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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.

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  • \$\begingroup\$ Are you looking for a number or for an expression? \$\endgroup\$
    – Eugene Sh.
    Sep 30, 2019 at 19:20
  • \$\begingroup\$ Am looking to calculate an actual number. \$\endgroup\$
    – user367640
    Sep 30, 2019 at 19:24
  • \$\begingroup\$ Then something is missing here. \$\endgroup\$
    – Eugene Sh.
    Sep 30, 2019 at 19:26
  • \$\begingroup\$ Show the whole question, if you know that damping factor that could help \$\endgroup\$
    – Voltage Spike
    Sep 30, 2019 at 19:28
  • \$\begingroup\$ 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. \$\endgroup\$
    – user103380
    Sep 30, 2019 at 19:48

1 Answer 1

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

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