Nonlinear Free Vibration in Flexure Beams with an Intermediate Rigid Element and a Tip Mass

A usual method in achieving a proper value for the ratio of constraint to degree of freedom stiffness in a compliant mechanism, is using an intermediate rigid element in its constitutive beams. This paper aims to study the nonlinear free vibration of a stiffened beam with a mass connected to its tip. Hamilton’s principle is used to find nonlinear partial differential equations governing behavior of the beam. The mode-shapes of the normalized and linearized system are then found analytically and verified via Abaqus simulations. Using a single mode approximation, the first mode-shape of the system is used along with the Lagrange equations to find governing ordinary differential equations of degree of freedom and degree of constraint dynamic. These equations are then solved numerically using MATLAB. The Discrete Fourier Transform of dynamic responses show that the degree of freedom dynamic contains a single dominant frequency, while the constraint dynamic contains three main harmonics. It is observed that dominant frequencies are essentially natural frequencies of the linearized system which are available in a closed form. The suggested analytical formulations as well as the proposed frequency analysis, is expected to provide an effective approach for analytical dynamic modeling of more complex compliant mechanisms.