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Spring-Mass System

When two physical elements are driven by a common force (yet have independent velocities, as we'll soon see is quite possible), they are formally in parallel. An example is a mass connected to a spring in which the driving force is applied to one end of the spring, and the mass is attached to the other end, as shown in Fig.7.11. The compression force on the spring is equal at all times to the rightward force on the mass. However, the spring compression velocity $ v_k(t)$ does not always equal the mass velocity $ v_m(t)$ . We do have that the sum of the mass velocity and spring compression velocity gives the velocity of the driving point, i.e., $ v(t)=v_m(t)+v_k(t)$ . Thus, in a parallel connection, forces are equal and velocities sum.

Figure 7.11: A mass and spring combined as one-ports in parallel.

Figure 7.12 shows the electrical equivalent circuit corresponding to Fig.7.11.

Figure: Electrical equivalent circuit of the parallel mass-spring combination driven by an external force, as diagrammed in Fig.7.11.

Figure: Impedance diagram for the force-driven, parallel mass-spring arrangement shown in Fig.7.11.

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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