Chapter 7.3, Problem 40P

Problems 39 and 40 illustrate Iwo types of resonance in a mass-spring-dashpot system with given external force F(t) and with the initial conditions $x\left(0\right)=x\text{'}\left(0\right)=0$ .

Suppose that m = 1, k = 9.04, c = 0.4, and $F\left(t\right)=6e^{-\frac{t}{5}}\cos 3t$. Derive the solution $x\left(t\right)=t{e}^{-\frac{t}{5}}\sin 3t$ .

Show that the maximum value of the amplitude function $A\left(t\right)=t{e}^{-\frac{t}{5}}$ is $A\left(5\right)=\frac{5}{e}$. Thus (as indicated in

Fig. 7.3.5) the oscillations of the mass increase in amplitude during the first s before being damped out as $t\to +\infty$