Solution of Nonlinear Hardening and Softening type Oscillators by Adomian's Decomposition Method

A type of nonlinearity in vibrational engineering systems emerges when the restoring force is a nonlinear function of displacement. The derivative of this function is known as stiffness. If the stiffness increases by increasing the value of displacement from the equilibrium position, then the system is known as hardening type oscillator and if the stiffness decreases by increasing the value of displacement, then the system is known as softening type oscillator. The restoring force as a nonlinear polynomial function of order three, can describe a wide variety of practical nonlinear situations by proper choosing of constant multipliers. In this paper, a spring-mass system is considered by the restoring force of the introduced type. Choosing suitable values for a, b and n, a hardening and softening type oscillators are constructed and related equations of motion are introduced as second order nonlinear differential equations. The equations are solved directly, using the Adomian’s decomposition method (ADM). In another approach, the equations are converted to systems of first order differential equations and then solved using the same method. The results show that the ADM gives accurate results in both approaches, beside it shows that converting the equation to a system of equations of lower order, tends to more accurate solutions when ADM applies.