Finding the natural frequency

I am trying to figure out the natural frequency and the time constants of a transfer function. Ignoring the numerator and using the denominator of the second order characteristic equation:

$$s^2+2\zeta\omega_ns + \omega_n^2$$

where $$\zeta=0.5,$$ $$\omega_n^2 = \frac{100}{\tau_1},$$

and $$2\zeta\omega_n = \frac{1+100\tau_2}{\tau_1}.$$

$$s^2+\frac{1+100\tau_2}{\tau_1}s + \frac{100}{\tau_1}$$

How do you find the value of natural frequency without knowing the tau constants? Is it even possible? It seems I have two equations with three unknowns? EDIT: Am looking for an actual number.

• Are you looking for a number or for an expression? Sep 30, 2019 at 19:20
• Am looking to calculate an actual number. Sep 30, 2019 at 19:24
• Then something is missing here. Sep 30, 2019 at 19:26
• Show the whole question, if you know that damping factor that could help Sep 30, 2019 at 19:28
• Without the values of $\tau_1$ and $\tau_2$ , I don't see how you can get a numeric value for $\omega_n$. There isn't enough information. I'm assuming $\tau$ is some time constant for a system, spring system, RC/RL circuit, etc. But perhaps the question simply wants you to solve for $\omega_n$ in terms of $\tau_1$ and $\tau_2$ and in that case you need to set a system of equations. You may be able to substitute your equations and have only $\tau_1$ but that's as far as you can probably get.
– user103380
Sep 30, 2019 at 19:48

How do you find the value of natural frequency without knowing the tau constants? Is it even possible?

No.
The problem is that there is 1 equation with 2 unknowns. You need to know either $$\\tau_1\$$ or $$\\tau_2\$$ as shown below.

Substituting $$\\zeta=0.5 \$$ in $$2\zeta\omega_n = \omega_n = \frac{1+100\tau_2}{\tau_1}$$

and using $$\omega_n^2 = \frac{100}{\tau_1}$$

gives 1 equation with 2 unknowns:

$$\sqrt { \frac{100}{\tau_1} } = \frac{1+100\tau_2}{\tau_1}$$ or $$10 \sqrt { \tau_1 } = 1+100\tau_2$$

If you know one of the unknowns, you can find rest using:

$$\tau_1 = \frac{ (1+100\tau_2 )^2}{100}$$

$$\tau_2 = \frac{ 10\sqrt{ \tau_1 } - 1 }{100}$$

and $$\omega_n = \sqrt { \frac{100}{\tau_1} } = \frac{100}{1+100\tau_2}$$