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Mass-Spring Oscillator Analysis, Continued

We can quickly verify that

$\displaystyle \zbox{e^{-at}u(t) \longleftrightarrow \frac{1}{s+a}}
$

where $ u(t)$ is the Heaviside unit step function which steps from 0 to 1 at time 0 .

By linearity, the solution for the motion of the mass is

\begin{eqnarray*}
x(t) &=& re^{-j{\omega_0}t} + \overline{r}e^{j{\omega_0}t}
= 2\mbox{re}\left\{re^{-j{\omega_0}t}\right\}
= 2R_r\cos({\omega_0}t - \theta_r)\\
&=& \frac{\sqrt{v^2_0 + {\omega_0}^2 x^2_0}}{{\omega_0}}
\cos\left[{\omega_0}t - \tan^{-1}\left(\frac{v_0}{{\omega_0}x_0}\right)\right]
\end{eqnarray*}

If the initial velocity is zero ($ v_0=0$ ), the above formula reduces to $ x(t) = x_0\cos({\omega_0}t)$ and the mass simply oscillates sinusoidally at frequency $ {\omega_0}=
\sqrt{k/m}$ , starting from its initial position $ x_0$ . If instead the initial position is $ x_0=0$ , we obtain

\begin{eqnarray*}
x(t) &=& \frac{v_0}{{\omega_0}}\sin({\omega_0}t)\\
\;\Rightarrow\; v(t) &=& v_0\cos({\omega_0}t).
\end{eqnarray*}


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Download Laplace.pdf
Download Laplace_2up.pdf
Download Laplace_4up.pdf

``The Laplace Transform'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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