A New Approach for Solving Nonlinear Oscillations with an m-Degree Odd-Power Polynomial Restoring Force

• DOI: 10.3390/dynamics5010009
• Publication Date: 2025-03-05T10:54:38Z
• Authors: Stylianos Vasilei...; Gamal M. Ismail; Vassilis Alimisis; Christos Dimas; Anna Malamou
• Keywords: Restoring force; Degree (music)

Solving nonlinear oscillations is challenging, as solutions to the corresponding differential equations do not exist in most cases. Therefore, numerical methods are usually employed calculate precise oscillation frequency. In addition, many interesting mathematical approaches leading approximate have also been developed. This paper focuses on a classic case of oscillator: oscillator with an odd-power polynomial restoring force. encompasses nearly all scenarios undamped oscillations. The idea combine two well-known strategies from literature: He’s approximation, which simple apply and valid for small amplitudes, analytical power-law forces. It shown that by combining these approaches, universal equation accurate any amplitude derived. Many tests proposed method’s accuracy presented using polynomials various degrees examples, such rotating pendulum, cubic–quintic Duffing oscillators, oscillators cubic harmonic novel ‘electrical analogue’ polynomial-type force introduced demonstrate this can be applied real industrial applications.