Page 135 - Aeronautical Engineer Data Book

Principles of flight dynamics 111
Moments of inertia
2
2
I = ∑ m(y + z ) Moment of inertia about
x
Ox axis
2
2
I = ∑ m(x + z ) Moment of inertia about
y
Oy axis
2
2
= ∑ m(x + y ) Moment of inertia about
I z
Oz axis
I
= ∑ m xy Product of inertia about
xy
Ox and Oy axes
= ∑ m xz Product of inertia about
I xz
Ox and Oz axes
I = ∑ m yz Product of inertia about
yz
Oy and Oz axes
The simplified moment equations become

I p – (I – I ) qr – I (pq + r) = L �
x
y
z
xz

2
2
q – (I – I ) pr – I (p – r ) = M
I y x z xz

r – (I – I ) pq – I (qr + p) = N
I z x y xz
7.5 Non-linear equations of motion
The generalized motion of an aircraft can be
expressed by the following set of non-linear
equations of motion:

g
c
m(U – rV + qW) = X a + X + X + X + X d
p

m(V – pW + rU) = Y + Y + Y + Y + Y d
c
p
a
g

m(W – qU + pV) = Z + Z + Z + Z + Z d �
g
p
a
c

I p – (I – I ) qr – I (pq + r) = L + L +
xz
y
a
x
g
x
+ L + L
L c p d

2
2
I q + (I – I ) pr + I (p – r ) = M + M +
a
g
y
z
x
xz
+ M + M
M c p d

I z r – (I – I ) pq + I (qr – p) = N + N +
y
a
g
x
xz
+ N + N
N c p d
7.6 The linearized equations of motion
In order to use them for practical analysis, the
equations of motions are expressed in their
linearized form by using the assumption that all
perturbations of an aircraft are small, and
about the ‘steady trim’ condition. Hence the
equations become: