Modeling Geometric Non-linearities in the Free Vibration of a Planar Beam Flexure with a Tip Mass

The objective of this work is to analytically study the nonlinear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton’s principal is utilized to derive the equations governing the non-linear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these non-linear partial differential equations are reduced to two coupled non-linear ordinary differential equations. These equations are solved analytically using combination of the method of multiple time scales and homotopy perturbation analysis. Parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed closed-form model. We expect that the qualitative and quantitative knowledge resulting from this effort will ultimately allow the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.