Chapter 8.4, Problem 31E

Aging Spring without Damping. In the mass-spring system for an aging spring discussed in Problem 30, assume that there is no damping (i.e., $b=0$), $m=1$, and $k=1$. To see the effect of aging, consider $\eta$ as a positive parameter.

a. Redo Problem 30 with $b=0$ and $\eta$ arbitrary but fixed.

b. Set $\eta =0$ in the expansion obtained in part (a). Does this expansion agree with the expansion for the solution to the problem with $\eta =0$? [Hint: When $\eta =0$, the solution is $x\left(t\right)=\cos t$.]

30. Aging Spring. As a spring ages, its spring “constant” decreases in value. One such model for a mass-spring system with an aging spring is

$m{x}^{″}\left(t\right)+b{x}^{\prime }\left(t\right)+k{e}^{-\eta t}x\left(t\right)=0$,

where $m$ is the mass, $b$ the damping constant, $k$ and $\eta$ positive constants, and $x\left(t\right)$ the displacement of the spring from its equilibrium position. Let $m=1\text{kg}$, $b=2\text{N-}\text{sec}/\text{m}$, $k=1\text{N}/\text{m}$, and $\eta =1{\left(\sec \right)}^{-1}$. The system is set in motion by displacing the mass $1\text{m}$ from its equilibrium position and then releasing it $\left(x\left(0\right)=1,x^{\prime }\left(0\right)=0\right)$. Find at least the first four nonzero terms in a power series expansion about $t=0$ for the displacement.