Experimental evidence of Schallamach wave-induced self-excitation in a belt-drive system

Yingdan Wu, Michael J. Leamy , Michael Varenberg

Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA

INTRODUCTION: Belt drives find application in manufacturing processes and power transmission at nearly every scale. Presently, properly installed and maintained belt-drive energy efficiency ranges from 85 to 95%¹. The efficiency varies with the operating torque of the drive and the resulting amount of belt slip incurred in the belt/pulley contact region. It is critical to understand belt-drive mechanics/dynamics and frictional behavior in order to achieve better belt-drive designs.

Recently, Wu et al.² experimentally tested a simple belt-drive system in an effort to detect the presence of detachment waves, including those known as Schallamach wave³. Schallamach waves frequently occur when an elastomer is in frictional contact with a more rigid surface. These waves appear as regions of lost contact which move across the contact area in the sliding direction.

This work reports further studies on the formation of Schallamach waves and related stick-slip contact dynamics in belt-drive systems, where an elastomer is in rolling contact with a rigid pulley. We study experimentally the dynamic behavior of a belt-drive system to explore the effect of operating conditions and system moment of inertia on the generation of detachment waves at the belt-pulley interface. Specifically, we report the frequency and amplitude of the frictional waves as functions of the applied torque, the operating speed, and the system moment of inertia. We find that the detachment waves lead to a self-excitation instability whereby the pulley undergoes oscillations that grow in time. Contrary to what was expected, this self-excitation behavior appears to be exacerbated by increasing pulley inertia.

METHODS:  The results reported herein were conducted on an experimental apparatus capable of measuring the tensions in both spans of a belt wrapped around a pulley. An encoder records the pulley angular displacement. The cases of driver and driven pulleys can be switched by changing the direction of the applied torque. A more detailed introduction to the apparatus is available in Ref. 2. The 400x5x2 mm belt specimens were produced by molding polydimethylsiloxane (PDMS). Each test was repeated at least 5 times. The temperature and relative humidity in the laboratory during the tests were 23°C and 35%, respectively.

RESULTS:  Fig. 1 shows the friction and angular velocity measured during the operation of the system under a driving speed of 3 mm/s and a net torque weight of 4 N in both the driven (Fig. 1a) and the driver (Fig. 1b) cases. The friction force was obtained by subtracting the slack side tension from the tight side tension of the belt. A zoomed-in frictional signal is correlated with their corresponding contact zone images for both cases. The angular velocities were approximated using a central difference scheme based on the angular displacement measured by the rotary encoder. The friction spectra of successive time periods were calculated via an FFT.

DISCUSSION: The Schallamach waves (detached fold among the contact zone; No. 3 image of contact zone in driver case) were only detected in the driver pulley case despite frictional instabilities being observed in both cases. Also, peaks in the friction signals of both cases have distinct shapes due to different relaxation mechanisms². Nevertheless, the friction force for both cases, integrating the stresses developing along the contact arc, captures global, large-scale stick-slip instabilities. As seen in the results for both driver and driven pulleys, angular velocity oscillations grow in time due to forcing caused by the stick-slip instabilities. We believe this growth is due to self-excitation fed by a feedback mechanism:

stick-slip instabilities excite rotational oscillations in the pulley, which in turn store a periodic belt tension pattern in the pulley 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. It is clear then that local frictional behavior affects large changes in the system’s global dynamics.

Figure 1 Friction and angular velocity measured during the operation with the friction spectra of successive periods and characteristic sequences of images representing evolution of the contact area (black region) in both the (a) driven and (b) driver cases.

From the friction spectra (Fig. 1), we observe two significant frequencies: a high frequency associated with stick-slip events, and a low frequency corresponding to the pulley rotation. The higher frequency tends to decrease with time. Given that the belt spans have changing frequencies with length, the evolution of the instabilities may be related to the changing span of the belt. As for the low frequency component, it remains constant (approximately that of a half revolution) and likely unrelated to the stick-slip events. From this point forward, we consider only the relatively high frequency component associated with the stick-slip instabilities. Furthermore, since the dynamic properties evolve in time as the self-excitation mechanism develops, we choose to focus on the last one revolution where the frictional events are most evident. Based on this regime, we found that a larger applied torque increases the threshold for slip relaxation in the belt-pulley interface, yielding more pronounced frictional oscillations with higher amplitude and lower frequency. Furthermore, the frequency of the frictional oscillations increases with increasing driving speed, whereas the amplitude of the oscillations shows little dependence on the driving speed. More importantly, and quite unexpectedly, increasing the system inertia can amplify the frictional instabilities, yielding larger and more detailed pulley oscillations.

REFERENCES:  1.Zhang, MINExpo. (2004), 2. Wu, Trib. Int. (2018), 3. Schallamach, Wear (1971)