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Spring and Free Mass

For this example, we have an external force $ f(n)$ driving a spring $ k$ which in turn drives a free mass $ m$ . Since the force on the spring and the mass are always the same, they are formally ``parallel'' impedances.

This problem is easier than it may first appear since an ideal ``force source'' (i.e., one with a zero source impedance) driving impedances in parallel can be analyzed separately for each parallel branch. That is, the system is equivalent to one in which the mass and spring are not connected at all, and each has its own copy of the force source. With this insight in mind, one can immediately write down the final wave-digital model shown in Fig.F.25. However, we will go ahead and analyze this case more formally since it has some interesting features.

Figure F.21 shows the physical diagram of the spring-mass system driven by an external force. The electrical equivalent circuit appears in Fig.F.22, and the first stage of a wave-variable conversion is shown in Fig.F.23.

Figure F.21: External force driving a spring which in turn drives a free mass sliding on a frictionless surface.
\includegraphics{eps/springmass}

Figure: Electrical equivalent circuit of the spring/mass system of Fig.F.21.
\includegraphics{eps/springmassec}

Figure: Intermediate wave-variable model of the mass and dashpot of Fig.F.22.
\includegraphics{eps/springmassdt}

Figure F.24: Wave digital model for the parallel combination of a wave digital mass $ m$ , wave digital spring $ k$ , and driving force $ f(n)$ . The corresponding port impedances are $ m$ , $ k$ , and 0 , respectively.
\includegraphics{eps/springmassjunc}

For this example we need a three-port parallel adaptor, as shown in Fig.F.24 (along with its attached mass and spring). The port impedances are 0 , $ k$ , and $ m$ , yielding alpha parameters $ \alpha_1 =2$ and $ \alpha_2=\alpha_3=0$ . The final result, after the same sorts of elementary simplifications as in the previous example, is shown in Fig.F.25. As predicted, a force source driving elements in parallel is equivalent to a set of independent force-driven elements.

Figure F.25: Wave digital filter model of an external force driving a mass through a spring. The mass force-wave components are denoted $ f^\pm _m$ , while those for the spring are denoted $ f^\pm _k$ .
\includegraphics{eps/wdspringmass}

From this and the preceding example, we can see a general pattern: Whenever there is an ideal force source driving a parallel junction, then $ \Gamma _1=\infty$ and all other port admittances are finite. In this case, we always obtain $ \alpha_1 =2$ and $ \alpha_i=0$ , $ i\neq 1$ .


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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
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