Page 303 - Aeronautical Engineer Data Book

Appendix 2:
Aircraft response transfer
functions

Table A2.1 Longitudinal response transfer functions
is elevator input.

2
4
3
Common denominator polynomial ∆(s) = as + bs + cs +
ds + e
˚
a mI y (m – Z ˚ )
w
˚ ˚
˚
˚
˚ ˚
˚
b I y (X u Z w˚ – X w˚ Z u ) – mI Y (X + Z w ) – mM w˚ (Z q +
u
˚
˚
mU e ) – mM q (m – Z ˚ )
w
˚
˚ ˚
˚
˚
˚ ˚
˚
˚
c I y (X u Z w˚ – X w Z u ) + (X M ˚ – X M u )(Z q + mU e )
w
u
w
˚
˚
˚
˚
˚
˚ ˚
˚ ˚
˚
+ Z u (X w˚ M q – X M ˚ ) + (X M – X M u )(m – Z ˚ )
w
q
q
w
q
u
˚ ˚
˚
˚
˚
˚ ˚
˚
+ m(M q Z w – M w Z q ) + mW e (M w˚ Z u – M u Z w˚ )
˚
2 ˚
+ m (M ˚ g sin – u e M w )
w
e
˚
˚
˚
˚
˚
d (X u M w – X w M u )(Z q + mU e )
˚
˚ ˚
˚
˚
˚
˚ ˚
˚
˚
+ (M u Z w – M w Z u )(X q mW e ) + M q (X w Z u – X u Z w )
˚ ˚ ˚ ˚ ˚ ˚
+ mg cos e (M w˚ Z u + M u (m – Z ˚ )) + mg sin e (X w˚ M uw
˚ ˚
˚
– X u M w + mM w )
˚
˚
˚
˚
˚
˚
+ mg sin (X w M u – X u M w ) + mg cos (M w Z u –
e
e
˚ ˚
M u Z w )
˚
˚
˚
˚ ˚
˚
e mg sin (X w M u – X u M w ) + mg cos (M w Z u –
e
e
˚ ˚
M u Z w )

2
2
Numerator polynomial N 3 (s) = as + bs + cs + d
˚
˚
˚
˚
a I y (X ˚ Z + X (m – Z ˚ ))
w
w
˚
˚
˚
˚
b X (–I y Z w + mU e ) – M q (m – Z ˚ ))
w
˚
˚
˚
˚
˚
˚
+ Z (I y X w – X w˚ M q + M ˚ (X q – mW e ))
w
˚
˚
˚
˚
˚
+ M ((X q – mW e )(m – Z ˚ ) + X w˚ (Z q + mU e ))
w
˚
˚
˚
˚
˚
˚
c X (Z w M q – (M w (Z q + mU e ) + mg sin e M w˚ )
˚
˚
˚
˚
˚
˚
+ Z (M w (X q – mW e ) – X M q – mg cos e M w˚ )
w
˚
˚
˚
˚
˚
+ M (X w (Z q + mU e ) – Z w (X q – mW e ) – mg cos (m
e
˚
˚
– Z ˚ ) – mg sin e X w˚ )
w
˚
˚
˚ ˚
˚
˚
d X M w mg sin – Z M mg cos + M (Z mg cos
e
w
w
e
˚
e – X w mg sin e )