Prediction of Dynamic Stick-Slip Events in Belt-Drives Using a High-Fidelity Finite Element Model

Many practical mechanical components such as tires, brakes and belts involve a compliant material such as rubber in frictional contact with a much stiffer material. When a compliant material comes into frictional contact with a much harder material with a nonzero relative tangential velocity between the two material surfaces, surface instabilities may occur at the leading or trailing edge of the compliant surface resulting in propagation of surface waves in the compliant material. Those frictional waves can cause abrasion of the compliant material, noise, and unwanted vibrations which can negatively affect the reliability and durability of the overall mechanical system. Adolf Schallamach [1,2] was the first to document and investigate those surface friction driven instabilities. He observed that when the resistance to the relative motion (caused by friction and/or adhesion) between the two bodies is relatively high, the relative motion between the two materials takes the form of waves of detachment which propagate across the soft material contact surface from the front to the rear of the contact area. A wave of detachment is a buckling bend in the rubber surface which moves relative to the hard surface in an inch-worm fashion and facilitates the relative motion between the two surfaces. This buckling fold is caused by the accumulation of shear and compressive stress and then sudden release of that stress. Those detachment waves were termed “Schallamach waves.” When the resistance to the motion between the two materials is low, the relative motion between the rubber and the hard material takes the form of true sliding. Many experimental [3–10], analytical [3,6,11], and computational [12] studies on stick-slip instabilities in general and Schallamach Waves in particular in rubber in different systems have been conducted since the 1970s. In Ref. [13], stick-slip instabilities in a single pulley belt-drive were studied using a high-fidelity computational finite element (FE) model. Then, the computational results were validated using the experimental results of Wu et al. [8–10]. In this article, the results of Ref. [13] are presented in a more concise and systematic way. In addition, further analysis, discussion, and conclusions based on the model's results are presented. Specifically, the FE model is used to investigate the effects of Coulomb friction coefficient, adhesion stress, and traction versus slip direction (i.e., driver versus driven pulley configuration) on the dynamic stick-slip instability events at the surface of the belt and to understand the triggers and conditions under which those instabilities occur.

The explicit-time integration computational FE model used to model the belt-drive used in this article was presented in Refs. [13–17]. The FE one-pulley belt-drive model is shown Fig. 1. Initially the belt-drive is at rest in the horizontal plane (i.e., no gravitational forces). The pulley is modeled as a rigid body with its angular velocity controlled using a rotational actuator along with a proportional-derivative (PD) controller. Forces $F1$ and $F2$ are applied to the two ends of the belt along the Y-direction in order to provide the low and high-tension belt span forces. The total simulation time is 5.6 s. The explicit time-step is 0.1 μs. Two configurations are modeled by setting the belt spans tension forces to the following values:

• Driver pulley: inlet force $F1=6.04 N$ and exit force $F2=1.2 N$⁠.

• Driven pulley: inlet force $F1=1.2 N$ and exit force $F2=6.04 N$⁠.

The desired pulley angular velocity is linearly increased from 0 to 1 rad/s counterclockwise in 1 s, then kept constant for both driver and driven pulley cases. For the driver case, the direction of the pulley torque is counterclockwise, which is the same as the pulley angular velocity direction. In the driven case, the direction of the torque is clockwise which is opposite to the pulley angular velocity direction (Fig. 2). The model's material and geometric parameters are shown in Table 1. Those parameters match those of the experimental belt-drive in Refs. [8] and [9]. The belt is discretized using 9200 brick elements with the belt's cross section containing 30 nodes (Fig. 3). Small elements are used near the belt-pulley contact surface since Schallamach waves occur near that surface. Then, the element height is progressively enlarged from the bottom to the top belt's surface. Note that the adhesion stress acts only in the normal contact direction and friction acts only in the tangential contact direction.

Sixteen runs are performed to study the effect of friction coefficient (μ) and normal adhesion stress (σ_adh) on the initiation of stick-slip instabilities for the belt-drive for both the driver and driven pulley cases (Table 2). Coulomb Friction coefficient values are set to the following values: 1.2, 2.0, 2.5, and 5.0. The adhesion stress is set to following values: 0 and 20 kPa. The following quantities are plotted for each simulation as a function of pulley angle: average axial stress (the average value in the belt cross section along the thickness of the belt), and inner radial and shear stresses at the belt-pulley contact surface. All the stresses are measured along the centerline of belt cross section. When contact instabilities are observed, five snapshots of the belt stresses are plotted to show the transient effects on the belt stresses. The graphs for runs 1-4 and 9-16 are shown in Figs. 4 to 9. In those figures, for the driven pulley, the belt's inlet is at -180 deg and the belt's outlet is at 0 deg. For the driver pulley, the belt's inlet is at 0 deg and the belt's outlet is at 180 deg. The range of time for the simulation results of interest is from about 3.6 to 5.2 s. Prior to 3.6 s, the pulley typically did not reach steady-state conditions. After 5 s, the right belt span length (Fig. 1) approaches zero leading to unsteady behavior at the belt's inlet.

In Sec. 3.1, the results for driver and driven pulley cases with no stick-slip instabilities effects are presented. In Sec. 3.2, the results for driver and driven pulley cases are presented when stick-slip instabilities are present. In Secs. 3.3 and 3.4, the effects of Coulomb friction coefficient and adhesion are respectively studied when stick-slip instabilities are present. Finally, in Sec. 3.5 the simulation and experiment stick-slip instability results are compared.

The plots for runs 1, 2, 3, and 4 show that when μ = 1.2 no stick-slip instabilities are present for both the driver and driven pulley cases and for both σ_adh = 0 kPa and σ_adh = 20 kPa (Fig. 4). This shows that true sliding is taking place for coefficient of friction values at or below 1.2. Note that the results for the driver and driven pulleys in Fig. 4 show qualitatively similar stress distributions as those presented in Ref. [14] for a belt that has stiff cords on its top surface. When no stick-slip instabilities are present (at μ = 1.2), Fig. 4 shows that the main effects of increasing σ_adh from 0 to 20 kPa are:

1. For the driver pulley: shifting the peak shear stress by about 22 deg (from 78 to 100 deg) toward the belt outlet.

2. For the driven pulley: increasing the size of the pseudo-stick zone [14] on the belt's inlet side by about 20 deg (from −50 to −30 deg).

Figures 5 and 6 show the time-histories of the pulley torque for the driver and driven pulley cases, respectively, for the different values of μ and σ_adh. Frictional stick-slip events can be detected through pulses and subsequent oscillations in the pulley's torque applied by the PD angular velocity controller in order to maintain a constant angular velocity for the pulley. For the driver pulley case, each pulse roughly corresponds to the start of a Schallamach wave as will be explained below. For the driven pulley case, these pulses correspond to belt stick-slip instabilities in the form of expansion waves at the belt's exit, which will be explained later in this section. Note that whenever there is a spike in the torque curve, snapshots of stress values in Figs. 7 to 9 were plotted around the highest peak within the range of 4.1 to 4.5 s.

Figure 5 shows that for the driver pulley for μ = 1.2, negligible oscillations are present. This shows that no stick-slip instabilities are occurring. At around μ = 2.0 stick-slip instabilities start to occur regardless of the value of σ_adh. But from Fig. 5 the amplitude of the torque oscillations due to Schallamach waves at μ = 2.0 is much larger for σ_adh = 20 kPa than for σ_adh = 0, which shows that, for the driver pulley, adhesion shifts the friction Coefficient threshold at which Schallamach waves occur to a lower value. Schallamach waves are observed in all cases for friction coefficient of μ ≥ 2.0.

The phenomenon of Schallamach waves on the driver pulley can be explained as follows. When the Coulomb friction coefficient is relatively high (μ ≥ 2.0), the belt enters the pulley at high tension and remains relatively stuck to the surface of the pulley since high friction prevents slipping. Thus, the belt tension (axial stress) remains relatively constant on the pulley until about 90 deg for μ = 2.5 (Figs. 7 and 8). The belt tension has to transition from high tension to low tension. In addition, due to the tension difference, the belt has to slip opposite to the direction of the applied torque and angular velocity on the driver pulley (i.e., belt speed should be slower than the pulley speed). Since the belt slip is restricted due to high friction, it slips by locally buckling (lifting off the pulley surface) and moving in an inchworm fashion opposite to the direction of torque and angular velocity (Fig. 10). However, this only occurs when sufficient surface axial strain has accumulated in the belt such that friction can no longer prevent the belt from suddenly slipping and releasing this axial strain as a torque pulse. Comparing the plots for μ = 1.2 (Fig. 4) and μ = 2.5 (Fig. 7) when σ_adh = 0, shows that the belt radial and axial stresses remain relatively constant until about 60 deg for μ = 1.2 and 90 deg for μ = 2.5. Also, the magnitude of the belt radial stress drops much faster to near zero and is much smaller from 90 to 180 deg for μ = 2.5 than for μ = 1.2. Small radial stresses facilitate belt buckling. Starting around the peak point of the belt shear at 80–100 deg until 180 deg (Fig. 7), the belt stresses with respect to time are no longer constant during steady-state operation due to the propagation of Schallamach waves from the outlet of the belt toward the inlet. This is consistent with the experimental results presented by Wu et al. [8–10], which indicated the presence of stick-slip instabilities in the form of Schallamach waves on the driver pulley.

Similar to the driver pulley case, Fig. 6 shows that for the driven pulley for μ = 1.2, negligible torque oscillations are observed which indicates that no stick-slip instabilities are occurring. At around μ = 2.0 stick-slip instabilities start to occur irrespective of the value of σ_adh. But from Fig. 6 the amplitude of the torque oscillations at μ = 2.0 is much larger for σ_adh = 0 than for σ_adh = 20 kPa, which shows that for the driven pulley, adhesion shifts the friction coefficient threshold at which stick-slip instabilities occur to a higher value. The stick-slip instabilities for the driven pulley are different from those on the driver pulley. While in the driver pulley stick-slip instabilities take the form of local belt surface buckling and back propagation of a Schallamach wave, in the case of a driven pulley the stick-slip instabilities take the form of expansion waves near the belt's outlet. Those waves briefly travel forward in the direction of the belt's outlet before expanding out in the high-tension span. This can be explained as follow. For relatively high coefficient of friction (μ ≥ 2.0), the belt enters the pulley at low tension and remains relatively stuck to the surface of the pulley since the high friction prevents slipping. Thus, the belt tension remains relatively constant on the pulley (Fig. 7). However, the belt tension has to transition from low tension to high tension. In addition, due to the tension difference, the belt has to slip opposite to the direction of the applied pulley torque and in the same direction as the pulley angular velocity on the driven pulley (i.e., belt speed is faster than the pulley speed). Since the belt slip is restricted due to high friction, it slips near the outlet of the belt by suddenly expanding out into the high-tension belt span. This is consistent with the experimental results presented by Wu et al. [8–10] for a driven pulley configuration indicating the presence of stick-slip instabilities (which are not Schallamach waves) near the belt's exit. The slip mechanism on a driven pulley can be thought of as the reverse of the Schallamach wave mechanism on a driver pulley. In case of Schallamach waves on a driver pulley the belt slips by buckling/compression waves that travel into the belt contact arc. In case of expansion waves on a driven pulley the belt slips by expansion/tension waves that travel into the high-tension belt span.

The Schallamach wave effects on a driver pulley can be observed in the time variations of the belt radial and shear stresses from the belt exit point (180 deg) to about an angle of about 40 deg on the driver pulley (Fig. 7 to Fig. 9). A radial stress of near zero indicates loss of contact between the belt and pulley due to propagation of Schallamach waves. From Figs. 8 and 9, it can be observed that at μ = 2.5 and σ_adh = 20 kPa, Schallamach waves typically initiated closer to the belt's outlet and extended over a shorter angular range than at μ = 5.0 and σ_adh = 20 kPa. In addition, Fig. 6 shows that the toque pulse due to Schallamach waves has lower magnitude at μ = 2.5 than at μ = 5.0. Figures 4 and 8 show the effect of increasing the friction coefficient from μ = 1.2 to μ = 2.5 on the driven pulley. As μ increases, the area of large belt axial and shear stresses magnitudes at the belt's outlet becomes smaller with higher stress magnitude. This high stress gradient at the belt's outlet seems to be the mechanism of initiation of the expansion stick-slip pulses for the driven pulley.

For the driver pulley case, the effects of adhesion on belt stresses, when stick-slip instabilities are present, can be seen in Figs. 7 and 8 which show the results at μ = 2.5. Those are:

• Similar to the no stick-slip instabilities case—μ = 1.2 (Sec. 3.1), increased adhesion leads to shifting of the belt stresses toward the belt outlet.

• The Schallamach waves with zero adhesion are sharp and can be clearly seen propagating. The Schallamach waves at σ_adh = 20 kPa are diffused and dissipate quicker—akin more to a wave of belt lift over the pulley rather than a sharp belt buckling pulse.

For the driven pulley configuration, the effects of adhesion on the belt stresses, when stick-slip instabilities are present, are:

• Comparing Figs. 7 and 8 for μ = 2.5 shows that similar to the no stick-slip instabilities case—μ = 1.2 (Sec. 3.1), increased adhesion leads is to increasing the size of the pseudo-stick zone [14] on belt inlet side by about 20 deg (from −40 to −20 deg).

• Increasing the adhesion leads to reducing the expansion waves amplitude on the driven pulley (Fig. 6).

For a driver pulley, the Schallamach wave phenomenon can be experimentally (visually) characterized by four parameters: minimum start of contact angle (θ_s), maximum Schallamach waves propagation angle (θ_p), average amplitude of the torque (or span force) pulses, and time between torque/force pulses (t_p) (Fig. 5). θ_s is found by starting from the belt's outlet and detecting the angle at which the belt starts to contact the pulley (penetration in the penalty contact model is greater than zero) (Fig. 11). θ_p is found by starting from an angle of 30 deg from the belt's inlet and detecting the smallest angle from 30 deg on the pulley where contact with the belt is lost. θ_p is measured from the belt's outlet to that contact loss point (Fig. 11). Schallamach waves (the belt surface buckling waves) propagate on the surface of the belt within the slip zone at the outlet side of driver pulley (Fig. 2) from θ_s to θ_p. In addition, we define a full contact intimacy angle (θ_f) which we use in the simulation to enable clearly counting the number of Schallamach wave pulses over time. θ_f is found by starting from an angle of 30 deg from the belt's inlet and detecting the point on the pulley where the a contact penetration in the penalty model is less than 100 μm which corresponds to a belt's radial stress of about 50 kPa (Figs. 7– 9). θ_f is measured from the belt's outlet to that point (Fig. 11). θ_s, θ_p, and θ_f can be calculated from the simulation's radial stress curves (Figs. 7–9). θ_s, θ_p, and θ_f are all time dependent.

The belt-drive computational model parameters in Table 1 closely match the one-pulley belt-drive experiment in Wu et al. [10]. Figure 12 shows the experiment belt inlet side and outlet side forces along with the friction force (difference between those two forces) and evolution of the contact area. White in a contact area image corresponds to loss of contact and black corresponds to contact between the belt and pulley. Therefore θ_s and θ_c can be estimated from the time sequence of the experiment contact area images. The experimental results in Wu et al. [10] for the driver and driven pulley cases are close to Runs 11 and 12, respectively (μ = 2.5 and σ_adh = 20 kPa). Similar to the simulation, Schallamach waves were only observed in the experiment on the driver pulley. Schallamach waves were characterized by Wu et al. [10] using three parameters: start of contact angle (θ_s), Schallamach wave propagation angle (θ_p) (Fig. 11), and time between pulses (t_p). Similar to the simulation, no Schallamach waves were observed in the experiment in the driven pulley case. Also, similar to the simulation, for the driven pulley case, the exit belt span experienced pulses, which were characterized by only one parameter, namely, the time between pulses (t_p).

For the driver pulley case in Fig. 12 we can count 9 to 10 Schallamach wave pulses in half of a pulley revolution. Since the pulley angular velocity is 1 rad/s, this corresponds to an average t_p of 0.31 to 0.35 s. Furthermore, the average θ_s and average maximum θ_p for the driver pulley case in the experiment were estimated to be about 11.5 and 41 deg, respectively. For the driven pulley configuration in Fig. 12 we can count 15 to 16 pulses in half a pulley revolution. This corresponds to an average t_p of 0.19 to 0.21 s.

The simulation's time-histories of the instantaneous θ_s and θ_p are shown in Fig. 13. From this figure, the average θ_s is about 10 deg and the average maximum θ_p is about 40 deg. Figure 14 shows the simulation time-history of the contact intimacy angle θ_f which is used to show the start and end of each Schallamach wave stick-slip event. In Fig.14, we can count for the driver pulley case 7 to 8 pulses over 2 s, which corresponds to an average t_p of 0.25 to 0.29 s. For the driven pulley the number of expansion wave pulses can be estimated from the torque time-history (Fig. 6). From time 4.1 s to 4.5 s, we can count 6 large pulses (ignoring the much smaller pulses) which corresponds to an average t_p of 0.067 s. A summary of the comparison of θ_s, θ_p and t_p between the experiment and the simulation is shown in Table 3 and Table 4.

Note that the Schallamach waves in both the experiment and simulation were diffuse and more akin to belt lift (Fig. 12) rather than a sharp buckling fold as in the case when σ_adh = 0 kPa in the simulation. Also, note that the amplitude of the experiment force pulses in the exit span (torque pulses in the simulation) for the driven pulley are about half those for the driver pulley (Fig. 12), which was also the case in the simulation (comparing Fig. 5 and Fig. 6).

The main contribution of Ref. [13] and this paper is that for the first time a finite element computational model is used to accurately predict the following stick-slip dynamic instability phenomena in belt-drives: Schallamach wave phenomenon for driver pulleys, and the expansion wave phenomenon for driven pulleys. In addition, the one-pulley belt-drive computational belt-drive model was used to systematically study the effects of Coulomb friction coefficient and adhesion stress on dynamic the stick-slip phenomena. A main conclusion of this paper is that large Coulomb friction coefficient (>2) is the main reason for the occurrence of stick-slip instabilities which include Schallamach waves. This agrees with the conclusion of Ref. [12]. Another conclusion is that adhesion stress affects the amplitude and sharpness of the stick-slip waves but is not the trigger of those waves.

The authors thank Professor Michael Leamy, Professor Michael Varenberg, and Mr. Yingdan Wu of the School of Mechanical Engineering at Georgia Institute of Technology for providing the one-pulley belt-drive experiment data and for many valuable discussions.

• National Science Foundation (No. 1562357; Funder ID: 10.13039/100000001).