Stability and limit cycles of parametrically excited, axially moving strings

Tension fluctuations are the dominant source of excitation in automotive belts. In particular designs, these fluctuations may parametrically excite large amplitude transverse belt vibrations and adversely impact belt life. This paper evaluates an efficient discrete model of a parametrically excited translating belt. The efficiency derives from the use of translating string eigenfunctions as a basis for a Galerkin discretization of the equations of transverse belt response. Accurate and low-order models lead to simple closed-form solutions for the existence and stability of limit cycles near parametric instability regions. In particular, simple expressions are found for the stability boundaries of the general n^th-mode principal parametric instability regions and the first summation and difference parametric instability regions. Subsequent evaluation of the weakly non-linear equation of motion leads to an analytical expression for the amplitudes (and stability) of non-trivial limit cycles that exist around the n^th-mode principal parametric instability regions. Example results highlight important conclusions concerning the response of automotive belt drives.