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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.18 shows the physical diagram and the electrical equivalent circuit is given in Fig.F.19.

Figure F.18: External force driving a spring terminated by a rigid wall.

Figure F.19: Electrical equivalent circuit of the compressed spring of Fig.F.18.

Figure F.20 depicts the insertion of an infinitesimal transmission line, and Fig.F.21 shows the result of converting the spring impedance to wave variable form.

Figure F.20: 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.

Figure F.21: Intermediate wave-variable model of Fig.F.19.

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.22.

Figure F.22: Wave digital spring driven by external force $ f(n)$ .

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.10b 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 © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University