Numerical Solutions for Nonlinear Vibrations of the Mathematical Pendulum Based on a Modified Frequency Formulation: Location Point Configurations

Abstract

This paper aims to obtain accurate and efficient numerical solutions for the nonlinear vibrations of a mathematical pendulum using a modified frequency formulation, with a focus on the optimal design of location point configurations. The nonlinear motion equation of the pendulum is transformed to match the framework of the modified frequency formulation, and detailed derivations are carried out for three distinct location point schemes. A systematic comparison is performed between the proposed method and conventional analytical approaches (including the homotopy perturbation method (HPM), the modified homotopy perturbation method (MHPM), and the harmonic balance method (HBM)), with the fourth-order Runge-Kutta (RK4) solution serving as the benchmark. The results show that the modified frequency formulation, especially with the optimized location point configuration, achieves higher calculation accuracy and more stable error growth under large-amplitude, strongly nonlinear vibration conditions. The proposed method can effectively capture the dynamic characteristics of the mathematical pendulum and shows promising application potential in Micro-Electro-Mechanical Systems (MEMS) dynamics and related engineering fields.