Analysis of Nonlinear Oscillation Models with External Forcing Using the Multiple Scales Method - Ayuni Kemala Safira; Aminatus Sa’adah; Rani Sulvianuri; Yoli Agnesia (Department of Mathematics, Universitas Negeri Padang (UNP), 2026)
Nonlinear effects accompanied by external forces can cause the behaviour of the system to become more complex and difficult to explain using linear analysis. Therefore, analytical methods are needed to obtain approximate solutions. This paper presents an analysis of approximate solutions to nonlinear oscillation models subject to periodic external forces. The analysis was conducted using the Multiple Scales Method, a perturbation technique for obtaining asymptotic solutions to nonlinear differential equations. This approach is carried out by introducing several time scales and developing solutions as series in ε. The differential equations that model the system are analysed to orders. and to obtain approximate solutions that describe the oscillation dynamics of the system. The analysis was performed under two main conditions: when the external force frequency approached the system's natural frequency (main resonance) and when the two were not close. In the non-resonance condition, several special cases were also examined: non-resonant, superharmonic resonance, subharmonic resonance, and low excitation frequency. The results show that first-order asymptotic solutions agree well with numerical solutions. The system response is influenced by parameters such as the amplitude and frequency of the external force, as well as the damping parameter. These findings support further research on more complex nonlinear systems and have practical applications in the design of vibration absorbers and rotating mechanical components to control resonance and improve system stability.