Lagrange Multiplier Method in Dynamic Finite Element Analysis of Frames with Nonlinear Multipoint Constraints Under Harmonic Loading

This research introduces the approach to finding the solution to the dynamic problem of the frames, loaded harmonic variation force, with nonlinear dependent multipoint boundary conditions using FEM. The establishing problem-solving model of frame, having a nonlinear functional equation of constraints, under variation force, requires mathematical technique to incorporate the nonlinear equations of constraints into the master dynamic equilibrium equation. For constructing a frame system's nonlinear FEM dynamic equation with nonlinear multipoint constraints, the imposition process is developed by employing the method of Lagrange multipliers. Using the method of Lagrange multipliers for converting a nonlinearly constrained problem, the nonlinear dynamic finite element system of the equation has been constructed. Solving technique for the FEM nonlinear dynamic equations of frames under harmonic excitation is constructed utilizing the Newmark time-integration technique combining the Newton–Raphson technique. According to the established incremental–iterative algorithm, the computing program has been written for solving the investigated problem. The testing results show the advanced possibility of the constructed algorithm in the analysis of the vibration of frames with a nonlinear functional equation of boundary conditions under a time-dependent load.