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Force Driving a Spring against a Wall

For this example, we have an external force $ f(n)$ driving a spring $ k$ which is terminated on the other end at a rigid wall. Figure F.16 shows the physical diagram and the electrical equivalent circuit is given in Fig.F.17.

Figure F.16: External force driving a spring terminated by a rigid wall.
\includegraphics{eps/forcespring}

Figure: Electrical equivalent circuit of the compressed spring of Fig.F.16.
\includegraphics{eps/forcespringec}

Figure F.18 depicts the insertion of an infinitesimal transmission line, and Fig.F.19 shows the result of converting the spring impedance to wave variable form.

Figure F.18: Intermediate equivalent circuit for the force-driven spring in which an infinitesimal transmission line section has been inserted to facilitate conversion of the spring impedance $ k/s$ into a wave-variable reflectance.
\includegraphics{eps/forcespringscat}

Figure: Intermediate wave-variable model of Fig.F.17.
\includegraphics{eps/forcespringdt}

The two-port adaptor needed for this problem is the same as that for the force-driven mass, and the final result is shown in Fig.F.20.

Figure F.20: Wave digital spring driven by external force $ f(n)$ .
\includegraphics{eps/forcespringwdf}

Note that the spring model is being driven by a force from a zero source impedance, in contrast with the infinite source impedance interpretation of Fig.F.8b as a compressed spring. In this case, if the driving force goes to zero, the spring force goes immediately to zero (``free termination'') rather than remaining fixed.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA