A method to solve the nonlinear eigenvalue problem of Timoshenko plane frames with rigid offsets and end releases

The study reported here makes use of the dynamic stiffness matrix, DSM, for Timoshenko beams, hence considering rotatory inertia and shear deflection, to obtain the exact solution of the associated free vibration problem. The natural frequencies and vibration modes in the presence of end releases and rigid offsets are all exactly obtained for different combinations of boundary conditions and in plane arrangements of members. This is possible by solving the resulted nonlinear and transcendental eigenvalue problem using two general methods, one based on the Wittrick–Williams algorithm adapted to handle end release and rigid offset, and the other on the power deflation secant method. The latter, offered here as a novelty, is proven to be an efficient technique judging from the examples given, using both algorithms and the finite element method. It is concluded that the power deflation method is a powerful tool to solve the nonlinear eigenvalue problem, leading to exact solutions of the free vibration of complex skeleton-like structures.