Basic Polynomial Model

Aerodynamic models rely on wind tunnel and flight test data for their accuracy. However, many references list formulas for estimating stability derivatives based on geometry and mass properties alone $^{\ref{ref:etkin},\ref{ref:mccormick},\ref{ref:datcom}}$. These stability derivatives can serve as the coefficients in a basic polynomial model.

Many stability derivatives are constant for any trim point or transient point of significance. The polynomial model does not require higher-order terms to represent such relationships. However, some stability derivatives (for example, $C_{X_\alpha}$) vary with flight condition. The model requires higher-order terms to account for the variation.

Obviously, not all forces depend on all variables. Linear stability analysis decouples longitudinal motion from lateral-directional motion. Many of these coupled terms are not needed. However, in flight simulation, there are some coupled terms. Some of these terms, such as $C_{X_\beta}\beta$ and $C_{X_{\delta_r}}\delta_r$, are fairly significant terms that stability analysis neglects to decouple the motions. Other terms, such as $C_{N_{\delta_a}}\delta_a$, depend on the lift coefficient, which is considered constant in stability analysis. The equivalent term in flight simulation is a mixed coupled term: $C_{N_{\delta_aC_Z}}\delta_aC_Z$.

There are several nonlinear terms. Arguably, the most important relationship is C_Z versus $\alpha$. While this relationship is linear for low angles of attack, the onset of stall destroys this linearity. Modeling stall requires higher order terms; probably at least fifth-order for a close match. The relationship between C_Mand $\alpha$ is also nonlinear beyond stall; if the flight simulator models C_Z beyond stall, it should model C_M beyond stall also. (Other relationships may lose their linearity beyond stall also; however, it is not considered worthwhile to model these in a simple polynomial model.)

Another nonlinear relationship is C_X versus $\alpha$. Basic airplane performance calculations treat this relationship as quadratic, which is sufficient for a polynomial model, also.

Equations 54-59 present a basic polynomial model, with some nonlinear terms.

      

$$\begin{array}[t]{l} \displaystyle C_{Z_0} + C_{Z_\alpha}\alpha + ... ..._q}\overline q+ C_{Z_{\delta_e}}\delta_e + C_{Z_{\delta_f}}\delta_f \end{array}$$ C_Z = (54)

$$\begin{array}[t]{l} \displaystyle C_{X_0} + C_{X_\alpha}\alpha + ... ...a_f}}\delta_f + C_{X_{\delta_e}}\delta_e + C_{X_{\delta_r}}\delta_r \end{array}$$ C_X = (55)

$$\begin{array}[t]{l} \displaystyle C_{M_0} + C_{M_\alpha}\alpha + ... ...e{\dot\alpha} + C_{M_{\delta_e}}\delta_e + C_{M_{\delta_f}}\delta_f \end{array}$$ C_M = (56)

$$\begin{array}[t]{l} \displaystyle C_{Y_0} + C_{Y_\beta}\beta + C_{Y_r}\overline r + C_{Y_{\delta_r}}\delta_r \end{array}$$ C_Y = (57)

$$\begin{array}[t]{l} \displaystyle C_{L_0} + C_{L_\beta}\beta + C_... ...overline p\\ + C_{L_{\delta_r}}\delta_r + C_{L_{\delta_a}}\delta_a \end{array}$$ C_L = (58)

$$\begin{array}[t]{l} \displaystyle C_{N_0} + C_{N_\beta}\beta + C_... ...C_Z \\ + C_{N_{\delta_r}}\delta_r + C_{N_{\delta_aC_Z}}\delta_aC_Z \end{array}$$ C_N = (59)

This model's domain of accuracy extends through angles of attack several degrees beyond stall. Beyond this, the accuracy diminishes rapidly. This model is accurate for moderate values of sideslip, angular velocity, and control deflection. This model doesn't consider the effects of Reynolds number on drag or maximum lift, nor does it consider compressibility effects. In short, the model's accuracy is quite limited. Even so, a low-speed, subsonic airplane will spend most of its flying time within this model's domain of accuracy.

The major benefit of the basic model is that its coefficients are simple to calculate. Most coefficients stem from basic stability and performance equations, while a simple polynomial curve fit determines the higher order terms in Equations 54 and 56.