Coupled longitudinal and transverse vibration of automotive belts under longitudinal excitations using analog equation method

Abstract

There are many studies on vibrations of moving belts originated from automotive belt drives. These studies, however, mainly consider the nonlinear effect of the transversal vibration of the belt under transversal excitation without considering the coupling effect between the transversal and longitudinal vibrations under longitudinal excitation. In this paper, a nonlinear model is built for a standing tensioned belt periodically excited longitudinally at one end with a known amplitude and frequency and with the other end clamped, which allows for coupling between the periodic longitudinal excitation and transversal vibration. The nonlinear system is solved using the Analog Equation Method. The two coupled nonlinear hyperbolic differential equations of the system are reduced to two uncoupled linear equations pertaining to the longitudinal and transverse deformation of a substitute beam with unit longitudinal and bending stiffness, respectively. The reduced equations are under a fictitious time-dependent load distribution with the same boundary and initial conditions. The fictitious forces are established numerically from the integral expression of the oscillation solution with the requirement that the equations of motion be satisfied at discreet time intervals. It is found that the transversal vibration appears as a stable and predictable beating phenomenon due to an internal resonance from the periodic energy transfer between the longitudinal excitation and the transversal vibration, which manifests itself as a standing wave.