Q33E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Vibrating Spring with Damping. Using the model for a vibrating spring with damping discussed in Example$\mathbf{3}$

$\left(a\right)$ Find the equation of motion for the vibrating spring with damping if  $\mathbf{m}\mathbf{=}\mathbf{10}\mathbf{ }\mathbf{kg}\mathbf{,}\mathbf{b}\mathbf{=}\mathbf{60}\mathbf{ }\mathbf{kg}\mathbf{/}\mathbf{sec}\mathbf{,}\mathbf{k}\mathbf{=}\mathbf{250}\mathbf{ }\mathbf{kg}\mathbf{/}{\mathbf{sec}}^2\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{ }\mathbf{m}\mathbf{,}$and  $\mathbf{y}\text{'}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{sec}$.

$\left(b\right)$ After how many seconds will the mass in part $\left(a\right)$ first cross the equilibrium point?

$\left(c\right)$ Find the frequency of oscillation for the spring system of part $\left(a\right)$. 

 $\left(d\right)$Compare the results of problems $\mathbf{32}$ and  $\mathbf{33}$ determine what effect the damping has on the frequency of oscillation. What other effects does it have on the solution?

Step-by-Step Solution

Answer

1.  The equation of motion for the vibrating spring with damping is: $\mathrm{y}\left(\mathrm{t}\right)={\mathrm{e}}^{-3\mathrm{t}}(0.3\cos \left(4\mathrm{t}\right)+0.2\sin \left(4\mathrm{t}\right))$
2.  The mass crosses the equilibrium at $\mathrm{t}=14$ seconds
3.  The frequency of the spring is  $\mathrm{f}=\frac{2}{\mathrm{\pi }}$ .
4.  If the damping increases amplitude and frequency decrease.

Step 1: Find the value of c 1 and c 2

From example $3$,

$\begin{array}{l} \mathrm{y}\text{'}\left(\mathrm{t}\right)={\mathrm{e}}^{-3\mathrm{t}}(-4{\mathrm{c}}_1\sin \left(4\mathrm{t}\right)+4{\mathrm{c}}_2\cos \left(4\mathrm{t}\right))-3{\mathrm{e}}^{-3\mathrm{t}}\left({\mathrm{c}}_1\cos \left(4\mathrm{t}\right)+{\mathrm{c}}_2\sin \left(4\mathrm{t}\right)\right)\\ \mathrm{y}\text{'}\left(0\right)={\mathrm{e}}^0(-4{\mathrm{c}}_1\sin \left(0\right)+4{\mathrm{c}}_2\cos \left(0\right))-3{\mathrm{e}}^0\left({\mathrm{c}}_1\cos \left(0\right)+{\mathrm{c}}_2\sin \left(0\right)\right)\\ 4{\mathrm{c}}_2-3(0.3)=-0.1\\ 4{\mathrm{c}}_2=-0.1+0.9\\ 4{\mathrm{c}}_2=0.8\\ {\mathrm{c}}_2=0.2\end{array}$

Therefore, the solution is  $\mathrm{y}\left(\mathrm{t}\right)={\mathrm{e}}^{-3\mathrm{t}}(0.3\cos \left(4\mathrm{t}\right)+0.2\sin \left(4\mathrm{t}\right))$.

Step 2: Find the value of time.

When the spring crosses the equilibrium $\mathbf{y}\left(\mathbf{t}\right)\mathbf{=}\mathbf{0}$ , so we have to find the $\mathbf{t}$

$\begin{array}{l}0.3\cos \left(4\mathrm{t}\right)+0.2\sin \left(4\mathrm{t}\right)=0\\ 0.3\cos \left(4\mathrm{t}\right)=-0.2\sin \left(4\mathrm{t}\right)\\ \tan \left(4\mathrm{t}\right)=-\frac{0.3}{0.2}\\ 4\mathrm{t}=-\arctan (1.5)\\ \mathrm{t}\approx \frac{}{}\\ \mathrm{t}\approx 14\end{array}$

So, at $\mathrm{t}=14$ seconds the mass crosses the equilibrium

Step 3: Find the value of frequency.

Here $\mathrm{\beta }=4$ , the frequency of the spring is $\mathbf{f}\mathbf{=}\frac{4}{2\pi }\mathbf{=}\frac{2}{\pi }$ .

Step 4: Comparing the problems 32 and 33

$\left(\mathrm{d}\right).$If the damping increases amplitude and frequency decrease.