Analytical formulas for the energy, velocity and displacement decays of purely nonlinear damped oscillators

Accurate formulas for obtaining the energy, velocity and displacement decay envelopes in purely nonlinear damped oscillators are presented here. These purely nonlinear oscillators have no linear stiffness components, and the new formulas derived in this paper are found to be highly accurate in determining the decay envelopes, regardless of the system's physical parameters and its initial condition. The mechanism of these new formulas for obtaining the decay envelopes of the nonlinear oscillators is found to be exactly the same as the mechanism of the well-known amplitude decay formulas used for the linear oscillator. Finding such formulas is significant to many scientific applications of these nonlinear oscillators, such as the passive targeted energy transfer through the stiffness-based nonlinear energy sinks (NESs). In these types of NESs, these formulas can be easily applied for system identification and dynamic analysis. In addition, they are expected to have a significant application in different scientific fields for such oscillators.