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Physical Audio Signal Processing
    Introduction to Lumped Models
       Impedance
          Ideal Mass

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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 L.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 [290],L.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. L.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: