How does one find the zeta value or the frequency of an over damped response curve. I am used to determining things by using the log-decrement equation. Does that apply here even though there is no oscillations?
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$\begingroup$ So there is no way to determine these values from the graph ? $\endgroup$– user15588486Commented Oct 6, 2021 at 0:48
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1$\begingroup$ Can you post a sample response? Why not use some optimization algorithm to fit a second order system step / impulse response to the obtained response? It may be a bit overkill though. $\endgroup$– AJNCommented Oct 6, 2021 at 2:33
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$\begingroup$ Over-damped response does not have a frequency. $\endgroup$– alephzeroCommented Oct 6, 2021 at 10:57
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2 Answers
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The logarithmic decrement is not applicable to the overdamped systems because there are no oscillations.
Also, there is not (at least to my knowledge) an equivalent to the logarithmic decrement approach to calculate the damping ratio $\zeta$ for overdamped systems.
Free vibration measurement
As AJN proposed what you can do is given the equation that describes the response (e.g. displacement). For example for free vibrations of a system:
$$x(t) = C_1 e^{\left(-\zeta+\sqrt{\zeta^2-1}\right) \omega_n t} + C_2 e^{\left(-\zeta-\sqrt{\zeta^2}-1\right) \omega_n t}$$
where:
- x: is the displacement
- $C_1, C_2 $ are integration constants
- $\zeta$ is the dampting ratio
- $\omega_n $ is the natural frequency (should be equal to $\sqrt{\frac{k}{m}}$)
- t is the time.
Given a set of measurements t, x(t)
| t | x(t) |
|---|---|
| 0 | 1 |
| 0.001 | 0.98 |
| 0.002 | ... |
you can use a non linear fit tool to obtain the parameters: $C_1, C_2, \zeta$.
Another approach is that you can also fit $\omega$ and the validate against the $\sqrt{\frac{k}{m}}$). This is good because, the nonlinear fit is not a trivial exercise, and it would be preferable if you have a benchmark/validation of your effort. (however, I stand with AJN that this is probably overkill).
measurement with forced response on steady state.
Alternatively, another approach for overdamped systems is to apply a forced vibration of known magnitude and frequency. At the steady state, measure the dynamic amplitude of the system, and calculate the Magnification or Transmissibility ratio and from that calculate the damping radio.
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Picking up from NMech's answer:
For the step response, integration constants can be expressed in terms of $\zeta$ and $\omega_0$
According to swarthmore.edu
Where $\omega_0 = \sqrt{\omega_1\omega_2}$ with $(\omega_1,\omega_2)$ being the two characteristic frequencies, and $K$ is the step amplitude
After that, I believe $\zeta$ can be extracted from any two points on the step response $(t_1,x_1)$ and $(t_2,x_2)$, with the procedure made independent of $K$ and $\omega_0$ by looking at the ratio $x_1/K$, $x_2/K$ and $t_2/t_1$ . For any ($x_1/K$, $x_2/K$), chosen arbitrarily, $t_2/t_1$ should correspond to $\zeta$. A table could then be generated with a spreadsheet.
