On the truly nonlinear oscillator with positive and negative damping

• Approximate solution for the oscillator with rational order nonlinearity and viscous damping is presented.
• Procedure for solving a second order strong nonlinear differential equation based on the Ateb function is developed.
• Novel results on the limit cycle motion of the damped truly nonlinear-van der Pol oscillator are obtained.

In this paper the oscillator with a restoring force of rational order of nonlinearity (named truly nonlinear oscillators) and damping (positive and negative) is considered. The mathematical model of the oscillator is a second order differential equation with nonlinear terms of integer or noninteger and also linear and nonlinear damping terms. The approximate solution of the generating differential equation (only the nonlinear deflection and linear viscous damping terms exist) in the form of the Ateb function is obtained using the harmonic balance method. Based on the generating solution the vibrations of the oscillator with damping are obtained by extending the method of time variable amplitude, frequency and phase. The special attention is given to the truly nonlinear damped-van der Pol oscillator. The interaction of the viscous and van der Pol damping on the motion of the truly nonlinear oscillator is investigated. The boundary for the limit cycle motion depending on the order of nonlinearity is analyzed. The interactive influence of the damping coefficients (positive and negative) and the order of nonlinearity on the frequency and the period of the vibration is also analyzed. Two numerical examples are considered. The approximate analytical solutions are compared with numerically calculated ones. The obtained results are in a good agreement.