A Multi-Variable Collaboration Method for Free Vibration of Timoshenko Beams in the Hamiltonian System

A Hamiltonian system-based new polynomial analysis method is proposed to study the free vibration characteristics of Timoshenko beams. It first introduces dual variables into the Hamiltonian system by using the motion equation of Timoshenko beams, and transforms the solution system from one class of variables to two classes of variables, namely, deflection, rotation angle, shear force, and bending moment. Subsequently, four variables are expressed in polynomial series and then substituted into the Lagrange equation, thus the problem of solving the natural frequency is transformed into solving the matrix-generalized eigenvalue by combining with the Rayleigh–Ritz method. The space coordinate is simulated as the time coordinate in traditional Lagrangian and Hamiltonian system, and the potential energy is expressed by dual variables by transformation through the Hamiltonian function. The method is also applied to Euler–Bernoulli beam and the frame structure, and the natural frequencies are obtained for various boundary conditions. The impact of variables with different polynomial terms on the natural frequencies is examined. Finally, the numerical results are compared by the analytical solutions, and some computational examples are carried out to validate the proposed multi-variable collaboration method. It is confirmed that the present calculation accuracy can be effectively improved by adjusting the number of terms for each variable. By means of the proposed method, it is convenient to obtain the analytical expressions for deflection, rotation angle, shear force, and bending moment. Therefore, it is feasible to structural modal analysis and internal force calculation.