Longitudinal Stability (fixed stick)

DWT automatically runs
multiple Cmarc
analyses in order to calculate the stability derivatives necessary for longitudinal
analysis. Parameters are automatically varied over a prescribed increment in order to
use first order differencing to calculate the derivatives. The calculation of the
necessary parameters requires two parameter variations, elevator deflection and the
angle of attack (alpha). In order for the longitudinal calculation to be most accurate, it
is best to perform the differencing at the trimmed condition. Once this point is established
(elevator deflection and aircraft angle of attack) at some given flight condition, alpha may
be varied in order to calculate the stability derivatives that are a function of alpha and the
elevator may be deflected in order to calculate derivatives that are a function of delta. Although
DWT does calculate derivatives that are a function of elevator deflection, at this time DWT
is limited to calculating the short and long term (fugoid) frequencies and damping in
the *stick fixed* condition. Future versions of DWT will include stick free dynamic analyses.

Elevator deflection is handled by what AeroLogic is calling the "tilt vector" approach. Tilt Vector is a numerical enhancement to Cmarc, which allows for the modification of the panel normal vector about a hinge line. It has been shown that this technique is accurate for movable surfaces such as elevators, rudders, ailerons, etc. up to approximately 15 degrees of deflection. This technique allows for deflecting control surfaces without having to modify the paneled model, which can be a time consuming task. Colored in red in the following figure are the panels

The first step in a longitudinal stability analysis once the model has been created in Loftsman/P is to go to the stability form. The delta symbol signifies differencing or the calculation of rates of change with respect to some variable. The following form is brought up. Each section of this form will be described more fully.

The first section is where the physical properties of the aircraft are defined as well as the flight condition for the analysis.

- W - Weight
- g - Acceleration of gravity
- Sh - Area of horizontal stabilizer
- S - Wing area
- c - Mean chord of the wing
- ch - Mean chord of the horizontal stabilizer
- lam - Ratio of tip chord to root chord of the wing
- Xcg - Center of Gravity location
- Xt - Location of horizontal tail quarter chord point
- b - Wingspan
- Ixx, Iyy, Izz, Ixy - Moments of inertia of the aircraft.
- e@x, e@y, e@z - Location where rate of change of downwash will be calculated
- Listbox - Press the load button and the list box will be populated with the models patches. Select the patches that correspond to the horizontal tail. Cl for the horizontal tail will be calculated.
- Cmq, Cmad - Button to calculate these two terms from Cl of the horizontal tail.

The next section is where some values are set for the calculation of the trimmed condition as well as the longitudinal derivatives.

If the "Calculate Trimmed condition" and "Use Calculated Trim" boxes are checked, DWT will calculate the elevator position and alpha that correspond to the flight condition of interest. The results of this series of Cmarc runs are "Base Alpha" and "Elev Def" which are the starting values of alpha and elevator deflection (baseline) to be used in the differencing process. Alpha Start, Alpha Stop, Alpha Inc and Ele Start, Ele Stop, Ele Inc must bracket the unique alpha and elevator deflection, i.e. extrapolation is not done. Multiple alpha and elevator deflections are permitted in this calculation to account for any nonlinear effects in order to arrive at a more accurate solution.

In the Cmarc bulk data deck, there are NNROT (see the Cmarc reference manual) specifications of tilt vectors. Ele Tilt indicates which corresponds to the elevator. DWT will be automatically varying the value of elevator deflection and thus needs to be able to distinguish the elevators from the rudders and ailerons. The form also needs the size of the increment of alpha and elevator deflection to be used in the differencing runs. Typically 1 degree is adequate.

The right side of the above form is where the results of the Cmarc runs are displayed.

- Cl - Lift curve slope
- Cd - Coefficient of drag. Note that this value is not well calculated by inviscid panel methods.
- Cla - Rate of change of Cl with respect to alpha.
- Cda - Rate of change of Cd with respect to alpha.
- Cma - Rate of change of Cm with respect to alpha.
- Cmq - Aircraft damping in pitch.
- Cmad - Rate of change of pitching moment coefficient with rate of change of angle of attack.
- Clah - Lift curve slope of the horizontal surface.
- de/da - Rate of change of downwash angle with respect to alpha.
- dCm/dCl - Rate of change of moment coefficient with respect to rate of change of lift coefficient.
- Cld - Rate of change of Cl with respect to elevator deflection. This value is only displayed and is not used in any calculations currently in DWT.
- Cmd - Rate of change of Cm with respect to elevator deflection. This value is only displayed and is not used in any calculations currently in DWT.
- dx - Increment used to calculate the neutral point.
- Neutral Point - CG location where dCm/dCl = 0.
- Load Coefs - Button that takes aircraft physical properties and stability derivatives and loads the stability matrix.