On Nonlinear Vibration of Nonuniform Beam with Rectangular Cross-Section and Parabolic Thickness Variation

Abstract

We present the exact solution of differential equation in the linear case of free bending vibrations of nonuniform beam with rectangular cross-section using the factorization method. This beam with constant width and parabolic thickness is a good approximation of the gear tooth profile. It permits a nonlinear bending vibrations study (moderately large curvatures) of the gear tooth (the cantilever beam case). The case of the beam with a sharp end is considered. We use the method of multiple scales to treat the governing partial-differential equations and boundary conditions directly. In the absence of internal resonance (weakly nonlinear systems) the nonlinear modes are taken to be perturbed versions of the linear modes. We determine the nonlinear planar mode shapes and natural frequencies of a gear tooth with a sharp end variation (the cantilever beam case).