# Determine Stiffness (k) and Mass (m) of a system from impulse response

I have a test sample (Piezo Diaphragm) which I am actuating with a voltage supplied to the sample to assess the impulse response.

When this voltage is supplied to the sample a deformation per unit voltage occurs and the diaphragm undergoes a deflection (displacement).

Once the voltage is removed, the diaphragm (sample) undergoes damped oscillations until it returns to its initial condition before the voltage was supplied.

The displacement over time is measured with a laser vibrometer and results in a damped oscillations plot. From this plot I am able to determine system characteristics using the log decrement method such as natural frequency and damping ratio.

My question is whether it is possible to determine system characteristics such as Stiffness (k) or Mass (m) solely using the displacement over time data and the damped oscillation plot?

• you can get the ratio, k/m, from the frequency of the ringing, assuming the 2nd order model is a good fit. That plus the damping ratio is usually sufficient for control purposes – Pete W Feb 27 at 19:55

## 1 Answer

The equation of motion after the impulse is $$m\ddot x + c\dot x + kx = 0$$ with initial conditions $$x = 0, \dot x = \dot x_0 \text{ when } t = 0.$$

You have found the frequency and damping ratio already, so you can rewrite the EOM as $$m(\ddot x + \beta \omega\dot x + \omega^2 x) = 0$$ where you know $$\beta$$ and $$\omega$$.

Scale your measured results to correspond to a unit impulse, and use the measured initial velocity $$\dot x_0$$ to find the value of $$m$$.

This is a simple example of the more general process where the system has multiple degrees of freedom and/or you make measurements are multiple positions, which is called "modal testing" or "experimental system indentification". A good overview is this book. You can probably find copies of it on line, or Google for the buzzwords given above.