Harmonic Oscillator with Variable Damping Coefficient

Damped harmonic oscillator with damping coefficient as a linear function of position coordinate is represented by nonlinear ordinary differential equation. Solution of this equation is studied using both two-timing perturbation method and numerical method. Both solutions match exactly eventhough the perturbation term is not infinitesimally small in the underdamped case. The rate of variation of energy is found to be very slowly when damping coefficient varies as a linear function position coordinate as compared with constant damping case. There exist only one critical point, the origin, which is asymptotically stable in the underdamped situation. In the critically damped and overdamped conditions the critical point behaves like stable node.