Dynamic Equations

The dynamic equations derive from Newton's Second Law, $\vec F = m \vec a$, and the analogous equation for angular motion. Resolving these equations along body axes, and including terms to account for centrifugal forces due to the rotating body reference frame, yields the dynamic equations of motion. For a symmetric airplane ( I_xy=I_yz=0), the dynamic equations are given by Equations 10-15.

      

$$\dot u$$ = rv - qw + X/m (10)

$$\dot v$$ = pw - ru + Y/m (11)

$$\dot w$$ = qu - pv + Z/m (12)

$$\dot p \frac{I_{xz}L'+I_{xx}N'}{I_{xx}I_{zz}-I_{xz}^2}$$ = (13)

$$\dot q \frac{M-(I_{xx}-I_{zz})rp+I_{xz}(p^2-r^2)}{I_{yy}}$$ = (14)

$$\dot r \frac{I_{zz}L'+I_{xz}N'}{I_{xx}I_{zz}-I_{xz}^2}$$ = (15)

The L' and N' from Equations 13 and 15 are given by

L' = L - (I_zz-I_yy)qr - I_xzpq

N' = N - (I_yy-I_xx)pq - I_xzqr

Equations 10-15 introduces forces and moments (X, Y, Z, L, M, N). Because of these terms, the simulator must calculate the resultant force and moment on the airplane before it can calculate the time derivatives.