Page 134 - Aeronautical Engineer Data Book

110 Aeronautical Engineer’s Data Book
Inverting gives:

�
�� 1 sin tan cos tan �� �

p

= 0 cos
–sin
q

0 sin sec cos sec
7.3 The generalized force equations r
The equations of motions for a rigid aircraft are
derived from Newton’s second law (F = ma)
expressed for six degrees of freedom.
7.3.1 Inertial acceleration components
To apply F = ma, it is first necessary to define
acceleration components with respect to earth
(‘inertial’) axes. The equations are:
1 2 2
a x = U – rV + qW – x(q + r ) + y(pq – r)
+ z(pr + q) �
2
2
1
a = V – pW + rU + x(pq + r) – y(p + r )
y
+ x(qr – p)
1
a = W – qU + pV = x(pr – q) + y(qr + p)
z
2
2
– z(p + q )
1
1
where: a , a , a 1 z are vertical acceleration
x
y
components of a point p(x, y, z) in the rigid
aircraft.
U, V, W are components of velocity along the
axes Ox, Oy, Oz.
p, q, r are components of angular velocity.
7.3.2 Generalized force equations
The generalized force equations of a rigid body
(describing the motion of its centre of gravity)
are:
m(U – rV + qW) = X � where m is
m(V – pW + rU) = Y the total mass
m(W – qU + pV) = Z of the body

7.4 The generalized moment equations
A consideration of moments of forces acting at
a point p(x, y, z) in a rigid body can be
expressed as follows: