Efficient eigensolution, dynamic response, and eigensensitivity of serpentine belt drives

Abstract

An efficient method for calculating the eigensolutions and dynamic response of a serpentine belt drive is presented. The model is a hybrid discrete-continuous one where the motions consist of rotations of the pulleys, rotation of the tensioner arm, and transverse vibrations of the continuum belt spans adjacent to the tensioner. The speed of solution results from discretization of the belt spans where the unusual feature is the use of Lagrange multipliers to enforce the geometric boundary conditions at the belt-tensioner interface. The method reduces the computational effort by several orders of magnitude compared to published methods using the same model. Also, it is not susceptible to numerical problems that hinder the published methods. The sensitivities of the belt drive natural frequencies to system parameters are also studied. The model parameters under consideration include belt longitudinal stiffness, tensioner spring stiffness, span tensions, belt transport speed, belt density, and pulley moments of inertia. Closed-form expressions for the eigensensitivities to these parameters are obtained using perturbation methods. These expressions are reduced to simple formulae related to the modal strain and kinetic energies. The eigensensitivities are readily determined, quantitatively and qualitatively, by inspection of the modal energy distributions.