Stability of the inverted pendulum system: An optimized equivalent linearization approach

Abstract

A unified framework for analyzing nonlinear vibrations in a two-degree-of-freedom (2DOF) system comprising an inverted pendulum (IP) mounted on a cart is proposed. Unlike existing approaches that rely on direct numerical integration or restrictive small-angle assumptions, this work introduces a novel combination of the Optimized Equivalent Linearization Method (OELM) and El-Dib’s frequency formula to systematically transform the non-autonomous governing equations into an autonomous form via an equivalent periodic forcing function, a treatment not previously applied to this class of coupled nonlinear oscillators. The linearized equations are examined under both common-frequency and two-frequency response regimes, yielding closed-form stability criteria that show strong agreement with full numerical solutions. Key factors governing stability are the applied force amplitude, vibration amplitude, and friction coefficient; increasing friction promotes instability while higher vibration amplitudes enhance dynamic stability, a counterintuitive finding with direct design implications. The framework is validated through comprehensive time-history analysis, demonstrating minimal errors and sustained periodic behavior in the absence of chaotic motion. The results apply to a broad class of engineering systems. The inverted pendulum on a cart serves as a canonical model for self-balancing robots, Segway-type vehicles, and bipedal locomotion systems, where maintaining upright stability under external excitation is critical. The findings also inform the design of vibration isolation platforms, active suspension systems, and offshore structures subjected to wave-induced loading, where controlled nonlinear oscillations must be managed to prevent structural failure. The stability maps generated by this framework offer practical design guidelines for tuning damping and forcing parameters in such systems. The study thereby broadens the analytical treatment of nonlinear oscillatory systems and provides concrete guidance for engineers in robotics, structural dynamics, and mechanical control.