Page 136 - Aeronautical Engineer Data Book

112 Aeronautical Engineer’s Data Book
m(u + qW ) = X + X + X + X p
g
a
e
c
m(v + pW + rU ) = Y + Y + Y + Y p �
c
g
e
a
e
m(w + qU ) = Z + Z + Z + Z p
a
c
g
e
I p – I r = L + L + L + L p
g
c
x
xz
a
I q = M + M + M + M p
y
g
a
c
I r – I p = N + N + N + N p
xz
z
a
c
g
A better analysis is obtained by substituting
appropriate expressions for aerodynamic,
gravitational, control and thrust terms. This
gives a set of six simultaneous linear differen­
tial equations which describe the transient
response of an aircraft to small disturbances
about its trim condition, i.e.:
˚ ˚ ˚ ˚
v
w
mu – X u u – X v – X w – X w
w
˚
˚
˚
–X p – (X – mW ) – X r + mg cos =
q
r
p
e q
e
˚
˚
˚
˚
X + X + X + X

˚
˚
˚
˚
˚
–Y u + mv – Y v – Y w – Y w – (Y +
w
w
u
p
v
mW )p
e
˚
˚
–Y q – (Y – mU )r – mg cos – mg
r
e
e
q
˚
˚
˚
˚
sin = Y + Y = Y + Y

e
˚
˚
˚
˚
–Z u – Z v + (m – Z w) w – Z w
u
w
w
v
˚
˚
˚
–Z p – (Z – mU ) – Z r + mg sin =
e q
e
r
p
q
˚
˚
˚
˚
Z + Z = Z + Z

˚
˚
˚
˚
–L u – L v – L w – L w
u
v
w
w
˚
˚
˚
˚
˚
+I p – L p – L q – I r – L r = L + L

r
xz
p
q
x
˚
˚
= L + L

˚
˚
˚
–M u – M v – M w
v
u
w
˚
˚
˚
˚
˚
–M w – M p – + I q – M q – M r = M

y
w
r
p
q
˚
˚
˚
+ M = M + M

˚
˚
˚
˚
–N u – N v – N w – N w
w
v
u
w
˚
˚
˚
˚
˚
I p – N p – N q + I r – N r = N + N

r
xz
z
q
p
˚
˚
= N + N