Oscillators with nonlinear elastic and damping forces

Generalization to the previous oscillators is done by introducing the nonlinear elastic and damping forces. The mathematical model of the system is a second order differential equation with nonlinear elastic and damping terms whose order is integer and/or noninteger. Cveticanin’s solving procedure is extended for solving such a strong nonlinear differential equation. The approximate solution obtained is a function of initial amplitude and initial phase. A damping coefficient and order of damping interaction with an elastic coefficient and order of elasticity for the generalized oscillators are also determined. Special attention is paid to obtain the relation between initial amplitude and phase, on the one hand, and initial displacement and velocity on the other hand. Correction to the frequency of vibration for the linear oscillators with nonlinear damping and for the pure nonlinear oscillators with linear damping is obtained and analyzed. Analytical results given in this paper are compared with numerically obtained ones and show a good agreement.