飛行力學《作業九》
2017/05/11 - 2017/05/17
The variations of u (t ) ,  (t ) , and  (t ) are governed by the following equation which is quoted from (1) in Homework 七:
A  X ( s)  B  Z ( s)
 s  Tu  Du

A    Lu
  M u
g
sU 0
s  sM 
2
D  g


 L  sU 0 ,
 M   sM  
 u 
X    
 
  
T
B0

 0
0 
  
0  Z   T

  e 
M  
(9.1)
Note that u ,  , and  appear in (1) as the function of s , and the equation is referred to as being in the s-dommain,
Two developments concerning (9.1) follow. Firstly, in order to compute u (t ) ,  (t ) , and  (t ) for the given functions of
 T (t ) and  e (t ) , (9.1) was converted into a time domain and computable formula. The mathematical process to accomplish
this task is given in Homework 八. Secondly, the variations of u (t ) can be isolated from that of  (t ) by converting (9.1),
which is a 3D dynamic equation, into two 2D reduced dynamic equations given as follows:
(A) The 2D equation for the pitching motion:
A1  X 1 ( s)  B1   e ( s),
 sU 0
A1   2
s  sM 
 L  sU 0 
,
 M   sM  
 0 
  
X 1   , B1   
 
M 
(9.2)
(B) The 2D equation for the variation in flight speed:
A2  X 2 ( s)  B 2  T ( s),
 s  Tu  Du
A2  
  Lu
g 
,
sU 0 
 u 
T 
X 2   , B 2    
 
0
(9.3)
Note that  in (9.3) equals  in value.
It was also true that the variations of u (t ) ,  (t ) , and  (t ) can be characterized by e st for 4 different values of s ;
specifically, s  p1 , s  p1 ' , s  p2 , and s  p2 ' . In addition, these values of s also satisfy the following equation:
(9.4)
d ( s)  det A  ( s  p1 )( s  p1 ' )( s  p2 )( s  p2 ' )
Because (9.2) and (9.3) are the 2D reduced equations if (9.1), the following equations also hold:
d 1 ( s)  det A1  s( s  p1 )( s  p1 ' ),
p1  p1
(9.5)
And
d 2 ( s)  det A2  ( s  p2 )( s  p2 ' ),
Note that
p1  p2
p2  p2
(9.6)
is assumed. This assumption on p1 and p2 is different from that made discussed in class. Because of
1
(9.5) and (9.5), the values of p1 can be computed from the roots of d 1 ( s) and that of p2 from the roots of d 2 ( s) .
In Homework 八, the s-domain 3D equation given in (9.1) was converted into time-domain so that the data of u (t ) ,  (t ) ,
and  (t ) can be computed. Likewise, the 2D reduced dynamic equations given in (9.5) and (9.6) can be separately converted
into time domain and computable formulas. These tasks are presented as follows.
(甲) The following equation is inferred from (9.5):
(9.7)
s  Φ1  Ψ1   Y 1 ( s)  Φ1  sY 1 ( s)  Ψ1  Y 1 ( s)  Ω1   e ( s)
Where
 U0

Φ1   1
  M 
 U0 
0 0  L 

0
0 , Ψ1  0  1
0 ,


1  M  
0 0  M  
0
  
  
X 1  q and
 
  
 0 
Ω1   0 
 
 M  
As a result, the following time domain equation can be written:
(9.8)
1
1
Y 1 (t )  F1  Y 1 (t )  G1   e (t ), F1  Φ1  Ψ1, G1  Φ1  Ω1
(乙) From (9.6), it is also inferred that:
1 0 
 T  Du
( sΦ2  Ψ2 )  X 2 ( s )  Φ2  s X 2 ( s )  Ψ2  X 2 ( s )  Ω2   T ( s ), Φ2  
 Ψ2   u

0 U 0 
  Lu
g
, Ω2  B 2
0 
(9.9)
As a result, the following time domain equation can be written:
1
1
X 2 (t )  F2  X 2 (t )  G2   T (t ), F2  Φ2  Ψ2 , G2  Φ2  Ω2
(9.`0)
Subsequently, both (9.8) and (9.10) can be converted into a computable formulas by means of the same mathematical
process that converts (5) into (8) in Homework 八.
Your homework are as follows:
(1) Construct d 1 ( s) and compute p1 . Compare the computed value of p1 with that of p1 that was computed in Homework 七.
(2) Construct the computable formula of (9.8). Then, compute the data of  (k )   (k ) for a step  e . Plot the computed
data and compare it with that that was computed in Homework 八 under the same input. In particular, what can you say
about the similarity and the differences of the two results.
(3) Construct d 2 ( s) and compute p2 . Compare p2 with that of p2 that was computed in Homework 七.
(4) Construct the computable formula of (9.10). Then, compute the data of u (k ) for a step  T . Plot the computed data and
2
compare it with that that was computed in Homework 八 under the same input. In particular, what can you say about the
steady state values of u and  .
3