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Research Papers

# Schallamach Wave-Induced Instabilities in a Belt-Drive SystemPUBLIC ACCESS

[+] Author and Article Information
Yingdan Wu

George W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: yingdanwu@gatech.edu

Michael Varenberg

George W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: michael.varenberg@me.gatech.edu

Michael J. Leamy

Fellow ASME
Professor
George W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: michael.leamy@me.gatech.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 15, 2018; final manuscript received November 20, 2018; published online December 17, 2018. Assoc. Editor: George Haller.

J. Appl. Mech 86(3), 031002 (Dec 17, 2018) (9 pages) Paper No: JAM-18-1534; doi: 10.1115/1.4042101 History: Received September 15, 2018; Revised November 20, 2018

## Abstract

We experimentally study the dynamic behavior of a belt-drive system to explore the effect of loading conditions, driving speed, and system inertia on both the frequency and amplitude of the observed frictional and rotational instabilities. A self-excited oscillation is reported whereby local detachment events in the belt–pulley interface serve as harmonic forcing of the pulley, leading to angular velocity oscillations that grow in time. Both the frictional instabilities and the pulley oscillations depend strongly on operating conditions and system inertia, and differ between the driver and driven pulleys. A larger net torque applied to the pulley generally intensifies Schallamach waves of detachment in the driver case but has little influence on other measured response quantities. Higher driving speeds accelerate the occurrence of frictional instabilities as well as pulley oscillations in both cases. Increasing the system's inertia does not affect the behavior of contact instabilities, but does lead to a steadier rotation of the pulley and more pronounced fluctuations in the belt tension. A simple dynamic model of the belt-drive system demonstrates good agreement with the experimental results and provides strong evidence that frictional instabilities are the primary source of the system's self-oscillation.

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## Introduction

Belt-drives are simple and economical machines for transmitting mechanical power in many engineering applications, such as in automotive front end accessory drives and continuously variable transmissions, manufacturing machines, household appliances, and magnetic tape data storage systems. Although properly installed and maintained belt-drives can preserve energy efficiency up to 95% [1], their efficiency and performance are affected by complex system dynamics arising primarily from excitation at the belt–pulley interface, fluctuations in the pulley angular velocities and span tensions, and belt misalignment. These, in turn, lead to energy loss, undesired vibration and noise, wear, and speed loss between the driver and driven pulleys. For these reasons, it is critical to understand belt-drive mechanics/dynamics for robust system design and energy efficiency.

The literature regarding belt-drive mechanics is extensive; the earliest investigations on belt-drive mechanics can be traced back to Leonard Euler's [2] study of a belt wrapped over a fixed pulley as well as Grashof's [3] analysis of the frictional mechanics of belt-drives under steady operation, yielding the classical belt creep theory. In the creep theory, the belt–pulley frictional contact region is governed by a Coulomb law, and the belt is treated as a flexible one-dimensional string. The theory predicts a single-slip arc in the exit region of the pulley and an adhesion arc in the remaining contact region. More recently, creep theory has been enhanced to include the effects of belt flexural stiffness [4] and radial and tangential belt inertia [46]. In contrast to the creep theory, Firbank [7] proposed a belt shear theory in which shear strains in the belt envelope dominate in determining drive behavior instead of longitudinal strains. It is notable that the symmetry in contact mechanics between the driver and driven pulley is broken by the presence of shear, coinciding with observations from experimental investigations [8]. Later investigations considered the influence of radial compliance [912] and bending stiffness/inertia [9,10,1214] on the belt-drive mechanics, and compared shear and creep theories [10,13]. Also, experiments were conducted to study the creep loss between the driver and driver pulleys [7,1517], tangential and normal forces acting on the pulley [1820], and strain/tension variation along the surface of a belt [8,21]. Recently, analytical and numerical models (e.g., based on finite element modeling) have extended analyses to unsteady operation due to harmonic excitation, with some also considering bending stiffness and one-way clutches [2229].

In the reviewed belt-drive studies, some variant of Coulomb friction is assumed to govern in the belt–pulley frictional contact region, implying sliding between the belt and pulley in the slip arc. It is now understood that, at least for low-speed operation using moderately thick and homogeneous belts with smooth pulleys, the large contact area formed by the belt–pulley interface may prevent sliding in the accepted sense. In 2018, Wu et al. [30] introduced experimental evidence that in such a simple belt-drive system, displacement between the belt and the pulley was accommodated primarily by detachment events (and not sliding), including Schallamach waves [3133], which are narrow lines of lost contact (e.g., due to surface buckling) that move across the contact region. They also found that rolling contact mechanics differ between driver and driven pulleys—although detachment has been found in both cases, Schallamach waves appear in the driver pulley only. These findings motivate further investigation to uncover the ranges of speed, load, belt material properties, pulley roughness, and other factors that may exhibit detachment vice sliding behavior.

In this paper, we continue the exploration of the simple belt drive system first reported in Ref. [30], studying the formation of detachment waves as functions of the applied torque, the operating speed (all speeds still limited to slow operation), and the system moment of inertia. In so doing, we uncover a self-excited instability in which periodic detachment waves lead to pulley oscillations which grown in time.

## Experimental Details

###### Apparatus and Belt Samples.

The experimental apparatus (Fig. 1) is capable of measuring tension in both belt spans (via force transducers) and the angular displacement of the pulley (via a rotary encoder). A replaceable flywheel attached to the pulley allows adjustment of the system inertia. A digital camera, fixed above the exiting side and angled approximately 8 deg as shown in Fig. 1, records the evolution of the belt–pulley contact zone on the trailing side. A releasing motor controls the speed of the driving stage, ensuring a near-constant speed. It is notable that the user can switch between the driver (solid lines in Fig. 1(a)) and driven configuration (dashed lines in Fig. 1(a)) by reversing the direction of the torque. A detailed description of the apparatus is available in Ref. [30].

The belts tested herein are cast of polydimethylsiloxane (PDMS, Sylgard 184, Dow Corning, Midland, MI) using a 10:1 mixture of Sylgard 184 prepolymer and its cross-linker cured for 14 h at 65 °C against a flat smooth template. PDMS is a transparent elastomer with a Young's modulus of 1.6 MPa (measured at rate of 0.01 s−1 with ES10 tensile test stand, Mark-10 Co., Copiague, NY). The belt extends 400 mm in length, 5 mm in width, and 2 mm in thickness. The travel distance of the driving stage is 300 mm, which is large enough (compared to the pulley radius, 10 mm) to achieve pronounced instabilities in system behavior and reach a near steady-state rolling condition. Note that PDMS was chosen for its transparency to enable observation of the contact area between the belt and pulley.

###### Operating Conditions.

Previous investigations on Schallamach waves in sliding find that the frequency and the amplitude of the detachment waves depend on sliding velocity and loading conditions [31,32,3436]. To study the same quantities in our belt drive system, we varied the stage driving velocity from 3 mm s−1 to 11 mm s−1 using incremental changes of 2 mm s−1.

Note that there is a distinction between the dead weight hanged to apply torque and the “net torque weight” effectively acting on the pulley. This difference arises due to the parasitic resistance from friction in the bearings and the rotary encoder. As a result, the cases of driver and driven pulley need different dead weights to achieve the same net torque applied to the pulley. Bearing this in mind, we adjusted the loading such that the net torque weight applied to the pulley ranged from 3.5 N to 5.5 N with an increment of 0.5 N for both driver and driven cases.

For all tests performed, the high tension (on the entering side for the driver case and on the exiting side for the driven case) remained 6 N, the maximum tension that our belt specimen can bear without failure. Tables 1 and 2 list detailed operating conditions for variation of driving speed and load, respectively. Each test was repeated at least five times. All statistical tests were performed using one-way ANOVA, all pairwise multiple comparison procedures (Holm–Sidak method), overall significance level 0.05, using the sigmaplot software package (Systat Software, Inc., San Jose, CA). The temperature and relative humidity in the laboratory during the tests were 23 °C and 35%, respectively.

## Experimental Results

###### Contact Mechanics and Instability Formation.

Figure 2 summarizes findings from our previous study [30] of a slowly rotating pulley in frictional contact with a flat belt. In both the driver and driven pulley cases, the transition from high to low tension (or vice-versa) occurs in a finite zone at the exit region of the belt–pulley contact zone. There, friction traction accompanies changes in tension. Due to the net tension force acting approximately at the centroid of the cross section, and the friction traction acting at the contact surface of the belt, a moment about the belt center arises, which tends to lift the belt from a driver pulley (Fig. 2(a)), while tending to do the opposite in a driven pulley (Fig. 2(b)).

In the event of belt detachment from the driver pulley, the frictional traction disappears and the traction moment relaxes, which brings the belt back in contact due to the restoring action of the tension force's radial component. This sequence leads to the generation of a surface fold (driver contact instability) that can travel along the interface in the backward direction until it closes due to increasing normal load at smaller angles. In the driven case, the belt has a tendency to become thinner due to the increasing tension. This thinning may pull the belt out of the contact, leading to local detachment at the contact area edge (driven contact instability). Once detached from the pulley, the belt does not attach again, and the contact area simply moves forward together with the pulley until the belt peels off again. Thus, the contact mechanics of the driving and the driven pulleys differ significantly.

###### Self-Excited Oscillation.

Contact instabilities formed in both driver and driven cases lead to oscillations in belt tension and pulley rotation. Figure 3 documents the friction force and angular velocity, as a function of pulley revolution, obtained experimentally at a driving speed of 3 mm s−1 using a net torque weight of 4 N in both the driven (Fig. 3(a)) and the driver (Fig. 3(b)) cases. The friction force is arrived at by taking the difference between the tight and slack side tensions measured by the force transducers with a resolution of 0.04 N. The angular velocity is computed via a central difference scheme from the angular displacement registered by the rotary encoder with a resolution of 8192 counts per revolution.

Subfigures in the top row clearly document growing oscillations, with similar frequency content, in both measured quantities. A zoomed-in plot (middle row) shows the correlation between the friction and angular velocity such that the period of the friction oscillations coincides with that of the pulley oscillations (each period denoted by two adjacent dashed–dottted lines). It is also evident that the friction signal exhibits additional, higher frequency content associated with belt detachment at the exit of the pulley. This is borne out by comparisons of the friction signal with the contact area in the exiting region of the belt–pulley interface (see numbered points on the friction force, middle subfigure, and corresponding numerals on the contact area snapshots), where black areas denote contact, and white areas denote loss of contact between the belt and the pulley. Similar to our previous study [30], Schallamach waves (see isolated detached pocket on third contact snapshot in the driver case) are detected only in the driver case.

The friction fluctuations were analyzed further via wavelet transform routines available in matlab and were decomposed into two primary components: (1) a high-frequency component (dashed line) associated with the detachment events at the belt–pulley interface (Fde); and (2) a low-frequency component (solid line) associated with the pulley oscillation (Fpo). Interestingly, the Fpo signals in the driver and driven cases exhibit distinct reverse saw-tooth shapes, and this becomes more evident as the system inertia increases (see Sec. 3.5). This result can be explained in the following way. Comparing the Fpo signal to the angular velocity oscillations (middle row in Fig. 3), we can see that the minute halts in the pulley rotation are associated with the force drop in the driver case and with the force rise in the driven case. In the driver case, when the pulley drives the belt, the pulley motion cessation leads to decrease of the difference between the tight and slack side tensions (the belt is relaxed). On the other hand, in the driven case, when the belt drives the pulley, the pulley motion cessation leads to increase of the difference between the tight and slack side tensions (the belt is loaded).

We believe the oscillation growth observed in both the friction force and the angular velocity is due to a positive feedback mechanism as follows: contact instabilities in the belt exit zone excite rotational oscillations in the pulley, which, in turn, store a periodic belt tension pattern in the belt entry zone. This tension pattern then serves as an additional excitation source when released at the exiting side of the belt, further destabilizing the pulley angular velocity. Thus, local contact instabilities induce large changes in the system's global dynamics. Given that the dynamic response evolves in time, in the parameter studies to follow, we chose to focus on the final pulley revolution where the self-excited oscillations are most evident.

The fluctuations in both the friction force and the angular velocity were decomposed into two components related to detachment events and pulley oscillations. Their corresponding frequencies and amplitudes are shown in Fig. 4. The fluctuations in angular velocity resemble those in friction force (especially in terms of frequency), which verifies the correlation between these two signals. The information on the fluctuations in the angular velocity associated with detachment events is missing for the driven case because of a low signal-to-noise ratio resulting in an inability to get reliable data.

The fluctuations in the friction force associated with detachment events exhibit higher frequency and lower amplitude than those associated with pulley oscillations in both the driver and driven cases (Figs. 4(a) and 4(b)). As the net torque weight increases in the driver case, the amplitude AF,de is nearly not affected (statistically significant difference is observed only between torque weights 3.5 and 5.5 N, and between 3.5 and 5 N), whereas its frequency fF,de increases. This increase is consistent with the hypothesis that a traction-induced moment tends to lift the belt from the pulley, while the belt remains attached until the traction moment reaches a certain threshold [30]. Increasing the net torque weight raises the traction applied to the belt and reduces the gap between the acting moment and the threshold value. This leads to less time needed to reach the threshold value for detachment, resulting in increase of the fluctuation frequency. In the driven case, however, the frequency fF,de and the amplitude AF,de do not depend on the net torque weight (no statistically significant effect is observed). In this case, the traction-induced moment acts in the opposite direction, pressing the belt against the pulley. The detachment at the contact edge happens as a result of the thinning and peeling of the belt [30], and these mechanisms are not directly affected by the frictional traction. The amplitude AF,de in the driver case is larger than that in the driven case because the scale of contact instabilities is much larger in the former case (Fig. 3).

The frequency of the friction force fluctuations associated with the pulley oscillations (fF,po) does not depend on the net torque weight (no statistically significant effect is observed). The torque weight adds inertia to the system, so, in principle, the frequency fF,po should decrease with increasing torque. The reason for not observing this effect can be an insufficient range of the torque change. A small difference between the frequencies fF,po in the driver and driven cases may result from a larger torque weight used in the driver case to attain the same net torque weight as in the driven case, which results in a larger system inertia. The amplitude of the friction force fluctuations associated with the pulley oscillations (AF,po) shows inconsistent step-like increase with increasing the net torque weight from 4 to 4.5 N, while maintaining statistically indistinguishable values otherwise. This can be related to some issue that went unnoticed during the tests and is worth verifying based on the analysis of the angular velocity oscillations as follows.

The frequencies of the angular velocity fluctuation fα,de and fα,po (Fig. 4(c)) are almost identical to those of the friction force fluctuations (Fig. 4(a)), because they characterize the same instabilities. Similar to AF,de in the driver case (Fig. 4(b)), the amplitude of the angular velocity fluctuation Aα,de (Fig. 4(d)) is nearly not affected by the torque increase (no statistically significant difference is observed between the torque weights 4, 4.5, 5 and 5.5 N). Analyzing the amplitude Aα,po, we also see almost no effect of the torque weight (statistically significant difference is observed only between the torque weights 4 and 5.5 N in the driver case, as well as between the torque weights 4 and 5.5 N, 4 and 5 N, and 3.5 and 5 N in the driven case), which resembles the results obtained for AF,po. Thus, based on a comparative analysis of the effect of torque, we can conclude that, to a first approximation, the only affected parameter is the frequency of Schallamach waves of detachment in the driver case.

###### Effect of Driving Speed.

In both the driver and driven cases, increasing driving speed accelerates the occurrence of contact instabilities (detachment events) and pulley oscillations regardless of the analyzed signal source, be it either friction force or angular pulley velocity (Figs. 5(a) and 5(c), respectively). Given that the formation of the stress pattern along the contact arc relies on the rotation of the pulley, the stress relaxation associated with detachment events at the belt–pulley interface takes less time as the pulley rotates faster. The pulley oscillations are mainly caused by contact instabilities, so when the latter occur more often, the former follow suit. Hence, all frequencies increase with increasing driving speed. It is also evident that the pulley oscillations in the driven case appear to be more sensitive to the driving speed, so the frequency fF,po increases more rapidly than that in the driver case. This can result from a larger system inertia in the driver case due to a larger torque weight used to obtain the same net torque as in the driven case.

The amplitude of the frictional force fluctuations associated with the pulley oscillations (AF,po, Fig. 5(b)) exhibits inconsistent step-like decrease with increasing the driving speed from 5 to 7 mm s−1, while maintaining otherwise statistically indistinguishable values in the driven case. In the driver case, the effect of the driving speed on the amplitude AF,po is also not statistically reliable (no statistically significant difference is observed between the speeds 5 and 9 mm s−1, 7 and 9 mm s−1, 7 and 11 mm s−1, 9 and 11 mm s−1). The amplitude of the frictional force fluctuations associated with the detachment events (AF,de, Fig. 5(b)) also does not demonstrate clear effect of the driving speed. In the driven case, no statistically significant difference between different amplitudes is observed at all, while in the driver case, only the amplitude obtained at the speed 3 mm s−1 differs from all other measurements.

The amplitude of the angular velocity fluctuations associated with the pulley oscillations (Aα,po, Fig. 5(d)) has shown no statistically significant effect of driving speed in either the driver or driven case. The effect of the driving speed on the amplitude of the angular velocity fluctuations associated with the detachment events (Aα,de, Fig. 5(d)) is also negligible, with only the amplitude obtained at the speed 3 mm s−1 being different from all other measurements. Thus, based on a comparative analysis of the effect of driving speed, we can conclude that, to a first approximation, while the amplitudes of either the contact instabilities or pulley oscillations are not affected, their frequencies grow with increasing the driving speed. Note that the frequency versus speed relationship in our findings compares well with that found in sliding cases where an increasing speed also leads to an increased Schallamach wave frequency [32].

###### Effect of System's Inertia.

As reported earlier, contact instabilities excite pulley oscillations, complicating the study of the contact mechanics. In an attempt to limit these oscillations, we increased the moment of inertia of the pulley by using two removable flywheels, whose moments of inertia are 9 and 99 times that of the pulley. These are referred to as small and large flywheels. Table 3 lists the conditions employed in assessing the effect of inertia.

An illustrative example of the effect of the pulley's moment of inertia on the friction force and angular velocity is presented in Fig. 6. Looking at the friction curves, we conclude that the effect of the pulley's moment of inertia is identified easier in the fluctuations associated with the pulley oscillations (see the wavelet decomposition subfigure in Fig. 3 for comparison), while the fluctuations associated with the contact instabilities seem to be less sensitive to this parameter. Interestingly, increasing the pulley's moment of inertia results in much more violent oscillations in the friction force, while the pulley oscillations become more restrained. This is explained by noting that more friction force (the difference between the tight and slack side tensions) is needed to move a heavier pulley, which has larger moment of inertia and hence rotates more steadily.

Analyzing the frequencies and amplitudes of the fluctuations in the friction force and the pulley's angular velocity (Fig. 7), we can draw similar conclusions. The frequencies of the fluctuations in both the friction force and the angular velocity associated with the pulley oscillation (fF,po and fα,po, respectively) decrease with increasing the pulley's moment of inertia in both the driver and driven cases, which is the expected result. The amplitude of the friction force fluctuations associated with the pulley oscillations (AF,po) grows with increasing the pulley's moment of inertia in both the driver and driven cases, which results from higher belt tension being required to move (or interfere with) the heavier pulley. The amplitude of the angular velocity fluctuations associated with the pulley oscillations (Aα,po) decreases (as expected) with increasing the pulley's moment of inertia, but the changes are less pronounced (no statistically significant difference is observed between small and large flywheels in the driver case, as well as between no and small flywheel, and between small and large flywheels in the driven case).

The frequencies of the friction force fluctuations and the angular velocity fluctuations associated with contact instabilities/detachment events (fF,de and fα,de, respectively) seem to be independent of the pulley's moment of inertia (no statistically significant difference is observed between any of the tested points in the driven case, as well as between force fluctuations with small and large flywheels, and between velocity fluctuations with no and large flywheel in the driver case). The amplitudes of the friction force fluctuations and the angular velocity fluctuations associated with contact instabilities/detachment events (AF,de and Aα,de, respectively) are nearly independent of the pulley's moment of inertia in all cases (statistically significant difference is observed only between force fluctuations with no and large flywheel in the driver case). The amplitude AF,de in the driver case is larger than that in the driven case because larger contact area is involved in the detachment events. Thus, based on a comparative analysis of the effect of the pulley's moment of inertia, we can conclude that, to a first approximation, while the force and velocity fluctuations associated with the contact instabilities are not affected, the force and velocity fluctuations associated with the pulley oscillations do depend on the pulley mass.

## Theoretical Model

To verify whether frictional instabilities can serve as a source of self-oscillation in our system, we developed a simple dynamic model designed in such way that friction fluctuations are used as a modulated input, and the pulley's angular velocity is computed as an output and then compared to experimental data. The following assumptions were made:

1. (1)The belt is uniform and perfectly flexible, and it stretches in a quasi-static manner; the two spans of the belt are hence treated as massless linear elastic springs coupled with a massless damper.
2. (2)The belt deformation along the belt width is decoupled from the belt deformation along the belt length.
3. (3)Belt extension s(t) resulting from detachment events is applied uniformly over the exiting portion of the belt.
4. (4)The torque weight is applied to the pulley through an inextensible string.
5. (5)The speed of the belt exiting span and the masses of the loading weights are taken from the experiment.

The diagrams of the model shown in Fig. 8 depict a lumped system with two degrees-of-freedom: the angular displacement α(t) of the pulley and the linear displacement y(t) of the tension mass M. The motion of the torque mass m follows directly from the pulley motion at the attachment point. The linear motion of the exiting span of the belt is denoted as x(t) and is prescribed using the constant speed employed in the experiment. Detachment-driven extension in the exit zone is approximated as s(t). The frequency and amplitude of s(t) is estimated based on experimental measurements with one set of loading parameters and then used for all other test points (see Sec. 5 for further details).

The elongation of the exiting span of the belt δ(t) can be defined as Display Formula

(1)$δt=xt−Rαt−s(t)$

where R denotes the radius of the pulley. The elongation of the entering span of the belt is the difference between (t) and y(t). Hooke's law yields the span stiffness values, k1(t) and k2(t), for the exiting and entering belt spans, respectively Display Formula

(2)$k1t=EAl01+x(t),andk2t=EAl02−y(t)$

where E, A, l01 and l02 denote the belt elastic modulus, cross-sectional area, and the initial lengths of the exiting and entering spans, respectively. Assuming quasi-static belt stiffness changes, we can derive the governing equations for our belt-drive system (driver pulley, Eq. (3a); driven pulley, Eq. (3b) as Display Formula

(3a)$I+mR200Mα¨y¨+Rc1+R2c2−Rc2−Rc2c2α˙y˙+[R2(k1+k2)−Rk2−Rk2k2]αy=mgR+Rk1(x(t)−s(t))−Mg$
Display Formula
(3b)$I+mR200Mα¨y¨+Rc1+R2c2−Rc2−Rc2c2α˙y˙+R2(k1+k2)−Rk2−Rk2k2αy=−mgR+Rk1(x(t)−s(t))−Mg$

where $c1$ denotes the damping coefficient associated with the pulley oscillations (losses in the belt-pulley contact and bearings) and $c2$ denotes the damping coefficient of the free spans of the viscoelastic belt. Both coefficients are assumed to be constant due to an approximately constant length of the contact arc in the first case and a constant total length of the belt in the second case.

## Comparison of Theoretical and Experimental Results

The belt extension s(t) in the detachment region is considered to correlate closely to the friction force fluctuations associated with detachment events (bottom row in Fig. 3). Hence, s(t) can be abstracted as a saw-tooth function for both the driver (Eq. (4a)) and driven case (Eq. (4b)). The frequency of the saw-tooth function is taken as the frequency fF,de measured in the case with small flywheel for both the driver and driven pulleys. The duty cycles (loading phase fractions) of the saw-tooth function are 2/3 and 1/3 for the driver and driven pulleys, respectively, taken in accord with the friction force fluctuations associated with detachment events. The amplitude of belt extension, As, cannot be defined based on our experimental data. To this end, we have determined As via trial-and-error method: the amplitude of the angular velocity fluctuations (Aα,po) is calculated by numerical integration of Eq. (3) (via matlab's ode45 routine), while As is varied until the predicted Aα,po matches the Aα,po measured with the small flywheel in both the driver and driven cases. These values are applied to the cases of no and large flywheels to see whether the model can be predictive. The expressions used to describe s(t) are Display Formula

(4a)$st=As3fF,de4πt,0≤t≤4π3fF,deAs3−3fF,de2πt,4π3fF,de≤t≤2πfF,de$
Display Formula
(4b)$st=As3fF,de2πt,0≤t≤2π3fF,deAs32−3fF,de4πt,2π3fF,de≤t≤2πfF,de$

All model parameters (Table 4) are taken from the experiment except for the two damping coefficients, c1 and c2, which were also chosen based on the trial-and-error method to obtain a good agreement between the theoretical and experimental results. The values c1 and c2, however, are verified to fall within a reasonable range for PDMS [37].

The equations of motion were numerically integrated using the ode45 function in matlab, while the real-time belt span lengths updated the stiffness values according to Eq. (2). Wavelet routines in matlab were used to postprocess the data from both the numerical model and the experiment. The model was also used to compute the quasi-static vibration modes (eigenfrequencies and eigenvectors) as a function of time (with the instantaneous stiffness values).

The development of the angular velocity fluctuations in the driver and driven cases is presented using wavelet scalograms in Figs. 9 and 10, respectively. Dotted white traces provide the time-varying natural frequencies obtained from the model eigen analysis. The corresponding vibration modes (at 1.8 min, as denoted by the red dash-dotted line) for the observed two frequencies are shown in Fig. 9(a). For the lower frequency of the dominant vibration mode, the oscillations of the pulley and the tension weight are similar in scale, while for the higher frequency of the secondary vibration mode, the tension weight oscillations dominate the system response. It is also evident in all plots that the contact instabilities (detachment events) excite the lower frequency vibration mode, whose magnitude grows in time.

The scalograms document strong agreement between the numerical model and the experimental results, and clearly explain the time-dependent frequencies observed in the experiment. Both the model and the experiment demonstrate a frequency band between the low and high natural frequencies of the angular velocity fluctuations, which is associated with the contact instabilities in the experiment and with the excitation source s(t) in the numerical model. This band is characterized by a fundamental frequency and higher harmonics in the experiment, while the model exhibits only a fundamental frequency due to a simple excitation function encoded in Eqs. (4a) and (4b). Despite these differences and an overall simplicity of the model, the theoretical results match closely the experiment, which provides strong evidence that frictional instabilities driven by unmodulated external power are the primary source of the studied system's self-oscillation.

As a final comparison, Fig. 11 details the computational and experimental results obtained at the end of the fourth revolution of the pulley. Consistent with the results shown in Figs. 7(c) and 7(d), the frequency and the amplitude of the computed angular velocity oscillation in both the driver and the driven cases decrease with increasing the system's inertia, while the maximum discrepancy between the theory and experiment is less than 10%. Thus, having a formal description of local contact instabilities, we can predict the global dynamic behavior of our belt-drive system.

## Conclusion

To summarize, we highlight the key findings as follows. A larger applied torque accelerates the occurrence of contact instabilities in the driver case, while all other studied system response quantities remain unaffected. Increasing the driving speed results in an increase in the frequencies of the contact instability occurence and the pulley's angular velocity oscillations, while their amplitudes are essentially unaffected. The former suggests that as transmitted power increases, more power dissipates at the interface, as expected. Surprisingly, increasing the pulley's inertia does not remediate the contact instabilities, but instead leads to more pronounced fluctuations in the belt tension. Crosschecking, we draw similar conclusions based on a simple dynamic model, which provides strong evidence that contact instabilities driven by unmodulated external power are the primary source of the system's self-oscillation. To this end, our main conclusion is that contact instabilities and, hence, the resulting global system's oscillation, most likely cannot be conditioned from outside and instead the main focus must be on the interface itself.

## Funding Data

• The National Science Foundation under (Grant No. 1562129, Funder ID. 10.13039/501100008982)

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Sorge, F. , 2007, “ Shear Compliance and Self-Weight Effects on Traction Belt Mechanics,” Proc. Inst. Mech. Eng., Part C, 221(12), pp. 1717–1728.
Sorge, F. , 2008, “ A Note on the Shear Influence on Belt Drive Mechanics,” ASME J. Mech. Des., 130(2), p. 024502.
Alciatore, D. , and Traver, A. , 1995, “ Multipulley Belt Drive Mechanics: Creep Theory vs Shear Theory,” ASME J. Mech. Des., 117(4), pp. 506–511.
Kong, L. , and Parker, R. G. , 2005, “ Microslip Friction in Flat Belt Drives,” Proc. Inst. Mech. Eng., Part C, 219(10), pp. 1097–1106.
Balta, B. , Sonmez, F. O. , and Cengiz, A. , 2015, “ Speed Losses in V-Ribbed Belt Drives,” Mech. Mach. Theory, 86, pp. 1–14.
Chen, T. , and Sung, C. , 2000, “ Design Considerations for Improving Transmission Efficiency of the Rubber V-Belt CVT,” Int. J. Veh. Des., 24(4), pp. 320–333.
Zhu, C. , Liu, H. , Tian, J. , Xiao, Q. , and Du, X. , 2010, “ Experimental Investigation on the Efficiency of the Pulley-Drive CVT,” Int. J. Automot. Technol., 11(2), pp. 257–261.
Firbank, T. , 1977, “ On the Forces Between the Belt and Driving Pulley of a Flat Belt Drive,” Design Engineering Technical Conference, Chicago, IL, Sept., pp. 1–5.
Kim, H. , and Marshek, K. , 1988, “ Belt Forces and Surface Model for a Cloth-Backed and a Rubber-Backed Flat Belt,” J. Mech., Transm., Autom. Des., 110(1), pp. 93–99.
Kim, H. , Marshek, K. , and Naji, M. , 1987, “ Forces Between an Abrasive Belt and Pulley,” Mech. Mach. Theory, 22(1), pp. 97–103.
Palmer, R. , and Jarvis, J. , 1980, “ Steady State Strains in Power Transmitting Flat Belts Made of Composite Material,” Strain, 16(4), pp. 156–161.
Leamy, M. J. , and Wasfy, T. M. , 2002, “ Transient and Steady-State Dynamic Finite Element Modeling of Belt-Drives,” ASME J. Dyn. Syst. Meas. Control, 124(4), pp. 575–581.
Wasfy, T. M. , and Leamy, M. , 2002, “ Effect of Bending Stiffness on the Dynamic and Steady-State Responses of Belt-Drives,” ASME Paper No. DETC2002/MECH-34223.
Leamy, M. , and Wasfy, T. , 2002, “ Analysis of Belt-Driven Mechanics Using a Creep-Rate-Dependent Friction Law,” ASME J. Appl. Mech., 69(6), pp. 763–771.
Leamy, M. J. , 2005, “ On a Perturbation Method for the Analysis of Unsteady Belt-Drive Operation,” ASME J. Appl. Mech., 72(4), pp. 570–580.
Kerkkänen, K. S. , García-Vallejo, D. , and Mikkola, A. M. , 2006, “ Modeling of Belt-Drives Using a Large Deformation Finite Element Formulation,” Nonlinear Dyn., 43(3), pp. 239–256.
Dufva, K. , Kerkkänen, K. , Maqueda, L. G. , and Shabana, A. A. , 2007, “ Nonlinear Dynamics of Three-Dimensional Belt Drives Using the Finite-Element Method,” Nonlinear Dyn., 48(4), pp. 449–466.
Čepon, G. , and Boltežar, M. , 2009, “ Dynamics of a Belt-Drive System Using a Linear Complementarity Problem for the Belt–Pulley Contact Description,” J. Sound Vib., 319(3–5), pp. 1019–1035.
Kim, D. , Leamy, M. J. , and Ferri, A. A. , 2011, “ Dynamic Modeling and Stability Analysis of Flat Belt Drives Using an Elastic/Perfectly Plastic Friction Law,” ASME J. Dyn. Syst. Meas. Control, 133(4), p. 041009.
Wu, Y. , Leamy, M. J. , and Varenberg, M. , 2018, “ Schallamach Waves in Rolling: Belt Drives,” Tribol. Int., 119, pp. 354–358.
Barquins, M. , 1985, “ Sliding Friction of Rubber and Schallamach Waves—A Review,” Mater. Sci. Eng., 73, pp. 45–63.
Fukahori, Y. , Gabriel, P. , and Busfield, J. J. C. , 2010, “ How Does Rubber Truly Slide Between Schallamach Waves and Stick-Slip Motion?,” Wear, 269(11–12), pp. 854–866.
Schallamach, A. , 1971, “ How Does Rubber Slide?,” Wear, 17(4), pp. 301–312.
Barquins, M. , and Courtel, R. , 1975, “ Rubber Friction and the Rheology of Viscoelastic Contact,” Wear, 32(2), pp. 133–150.
Barquins, M. , and Roberts, A. D. , 1986, “ Rubber-Friction Variation With Rate and Temperature—Some New Observations,” J. Phys. D-Appl. Phys., 19(4), pp. 547–563.
Best, B. , Meijers, P. , and Savkoor, A. R. , 1981, “ The Formation of Schallamach Waves,” Wear, 65(3), pp. 385–396.
Lin, T. R. , Farag, N. H. , and Pan, J. , 2005, “ Evaluation of Frequency Dependent Rubber Mount Stiffness and Damping by Impact Test,” Appl. Acoust., 66(7), pp. 829–844.
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## References

Zhang, S. , and Xia, X. , 2011, “ Modeling and Energy Efficiency Optimization of Belt Conveyors,” Appl. Energy, 88(9), pp. 3061–3071.
Euler, M. L. , 1762, “ Remarques Sur L'effect du Frottement Dans L'equilibre,” Mémoires De L'Académie Royale Des Sci., 18, pp. 265–278.
Grashof, F. , 1890, Theoretische Maschinenlehre, L. Voss, Leipzig, Germany.
Kong, L. , and Parker, R. G. , 2005, “ Steady Mechanics of Belt-Pulley Systems,” ASME J. Appl. Mech., 72(1), pp. 25–34.
Bechtel, S. , Vohra, S. , Jacob, K. , and Carlson, C. , 2000, “ The Stretching and Slipping of Belts and Fibers on Pulleys,” ASME J. Appl. Mech., 67(1), pp. 197–206.
Rubin, M. , 2000, “ An Exact Solution for Steady Motion of an Extensible Belt in Multipulley Belt Drive Systems,” ASME J. Mech. Des., 122(3), pp. 311–316.
Firbank, T. , 1970, “ Mechanics of the Belt Drive,” Int. J. Mech. Sci., 12(12), pp. 1053–1063.
Della Pietra, L. , and Timpone, F. , 2013, “ Tension in a Flat Belt Transmission: Experimental Investigation,” Mech. Mach. Theory, 70, pp. 129–156.
Gerbert, G. , 1991, “ Paper XII (i) On Flat Belt Slip,” Tribol. Ser., 18, pp. 333–340.
Gerbert, G. , 1996, “ Belt Slip—A Unified Approach,” ASME J. Mech. Des., 118(3), pp. 432–438.
Sorge, F. , 2007, “ Shear Compliance and Self-Weight Effects on Traction Belt Mechanics,” Proc. Inst. Mech. Eng., Part C, 221(12), pp. 1717–1728.
Sorge, F. , 2008, “ A Note on the Shear Influence on Belt Drive Mechanics,” ASME J. Mech. Des., 130(2), p. 024502.
Alciatore, D. , and Traver, A. , 1995, “ Multipulley Belt Drive Mechanics: Creep Theory vs Shear Theory,” ASME J. Mech. Des., 117(4), pp. 506–511.
Kong, L. , and Parker, R. G. , 2005, “ Microslip Friction in Flat Belt Drives,” Proc. Inst. Mech. Eng., Part C, 219(10), pp. 1097–1106.
Balta, B. , Sonmez, F. O. , and Cengiz, A. , 2015, “ Speed Losses in V-Ribbed Belt Drives,” Mech. Mach. Theory, 86, pp. 1–14.
Chen, T. , and Sung, C. , 2000, “ Design Considerations for Improving Transmission Efficiency of the Rubber V-Belt CVT,” Int. J. Veh. Des., 24(4), pp. 320–333.
Zhu, C. , Liu, H. , Tian, J. , Xiao, Q. , and Du, X. , 2010, “ Experimental Investigation on the Efficiency of the Pulley-Drive CVT,” Int. J. Automot. Technol., 11(2), pp. 257–261.
Firbank, T. , 1977, “ On the Forces Between the Belt and Driving Pulley of a Flat Belt Drive,” Design Engineering Technical Conference, Chicago, IL, Sept., pp. 1–5.
Kim, H. , and Marshek, K. , 1988, “ Belt Forces and Surface Model for a Cloth-Backed and a Rubber-Backed Flat Belt,” J. Mech., Transm., Autom. Des., 110(1), pp. 93–99.
Kim, H. , Marshek, K. , and Naji, M. , 1987, “ Forces Between an Abrasive Belt and Pulley,” Mech. Mach. Theory, 22(1), pp. 97–103.
Palmer, R. , and Jarvis, J. , 1980, “ Steady State Strains in Power Transmitting Flat Belts Made of Composite Material,” Strain, 16(4), pp. 156–161.
Leamy, M. J. , and Wasfy, T. M. , 2002, “ Transient and Steady-State Dynamic Finite Element Modeling of Belt-Drives,” ASME J. Dyn. Syst. Meas. Control, 124(4), pp. 575–581.
Wasfy, T. M. , and Leamy, M. , 2002, “ Effect of Bending Stiffness on the Dynamic and Steady-State Responses of Belt-Drives,” ASME Paper No. DETC2002/MECH-34223.
Leamy, M. , and Wasfy, T. , 2002, “ Analysis of Belt-Driven Mechanics Using a Creep-Rate-Dependent Friction Law,” ASME J. Appl. Mech., 69(6), pp. 763–771.
Leamy, M. J. , 2005, “ On a Perturbation Method for the Analysis of Unsteady Belt-Drive Operation,” ASME J. Appl. Mech., 72(4), pp. 570–580.
Kerkkänen, K. S. , García-Vallejo, D. , and Mikkola, A. M. , 2006, “ Modeling of Belt-Drives Using a Large Deformation Finite Element Formulation,” Nonlinear Dyn., 43(3), pp. 239–256.
Dufva, K. , Kerkkänen, K. , Maqueda, L. G. , and Shabana, A. A. , 2007, “ Nonlinear Dynamics of Three-Dimensional Belt Drives Using the Finite-Element Method,” Nonlinear Dyn., 48(4), pp. 449–466.
Čepon, G. , and Boltežar, M. , 2009, “ Dynamics of a Belt-Drive System Using a Linear Complementarity Problem for the Belt–Pulley Contact Description,” J. Sound Vib., 319(3–5), pp. 1019–1035.
Kim, D. , Leamy, M. J. , and Ferri, A. A. , 2011, “ Dynamic Modeling and Stability Analysis of Flat Belt Drives Using an Elastic/Perfectly Plastic Friction Law,” ASME J. Dyn. Syst. Meas. Control, 133(4), p. 041009.
Wu, Y. , Leamy, M. J. , and Varenberg, M. , 2018, “ Schallamach Waves in Rolling: Belt Drives,” Tribol. Int., 119, pp. 354–358.
Barquins, M. , 1985, “ Sliding Friction of Rubber and Schallamach Waves—A Review,” Mater. Sci. Eng., 73, pp. 45–63.
Fukahori, Y. , Gabriel, P. , and Busfield, J. J. C. , 2010, “ How Does Rubber Truly Slide Between Schallamach Waves and Stick-Slip Motion?,” Wear, 269(11–12), pp. 854–866.
Schallamach, A. , 1971, “ How Does Rubber Slide?,” Wear, 17(4), pp. 301–312.
Barquins, M. , and Courtel, R. , 1975, “ Rubber Friction and the Rheology of Viscoelastic Contact,” Wear, 32(2), pp. 133–150.
Barquins, M. , and Roberts, A. D. , 1986, “ Rubber-Friction Variation With Rate and Temperature—Some New Observations,” J. Phys. D-Appl. Phys., 19(4), pp. 547–563.
Best, B. , Meijers, P. , and Savkoor, A. R. , 1981, “ The Formation of Schallamach Waves,” Wear, 65(3), pp. 385–396.
Lin, T. R. , Farag, N. H. , and Pan, J. , 2005, “ Evaluation of Frequency Dependent Rubber Mount Stiffness and Damping by Impact Test,” Appl. Acoust., 66(7), pp. 829–844.

## Figures

Fig. 1

The experimental apparatus: (a) schematic and (b) system as built

Fig. 2

Schematic of the contact behavior in the driver (a) and driven (b) cases

Fig. 3

Friction, angular velocity and characteristic images representing evolution of the contact area (shown in black) in (a) the driver and (b) the driven cases. A wavelet decomposition exhibits the two primary components of the friction signal: fluctuations associated with detachment events (Fde) and fluctuations associated with pulley oscillations (Fpo).

Fig. 4

The frequency (f) and amplitude (A) of the fluctuations in the friction force (F), (a) and (b), respectively, and in the angular pulley velocity (α), (c) and (d), respectively, associated with detachment events (de) and pulley oscillations (po), and presented as a function of loading conditions in both the driver and driven cases. The error bars show standard deviation.

Fig. 5

The frequency (f) and amplitude (A) of the fluctuations in the friction force (F), (a) and (b), respectively, and in the angular pulley velocity (α), (c) and (d), respectively, associated with detachment events (de) and pulley oscillations (po), and presented as a function of driving speed in both the driver and driven cases. The error bars show standard deviation.

Fig. 6

Friction force obtained with and without flywheels in the driver (a) and driven (b) cases

Fig. 7

The frequency (f) and amplitude (A) of the fluctuations in the friction force (F), (a) and (b), respectively, and in the angular pulley velocity (α), (c) and (d), respectively, associated with detachment events (de) and pulley oscillations (po), and presented as a function of the relative pulley's moment of inertia in both the driver and driven cases. The error bars show standard deviation.

Fig. 8

Diagrams of a simple belt-drive system defined for the driver and driven cases

Fig. 9

Wavelet scalograms of the angular velocity fluctuations obtained from (a) the experiment and (b) the numerical model for the driver pulley equipped with no, small, and large flywheels

Fig. 10

Wavelet scalograms of the angular velocity fluctuations obtained from (a) the experiment and (b) the numerical model for the driven pulley equipped with no, small, and large flywheels

Fig. 11

The frequency (a) and the amplitude (b) of the angular velocity oscillations obtained from the experiment and numerical model as a function of the relative pulley's moment of inertia in the driver and driven cases

## Tables

Table 1 Experimental conditions for variation of driving speed
Table 2 Experimental conditions for variation of load
Table 3 Experimental conditions for variation of the moment of inertia of the pulley
Table 4 Parameters in the dynamic model

## Errata

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