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Force-Driven Mass Analysis, Continued

If the applied external force $ f(t)$ is zero, we obtain

$\displaystyle X(s)
= \frac{x(0)}{s} + \frac{{\dot x}(0)}{s^2}
= \frac{x(0)}{s} + \frac{v(0)}{s^2}.
$

Since $ 1/s$ is the Laplace transform of the Heaviside unit-step function

$\displaystyle u(t)\mathrel{\stackrel{\mathrm{\Delta}}{=}}\left\{\begin{array}{ll}
0, & t<0 \\ [5pt]
1, & t\ge 0 \\
\end{array} \right.,
$

we find that the position of the mass $ x(t)$ is given for all time by

$\displaystyle x(t) = x(0)\,u(t) + v(0)\,t\,u(t).
$

In summary, we used the Laplace transform to solve for the motion of a simple physical system (an ideal mass) in response to initial conditions (no external driving forces).


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Download Laplace.pdf
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``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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