A Comparative Study of the Pendulum Equation Using Two Analytical Methods

This paper presents a comprehensive comparison between the modified harmonic balance method (MHBM) and He’s frequency formulation (HFF) for solving the nonlinear dynamics of an excited pendulum constrained by a crank-shaft-slider mechanism (CSSM). The pendulum’s motion, influenced by the CSSM, introduces strong nonlinearity, making the analysis challenging using conventional methods. The harmonic balance method (HBM), a widely used technique for approximating solutions to nonlinear differential equations, often becomes cumbersome due to its complex nature, particularly when higher-order approximations are necessary. This complexity leads to the emergence of a set of intricate nonlinear complex equations that are not easy to solve. To overcome the limitations of HBM, a modified form of the HBM is adopted in this study. The performance of the MHBM is thoroughly evaluated by comparing the results with those obtained using HFF. Additionally, the numerical solutions are obtained using the Runge–Kutta fourth-order method, serving as a benchmark to assess the accuracy of both analytical approaches. Remarkably, the first approximation provided by the MHBM exhibits superior accuracy compared to the solutions obtained from the HFF method and other existing analytical methods. The results highlight the potential of MHBM as a reliable and accurate method for solving nonlinear differential equations in various engineering applications, particularly in systems exhibiting strong nonlinearity.