Dynamic instability analysis of high-speed flexible four-bar mechanisms

This paper presents an efficient and accurate procedure for determining values of critical running speeds that cause a high-speed flexible mechanism to become dynamically unstable because of parametric resonance. The finite element method is employed to model each unconstrained flexible link for axial and lateral deformations. The global equations of motion for the entire mechanism are formulated using the augmented Lagrange equations. To solve the equations of motion having periodically varying mass, gyroscopic and stiffness matrices, eigenanalysis of the mechanism is first performed at each fast Fourier transform configuration. Harmonic compositions of all natural frequencies and their corresponding modal vectors are then obtained. With the help of the modal superposition method, the equations of motion of large dimensions in terms of nodal variables are reduced to much smaller dimensions in terms of modal variables. The harmonic balancing method is subsequently used to obtain a periodic solution to the reduced equations of motion, and to transform the dynamic stability problem into a standard eigenvalue problem for which values of all critical running speeds may be readily computed.