Chapter 4.4, Problem 27P

Use the differential equation derived in Problem 19 of section 4.1 to determine the period of vertical oscillations of a cubic block floating in a fluid under the stated conditions.

Problem 19, Section 4.1

Duffing’s Equation

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.

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.