The Periodic Transport
199
x
a
I
3
2
1
FIGURE 8.8: Behavior of a mismatched beam.
Thus
(x|δ)
(a|δ)
=
1 − (x|x) −(x|a)
−(a|x) 1 − (a|a)
D
D
,
(8.3)
D
D
=
1 − (x|x) −(x|a)
−(a|x) 1 − (a|a)
−1
(x|δ)
(a|δ)
,
when
det
1 − (x|x) −(x|a)
−(a|x) 1 − (a|a)
= 2 − (x|x) − (a|a) = 0.
Note that when tr ˆ
M < 2, which is satisfied if only stable motion is considered,
D, D
are uniquely determined.
D
D
=
1
2 − (x|x) − (a|a)
1 − (a|a) (x|a)
(a|x) 1− (x|x)
(x|δ)
(a|δ)
=
1
2 − (x|x) − (a|a)
(1 − (a|a))(x|δ) + (x|a)(a|δ)
(1 − (x|x))(a|δ) + (a|x)(x|δ)
.
From the point of view of the computation, a slightly different form of the
same result helps to develop an algorithm that can be applied to arbitrary
order with the help of the Differential Algebraic (DA) technique. Eq. (8.2)
can be rewritten as
⎛
⎝
1 0 0
0 1 0
0 0 0
⎞
⎠
⎛
⎝
D
D
1
⎞
⎠ +
⎛
⎝
0
0
1
⎞
⎠ =
⎛
⎝
(x|x) (x|a) (x|δ)
(a|x) (a|a) (a|δ)
0
0
1
⎞
⎠
⎛
⎝
D
D
1
⎞
⎠ ,
199
x
a
I
3
2
1
FIGURE 8.8: Behavior of a mismatched beam.
Thus
(x|δ)
(a|δ)
=
1 − (x|x) −(x|a)
−(a|x) 1 − (a|a)
D
D
,
(8.3)
D
D
=
1 − (x|x) −(x|a)
−(a|x) 1 − (a|a)
−1
(x|δ)
(a|δ)
,
when
det
1 − (x|x) −(x|a)
−(a|x) 1 − (a|a)
= 2 − (x|x) − (a|a) = 0.
Note that when tr ˆ
M < 2, which is satisfied if only stable motion is considered,
D, D
are uniquely determined.
D
D
=
1
2 − (x|x) − (a|a)
1 − (a|a) (x|a)
(a|x) 1− (x|x)
(x|δ)
(a|δ)
=
1
2 − (x|x) − (a|a)
(1 − (a|a))(x|δ) + (x|a)(a|δ)
(1 − (x|x))(a|δ) + (a|x)(x|δ)
.
From the point of view of the computation, a slightly different form of the
same result helps to develop an algorithm that can be applied to arbitrary
order with the help of the Differential Algebraic (DA) technique. Eq. (8.2)
can be rewritten as
⎛
⎝
1 0 0
0 1 0
0 0 0
⎞
⎠
⎛
⎝
D
D
1
⎞
⎠ +
⎛
⎝
0
0
1
⎞
⎠ =
⎛
⎝
(x|x) (x|a) (x|δ)
(a|x) (a|a) (a|δ)
0
0
1
⎞
⎠
⎛
⎝
D
D
1
⎞
⎠ ,