Time-varying frequency formula for the purely nonlinear damped oscillator

The frequency of the purely nonlinear and nonconservative oscillator is a time-varying quantity due to the presence of damping. For the nonlinear oscillator addressed here, only cubic-power stiffness nonlinearity is considered. The nonlinear frequency of the conservative nonlinear oscillator is dependent on the initial energy induced into the system. However, for the non-conservative and purely nonlinear oscillator, the instantaneous frequency is dependent on the instantaneous energy of the system. Consequently, the exact amplitude decay formula obtained in a recent publication for such oscillator is accurately applied here to obtain an accurate analytical formula for the time-varying frequency of the considered system. Excellent agreement between the results obtained by the new time-varying frequency formula presented here and both numerical simulation and wavelet transform has been clearly observed. This analytical formula is found to be accurate in identifying the instantaneous frequency change of the system regardless of its physical parameters and the initial input energies.