The short period mode is one of the two symmetric ( up / down ) eigen - modes of an aircraft. This mode is controlled by the elevator.
The eigen - frequency of the mode is one of the two ( complex ) eigen - values of the full, 4th order system matrix for the symmetrical motions. The "shape" of the mode ( the motion pattern ) is given by the accompanying eigen - vector.
The full symmetrical equations can be simplified
in two ways : one which brings out the key elements
of the
The animation gives a first impression of the mode's shape, and of its real time frequency. The mode is often described as a pitch angle oscillation, but the actual motion is also, and in some ways primarily, a vertical bobbing up and down relative to the nominal, horizontal flight path.
Consult Claude about attaching the mouse to an id=. . set by an svg.innerHTML.
Figure 1 : Short period oscillation of a B747 at cruising altitude.
jawel
The mode is surprisingsly closely related to the motion of a weathercock, or more precisely that of a dart arrow.
The main difference is that the airplane has much more up-and-down ( in the arrow, sideways ) motion, because the restoring aerodynamic force applies closer to the center of mass, giving larger sideways excursions.
TODO Discuss the aerodynamic force due to α
- as a result of θ
- as a result of w' where w' is the true, inertial vertical velocity.
In terms of the conventional variables γ , θ and α (note that γ is positive up, while w and w' are down), we have for the descent velocity w' :
w' = − V . γ = − V . ( θ - α ) = + V . ( α - θ )
α = θ + w' / V
We note that θ by itself does not generate any aerodynamic forces or moments. These will only come from α, and to a smaller degree from q.
In the pitch angle oscillation shown in the animation, the aerodynamic force and moment has an α component proportional to θ, and a component due to w'.
For the component due to θ, the moment and the force are in phase with each other and with θ. They act like a spring force trying to center θ. This is a perfeclty undamped weathervane which oscillates around a center of percussion some way ahead of the airplane. Th distance is given by ******.
However, the force and moment contribution due to the vertical, parallel heave velocity are 90°ree; out of phase with the pitch angle oscillation. This force is proportional and opposite to the actual vertical velocity due to the derivative of θ, and therefore it represents a damping.
Maybe I need to move the spring attachment point in the animation up and down for this.
TODO - transform A numerically and symbolically to the state w'.
- look up Gerlach's (38b) for the state in γ, which equals w'/V.
If w' = 0 then θ = α and there is no CZα damping.
Since CZα is negative the force from the truly vertical w' is opposed to w', so it is a damping.
CZα is on the system diagonal ( trace ), just like Cmq
The Iyy does not change with θ, so there is no real reason to use body-fixed velocities.
Maybe put this idea on a separate page.
TBW.
Etkin gives the full order system equations for the B747 in cruising flight at altitude. These can be used to find the "exact" eigenvalues and eigenvectors for that flight condition. The resulting values for the short period mode are :