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Dashpot

The elementary impedance element in mechanics is the dashpot which may be approximated mechanically by a plunger in a cylinder of air or liquid, analogous to a shock absorber for a car. A constant impedance means that the velocity produced is always linearly proportional to the force applied, or $ f(t) = \mu v(t)$ , where $ \mu $ is the dashpot impedance, $ f(t)$ is the applied force at time $ t$ , and $ v(t)$ is the velocity. A diagram is shown in Fig. 7.1.

Figure 7.1: The ideal dashpot characterized by a constant impedance $ \mu $ . For all applied forces $ f(t)$ , the resulting velocity $ v(t)$ obeys $ f(t) = \mu v(t)$ .
\includegraphics[scale=0.9]{eps/ldashpot}

In circuit theory, the element analogous to the dashpot is the resistor $ R$ , characterized by $ v(t) = R i(t)$ , where $ v$ is voltage and $ i$ is current. In an analog equivalent circuit, a dashpot can be represented using a resistor $ R = \mu$ .

Over a specific velocity range, friction force can also be characterized by the relation $ f(t) = \mu v(t)$ . However, friction is very complicated in general [423], and as the velocity goes to zero, the coefficient of friction $ \mu $ may become much larger. The simple model often presented is to use a static coefficient of friction when starting at rest ($ v(t)=0$ ) and a dynamic coefficient of friction when in motion ( $ v(t)\neq 0$ ). However, these models are too simplified for many practical situations in musical acoustics, e.g., the frictional force between the bow and string of a violin [311,551], or the internal friction losses in a vibrating string [73].


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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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