Flight Dynamicsdraft

You guys have the supplementary notes in PDF form for the aircraft modes of motion, and the last bit is best taught by way of an example.

Final Examples

Aircraft Modes from Stability Derivatives

Consider the following aircraft data for the Pipe Cherokee PA-28-180 flying at 50m/s.

The following example has been completed using Python, and I’m a better coder than when I first wrote the examples in MATLAB nearly three years ago, so my codes look more like programs than a list of calculator instructions.

I know that this can make things seem less accessible to some, but it’s smarter to keep data in a class like below.

Longitudinal Modes

The two eigenvalues are $-2.4663\pm3.4056j$ and $-0.0279\pm0.2452j$, and we can easily discriminate between the two modes because of the respective size. Recall that the damped natural frequency is given by the imaginary part of the eigenvalue, so we can see:

Since both modes are stable, the time to half amplitude can be found from the real part of the eigenvalue

The undamped natural frequency, $\omega_n$, is given by the absolute value of the eigenvalue,

and hence the damping ratio, $\zeta$ can be determine either from comparison with the standard form characteristic equation and

$\zeta = -\frac{\Re(\lambda)}{\omega_n}$

or from the definition of the damped natural frequency

$\zeta = \sqrt{1-\frac{\omega_d^2}{\omega_n^2}}$

the two methodologies obviously are, and were always going to be equivalent, but it’s nice to show redundancy in methods.

Lateral Directional Modes

From the A matrix for lateral directional motion the eigenvalues

$-0.3468\pm3.3718j$, $0.0$,
$-2.7823$, $0.0194$

are found. The first value is the only complex pair, and therefore has to be the Dutch Roll mode (the only oscillatory lateral/directional mode). We can see that is stable (negative real part) and the following can be yielded using the same as before:

For the remaining eigenvalues, one simply denotes the aircraft’s ability to yaw - the zero value shows that the aircraft has directional freedom and no real concept of heading stability (you could at most say the aircraft possesses neutral heading stability). The other two eigenvalues are real-only; $-2.7823$ and $0.0194$.

We know that one denotes the spiral mode and one denotes the roll mode. The absolute magnitude of the damping indicates that $-2.7823$ is the roll mode, whilst it can also be seen that $0.0194$ is unstable so cannot be the roll mode.

In summary:

Extracting Aircraft Modes from Data

Say you’ve been given some data for an aircraft subjected to a roll disturbance in a datafile - the information you have is roll attitude vs time in a text file.

The data is the free-response of the aircraft, just as seen in the examples following the impulsive forcing of the aircraft.

Plotting these data vs time is the first step, and we can see:

Note that the file itself doesn’t actually tell you the units of the roll attitude, but for determination of the modal qualities, it doesn’t acutally matter. The y axis values are only used to get the damping ratios.

You can determine that this file contains information about the lateral-directional modes since we’re looking at roll attitude. Furthermore, you can see:

• Spiral mode and Dutch roll mode are superposed on the data.
• The Dutch roll mode is stable (decreasing amplitude).
• The Spiral mode is unstable (increasing amplitude).
• The Roll mode is not visible.

This is the same as the example went over in class except the spiral mode is unstable. The theory is the same, though. We need to remove the effect of the spiral mode on the roll mode. You can do this by hand, or using computational methods.

The first step is to graphically separate the two modes. We will look at what the starting value of the spiral mode is. Visually, this is the intercept if we were to imagine the curve without the Dutch roll mode.

The spiral roll attitude is of the form

where $\phi_{0}$ is intercept on the y-axis if the Dutch roll were not present.

We can see that if we extend the curve to the y-axis, it looks like it would intercept around 1.2 degrees, and has the two values given above. We can try this number and see if it works well by looking at the final value on the curve. That is, if we hvae $\phi_0=1.2$, then use the fact that we know $\phi(80s)=10$ and solve for $\lambda_{sp}$

Below shows two methods - one is guessing, which you can perform by trial and error and see which is a good fit through different data points, and the other is a means to fit the curve. That’s just there for advanced folks.

With the eigenvalue for the spiral mode now known, it can be removed from the data to look at the Dutch Roll on its own.

If you were doing this by hand, you could look at the peaks of the DR on its own, but it’s relatively easy to do this on the whole array. This will be completed with the least-squares fit and the guessed values from above.

The damped natural frequency is available from the red plot. The logic used to display them on the graph isn’t necessary for you guys to do. You can do this by inspection of the graph - but it’s easier to do it as an example this way.

The damped natural frequency is:

The damping ratio is given by the exponential decay:

The exponential decay ‘envelope’ is pretty good, but it neither tends to actual zero, which it should, not does it really go through the actual peaks. This is because the spiral mode hasn’t been fully removed from the data above due to the guess not being quite correct.

Now this will be repeated with the fitted values in place of the guessed ones for the Spiral mode:

and, again, the damping ratio: