is defined as the ratio between the power of the driven pulley and the driver pulley M ω VL( )1(()/+ TL EA) η ==22 22 = 22 (5.53) ω + M11VTEA 1(0) 1 ( 1 (0) / )

Figure 5.8 shows that increasing the bending stiffness significantly decreases the maximum transmissible moment and increases the power efficiency η . The significant overestimation of the maximum transmissible using the string model can lead to poor performance and unanticipated full belt slip, especially for belts with appreciable bending stiffness. Bending stiffness reduces the effects of belt speed C on the steady motion of the system because variations of the non-dimensional variables over the same range of C are decreased for increasing bending stiffness.

143

2.5 1

EI=0.05 0.99 2 η EI=0 N.m2 EI=0.0015 1.5 0.98 EI=0.015

1 EI=0.015 0.97 EI=0.0015 EI=0 N.m2 Power efficiency 0.5 0.96 EI=0.05 Maximum moment (Mmax)

0 0.95 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Dimensionless speed C Dimensionless speed C

1.2 3 2.8 ini ini 1 EI=0.05 2.6 /T /T 2.4 0.8 EI=0.0015 2.2 2 t_midspan s_midspan EI=0 N.m EI=0.015 0.6 EI=0.0015 2 EI=0 N.m2 1.8 EI=0.015 0.4 1.6 1.4 0.2 EI=0.05 Tension (tight) T

Tension (slack) T 1.2 0 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Dimensionless speed C Dimensionless speed C

= Figure 5.8. Comparison of maximum transmitted moment M maxMTR 2 _ max / ini 2 , power η efficiency , Tt _ midspan/T ini , and Ts _ midspan/T ini between the string and beam models for the belt drive in Table 5.3.

144

CHAPTER 6

SUMMARY AND FUTURE WORK

6.1 Summary

This work analytically investigates several key issues encountered in analyzing serpentine belt drives. The main results are summarized for each specific topic.

1. Approximate eigensolutions of axially moving beam with small bending stiffness

Perturbation techniques for algebraic equations and the phase closure principle are combined to analyze the free vibration of axially moving beams with small bending stiffness. Closed-form approximate natural frequencies and vibration modes are derived based on the propagation and attenuation properties of the constituent waves. Uniformly valid approximate eigenfunctions are obtained. Different combinations of boundary conditions can be readily handled. Boundary layers, when present, are incorporated via

145 evanescent waves. When the axial speed of the beam is zero, the solutions converge to known solutions for these non-gyroscopic systems. The approach is straightforward, suited for different boundary conditions, and has accessible physical explanation.

2. Modeling and steady state analysis of serpentine belt drives with bending stiffness

A model of entire serpentine belt systems including belt bending stiffness is established and a steady state analysis is performed. A numerically exact solution is presented to determine the span and tensioner steady state. This requires a novel transformation of the governing equations to a standard ODE form readily accepted by general-purpose BVP solver codes. A closed-form analytical solution is also developed for the case of small bending stiffness using singular perturbation. A coupling indicator

( Γ ) is defined based on the steady state to quantify the undesirable vibration coupling between pulleys and spans. The perturbation solution analytically exposes the effects of the design variables on the steady state and coupling in terms of simple expressions. The major conclusions of this part include: 1) Belt bending stiffness and span tensions strongly influence the steady state deflections and pulley-span coupling. 2) Speed has a much smaller effect on this coupling due to the automatic tension compensation ability of the tensioner assembly when properly designed. 3) The effects of tensioner spring stiffness and longitudinal belt stiffness on steady state are small over a wide range of variation. 4) The vibration coupling between the pulleys and spans is determined mainly by the steady state span curvatures. A defined coupling indicator captures the magnitude of coupling for each span. 5) System geometry, especially the pulley radius to span length 146 ratios, significantly affects steady state and coupling. This suggests that an effective, low cost solution to troubleshoot coupling problems is increasing the radii of pulleys that bound the problem span.

3. Dynamic analysis of serpentine belt drives with bending stiffness

Free vibration about above non-trivial steady state that result from belt bending stiffness are examined. A mathematical reformulation of the governing equations leads to an extended operator form that has the mathematical structure of a conservative gyroscopic system. In contrast to prior formulations, the mixed continuum/discrete boundary conditions at the interface between the belt and the tensioner pulley are replaced by trivial boundary conditions for all spans, including those adjacent to the tensioner. This transformation admits an efficient spatial discretization using Galerkin’s method applied to the structured extended operator form. The method is numerically robust and free of missing/false natural frequency concerns, while at the same time preserving the conservative gyroscopic character of the discretized model. Dynamic response calculations using the discretized model follow naturally.

Belt bending stiffness introduces a linear coupling between the belts bounded by fixed pulleys and the rest of the system. For appreciable bending stiffness (or low tension), all modes are spatially distributed and involve transverse deflections of all spans in addition to the pulley rotations, in contrast to zero bending stiffness models where the modes divide into rotational pulley and transverse span modes. This modal coupling provides a mechanism whereby pulley rotation, which is directly excited by engine torque/speed fluctuations, couples to transverse vibration of all spans, including those

147 bounded by fixed centers. This provides an explanation for the span vibrations observed in practice that potentially lead to noise and belt fatigue failure. This mechanism applies at engine idle speeds where parametric excitation mechanisms based on higher frequency excitation [14,15,29] do not apply.

While the natural frequencies generally increase with bending stiffness, the changes are not monotonic. For small bending stiffness, some natural frequencies initially decrease. This unusual phenomenon results because the system steady state changes with bending stiffness in a way that tends to increase compliance for deflections about steady state.

Belt speed has reduced effect on the natural frequencies as bending stiffness increases within practical ranges. In contrast to the string model where only transverse dominant modes are affected by speed, all natural frequencies change with speed as bending stiffness induced modal coupling increases. Unlike single span moving string and beam models, serpentine drive natural frequencies do not decrease monotonically with speed.

For systems with small bending stiffness, changing the tensioner orientation to

η increase the tensioner effectiveness increases the natural frequencies of rotationally dominant modes while having little influence on the natural frequencies of transversely dominant modes. For systems of large bending stiffness, tensioner orientation influences

η all natural frequencies and, for the example system, tends to decrease them as increases.

148

4. Steady state contact mechanics of belt-pulley systems

Belt bending stiffness is included in the steady state analysis of belt-pulley systems where belt inertia is also modeled. An iterative solution is presented to determine the deflections of the belt, belt-pulley contact points, span lengths, and the distributions of speed, tension, and friction along the belt. Inclusion of bending stiffness leads to initially unknown belt-pulley contact points, yielding a governing boundary value problem (BVP) with undetermined boundaries. This requires a transformation of the governing equations to a standard ODE form with fixed boundaries. This form is readily accepted by general- purpose BVP solver codes. The main conclusions of this part include: 1) Inclusion of bending stiffness leads to non-uniform speed and tension in the free spans and reduces the belt wrap angles on pulleys, especially for small radii. Span tensions, which directly impact belt life, increase markedly with bending stiffness. 2) Bending stiffness decreases the wrap angles, causes earlier full slip of the belt on the pulleys, increases the power efficiency η , and decreases the maximum transmissible moment. Some of these effects are pronounced for appreciable bending stiffness and may cause poor performance in systems designed based on string model analysis. 3) The effects of belt speed on the steady motion are reduced as the bending stiffness increases.

149

6.2 Future Work

The future work in the following challenging areas is recommended to advance the modeling and understanding of general belt-pulley systems, where serpentine belt drives are one application.

1. Modeling of belt-pulley systems considering belt radial stiffness

In addition to belt bending stiffness, other mechanisms may also cause dynamic interaction between the belt and pulleys. For example, radial belt stiffness may induce coupling between the belt and pulleys, especially for thick belts. Radial stiffness was first considered by Gerbert when dealing with flat belts [70]. Only the steady state problem is considered there; dynamic vibration has not been investigated.

H T x T

R r x Cords r R Elastomer

T T

Figure 6.1. Model considering the radial stiffness of the belt.

150

The cross-section of the multi-ribbed belt consists of two parts: the high tension cords and the elastomer compression layer surrounding the cords (Figure 6.1). The cords bear most of the span tension, and the elastomer enables the belt to grip the pulley through friction and deliver power from the crankshaft. In the contact zones between the belt and the pulley, the cords lie on an elastic cushion layer, as seen in Figure 6.1. The elastic cushion layer (elastomer) experiences compressive and shearing deformations.

The lower surface of the elastomer is assumed to stick to the pulley due to friction, and potential energy is stored in the cushion layer sandwiched between the cords and pulley.

Through shearing of the elastomer, the pulley rotations interact with the longitudinal motions of the cords in the contact zones.

For the curved cords in the contact zones, the longitudinal and transverse vibrations of cables are linearly coupled with each other due to the curvature, unlike for straight continua [71,72]. Thus, in the contact zones, the transverse motion of the cords is coupled with the pulley rotations as well as the cord longitudinal motions. Furthermore, for the free span cords between two pulleys, both the longitudinal and transverse vibrations can be excited by the vibrations of the cords in the contact zones due to the continuity of longitudinal and transverse displacements at the belt-pulley boundaries.

For example, for the two pulley system, if treating the cords as tensioned strings, there are four segments of strings: two contact zones and two free spans. The degrees of freedom of the system include the transverse and longitudinal displacements of the four string segments (continuous variables) and the rotations of the two pulleys (discrete variables). The elastomer is modeled as an elastic foundation that stores potential energy

151 by experiencing compressive and shear deformations. Hamilton’s principle can be used to derive the equations of motion. Because the belt speed can be high, the model will be a hybrid gyroscopic continuous-discrete system.

A key feature of this belt-pulley mechanism is that the cords in the contact zones can have transverse vibration due to the elastic foundation between the cords and pulley.

Through the curvature of the cords in the belt pulley contact zones and the continuity of the longitudinal and transverse displacement at the free span interface, the transverse vibrations of the free span belt and the pulley rotations are coupled. This coupling mechanism will be more important when the stiffness of the cushion layer is smaller.

Generally, the thicker the belt is, the smaller the radial stiffness of the elastomer [70].

Similar to the moving beam model in chapters 3 and 4, the steady analysis must be investigated first, and then free and forced vibration analysis about the steady solution can be conducted. To solve the corresponding eigenproblem, computational methods like those used in the previous moving beam model can be used [59] (e.g., the extended operator form and spatial discretization through reformulation). Modal coupling between the cords and pulleys is expected. The effects of various design parameters on this belt- pulley coupling, like the thickness of the belt, the curvature (radius) of the pulleys, the belt speed, and so on, need be investigated.

152

2. Steady state contact mechanics for V-belt/ribbed-belt drive systems

The model discussed in chapter 5 and those in [45,46] address the steady state mechanics between the pulleys and the flat belt. There are no grooves on the surface of the pulleys. In the belt-pulley contact zones, it is assumed that the belt’s radius is exactly the same as that of the pulleys, and the friction is only along the tangential direction of the pulley surface.

On the contrary, for V-belts (or ribbed belts as used in serpentine belt drives), the belt seats in grooves on the surface of the pulleys. As a result, the curvature (pitch radius) of the belt varies even in the belt-pulley contact zones, see Figure 6.2. Hornung [73] seems to be the first person to mention this phenomenon in his dissertation, but his discussion was limited to a qualitative physical description and no mathematical model was established. With the availability of modern general-purpose BVP solvers, and based on the available experience for the flat belt, this problem is expected to be solvable. A

Coulomb law can still be used to describe this frictional force. Due to the varying pitch radius of the belt on the pulleys, however the friction force is 2-dimensional and its actual direction is not known a priori, which makes the problem more challenging. Also, the belt-pulley contact zones should be divided into four different zones (seating, adhesion, sliding, and unseating), as opposed to two (adhesion and sliding) for the flat belt case.

The boundaries of all these different zones are to be determined from the compatibility condition (like that used in chapter 5) and the continuity conditions for friction force, belt

153 radius, tension, and speed at the boundary points. This model is an improvement of that in chapter 5 and can better predict system performance criteria such as power efficiency and maximum transmissible moment for V-belts and ribbed belts.

In addition, the discussion up to now focuses on fixed pulley systems. This can be further extended to serpentine belt drives including a tensioner assembly. Consideration of the tensioner arm may further complicate the steady state analysis because the rotational position of the tensioner arm adds another degree of freedom.

Finally, this steady state mechanics analysis (based on fewer assumptions compared with those adopted in chapters 3 and 4) can be extended to dynamic analysis, which if implemented, would be a major advancement in the field of general belt-pulley systems.

Tight Span Unseating

V1(S1) Sliding Seating ) Sliding Adhesion δ2 σ1 N β β 1 2 α 1 M1 M2 Driven pulley α Driver pulley 2 δ1

σ2 Adhesion Unseating N' β N' V (S ) 2 2 Seating Slack Span F=2 µ N' (2-D)

Figure 6.2. Steady motion of an extensible belt considering bending stiffness.

154

3. Friction-induced vibration of belt-pulley systems

Under certain conditions, large span transverse motions are accompanied by squeal/chirp sounds in belt-pulley systems, which indicate that relative stick-slip motions exist in the contact interface between the pulleys and the spans. These squeal/chirp sounds and the large transverse vibration of the spans are often perceived as disturbing noises by vehicle occupants. Reducing these noises is important to the design of the serpentine belt system. However, the underlying mechanism responsible for this phenomenon has not been examined.

FR X X

B A m FR Vo V 0 Vo X Vr =Vo-X

V0 Xs a) b) c)

Figure 6.3. a) Self-sustained system, b) friction force characteristics, c) limit cycle.

155

Squeal/chirp is related to a broad class of vibration referred to as self-excited or self-sustained vibrations. In this case, the mechanism is dry friction. Friction-induced vibrations occur in a variety of mechanical systems including squealing brakes, chattering machine tools, squeaking doors, and musical instruments like violins. Figure 6.3 shows a simple friction-induced vibratory system. In such a system, the negative slope between friction force and relative speed between the mass and belt in a local speed range near equilibrium is necessary to initiate self-excited vibration. For most dry friction cases, the negative slope comes from the difference between the static and kinetic friction coefficient [60]. Rayleigh used a similar system to explain the vibration of a bowed string.

For the grooved pulleys and multi-ribbed belt used in the serpentine belt system, the interfacial friction between the belt and the pulley consists of tangential and radial components (Figure 6.4). Former research has focused on tangential slip, which may induce longitudinal vibration of the belt [74,75]. In reality, however, friction is two- dimensional and has a radial component, especially in the seating and unseating sections of the belt-pulley contact zones due to the variation of span tensions when the belt enters and exits the pulley grooves. As shown in Figure 6.4, the belt slides over the sliding arc not only in the tangential direction but also in the radial direction. In the middle of the sliding arc, the distance of the belt from the bottom of the groove is H ' while at the exit point, this distance is HH< ' . Recent experiments [74-76] also indicate some relations between radial sliding and transverse vibration of the spans, but no rigorous dynamic model has been established to describe these phenomena. More insight into this friction-

156 induced coupling between the belt and pulley can be sought. As the first step, the belt can be modeled as a moving string because bending stiffness is not necessary for this belt- pulley coupling. Later, more effects (like bending stiffness) might be incorporated.

Questions to be considered include: 1) Under what conditions will this coupling mechanism be dominant? 2) What are the effects of various design parameters on this coupling mechanism? 3) What analysis tools are effective for the friction model? 4) How can proper design reduce this belt-pulley coupling?

H Groove Belt FT H' τ

Driven Pulley β Friction

Figure 6.4. Two-dimensional friction between belt and grooved pulley.

157

A key characteristic of this model is that this belt-pulley mechanism is not driven by the oscillation of the pulleys. Even if there is no pulsation excitation from the crankshaft, meaning the pulleys have constant rotational speed, this belt-pulley coupling mechanism is active. On the other hand, experimental data shows that large transverse vibration of some spans in serpentine belt systems is sometimes accompanied by large rotational oscillations of the two adjacent pulleys. For this kind of coupled vibration between the pulley and belt, the friction force may also play an important role. Separate modeling needs to be established to address this phenomenon, and more investigation can be done along this research direction.

158

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