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 $f(t) = \mu v(t)$ .

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 [311,553], or the internal friction losses in a vibrating string [73].