Nonlinear Coupled Vibration Response of Serpentine

R. S. Beikmann Noise and Vibration Center, Nortti American Operations, General Motors Corporation, IVIilford, IVII 48380

N. C. Perkins Associate Professor. IVIem. ASME

A. G. Ulsoy Professor. Fellow ASME Mecfianical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, Ml 48109

Nonlinear Coupled Vibration Response of Serpentine Belt Drive Systems This theoretical and experimental study identifies a key nonlinear mechanism that promotes strongly coupled dynamics of serpentine belt drive systems. Attention is focused on a prototypical three-pulley system that contains the essential features of automotive serpentine drives having automatic (spring-loaded) tensioners. A theoretical model is presented that describes pulley and tensioner arm rotations, and longitudinal and transverse belt vibration response. A recent investigation demonstrates that infinitesimal belt stretching creates a linear mechanism that couples transverse belt vibration to tensioner arm rotation. Here, it is further demonstrated that finite belt stretching creates a nonlinear mechanism that may lead to strong coupling between pulley/tensioner arm rotation and transverse belt vibration, in the presence of an internal resonance. Theoretical and experimental results confirm the existence of this nonlinear coupling mechanism. In particular, it is shown that very large transverse belt vibrations can result from small resonant torque pulses applied to the crankshaft or accessory pulleys. These large amplitude transverse vibrations are particularly sensitive to seemingly small changes in the rotational mode characteristics.

Introduction Since 1979, conventional V-belt drive systems have been largely replaced by serpentine belt drives with dynamic tensioners (see Fig. 1), utilizing a thin multirib belt. Noise and vibration levels are generally reduced (Cassidy et al., 1979), but considerable interest still exists in (1) better understanding the special characteristics of the serpentine drive, and (2) producing effective tools for predicting vibration response. Two fundamental types of vibration occur in belt drive systems (Houser and Oliver, 1975): (1) rotational motion* and (2) transverse motion. During rotational motion, the accessories rotate about their spin axes, and the belt spans simply stretch. During transverse motion, the belt spans vibrate transversely, similar to a vibrating string. The translating belt is an example of an axially moving material (Wickert and Mote, 1988). The rotational modes of a serpentine drive are determined in recent studies by Hawker (1991) and Hwang et al. (1993), who examine serpentine drives with dynamic tensioners. Barker et al. (1991) analyze the transient response of a similar system under rapid engine acceleration, to predict the onset of belt slip. These studies ignore transverse belt dynamics. Ulsoy et al. (1985) examine nonlinear transverse belt vibration response in the subsystem composed of a tensioner and the adjacent belt spans. Dynamic tensions in the spans are prescribed by torque variations in the adjacent driven accessories. The dynamic tensions parametrically excite transverse vibration, leading to Mathieu-type instabilities. The investigations above assume linearly independent rotational and transverse belt motions. However, Beikmann et al. (1992, 1996) demonstrate that the transverse motion of certain spans is linearly coupled to pulley rotation, through the tensioner arm rotation. A three pulley serpentine drive is examined, and the coupled eigenvalue problem governing linear vibration

response is solved in closed form. The coupled eigensolutions describe modes that are generally dominated by either the rotational or transverse motion. The rotationally dominant modes are readily excited in serpentine drives through crankshaft and accessory torque pulses. The objective of the present investigation is to examine the strong nonlinear mechanisms that couple rotationally dominant modes and transversely dominant modes, using theoretical and experimental methods. To this end, a theoretical model is presented that is a nonlinear extension to that in Beikmann et al. (1992, 1996). Using the eigensolutions obtained therein, the nonlinear vibration model is discretized and the coupled vibration response is evaluated numerically. Numerical results focus on a fundamental class of nonlinear forced responses involving a 1:2 internal resonance, which is also observed in an accompanying experimental study. Theoretical Model Figure 1 shows a prototypical serpentine drive system, containing (1) a driving pulley, (2) a driven pulley, and (3) a dynamic tensioner. A complete derivation of the theoretical model governing this prototype system and the discretization is detailed by Beikmann (1992), and reviewed in Beikmann et al. (1992, 1996). Presently, a summary of the key steps is provided. Hamilton's principle is used to derive the nonlinear equations of motion about the equilibrium state of the system operating at a steady speed (Beikmann, 1992). The nonlinearities considered are geometric, and describe finite stretching of the belt spans. The computation of the equilibrium state follows that in Beikmann et al. (1991) and Beikmann (1992). The nonhnear equations of transverse motion w, {x, t) for the three belt spans are (Wickert and Mote, 1988) m(W,-,„ -I- 2cWi,^,) - PiiWi^:, = P^Wij

Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 1994. Associate Technical Editor: K. W. Wang. * Note that in the automotivefield,the rotational modes are commonly referred to as "torsional" modes. This is a misnomer, however, since these modes do not induce torsional strain energy in any component. This paper uses the term "rotational" instead of "torsional" for this type of vibration.

1,2,3

(1)

where c is the constant belt translation speed, i, is the span number, and the subscripts, x, and, t, denote partial derivatives with respect to span coordinate, x, and time, t. Here, P„ is the tractive tension component in span i at equilibrium. This component provides the normal force between the pulley and OCTOBER 1996, Vol. 1 1 8 / 5 6 7

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P2

-k\ cos i//i - * 2

0 0 m 0 0 m

and m, 0 0 0

k\ + /C2

PcliWxxx PlUNL ~ PiliM. + Fd frfl PitlNL ~ PjlNL -PdXNL COS l/^i + Pj2NL COS l//2 Pd^NL ~ Pd2NL + fr,i4 rf

(18)

where F^, = M^i/ri, in which M,;; represents an applied dynamic moment on pulley /.

(11) First Order Form and Discretization

consists of

A numerical solution is pursued, in which the equations of motion are first discretized using the modes of the linearized model (Beikmann et al., 1996). Prior to discretization, the equations of motion (5) are expressed in first-order (state equation) form:

2mc • dx 0

Grc —

2mc — dx

(12) AU -I- BU = X(f) where the state vector U is defined as

2mc • dx

W(r) W(0

U(/)

and

G„

0 0 0 0 —mc sin i/zi I/, mc sin i//2 to 0 0

0 0 0 0.

(13)

KcC

KcD

_ Koc

KDD

"M

(14)

0

0" K

X(t)

i -p

0

Kc

0

0

^ ^" dx'

(15)

0

dx'

and

KDC

—

0

0

0

0

0

0

Pi\ sin Vi

9JCI

n 1^2

— dX2 0

0

(16)

G

K

-K

0

(21)

{?

(22)

contains all nonlinear and nonhomogeneous terms. The symmetry of K and skew-symmetry of G follow from the inner product definition in Eq. (28). That is, (AY, Z ) = = -Prfiw,

Ch

+

{uJrUj,WiXx)w't{x) +

Wir{x)\vt{x))dx

dxi Jo

Jo

+ uj.uj,xJxf

Pd2l^r I I'D

+ X^X? (28)

where * denotes complex conjugation. A finite eigenfunction expansion is sought for the state vector:

W2j^2rdX2

W,„V^rdX

W2„V2rdX2 + PdzOJr

lo Jo

is the vector of modal coordinates. Substituting (29) into (19) yields A P V ( 0 + B P V ( 0 = X(r)

(30)

Forming the inner product of (30) with P yields

where the constants Q « and Dy^, depend on the eigenfunctions, Fig. 2 Resonant response of rotationally dominant mode and are detailed in Beikmann (1992). The dynamic tension components are also expressed in terms of the modal coordinates, PdiL being linearly dependent and P^tiNL being quadratically dependent on the modal coordinates. Similar modal equations u),. Similarly, quadratic terms involving the product ^1^2 appear in Eq. (40), and are resonant when 0J2 = 2wi. Thus, the quagoverning mode 2 are obtained by iterating subscripts. The modal equations for mode 3 are of a generically different dratic nonlinearities may produce a 1:2 type internal resonance (Nayfeh and Mook, 1979), which, in this case, could excite and simpler form: strongly coupled response of the first two modes. Similar conclusions hold for internal resonances between mode 1 (rotationally dominant) and mode 3 (transversely dominant). = {Pd3L + /".S/VL 1(6^3331 + ViDi,,,) " 2^3^36 (42) ?73 + W3^3 = { PdiL + PdiNL 1 ( 6 ^ 3 3 3 2 + 77303332)

(43)

Note the introduction of modal damping ^^ in Eqs. (40) and (42). The use of modal damping is an approximation, and included as a further means to limit the simulated vibration response. The modal damping coefficients were obtained experimentally from free (decay) response data (Beikmann, 1992). Note on Internal Resonances Inspection of the modal equations reveals the conditions leading to internal resonance, where vibration in one mode excites another mode in a nonlinear manner (and vice versa). Under these conditions, the participating modes may exchange substantial energy through key nonlinear (resonance) terms (Nayfeh and Mook, 1979). This is a particularly noteworthy coupling mechanism for serpentine belt drives. In accessory drive applications, the belt is often excited by crankshaft and accessory torque pulses which initially drive rotationally dominant modes. When driven at an appropriate frequency and sufficient amplitude, these rotational modes may, in turn, excite transversely dominant modes, resulting in large amplitude transverse belt vibration. These large amplitude oscillations are the subject of a focused study (Mockensturm et al., 1994) which considers the one-way coupling (parametric excitation) from rotational vibration to transverse vibration. Consider, for instance, the terms proportional to Pa\ in (40). Evaluation of these terms produces the quadratic nonlinear term ^2^12^1211- If mode 2 is excited near resonance (i.e., C2 = ^ cos ujit), then the above term is resonant in (40) when 2w2 =

Numerical and Experimental Results Example results are presented to illustrate the forced vibration response characteristics of an internally resonant, two degreeof-freedom approximate model (mode 2 in the previous formulation is omitted). First, numerical time simulations for the analytical system (physical properties shown in Table 1) are presented. Figures 2 and 3 show numerical results obtained under resonant excitation (excitation frequency w = 261.0 Hz) of the lowest rotationally dominant mode (mode 1, natural frequency 261.0 Hz, 1.0 percent damping), in the presence of an internal resonance with mode 3. The excitation is applied to pulley 4, and has amplitude 0.1356 N-m. Mode 3 is a fixed-fixed span 3 mode (natural frequency 130.5 Hz, 0.3 percent damping), such that ujjoj} = 2. In this simulation, mode 1 is assigned trivial initial conditions, while niode 3 is assigned a small, non-zero initial displacement. Figure 2 shows that, initially, the amplitude of mode 1 (externally excited near resonance) grows linearly, and then saturates. Figure 3 shows that, once mode 1 has reached a sufficient amplitude, mode 3 is excited (by an internal resonance), and initially grows exponentially, eventually reaching a limiting amplitude (not shown on short time scale of Fig. 3).

Table 1 Component properties for the numerical simulations

Rotational inertias; J 1=0.003512 kg-rn^, J2=0.000293 kg-m^, J3=0.000293 kg-rn^, J4= 0.05853 kg-ni2, tensioner pulley mass: 0.2268 kg, m=0.0893 kg/m. Radii: ri=0.00508 m, 12=0.0381 m, i3=0.00508 m, and r4=0.0762 ni. Spin axis coordinates (meters): (Xi,Zi)=(0.3048,0.0), (X2,Z2)=(0.1778, -0.0127), and (X4,Z4)=(0.0,0.0). Tensioner pivot coordinates (meters); (X3,Z3)= (0.127, -0.0127). Belt modulus; EA=88964 N.

Time (sec)

Tensioner spring constant: ki=54.38 N-m/rad. Reference (rest) tension: 444.8 N, applied loads Msi=13.558 N-m, Ms2=Mifl=0, Mrt=-20.337 N-m.


Fig. 3 Response of transversely dominant mode (span 3) excited by an internal resonance

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Fig. 5(a)

Span 3 transverse vibration mode (33.0 Hz)

BedDlate

AmDli'f 1er

Fig. 5{b)

Amplifier

Rotational vibration mode for baseline system (62.5 Hz)

Oscilloscope

/WW"

^ Fig. 5(c) Rotational vibration mode for modified system (58.0 Hz). A mass of 76 grams has been added to the tensioner pulley center. Fig. 4

Experimental test setup

Experimental testing was conducted to confirm the existence of the internally response described above. The experimental setup is schematically illustrated in Fig. 4. Physical parameters for two systems experimentally tested are listed in Table 2. The two systems are identical, except that the modified system has a modest 76 gram mass added to the tensioner pulley center. The following observations are derived from exciting the experimental systems with the electro-dynamic shaker, while the system is at rest (c = 0 ) . A sinusoidal tangential force is applied with an electro-dynamic shaker near the outer perimeter of pulley 4. The experimental system has two modes of particular interest. The first is a transverse mode of span 3, with natural frequency W3 = 33.0 Hz [ Fig. 5 ( a ) ] . The second is a rotationally dominant mode, which has a natural frequency of Wi = 62.5 Hz in the baseline system [Fig. 5(b)]. Thus, UJ^/UJ, is near the 1:2 commensurable ratio required for internal resonance. The modified system has Wi = 58.0 Hz [Fig. 5 ( c ) ] , and thus is rather detuned from the 1:2 ratio. Figure 6 presents experimental results obtained for the baseline system. Shown is the measured steady state response amplitude of the fixed-fixed span mode (mode 3), as a function of excitation amplitude, resulting from near resonant excitation of the rotationally dominant mode (excitation frequency 66.0 Hz). Figure 6 shows that, as the excitation amplitude increases from very small values, the span 3 response is negligible up to a

certain threshold excitation level. Beyond this level, the amplitude of the span 3 response suddenly jumps (by more than 50 dB) to that of a large and visible limit cycle. Furthermore, there is a range of excitation amplitudes wherein the vibration response amplitude depends on initial conditions. One observes that for these excitation amplitudes, the span 3 mode response is negligibly small when started from a quiescent (unplucked) state, but attains the large-amplitude limit cycle response when perturbed by plucking.

30 20 10 0

e

m E

\in pluokec

\ Plu >k8d

\

>

"s

-10 -20 -30 -40

Table 2 Component properties for the numerical test systems

•50

Rotational inertias: Ji=0.07248 l;g-m2 J2=0.0002CG kg-m2, J3=0.001165 kg-m^, J4=

-60

\,

X

0.000293 kg-m2, tensioner pulley mass: 0.302 kg (baseline), 0.37? kg (modified), and m=0.1029 kg/m. Radii: ri=0.0889 m, r2=0.0452 m, t3=0.097 m, and r4=0.02697 m. Spin axis coordinates (meters): (Xi,Zi)=(0.S525,0.S556), (X2,Z2)=(0.3477, 0.05715), and (X4,Z4)=(0.0, 0.0). Tensioner pivot coordinates (meters); (X3,Z3)= (0.2508, 0.0635). Belt modulus: EA= 170000 N. Tensioner spring constant: 1^54.37 N-m/rad.

572 / Vol. 118, OCTOBER 1996

-70

5 10 15 20 25 Input Force dB N (rms)

30

35

Fig. 6 Span 3 response amplitude of fixed-fixed span 3 mode (33 Hz) to near-resonant excitation (66 Hz) of rotationally dominant mode (62.5 Hz). Data collected for plucked O and unplucked x conditions.

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The threshold levels are somewhat difficult to determine experimentally, due to slow growth/decay rates of the (parametrically) excited span 3 response. In the neighborhood of the threshold excitation, the transient response may persist for 3 5 minutes prior to achieving a steady state value. This indicates that standard diagnostic vehicle tests (e.g., engine speed sweeps and ' 'order-tracking'') may be ineffective in finding these resonances. In such tests, it is possible that insufficient "dwell time" would be spent at the critical engine speed (excitation frequency) to generate the steady state response. Nevertheless, such resonances may plague vehicles during sustained operation at particular steady engine speeds. It is well known that, for a parametrically excited system, an increase in the excitation amplitude produces an increase in the frequency range (instability region) leading to principle parametric instability. This fact is also illustrated in Fig. 7, which shows the measured frequency response of the fixedfixed span 3 mode. Note that a 6 dB increase in the excitation level roughly doubles the instability region. The data of Fig. 8 demonstrate the controlling influence of external and internal tuning/detuning (between rotational and transverse modes). The upper frequency response curve is that previously shown in Fig. 6 (unplucked condition), and the second curve is that obtained after adding the small mass to the tensioner arm. As mentioned, this additional mass reduces the natural frequency of the rotationally dominant mode from 62.5 Hz to 58.0 Hz, increasing the detuning from external resonance (excitation is still at 66 Hz), and also increasing the internal detuning further away from a 1:2 natural frequency ratio. This single and modest change significantly reduces the dynamic tension component in span 3 for a given force excitation amplitude. As a result, the excitation amplitude required to initiate the internally resonant response of span 3 increases by 15 dB.

Summary and Conclusions The nonlinear vibration response of the serpentine drive with coupling between discrete (pulleys and tensioner arm) and distributed (belt spans) components is examined, using theoretical and experimental methods. A modal expansion is used to obtain

30 20 10

^

ei.s Hi

>

0 se.o Hz

-10 -20 •a 1/1 I— h4

e

-30 -40 -50 -60

c^

/J

-70 -6

0

5

10

15

20

25

30

35

Excitation Amplitude, dB N (rms) Fig. 8 Response amplitude of fixed-fixed span 3 mode (33 Hz) to nearresonant excitation (66 Hz) of rotationally dominant mode: x = 62.5 Hz, O = 58.0 Hz.

a low-order discrete model for nonlinear forced response. This model captures the two-way coupling between all mode-pairs. A strong nonlinear mechanism is identified, which couples rotationally dominant (longitudinal belt) modes and transversely dominant (lateral belt) modes. The rotationally dominant modes generate dynamic tension fluctuations which may excite large transverse belt vibration in the presence of a 1:2 internal resonance. The theoretical and experimental results both exhibit this coupling mechanism, and provide the following insights: (1) large transverse belt response is excited through the internal resonance, provided the excitation amplitude exceeds a threshold level that is frequency dependent, (2) increasing the excitation amplitude widens the excitation frequency range leading to large transverse belt response, (3) long duration transients ( 3 - 5 minutes), make these resonances difficult to detect using standard laboratory diagnostic test procedures, and (4) small changes in the natural frequency of a rotationally dominant mode may drastically alter transverse belt response. Acknowledgment The authors thank the Noise and Vibration Center of North American Operations, General Motors Corporation, for support of this research. References

-g - 4 0 -60 -60

66

66.4

66.8

66.2

66.6

E x c i t a t i o n Frequency (Hz) Fig. 7 Frequency response of span 3 mode. Amplitude of span 3 mode vs. excitation frequency, for two excitation amplitudes.


Barker, C. R„ Oliver, L. R., and Breig, W. F., 1991, "Dynamic Analysis of Belt Drive Tension Forces During Rapid Engine Acceleration," SAE Paper No. 910687. Beikmann, R. S., 1992, "Static and Dynamic Behavior of Serpentine Belt Drive Systems; Theory and Experiment," Ph.D. Dissertation, The University of Michigan, Ann Arbor, MI. Beikmann, R. S., Perkins, N, C , Ulsoy, A. C , 1991, "Equilibrium Analysis of Automotive Serpentine Belt Drive Systems Under Steady Operating Conditions,'' Proceedings of the ASME Midwestern Mechanics Conference, Rolla, MO, October 6 - 8 , pp. 533-534. Beikmann, R. S., Perkins, N. C , and Ulsoy, A. G., 1992, "Free Vibration Analysis of Automotive Serpentine Belt Accessory Drive Systems," Proceedings of the CSME Forum. "Transport 1992+," Montreal, Canada, June 1-4. Beikmann, R. S., Perkins, N. C , and Ulsoy, A. G., 1996, "Free Vibration of Serpentine Belt Drive Systems,'' ASME JOURNAL OF VIBRATION AND ACOUSTICS, July.

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Doyle, E., and Homung, K. G., 1969, "Lateral Vibration of V-Belts," ASME Paper Afo. 69-VIBR-24. Gaspar, R. G. S., and Hawker, L. E., 1989, "Resonance Frequency Prediction of Automotive Serpentine Belt Drive Systems By Computer Modeling," ASME Design Engineering Division (Publication) DE v 18-2, pp. 13-16. Hawker, L. E., 1991, "A Vibration Analysis of Automotive Serpentine Accessory Drive Sy,stems," Ph.D. Dissertation, University ofWindsor, Ontario, Canada. u I-, r, J ^1. ,r.-,c ..^ru .• f ^7 D 1. A • c •• J u Houser, D. R., and Oliver, L., 1975, Vibration of V-Belt Drives Excited by . ^ , . - , ; , - ' , , ' ,. J. , y , „ , ,j^ ./ Lateral and Torsional Inpu s Proceedings of the Fourth World Congress on the Theory of Machines and Mechanisms, Vol. 4, Newcastle Upon Tyne-England, Sept. 8-13. Hwang, S. J., Perkins, N. C , Ulsoy, A. G., and Meckstroth, R., 1994, "Rotational Response and Slip Prediction of Serpentine Belt Drive Systems," ASME JOURNAL OF VIBRATION AND ACOUSTICS, Vol. 116, No. 1, pp. 71-78. Meirovitch, L., 1974, "A New Method of Solution of the Eigenvalue Problem for Gyroscopic Systems," AIAA Journal. October, pp. 1337-1342.

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Meirovitch, L., 1975, "A Modal Analysis for the Response of Linear Gyroscopic Systems," ASME Journal of Applied Mechanics, June, pp. 446-450. Mockensturm, E. M., Perkins, N. C , and Ulsoy, A. G., 1994, "Limit Cycles and Stability of a Parametrically Excited Axially Moving String," ASME JOURNAL OF VIBRATION AND ACOUSTICS, in press. Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations. John Wiley ' " 1 ^ ™ ' \ ' ^ ^ * ^ ° * ' ^ • ^ • \ P P ; , ^ ' * - ' ° ' J ' ^ ^ - ^ ™ v ' ' t 3 - t ^ L ..,. • ,„ , Ulsoy, A. G., Whitese , J. E., and Hooven, M. D., 1985, "Design of BeltT • c . t n • c» u i v .. Acmr- i t, Tensioner Systems for Dynamic Stability, ASME JOURNAL OP VIBRATION, ACOUSTICS, STRESS, AND RELIABILITY IN DESIGN, Vol. 107, No. 3, July, pp. 2 8 2 jqn Wickert, J. A., and Mote, C. D., Jr., 1988, "Current Research on the Vibration ^^^ Stability of Axially-Moving Materials," Shock and Vibration Digest, Vol. 20, No. 5, May, pp. 3-13. Wickert, J. A., and Mote, C. D., Jr., 1990, "Classical Vibration Analysis of Axially Moving Continua," ASME Journal of Applied Mechanics, September, pp. 738-744.

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