Abstract
This paper is concerned with the stability and the steady state response of the main parametric resonance vibrations of a simply supported vertical beam. The beam carries a concentrated mass and is restrained at one end and subjected to a periodic axial displacement excitation at the other end. This system can be looked upon as a dynamic model of the vibrations of an engine valve mechanism. Non-linear terms arising from moderately large curvatures, longitudinal inertia of the beam elements and concentrated mass are included in the equation of motion. By using the one mode approximation and applying Galerkin's method, the governing partial differential equation is reduced to a non-linear ordinary differential equation with a periodic coefficient. The boundaries of the main parametric instability region of the linearized equation are obtained. The harmonic balance method is applied to solve the equation and an analytical expression for the dynamic response in the vicinity of the main parametric resonance is derived. The effects of various parameters on the boundaries of the instability region and the dynamic response are investigated.
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