Frequency-Response Matching Using
Digital Filter Design Methods

Given force inputs and velocity outputs, the frequency response of an ideal mass was given in Eq.(7.1.2) as

$$\Gamma_m(j\omega) \eqsp \frac{1}{m j\omega},$$

and the frequency response for a spring was given by Eq.(7.1.3) as

$$\Gamma_k(j\omega) \eqsp \frac{j\omega}{k}.$$

Thus, an ideal mass is an integrator and an ideal spring is a differentiator. The modeling problem for masses and springs can thus be posed as a problem in digital filter design given the above desired frequency responses. More generally, the admittance frequency response ``seen'' at the port of a general $N$ th-order LTI system is, from Eq.(8.3),

$$H(s) \isdefs \frac{B(s)}{A(s)} \isdefs \frac{b_M s^M + \cdots b_1 s + b_0}{a_N s^N + \cdots a_1 s + a_0} \protect$$ (9.14)

where we assume $M<N$ . Similarly, point-to-point ``trans-admittances'' can be defined as the velocity Laplace transform at one point on the physical object divided by the driving-force Laplace transform at some other point. There is also of course no requirement to always use driving force and observed velocity as the physical variables; velocity-to-force, force-to-force, velocity-to-velocity, force-to-acceleration, etc., can all be used to define transfer functions from one point to another in the system. For simplicity, however, we will prefer admittance transfer functions here.