This study presents a numerical and analytical investigation of the nonlinear smooth-discontinuous (SD) oscillator. The nonlinear restoring force is approximated by a fifth-order polynomial using a Taylor series expansion to simplify the governing equation while preserving the essential nonlinear characteristics of the system. To analyze the oscillator dynamics, two analytical approaches are applied, namely the spreading residue harmonic balance method (SRHPM) and the Multiple Scales Method (MSM). The obtained analytical solutions are validated through comparison with numerical simulations carried out using the classical fourth-order Runge–Kutta scheme. The results reveal a strong agreement between the analytical and numerical solutions, confirming the capability of both SRHPM and MSM to accurately describe the nonlinear oscillatory response of the SD system.
Alluhydan et al. (Thu,) studied this question.