Response of a stable second-order transfer function to a unit sine wave input

Question

In the lecture slides from my university it says that:

"The response of a stable second-order transfer function to a unit sine wave input is:"

$$Y(s)=\frac{1}{s^2+2\zeta\omega_n+\omega_n^2}*\frac{\omega}{s^2+\omega^2}$$

Isn't this missing an \$ω_n^2\$ in the numerator? Since the standard form of a second order transfer function is:

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

and the laplace transform of the sine wave input is:

$$\frac{\omega}{s^2+\omega^2}$$

Is this a mistake in the lecture slides?

Answer 1

You are probably right, but it depends on how you define the transfer function. The sine part is right, while as you can see your \$H(s)\$ is not adimensional, it's something like \$s^2\$, that is pretty strange for a transfer function[^seconds].

You are safe assuming that's a slide mistake. For the future keep in mind that checking the measurement units is always a great idea.

[^seconds]: the s is for seconds, not for the s variable.

Answer 2

If the DC gain (= steady state gain) of your transfer function should be unity, then yes, there should be a wn^2 in the numerator of the TF. But there's nothing essentially wrong with the TF as it stands, it could legitimately have a DC gain of 1/wn^2 (but include the missing 's' in the denominator!).