This paper focuses on the dynamic finite element analysis of the frames having nonlinear multi-freedom boundary constraints under harmonic loading. The establishing solving equation of the frame system with nonlinear multi-freedom constraints requires mathematical technique to incorporate the nonlinear multi-freedom constraints into the master stiffness matrix of the frame for dynamic analysis. This work proposes a way to employ the Lagrange multipliers method for constructing the nonlinear dynamic finite element equation of a frame system with nonlinear multi-freedom constraints. Using the Lagrange multipliers method to convert a constrained problem into an unconstrained problem, the nonlinear dynamic finite element system of equation is constructed. For solving the nonlinear dynamic system of equations of frames under harmonic excitation is employed combining the Newmark integration method and the Newton Raphson iteration method. The established incremental iterative algorithm based on combining these methods is used to write the calculation program for investigating the vibration of the plane frames with nonlinear multi-freedom constraints subjected to harmonic load. The numerical test results show the efficiency of the proposed algorithm in the dynamic analysis of frames with nonlinear multi-freedom constraints under harmonic excitation
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