Flight Test Data: Estimating Frequency and Damping

This is a short discussion about estimating frequency and damping parameters from flight test data.  Such data often suffers from some amount of contamination and often less that one cycle of the oscillation.  This can make frequency and damping estimates difficult.  

The example data plotted below is a helicopter's pitch response to a disturbance.  The longitudinal cyclic and collective were fixed while this data was recorded.  The longitudinal cyclic was pulsed before recording to excite the helicopter’s phugoid mode.  The values we’ll estimate are useful in understanding this helicopter’s longitudinal dynamic stability.

Plot of phugoid flight test data

We assume this data roughly follows a damped oscillation equation \(\theta (t) \approx Ae^{-\omega_n \zeta t}\cos (\omega_d t + \varphi ) + C\), where \(\omega_n\) is the undamped natural frequency, \(\zeta\) is the damping ratio, \(\omega_d\) is the damped frequency and \(\varphi\) is a phase offset.  (In this case the helicopter was initially trimmed to 0deg pitch so we’ll assume \(C=0\) henceforth.)

We’ll often assume that at least two extrema were recorded.  In this case \(v_1=5deg\) at \(t_1=6s\) and \(v_2=-2.3deg\) at \(t_2=28s\).  Using these four values, we’ll estimate the undamped natural frequency and damping ratio.

Step 1: Shift the values to be centered about 0.  E.g. if the helicopter was trimmed at \(\theta_0\) before the disturbance, subtract \(\theta_0\) from each recorded pitch value.

Step 2: Pick out two extrema \(v_1,v_2\) at times \(t_1, t_2\) from the shifted data, as described above. 

Step 3: Solve for the product \(p=\omega_n \zeta \) using the ratio \(\frac{\theta (t_1)}{\theta (t_2)}\).  This simplifies things because the \(\cos\) term is just \(\pm 1\) and the amplitude cancels out \( \frac{\theta (t_1)}{\theta (t_2)} = \frac{e^{-pt_1}}{e^{-pt_2}} = \frac{|v_1|}{|v_2|} \Rightarrow p(t_2-t_1) = ln|\frac{v_1}{v_2}|\).

Example: With our data above \(p = ln|\frac{v_1}{v_2}|/(t_2-t_1) \approx .0338\)

Step 4: Estimate the damped frequency \(\omega_d \approx \pi/(t_2-t_1) \).  This is assuming \(v_1, v_2\) are adjacent extrema (and therefore one is a min and the other is a max and they occur one half cycle apart).

Example:  With our data above \(\omega_d \approx 0.143 Hz \).

Step 5: Solve for the natural frequency \( \omega_n =\sqrt{\omega_d^2+p^2}\).  This follows from the damping ratio equation \(\zeta^2 = 1 - \frac{\omega_d^2}{\omega_n^2}\).

Example: Plugging in our example values above yields \(\omega_n \approx .147 Hz\).

Step 6: Solve for the damping ratio \(\zeta\) using the equation provided in step 5.

Example:  Here \(\zeta \approx 0.23\).

The curve estimated by the above approach is shown below alongside the real data.

Plot of estimated phugoid flight test data