B.2.5 Characteristics of longitudinal dynamic (II)
--- Complete longitudinal motion --B.2.5.1 The characteristic solution
○Complete set of the perturbed longitudinal motion equation is:
Lu
sU 0  L
 sU 0   u( s)   0 



   0  ( s)
s

D

T
D

g
g

(
s
)
u
u

e






Mu
sM  M sM  s2   ( s)   M 
○Determinant of the 3  3 matrix will be the characteristic polynomial of the
solution, which can be obtained as follows:
Q( s )  s4  a1s3  a2 s2  a3s  a4
L
a1  U  Du  Tu  M  M ,
with


0
L
L
a2  ( Du  Tu ) U  M  M  ( D  g ) Uu  M U  M ,
0
0
0

a3  ( D  g ) M  M g
and

Lu
U0

L

L

 ( Du  Tu ) M U  M  M uD ,
L

0
a4  g M u U  M Uu .
L
0
0
61
○The characteristics of the motion is established by the root(s) of Q( s ) .
1. For most aircraft, the roots of Q( s ) forms two complex conjugate pairs;
namely, Q( s )  ( s  p1 )( s  p1 )( s  p2 )( s  p2 ) .
2. The two characteristic modes, p1 and p2 , differ widely, both in real
and in imaginary terms.
--- The one mode with larger natural frequency is associated with the short
period motion of the longitudinal dynamics. This mode of motion involves
mainly the pitch maneuver.
--- The other mode, that with smaller natural frequency, is associated with the
Phugoid motion of the longitudinal dynamics. This mode of motion involves
mainly the speed and altitude variations.
3. In general, analytic solution of p1 and p2 in terms of the longitudinal
derivatives is difficult to obtain. However, due to the wide difference
between p1 and p2 , the aircraft maneuver in association with the two
characteristic modes can be separated into two simplified longitudinal
motions. From these simplified longitudinal analysis, approximate
analytic solutions of p1 and p2 can be estimated.
--- We can estimate the short period mode from the simplified motion of B.2.4.2,
and the Phugoid mode from another simplified motion, B.2.4.3.
--- In some way, we can think of the two characteristic modes as uncoupled.
62
4. However, numerical computation of p1 and p2 is always possible for
any particular polynomial Q( s ) :
--- For the longitudinal derivatives of Mohawk, we have:
Q( s )  s4  3.6265s3  5.6086 s2  1.4781s  0.2333
--- Numerical solution of this Q( s ) results in
p1  0.142  j 0.1745 and p2  1.671  j1.3475.
--- For comparison, the characteristic modes of two simplified longitudinal
motion as also computed as follows:
1. From B.2.4.3, the pitching and vertical freedom analysis, the
approxi-mate characteristic polynomial for the short period motion is:

L
 
L

s 2  U  M  M s  M  U M  s 2  3.319s  4.5846,
0
0
and its characteristic roots are  1.66  j1.353.
--- In engineering sense, these roots approximate p2 fairly well.
2. From B.2.4.5, the horizontal and vertical freedom analysis, the
approximate characteristic polynomial for the Phugoid motion is:
s 2  ( Du  Tu ) s g Uu  s 2  0.3058s  0.074 ,
0
L
and its characteristic roots are  0.153  j 0.225 ,
--- In engineering sense, these roots approximate p1 fairly well.
63
B.2.5.2 Effects of the variation of parameters
B.2.5.2.1 Forewords
○It is desirable to investigate the change in the roots of Q ( s ) due to the
variation of the aerodynamic derivatives.
○We can link aerodynamic derivatives with aircraft parameters, so results
from this analysis can give us cue for modifying the aircraft design.
B.2.5.2.2 Variation of M
○Assume that a variation, M , occurs in M so that the equation set of
the perturbed longitudinal motion becomes as follows:
Lu
sU 0  L
 sU 0   u( s )   0 



   0  ( s )
s

D

T
D

g
g

(
s
)
u
u

e






Mu
sM  M  M sM  s2   ( s )   M 
○By expanding this determinant, the new characteristic polynomial of the
~
solution, denoted as Q ( s) , will contain the original Q ( s ) plus additional
terms having M as factor. Namely,
Lu
~
2
Q( s )  Q( s )  M  ( s ),  ( s )  s  ( Du  Tu )s 
g.
U0
64
○We will compare this equation with a feedback system synthesis.
1. For the feedback system shown,  ( s) and
 ( s)
k
 ( s)
( s)
 ( s ) are polynomials, and H ( s ) 
is
( s)
the open-loop system transfer function.
k H ( s)
k ( s )

2. The closed -loop transfer function will be
.
1  k H ( s )  ( s )  k ( s )
--- The so-called closed-loop poles are in fact the roots of the denominator
polynomial ( s)  k ( s) .
 ( s) 
3. We can see that  ( s )  k ( s )   ( s )1  k
  ( s )1  k H ( s ),

 ( s ) 
and we use the root locus analysis of 1 k H ( s) to investigate the
change in the roots of ( s)  k ( s) due to variation in k .
 ( s)
~
4. Likewise, we can write Q( s )  Q( s )1  M   .

Q( s ) 
---  ( s ) Q( s ) will play the role of a fictitious open-loop transfer function,
and M a fictitious feedback gain.
~
--- The change in the roots of Q ( s ) due to variation in M can then be
analyzed by using roots locus theory.
65
○An illustration using the longitudinal derivatives of Mohawk:
2
1. Entering values for Mohawk, and we have  ( s )  s  0.3058 s  0.074 .
--- Roots of  ( s ) :  0.153  j 0.225 .
2. We already have Q( s )  s  3.6265s  5.6086 s  1.4781s  0.2333.
.  j 0175
.
.
--- Roots of Q( s ) : p1  014
and p2  1.67  j1348
.
3. Therefore, the fictitious open-loop transfer function has 2 zeros and 4 poles.

--- If we consider M  0 , then 180 condition applies.
4
3
2

--- Conversely, 0 root locus must be drawn for the case with M  0 .

M  0 ( 180 locus)
M  0 ( 0 locus)
s= +j
j

--- Decrease in M does not change p1 much, but hurt p2 a lot.
66
B.2.5.2.3 Variation of M u
○This time, the variation term is M u , and the equation set becomes
Lu
sU 0  L
 sU 0   u( s )   0 



   0  ( s )
s

D

T
D

g
g

(
s
)
u
u

e





 M u  M u sM  M sM  s2   ( s )   M 
g L
~
○Then, Q ( s )  Q ( s )  M u u ( s ), u ( s )  D ( s 
).
U0D

. ( s  111
. )  One zero at 111
--- For Mohawk, u ( s )  2912
. .
--- The fictitious system has 4 poles and 1 zero, 3 asymptotes will appear.

M u  0 ( 180 locus)
M u  0 ( 0 locus)
s=+j

j

67
B. 2.6 Perturbed longitudinal response to control input
B.2.6.1 The input to output transfer functions
○Solution to the complete longitudinal motion:
1
sU 0  L
 sU 0   0 

D  g
 g   0  e ( s )


2
sM  M sM  s   M 
○We can expand the matrix inversion to result in:
N ( s)
N ( s)
N ( s)
u( s )  u  e ( s ),  ( s )    e ( s ), and  ( s )    e ( s )
Q( s )
Q( s )
Q( s )
--- Q( s ) , N u ( s ) , N ( s ) and N ( s ) are polynomials; their coefficients
Lu
 u( s )  
 ( s )   s  D  T
u
u

 
Mu
 ( s )  
depends on the longitudinal derivatives of the aircraft.
--- Using data of Mohawk, we can compute that: N u ( s )  207.35s  233.3 ,
2
2
N ( s )  712
. s  2177
. s  0.523 , N ( s )  712
. s  9.32 s  2.235 .
and Q( s)  s  3.6265s  5.6086s  1.4781s  0.2333.
4
3
2
N ( s)
N u ( s ) N ( s )
,
and  , are referred to as the
Q( s ) Q( s )
Q( s )
transfer functions of input  e to outputs u ,  and  , respectively.
○Rational functions,
68
--- For Mohawk, these transfer functions are:
u( s ) 
207.35s  233.3
 e ( s)
s 4  3.6265s3 5.6086 s 2 1.4781s  0.2333
 ( s) 
712
. s2  218
. s  0.523
 e ( s)
s4 3.6265s3 5.6086 s2 1.4781s  0.2333
and
 ( s) 
712
. s2 9.32 s  2.235
 e ( s),
4
3
2
s 3.6265s 5.6086 s 1.4781s  0.2333
B.2.6.2 Input responses of the system
○In general, the response of u ,  and  to any particular  e input can be
computed using technique outlined in B.2.3.3.
○Example --- Angle of attack response to a step elevator input,  e ( s )  1 / s :
712
. s2  218
. s  0.523
1
 ( s)  4

s  3.6265s3  5.6086 s 2 1.4781s  0.2333 s
0.377  j 0.248
0.377  j 0.248
0.763 j 0.952
0.763 j 0.952
2.28
 s  014




.  j 0175
.
s  014
.  j 0175
.
s 1.67 j1348
.
s 1.67 j1348
.
s
014
. t
. t  0.496 sin 0175
. t
--- Hence,  ( t )  e
0.754 cos 0175
 e1.67t 1526
.
cos1348
. t 1904
.
sin 1348
. t   2.28 .
--- Responses of u( s ) and  ( s ) can be computed the same way.
69