Exact dynamic stiffness method for dynamic analysis of complicated damped plane beam structures with generalized boundary conditions

Closed-form dynamic stiffness (DS) formulations are developed for exact dynamic analysis of complicated damped beam structures with internal hinges and generalized boundary conditions (BCs). The damping effects are considered by introducing both the strain rate dependent viscous and the velocity-dependent viscous damping factors. A range of theories are adopted for the beam element, that is, the classical and Rayleigh-Love theories for the axial vibration and the Euler–Bernoulli and Timoshenko theories for the bending vibration. Based on the governing differential equations considering the damping effect, the elemental DS matrices are derived by substituting the exact shape functions into the generalized displacement and force BCs. Then, a hinge joint for a beam element is simulated by implementing an end release. Next, assembling the elemental DS matrices and introducing generalized BCs leads to the final DS matrix used for dynamic analysis of beam structures. The complex transcendental frequency equation of the beam structures considered herein is solved by a global complex roots and poles finding algorithm with a self-adaptive mesh generator. The proposed method is verified against the finite element method. The results obtained by using different theories are also compared. This research provides an exact tool for dynamic analysis of complicated damped beam structures.