Chapter 4.1, Problem 19P

Duffing’s Equation

(a) For the spring-mass system shown in Figure $\mathrm{4.1.10}$ derive the differential equation for the displacement $x(t)$ of the mass from its equilibrium position using the force law $F_s\left(\Delta x\right)=-k\Delta x-\epsilon {\left(\Delta x\right)}^3$, where $\Delta x$ represent the change in length of the spring from its natural length. Assume that damping forces and external forces are not present.

(b) Find the linearized equation for the differential equationderived in part (a) under the assumption that, in its dynamicstate, the maximum displacement of the mass from its equilibrium position is small.