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 [419], 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 [308,549], or the internal friction losses in a vibrating string [73].


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Memoryless Nonlinearities