Analysis of Tensioner Induced Coupling in Serpentine Belt Drive Systems

Transcription

1 of Tensioner Induced Coupling in Serpentine Belt Drive Systems Copyright 2007 SAE International R. P. Neward and S. Boedo Department of Mechanical Engineering, Rochester Institute of Technology ABSTRACT A primary concern in the design of serpentine belt drive systems is resonant strand vibrations induced from engine excitation. Two analysis approaches to investigate the system vibrational response have been reported in the literature. The first, denoted as the "decoupled analysis" approach, employs longitudinal belt stiffness and takes into account only pulley rotation and tensioner displacement as system degrees of freedom. Transverse belt vibration (normal to belt travel) on all belt strands is decoupled from the analysis. The second, denoted as the "coupled analysis" approach, combines transverse tensioner strand belt motion with pulley rotation and tensioner displacement. Transverse belt vibration on strands between fixed pulleys remains decoupled from the system. This paper provides apparently the first cross comparison of these two analysis techniques on three distinct serpentine belt system configurations reported in the literature. Two of the belt drive systems in this study involve numerical models only, and a third published work involves experimental results was used to confirm the accuracy of both analysis techniques. It was observed that the coupled and decoupled analyses did not agree well for low order modes, and only the coupled formulation was able to predict experimentally determined mode shapes. Both analyses indicated that natural frequenices associated with rotational modes were insensitive to crankshaft speed. Natural frequencies associated with transverse belt span modes as computed with the coupled analysis were less sensitive to changes in crankshaft speeds. INTRODUCTION An engine front-end subsystem utilizing a single V- ribbed belt is known as a serpentine belt drive due to the long and elaborate path that the belt must follow. The advantages of using a serpentine belt drive setup over multiple V-belts include compactness, ease of belt replacement, length of belt life, and the need to employ only a single tensioner mechanism. The function of the belt tensioner is to maintain constant tractive tension throughout the entire belt drive system in the presence of belt wear, assembly variation, and deviation in belt length due to changes in accessory torques, belt speed, and belt temperature. The simplest type of tensioner is comprised of an idler pulley pinned to a rigid moment arm which pivots about a fixed point where a coil spring provides the tensioning load. A primary concern in the design of serpentine belt drive systems are resonant strand vibrations induced from engine excitation. Two analysis approaches to investigate the system vibrational response have been reported in the literature. The first, denoted as "decoupled analysis", employs longitudinal belt stiffness and takes into account only pulley rotation and tensioner displacement as system degrees of freedom. Transverse belt vibration (normal to belt travel) on all belt strands is decoupled from the analysis. The second, denoted as the "coupled analysis", combines transverse tensioner strand belt motion with pulley rotation and tensioner displacement. Transverse belt vibration on strands between fixed pulleys remains decoupled from the system. The basis for the work that has been done in the field of serpentine belt drives is research on the vibration characteristics of axially moving material. Mote [1] studied the vibrational characteristics of band saws and found that the band saw natural frequencies depend upon the relative motion of the band pulley axis. Modeling and analyzing serpentine belt drive systems with a dynamic tensioner was first accomplished by Ulsoy et al. [2] in which they used a mathematical model to examine the transverse vibration and stability of coupled belt-tensioner systems. Gasper and Hawker [3] and Hawker [4] developed a system of governing equations for a serpentine belt drive system and introduced a solution technique that resulted in the eigenvalues and eigenvectors of the complete system which was validated by experimentation. Hwang et al. [5] derived a nonlinear model that governs the longitudinal response of the belt spans in correlation with the rotational response of the crankshaft and accessory pulleys. Solution of the equilibrium equations leads to the tension-speed relationship for the serpentine belt drive system. The overall equations of motion are then linearized about the equilibrium state allowing the

2 rotational mode characteristics to be obtained from the associated eigenvalue problem. Kraver et al. [6] extended the modal vibration analysis to include a viscous belt span and coulomb tensioner arm damping. In order to validate the model, their results were compared against Hwang et al. [5] and found be in good agreement. Using dry friction damping within their model, as opposed to the more commonly used viscous damping, Leamy and Perkins [7] were able to capture the primary and secondary resonances within the belt drive system. All of the works mentioned previously assume that the linear response of belt drives is composed of the superposition of independent transverse and longitudinal modes. However, as shown by Beikmann et al. [8-11], there exists a linear coupling between the transverse and rotational modes in the spans adjacent to the automatic tensioner. This linear coupling is created by the rotational degree of freedom of the tensioner arm. In addition, there exists a nonlinear coupling mechanism between rotational and transverse modes arising from the finite stretching of the belt. This coupling can become greatly magnified under conditions leading to internal or autoparametric resonance. Beikmann also introduced a tensioner support constant η which is an indicator of the systems ability to maintain tractive tension (despite load and speed variations), and an indicator of the stability of the reference equilibrium state. Zhang and Zu [12, 13] employed the three pulley model developed by Beikmann to extend the understanding of serpentine belt drive behavior. Belt bending stiffness, which is assumed to be negligible in the previous works, was studied extensively by Wasfy and Leamy [14] and Kong and Parker [15, 16]. In these works, belt bending stiffness is introduced into the models which require additional techniques to solve for the vibration characteristics. serpentine belt drive sample application suitable for cross-comparison. Experimental results from a third serpentine belt drive system reported by Beikmann [8] will be used to confirm the accuracy of both analysis techniques. A parametric engine speed study will be presented to assess the impact of each analysis method on the predicted natural frequencies and associated mode shapes of each belt drive system. SYSTEM GEOMETRY Figure 1 shows the geometry of the serpentine belt drive system employed in both decoupled and coupled analysis approaches. The system is comprised of n pulleys with radii R j, j = 1, n, connected by a single serpentine belt represented by n belt strands between adjacent pulleys. The rotational axes of the n-1 pulleys are fixed, while the nth tensioner pulley is attached to a movable spring-loaded tensioner arm. Pulley 1 is attached to the engine crankshaft rotating at fixed angular velocity ω 1, and the belt is thus driven by the crankshaft at a fixed linear speed V. The absolute rotation of the jth pulley is defined as Θ j (t) = ω j t + θ j (t) (1) where pulley angular velocity ω j is given by with ω j = V / R j (2) V = R 1 ω 1 (3) θ t Parker [17] formulated an efficient method for calculating the eigensolutions and dynamic responses of coupled serpentine belt drive systems. The speed of solution is drastically reduced and the numerical problems that hinder other published methods are eliminated. However, by coupling the rotational and transverse motions in the spans adjacent to the tensioner, the system of equations becomes significantly more complex than modeling the motions as uncoupled. As a result, the solution techniques required to solve these models demand considerably more computational power. P n Y S n w i-1 P i-1 θ 1 V This paper provides apparently the first crosscomparison of decoupled and coupled analysis techniques on three different serpentine belt systems. The decoupled and coupled analysis approaches are taken from the representative studies of Hwang et al. [5] and Parker [17], respectively. These particular papers were chosen based on the fact that each captures the essential features of the assumptions inherent in each analysis approach, and each paper provides a distinct X P 1 w i θ i R i P 2 P 3 S 2 ω 1 S 1 Fig. 1 Belt drive system geometry Assuming crankshaft angular velocity ω 1 is constant, phase angle θ j (t) represents the angular motion of the jth pulley as viewed by an observer rotating at the respective (constant) angular velocity of the jth pulley.

3 In other words, θ j (t) represents the motion of the jth pulley as observed by a stroboscope tuned to the respective frequency ω j. Transverse belt displacement on the ith belt strand is represented by a function W i (x i,t), i = 1, n, where x i is the spatial coordinate as measured between the two connected pulleys. Assuming crankshaft angular velocity ω 1 is constant, W i (x i,t) represents the transverse motion of the ith belt stand moving with an axial velocity of V. Under conditions of fixed crankshaft angular velocity ω 1 and no external disturbances, the belt drive system assumes an equilibrium configuration θ e [θ 2 θ 3 θ n θ t ] T e, W i (x i,t) = 0, where phase angles take on constant values and strand transverse motions are zero. Small disturbances from equilibrium arise primarily from crankshaft and camshaft torsional excitations, and this paper examines the natural frequencies and associated mode shapes of the linearized belt drive system about a specified equilibrium configuration. DECOUPLED FORMULATION The theoretical formulation for the decoupled analysis of the serpentine belt drive system is taken directly from Hwang et al. [5] and is outlined here in condensed form for completeness. The assumptions used in developing the governing equations include: -- The belt does not slip on the pulleys. -- The belt is uniform, perfectly flexible, and stretches in a quasi-static manner. -- Transverse belt response on all strands are decoupled from longitudinal belt response. -- The crankshaft motion and any torque inputs from accessories are prescribed (either zero or determined from experiments). -- The tensioner executes small motions about some steady state position. Moreover the tensioner mechanism is designed to be dissipative and is the dominant source of dissipation. This dissipation is assumed to be linear viscous damping, and dissipation in the belt and fixed pulleys is assumed to be negligibly small. The linearized equations of motion for the decoupled system take on the form M D 2 z + C D z + K z = 0 (4) where the degrees of freedom z [ ε 2 ε n ε t ] T represent the set of small pulley phase motion and tensioner arm rotation about an equilibrium configuration, with (time) differential operator D d/dt. Mass, damping, and stiffness matrices M, C, and K depend primarily upon strand stiffness, component polar moment of inertia, tensioner rotational stiffness, and equilibrium strand tensions, all of which depend on crank angular velocity ω 1. The equilibrium system configuration θ e and system vibrational response given by equation (4) are both relative to a reference tension (constant on all strands) obtained by setting V = 0. Defining z = Z exp(iλt), the natural frequencies λ and mode shapes Z of the system can be found in the usual manner from the eigenvalue problem [ λ 2 M - iλ C + K ] Z = 0 (5) The m = 1, natural frequencies associated with all belt strand transverse vibration (including those attached to the tensioner) are assumed completely decoupled from equation (5) and are given by Abrate [18] as λ belt = (mπ/l j )[(P j - ηρv 2 ) /ρ) 1/2 x [1 - (1-η)V 2 /c 2 ](1 + ηv 2 /C 2 ) -1/2 (6) where, for the jth strand, L j is the belt strand length, ρ is belt strand density, and P j is the equilibrium strand tension at engine speed ω 1. Parameter η depends upon the relative stiffness of the pulley supports and axial belt stiffness, and c = (P j / ρ) 1/2. COUPLED FORMULATION The theoretical formulation for the coupled analysis of the serpentine belt drive system is taken directly from Parker [17] and is outlined here in condensed form for completeness. Key assumptions preserved in this coupled formulation include: --The belt properties and belt speed are uniform. --Belt bending stiffness is negligible. -- Damping is not modeled. -- Belt-pulley wedging and belt slip at the belt-pulley interfaces are not considered. Referring to Figure 1, the transverse belt strand displacements W i-1 (x,t), W i (x,t) attached to the tensioner pulley are now assumed coupled to pulley rotations and tensioner pulley translation. Transverse strand vibration is governed by the partial differential equation ρ ( 2 W j / t 2 ) 2ρV ( 2 W j / x t) (P j ρv 2 ) ( 2 W j / x 2 ) = 0 j = i-1, i (7) Boundary conditions on each strand require zero belt transverse displacement at points attached to the fixed pulleys and displacements which are constrained to move with the center motion of the tensioner pulley.

4 Belt strand transverse displacement for those strands attached to the tensioner is represented by a linear combination of r orthogonal mode shapes α r (ξ i-1 ) and γ r (ξ i ) of the form W i-1 (x i-1,t) = R 1 Σ a r (t) α r (ξ i-1 ) (8) W i (x i,t) = R 1 Σ b r (t) γ r (ξ i ) (9) where ξ i = x i /L i and mode shapes α r and γ r satisfy the belt strand boundary conditions. Employing this modal formulation for the belt strands, incorporating reference strand tension, and linearizing the system about an equilibrium engine speed, the coupled analysis as described by Parker [17] once again takes the form M D 2 z + C D z + K z = 0 (10) Table 1 Belt Drive System: Case Study 1 Pulley Type Center Pulley Location (x,y) Radius 1 Crankshaft 0, Air Cond , Power Strg. 252, Idler 90.3, Alternator 86, Water Pump 0, Tensioner 151.2, where z [ a 1 a r b 1 b r ε 2 ε n ε t ] T now includes r modal contribution factors for each of the belt strands attached to the tensioner. The natural frequencies and mode shapes of this system will now involve coupling of pulley phase angles and modal contribution factors. The natural frequencies of transverse vibration for belt strands which are attached to fixed pulleys remain decoupled from the system and are given by equation (6) above. Tensioner Arm Pivot Location (x,y) Tensioner Arm Effective Length L t Tensioner Arm Installation Angle θ 0 (deg) 142, CASE STUDY 1 Table 1 provides data for a sample serpentine belt drive system denoted as Case Study 1, taken from Hwang et al. [5] based on the setup of an actual engine. The geometric configuration of the belt drive system representing Case Study 1 is shown in Figure 2. P6 P1 S5 Y P5 S4 P4 X S6 θ 0 S7 P7 S1 S3 P3 P2 Fig. 2 Belt drive system: Case Study 1 S2 Table 2 provides a set of mode shapes starting at the lowest natural frequency obtained from both decoupled and coupled analyses at an crankshaft speed of rev/min. (Modes and natural frequencies for transverse modes on belt strands between fixed pulleys are not included here.) In either analysis, each resulting mode shape for the system is dominated by a single rotational or transverse vibration on a single pulley or tensioner strand. Thus, the identification "P5 - Rotational -1" indicates that this mode shape is essentially dominated by rotational motion of pulley 5, and that it is the first mode shape encountered with this property as computed in the decoupled analysis. For this case, relative amplitudes on the remaining pulleys are small compared to pulley 5. (The second mode shape dominated by rotational motion of pulley 5 would be identified as "P5 - Rotational -2" and so on.) As a check, the relative amplitudes of pulley rotations corresponding to each mode shape obtained from the decoupled analysis were found to match well with those provided in the decoupled analysis by Hwang et al. [5]. Table 3 compares the corresponding natural frequencies obtained for each mode shape as computed from the decoupled and coupled analyses at a crankshaft speed of rev/min. Results from the decoupled analysis compare well with those obtained by Hwang et al. [5]. The mode shapes obtained from the coupled analysis are similar to those obtained from the decoupled analysis, as indicated by modal assurance values near 1 [19]. The corresponding natural frequencies found in the coupled analysis, however, are generally greater,

5 especially for mode shape 3 characterized by coupled rotation of the tensioner arm (TA) and tensioner pulley 7. Figure 3 shows that the natural frequencies associated with mode shapes 4, 5, 8, and 9 representing transverse motion of tensioner strands decrease with engine speed for both decoupled and coupled analyses. The reduction in natural frequency is less pronounced in the coupled analysis, indicating that the transverse belt motion acts as a stiffening mechanism on those respective modes. Table 2 Mode Shapes: Case 1 (P = pulley, TA = tensioner arm, S = belt span) Mode Shape 1 P5 Rotational 1 2 P2 Rotational 1 3 TA Rotational 1 4 S7 Transverse 1 5 TA/P7 Rotational 1 6 S6 Transverse 1 7 P3 Rotational 1 8 S7 Transverse 2 9 S6 Transverse 2 10 P7 Rotational 2 11 P4 Rotational 1 (a) decoupled analysis Table 3 Natural Frequencies: Case 1 (MA = modal assurance criterion) Mode Decoupled (Hz) Coupled (Hz) MA Hwang et al. [5] (b) coupled analysis Fig. 3 Natural frequencies associated with transverse modes on belt spans attached to the tensioner: Case Study 1 The natural frequencies for the remaining rotational mode shapes are essentially unaffected by crankshaft speed for both decoupled and coupled analyses. The decoupled results are again in agreement with trends obtained by Hwang et al. [5]. Figure 4 shows the effect of engine speed on the natural frequencies associated with mode shapes on the belt spans between fixed pulleys as computed from equation (6). Note that these natural frequencies fall in the range of those found in the system analyses.

6 P5 S4 S5 S6 P7 θ 0 P4 S3 P3 S7 S2 P6 Y P1 X P2 S1 Fig. 4 Natural frequencies associated with transverse modes on belt spans between fixed spans: Case Study 1 CASE STUDY 2 Table 4 provides data for a sample serpentine belt drive system denoted as Case Study 2, taken from Parker [17] based on an automotive system that was experiencing a noise and vibration problem. The geometric configuration of the belt drive system representing Case Study 2 is shown in Figure 5. Table 4 Belt Drive System: Case Study 2 Pulley Type Center Location (x,y) Pulley Radius 1 Crankshaft 0, Air Cond , Alternator 231.7, Idler 79.6, Power Strg , Water Pump -200, Tensioner -45.1, Tensioner Arm Pivot Location (x,y) Tensioner Arm Effective Length L t Tensioner Arm Installation Angle θ 0 (deg) 33, Fig. 5 Belt drive system: Case Study 2 Table 5 provides a set of mode shapes starting at the lowest natural frequency obtained from both decoupled and coupled analyses at an crankshaft speed of 680 rev/min. (Modes and natural frequencies for transverse modes on belt strands between fixed pulleys are not included here.) In this case, modes 2 and 11 exhibit dominant rotational motions on both pulleys 6 and 7, identified as "P6/P7". Table 5 Mode Shapes: Case 2 (P = pulley, TA = tensioner arm, S = belt span) Mode Shape 1 P3 Rotational 1 2 P6/P7 Rotational 1 3 P2 Rotational 1 4 P5 Rotational 1 5 S6 Transverse 1 6 S7 Transverse 1 7 P4 Rotational 1 8 S6 Transverse 2 9 S7 Transverse 2 10 P4 Rotational 2 11 P6/P7 Rotational 2 Table 6 compares the corresponding natural frequencies obtained for each mode shape as computed from the decoupled and coupled analyses at a crankshaft speed of 680 rev/min. Results from the coupled analysis compare well with those obtained by Parker [17]. However, low modal assurance values indicate that mode shapes 2, 3, and 4 obtained from the decoupled analysis are substantially different from those obtained

7 from the coupled analysis. The influence of tensioner strand coupling on the resulting system response is thus quite strong for this case study, as its inclusion substantially alters three of the predicted mode shapes. For the remaining set of mode shapes, the corresponding natural frequencies found in the coupled analysis are once again generally greater than those predicted using the decoupled analysis. Y P3 S2 X θ 0 S3 P2 S1 P1 Table 6 Natural Frequencies: Case 2 (MA = modal assurance criterion) Mode Decoupled Coupled MA Parker [17] (Hz) (Hz) The natural frequencies associated with mode shapes 5, 6, 8, and 9 representing transverse motion of tensioner strands exhibit the same trend with engine speed for both decoupled and coupled analyses as that found in Case Study 1. Moreover, the natural frequencies for the remaining rotational mode shapes are once again essentially unaffected by engine speed for both decoupled and coupled analyses. The coupled results are again in quantitative agreement with trends obtained by Parker [17]. The effect of engine speed on the natural frequencies associated with mode shapes on the belt spans between fixed pulleys follow the same trends as in Case Study 1, and these natural frequencies also fall in the range of those found in the system analyses. CASE STUDY 3 Table 7 provides data for a sample serpentine belt drive system denoted as Case Study 3, taken from Beikmann [8] based on an experimental test stand. The geometric configuration of the belt drive system representing Case Study 3 is shown in Figure 6. Although the system presented here is much smaller than those presented in the previous two case studies, it contains all the necessary components critical to a serpentine belt drive system including a driving pulley, automatic tensioner, and a driven pulley. Fig. 6 Belt drive system: Case Study 3 Table 7 Belt Drive System: Case Study 3 Pulley Type Center Location (x,y) Pulley Radius 1 Crankshaft 552.5, Tensioner 347.7, Idler 0, Tensioner Arm Pivot Location (x,y) Tensioner Arm Effective Length L t Tensioner Arm Installation Angle θ 0 (deg) 250.8, Table 8 provides a set of mode shapes starting at the lowest natural frequency obtained from both decoupled and coupled analyses at zero crankshaft speed. Table 9 compares the corresponding natural frequencies obtained for each mode shape as computed from the decoupled and coupled analyses at zero crankshaft speed. It is observed that decoupled formulation fails to capture the experimentally observed tensioner arm rotational mode shape. The coupled analysis predicts both the transverse span 2 and the tensioner arm rotational mode shapes, and the corresponding natural frequencies agree well with experiment. The higher order modes and corresponding natural frequencies are captured equally well by the decoupled and coupled formulations. Table 10 shows that the natural frequency predicted by equation (6) for transverse vibration of the span between fixed pulleys agrees very well with experiment.

8 Table 8 Mode Shapes: Case 3 (P = pulley, TA = tensioner arm, S = belt span) Mode Shape 1 S2 Transverse 1 2 P3 Rotational 1 3 S2 Transverse 2 4 S1 Transverse 1 5 P3 Rotational 2 6 S1 Transverse 2 7 P2 Rotational 1 8 TA Rotational 1 Table 9 Natural Frequencies: Case 3 (MA = modal assurance criterion) Mode Decoupled Coupled MA Beikmann [8] (Hz) (Hz) Table 10 Natural Frequency Comparison on Fixed-Fixed Tensioner Strand: Case 3 Equation (6) 31.9 Hz Experimental 33 Hz (Biekmann [8]) CONCLUSIONS Despite the comparative simplicity of the decoupled analysis, the results obtained from the three case studies clearly indicate that rotational pulley motion is indeed coupled to the transverse motions of the spans adjacent to the tensioner. The lowest ordered natural frequencies and associated mode shapes are clearly influenced by this coupling, while the decoupled analysis is adequate for higher ordered modes. The coupled analysis is shown to produce the more accurate results based on the comparison to the experimental data. A parametric study showed that natural frequencies associated with pulley rotations were insensitive to changes in crankshaft speed, regardless of whether the decoupled or coupled analysis methods was used. For the belt spans adjacent to the tensioner, the natural frequencies decreased with increasing engine speed, but this decrease was mitigated in the coupled analysis. Experimental data which allows for a more complete validation of analysis methods is clearly lacking, as least in published form. It is hoped this paper will provide some guidance for future papers related to testing. REFERENCES 1. Mote, C.D. A Study of Band Saw Vibrations. Journal of The Franklin Institute 279 (1965): Ulsoy, A.G., Whitesell, J.E., Hooven, M.D. Design of Belt-Tensioner Systems for Dynamic Stability. ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design (1985): Gasper, R.G.S., Hawker, L.E. Resonance Frequency Prediction of Automotive Serpentine Belt Drive Systems by Computer Modeling. Machinery Dynamics - Applications and Vibration Control Problems, September ASME, Hawker, L.E. "A Vibration of Automotive Serpentine Accessory Drive Systems." Diss., University of Windsor, Windsor, ON, Canada, Hwang, S.J., Perkins, N.C., Ulsoy, A.G., Meckstroth, R.J. Rotational Response and Slip Prediction of Serpentine Belt Drive Systems. ASME Journal of Vibration and Acoustics 116 (1994): Kraver, T.C., Fan, G.W., Shah, J.J. Complex Modal of a Flat Belt Pulley System With Belt Damping and Coulomb-Damped Tensioner. ASME Journal of Mechanical Design 118 (1996): Leamy, M.J., Perkins, N.C. Nonlinear Periodic Response of Engine Accessory Drives With Dry Friction Tensioners. ASME Journal of Vibration and Acoustics 120 (1998): Beikmann, R.S. "Static and Dynamic Behavior of Serpentine Belt Drive Systems: Theory and Experiment." Diss., University of Michigan, Ann Arbor, MI, 1992.

9 9. Beikmann, R.S., Perkins, N.C., Ulsoy, A.G. Free Vibration of Serpentine Belt Drive Systems. ASME Journal of Vibration and Acoustics 118 (1996): Beikmann, R.S., Perkins, N.C., Ulsoy, A.G. Nonlinear Coupled Vibration Response of Serpentine Belt Drive Systems. ASME Journal of Vibration and Acoustics 118 (1996): Beikmann, R.S., Perkins, N.C., Ulsoy, A.G. Design and of Automotive Serpentine Belt Drive Systems for Steady State Performance. ASME Journal of Mechanical Design 119 (1997): Zhang, L., Zu, J.W. Modal of Serpentine Belt Drive Systems. Journal of Sound and Vibration 222 (1999): Zhang, L., Zu, J.W. One-To-One Auto-Parametric Resonance in Serpentine Belt Drive Systems. Journal of Sound and Vibration 232 (2000): Wasfy, T.M., Leamy, M. Effect of Bending Stiffness on the Dynamic and Steady-State Responses of Belt Drives. ASME Design Engineering Technical Conferences and Computer and Information in Engineering Conference ASME, Kong, L., Parker, R.G. Coupled Belt-Pulley Vibration in Serpentine Drives with Belt Bending Stiffness. ASME Journal of Applied Mechanics 71 (2004): Kong, L., Parker, R.G. Equilibrium and Belt-Pulley Vibration Coupling in Serpentine Belt Drives. ASME Journal of Applied Mechanics 70 (2003): Parker, R.G. Efficient Eigensolution, Dynamic Response, and Eigensensitivity of Serpentine Belt Drives. Journal of Sound and Vibration 270 (2004): Abrate, S. Vibrations of Belts and Belt Drives. Mechanism and Machine Theory 27 (1992): Ewins, D.J. Modal Testing: Theory and Practice. England: Research Studies Press Ltd., 1986.