Impedance

Impedance is defined for mechanical systems as force divided by velocity, while the inverse (velocity/force) is called an admittance. For dynamic systems, the impedance of a ``driving point'' is defined for each frequency $ \omega $, so that the ``force'' in the definition of impedance is best thought of as the peak amplitude of a sinusoidal applied force, and similarly for the velocity. Thus, if $ F(\omega)$ denotes the Fourier transform of the applied force at a driving point, and $ V(\omega)$ is the Fourier transform of the resulting velocity of the driving point, then the driving-point impedance is given by

$\displaystyle R(\omega) \isdef \frac{F(\omega)}{V(\omega)}.
$

In the lossless case (no dashpots, only masses and springs), all driving-point impedances are purely imaginary, and a purely imaginary impedance is called a reactance. A purely imaginary admittance is called a susceptance. The term immittance refers to either an impedance or an admittance [35]. In mechanics, force is typically in units of newtons (kilograms times meters per second squared) and velocity is in meters per second.

In acoustics [317,318], force takes the form of pressure (e.g., in physical units of newtons per meter squared), and velocity may be either particle velocity in open air (meters per second) or volume velocity in acoustic tubes (meters cubed per second) (see §B.7.1 for definitions). The wave impedance (also called the characteristic impedance) in open air is the ratio of pressure to particle velocity in a sound wave traveling through air, and it is given by $ R=
\sqrt{\gamma P_0\rho } = \rho c$, where $ \rho$ is the density (mass per unit volume) of air, $ c$ is the speed of sound propagation, $ P_0$ is ambient pressure, and $ \gamma_c = 1.4$ is the ratio of the specific heat of air at constant pressure to that at constant volume. In a vibrating string, the wave impedance is given by $ R= \sqrt{K\epsilon }=\epsilon
c$, where $ \epsilon $ is string density (mass per unit length) and $ K$ is the tension of the string (stretching force), as discussed further in §C.1 and §B.5.2.

In circuit theory [110], force takes the form of electric potential in volts, and velocity manifests as electric current in amperes (coulombs per second). In an electric transmission line, the characteristic impedance is given by $ R=\sqrt{L/C}=Lc$ where $ L$ and $ C$ are the inductance and capacitance, respectively, per unit length along the transmission line. In free space, the wave impedance for light is $ R=\sqrt{\mu_0/\epsilon_0} =\mu_0 c$, where $ \mu_0$ and $ \epsilon_0$ are the permeability and permittivity, respectively, of free space. One might be led from this to believe that there must exist a medium, or `ether', which sustains wave propagation in free space; however, this is one instance in which ``obvious'' predictions from theory turn out to be wrong.

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].


Ideal Mass

Figure: The ideal mass characterized by $ f(t) = m \protect\dot v(t) = m{\ddot x}(t)$.
\includegraphics[scale=0.9]{eps/lmass}

The concept of impedance extends also to masses and springs. Figure 7.2 illustrates an ideal mass of $ m$ kilograms sliding on a frictionless surface. From Newton's second law of motion, we know force equals mass times acceleration, or

$\displaystyle f(t) = m a(t) \isdef m \dot v(t) \isdef m \ddot x(t).
$

Since impedance is defined in terms of force and velocity, we will prefer the form $ f(t) = m \dot v(t)$. By the differentiation theorem for Laplace transforms [284],8.1we have

$\displaystyle F(s) = m [s V(s) - v(0)].
$

If we assume the initial velocity of the mass is zero, we have

$\displaystyle F(s) = m s V(s),
$

and the impedance $ F(s)/V(s)$ of the mass in the frequency domain is simply

$\displaystyle R_m(s) \isdef m s.
$

The admittance of a mass $ m$ is therefore

$\displaystyle \Gamma_m(s) \isdef \frac{1}{ms}
$

This is the transfer function of an integrator. Thus, an ideal mass integrates the applied force (divided by $ m$) to produce the output velocity. This is just a ``linear systems'' way of saying force equals mass times acceleration.

Since we normally think of an applied force as an input and the resulting velocity as an output, the corresponding transfer function is $ H(s) = \Gamma(s) = V(s)/F(s)$. The system diagram for this view is shown in Fig. 7.3.

The impulse response of a mass, for a force input and velocity output, is defined as the inverse Laplace transform of the transfer function:

$\displaystyle \gamma_m(t) \isdef {\cal L}^{-1}\left\{\Gamma_m(s)\right\} = \frac{1}{m}u(t)
$

In this instance, setting the input to $ \delta(t)$ corresponds to transferring a unit momentum to the mass at time 0. (Recall that momentum is the integral of force with respect to time.) Since momentum is also equal to mass $ m$ times its velocity $ v(t)$, it is clear that the unit-momentum velocity must be $ v(t)=1/m$.

Figure 7.3: Input/output description of a general impedance, with force $ F(s)$ as the input, velocity $ V(s)$ as the output, and admittance $ \Gamma (s)$ as the transfer function.
\includegraphics[scale=0.9]{eps/lblackbox}

Once the input and output signal are defined, a transfer function is defined, and therefore a frequency response is defined [485]. The frequency response is given by the transfer function evaluated on the $ j\omega $ axis in the $ s$ plane, i.e., for $ s=j\omega$. For the ideal mass, the force-to-velocity frequency response is

$\displaystyle \Gamma_m(j\omega) = \frac{1}{m j\omega}
$

Again, this is just the frequency response of an integrator, and we can say that the amplitude response rolls off $ -6$ dB per octave, and the phase shift is $ -\pi /2$ radians at all frequencies.

In circuit theory, the element analogous to the mass is the inductor, characterized by $ v(t) = L di/dt$, or $ V(s) = Ls I(s)$. In an analog equivalent circuit, a mass can be represented using an inductor with value $ L=m$.


Ideal Spring

Figure 7.4 depicts the ideal spring.

Figure 7.4: The ideal spring characterized by $ f(t) = k x(t)$.
\includegraphics[width=3in]{eps/lspring}

From Hooke's law, we have that the applied force is proportional to the displacement of the spring:

$\displaystyle f(t) \eqsp k\, x(t) \isdefs k \int_0^t v(\tau)d\tau
$

where it is assumed that $ x(0) = 0$. The spring constant $ k$ is sometimes called the stiffness of the spring. Taking the Laplace transform gives

$\displaystyle F(s) = k V(s)/s
$

so that the impedance of a spring is

$\displaystyle R_k(s) \isdef \frac{k}{s}
$

and the admittance is

$\displaystyle \Gamma_k(s) \isdef \frac{s}{k}
$

This is the transfer function of a differentiator. We can say that the ideal spring differentiates the applied force (divided by $ k$) to produce the output velocity.

The frequency response of the ideal spring, given the applied force as input and resulting velocity as output, is

$\displaystyle \Gamma_k(j\omega) = \frac{j\omega}{k}
$

In this case, the amplitude response grows $ +6$ dB per octave, and the phase shift is $ +\pi /2$ radians for all $ \omega $. Clearly, there is no such thing as an ideal spring which can produce arbitrarily large gain as frequency goes to infinity; there is always some mass in a real spring.

We call $ v(t)$ the compression velocity of the spring. In more complicated configurations, the compression velocity is defined as the difference between the velocity of the two spring endpoints, with positive velocity corresponding to spring compression.

In circuit theory, the element analogous to the spring is the capacitor, characterized by $ i(t) = C dv/dt$, or $ I(s) = Cs V(s)$. In an equivalent analog circuit, we can use the value $ C = 1/k$. The inverse $ 1/k$ of the spring stiffness is sometimes called the compliance of the spring.

Don't forget that the definition of impedance requires zero initial conditions for elements with ``memory'' (masses and springs). This means we can only use impedance descriptions for steady state analysis. For a complete analysis of a particular system, including the transient response, we must go back to full scale Laplace transform analysis.


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Nonlinear Elements