how to derive the canonical form of a transfer second order equation?

Question

How to derive the canonical form of the second order transfer function??

$$\frac{(\omega_n)^2}{s^2+2\zeta\omega_ns + (\omega_n)^2}$$

Answer 1

The canonical example for this is the spring and damper system you can read about at wikipedia. If a mass $m$ is attached to an ideal, massless, perfrectly elastic spring and displaced by a length $x$, the force on the spring is $-kx$ where $k$ is a constant. Thus, applying Newton's second law, we find:

$$m\ddot{x} = -kx$$

Now imagine a dissipative term proportional to the velocity $-\gamma \dot{x}$. This is essentially like including a viscous damper on the spring so the overall equation is

$$m\ddot{x} + \gamma\dot{x} + kx = 0,$$

Which is the correct form for the characteristic equation. Now we need to get the coefficients right:

Recall that $s = j\omega$ so the dimensions of $s$ are Hz. We can introduce the natural frequency as the parameter for which $\ddot{x} = -\omega_n^2x$, and a dimensionless number $\zeta$ for which $\gamma = \omega_n\zeta$ so that the units of the terms all match and the dissipative term can be analyzed as a ratio of the damping coefficient to the natural frequency. Since any transfer function is trivial for the free response (by definition) we simply need to add a coefficient for the forcing term, which is typically $K$ but also often written as $\omega^2_n$ so that oscillations about the natural frequency are rejected in the closed loop by default. The latter form in particular is a bit nicer because it's clear the overall TF is dimensionless and no ambiguous gain, which might be interpreted to be from some $P$-controller, etc. is accidentally inferred.