Full nonlinear moment and force equations for a fixed aircraft in body axes, assuming mass symmetry about the x–z plane (Ixy = Iyz = 0), so only the product of inertia Ixz survives.
Body-axis convention — origin at the CG: x out the nose, y out the right wing, z downward (right-hand rule). Positive p, q, r are rotation rates about x, y, z.
01
Moment equations
L=Ixxp−Ixzr+qr(Izz−Iyy)−Ixzpq
Rolling moment: roll acceleration, cross-inertia yaw coupling, and inertial coupling from pitch–yaw and roll–pitch rate products.
M=Iyyq−pr(Ixx−Izz)+Ixz(p2−r2)
Pitching moment: pitch acceleration, roll–yaw gyroscopic coupling, and the product-of-inertia term from unequal roll/yaw rates.
N=−Ixzp+Izzr+pq(Iyy−Ixx)+Ixzqr
Yawing moment: yaw acceleration, cross-inertia roll coupling, and inertial coupling from roll–pitch and pitch–yaw rate products.
02
Force equations
Fx=m(u+qw−rv)
Axial force along the body x-axis, including centripetal terms from yaw–sway and pitch–heave coupling.
Fy=m(v+ru−pw)
Side force along the body y-axis, from yaw–surge and roll–heave coupling.
Fz=m(w+pv−qu)
Normal force along the body z-axis, from roll–sway and pitch–surge coupling.
03
Nomenclature
L, M, NRolling, pitching, yawing moments (body axes)
F•x, y, zForce components along x, y, z
p, q, rRoll, pitch, yaw rates (about x, y, z)
u, v, wBody-axis velocity components along x, y, z