AT1 SS 1994 Aufgabe 1:
a) :
1 ; 1
1 1
: (2) in (3) (1)und
: (1) in (3)
(3) (2)
(1)
) ( 2 ) ( 1
) ( 2 ) ( 1
) ( 2
) ( 2 ) ( 1
) ( 1 )
( ) ( ) (
) ( 1 ) ( )
( 2
) ( 1 ) ( 2 ) (
) ( )
( 2
) ( 1 ) ( ) ( 1 ) ( ) ( 2
) (
) ( 2
) ( ) (
) ( 2
) ( )
( 2 )
(
) ( ) ( 1 ) (
s s
s s
s s s
s s
s s
s s s
s s
s
s s
s s s s s s
s s s
s s s
s
s s s
F F
F F F
F F
F Y
F X
F F Y
F X F
F X F F X
F Y X
F Y X A
F B X F
B X
X F A B
B Y A
+ +
= ⋅ +
= +
=
⋅
=
+ +
⋅
⋅ −
−
⋅
=
−
=
⇒ =
⋅
=
−
⋅
=
−
=
b) :
; )
1 ( ) 1
1 ( )
1 ) (
1 1 (
) 1
(
2 1
2 1 2
1 2 1 2
1
2 1
) (
s K T s s
K
K K T
s s s K K T s
K K s
K T s
s K
s K T s
s K Fs
⋅ +
⋅ +
⋅
+
= ⋅
⋅ +
⋅ +
⋅ +
⋅ +
= ⋅
⋅ ⋅ + + ⋅
⋅ ⋅ +
⋅
=
AT1 SS 1994 Aufgabe 2:
a) :
sec ; 51 1 , sec 2
03 1 , 2
2 4
2 2 2
2 2
0
) (
2 x
x 1 2
: Feder
: Dämpfer
1 1
2
1 1 2
2 1
1 1 2
2 a
a
e a
a
e a
a
e a
a a
d c
c A
a e c
a c
a d
a a
x x
x
d x x c
d c d x c
x c c x
x c x
d
a F a F a F M
x x c F x
c F x
d x F
a x
a x v
dt v c F
v d F
⋅
⋅
=
⋅
⋅
+
⋅ ⋅
=
⋅
+ ⋅ +
⋅
=
⋅ +
⋅
⋅ +
⋅
⋅
⋅ ⇒
⋅ +
⋅
⋅ +
⋅
−
=
=
− ′
⋅
=
⋅
=
⋅
=
⋅
′ =
⋅ ⇒
′ =
=
⋅
⋅
=
⋅
=
∑
∫
&
&
&
&
&
b) :
; 1
23 , 1
sec) 5 , 0 1 ( 23 1 , sec) 1 5 , 0 1 (
23 , 1 1
sec 5 , 0 1
23 , 1 sec
03 1 , 2
sec 51 1 , 2
sec 5 , 0 )
(
) ( )
(
) (
) ( ) (
−
⋅
=
⋅
⋅ +
⋅ ⋅
⋅ =
⋅ +
= ⋅
⋅
=
⋅
⋅
= +
⋅ +
= ⋅
=
− ⋅t t
s s
s e
s a s
e h
s s s
F s H s
s s X
F X
c) :
AT1 SS 1994 Aufgabe 3:
a) :
( ) ( )
( )
(
0,6 3 21,2 2,6)
;1 3 6
, 2 2 , 2 6
, 0
2 3
) (
) ( ) (
) ( )
( 2
3
+
⋅ +
⋅ +
+
= ⋅
=
⋅ +
⋅
=
⋅ +
⋅ +
⋅ +
s s
s
s Z
F X
Z s X
s s
s
s s s
s
s s
b) :
( )
( )
( )
( )
( )
( ) ( )
1 ; 3 6
, 2 2 , 2 6
, 0
1 3 6
, 2 2 , 2 6
, 0
1 1 3
6 , 2 2 , 2 6
, 0
1 3 1
6 , 2 2 , 2 6
, 0
1 3
2 3
2 3
2 3
) ( ) (
) ( )
( ) (
2 ) 3
(
+
⋅
⋅ + +
⋅ +
⋅ +
+
= ⋅ + ⋅
⋅ +
⋅ +
+ + ⋅
+
⋅ +
⋅ +
+
⋅
⋅ =
= +
=
+
⋅ +
⋅ +
+
= ⋅
s K
s s
s
s s K
s s
s
s s
s
s F
F F F
K F
s s
s F s
R R s
R s s
s s s
z
R s R
s s
c) :
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
; 6 , 1 0
6 , 1
1 6
2 6 0 3
2 2
. 2
0 1
0 6
, 2 3
2 , 2 6
, 0 1
0 1
3 0 2 1
0 1
2 3
2 3
) ( ) (
>
<
>
⋅ +
>
⋅
⋅ +
⋅
>
⋅
>
= + +
⋅
⋅ + +
⋅ +
⋅
=
⋅ +
R R
R R
R
R R
s R s s
K K
K , , K ,
a a a a . K
a a
a a
K s
K s
s F F
AT1 SS 1994 Aufgabe 4:
a) :
); 5 1 ( ) 1 (
5 )
1 ) (
1 ( 1 5
) 1 (
5 1
) 1
(
2 2
) (
) (
) ( )
(
) ( ) ( ) (
) ( )
(
) ( ) (
) (
R R s
R s s
s z
s s s R s
R s s s
s
s s
s s
R s
K s s s
s K s F
F F Z
F X
F Z F F X
F X F Z
X
F A X
W X B
F B Z A
⋅ + +
⋅
= + +
⋅ + ⋅
+
= +
⋅
= +
=
⋅
=
⋅ +
⋅
⋅
−
=
⋅
=
−
=
⋅
−
=
b) :
1 ; 5
5 )
1 ( ) 5 1 ( 1
) 1 ( ) 5 1 ( 1
) (
) 1
(
2 2
) (
) (
) ( )
(
) ( )
( )
(
) ( )
(
) ( ) (
) (
s K
K s s
K s s K
F F
F F W
F X
W F F F
F X
F W F F X
X
F A X
W X B
F B Z A
R R
R R
s R
R s s
s W
s s R s
R s
R s R s s
s
s s
s s
R s
+ +
⋅
− ⋅
=
⋅ + +
⋅ +
⋅ + +
⋅
−
⋅ = +
− ⋅
=
=
⋅
⋅
−
=
⋅ +
⋅
⋅
−
⋅
−
=
⋅
=
−
=
⋅
−
=
c) :
; 8 , 5 3
1 05 1 , 1 0
1
) 0 ( ) 0 (
⇒ =
⋅
= +
⋅ =
= +
→
→
R R
s s s R
K K F
R F d) :
; 95 , ˆ 1
95 , 1 0 8 , 3 5
8 , 3 5
) 1
ˆ (
) (
) 0 (
) 0 ( )
(
⋅
=
− + =
⋅
− ⋅
=
−
⋅
=
∞
→
→
→
∞
→
w e
F
F w
e
t s w
s w t
AT1 SS 1994 Aufgabe 5:
a) :
1 ; 2
, 1
2
2
0 02 22
2 0
+
⋅
= +
⋅
⋅
⋅
⋅ +
= ⋅
s s
s D
s
F
sK
sω ω
ω
b) :
; 3 8 , 0
; 2 , 1
; 6 , 0 2 1
2 , 1 2
1 2
] 2 2 [ ] 2
[ 1 2 , 1
2 2 2
2 2 2
2 2 2
1 2 , 1
) 1 1 (
1 1 2
1 2
1 2 , 1
1 1 2
1 2 , 1
1 1 2
1 1
1 1 2
1 2 , 1
1 1 2
1 1 1
2 , 1 1 2
1 1 1
2 , 1
2 1
1 1
2 2
2 2
2 2
2 2
=
=
=
⋅ =
=
⋅
=
⋅
⋅
⋅ +
⋅
⋅ +
⋅
⋅
⋅
= +
⋅ +
⋅
⋅
⋅
⋅
⋅ +
⋅ + ⋅
⋅
⋅ +
⋅
⋅
⋅
⋅ + + ⋅
⋅
=
⋅
⋅
⋅
⋅ + + ⋅
⋅ + +
⋅ +
+
⋅
+ ⋅
+ ⋅
⋅
⋅
=
+ ⋅
+ ⋅
⋅
⋅ + +
⋅ +
+ ⋅
+ ⋅
⋅
⋅ + +
⋅ +
+ ⋅
+ ⋅
⋅
⋅ + =
+ ⋅
+ ⋅
⋅
⋅ + +
⋅ +
+ ⋅
+ ⋅
⋅
⋅
=
+ ⋅
+ ⋅
⋅ + ⋅
⋅ + +
+ ⋅
+ ⋅
⋅ + ⋅
⋅
= +
⋅ +
= ⋅
+ ⋅
+ ⋅
⋅
=
v n R
n R R
v R
n R R
v R
v R
n R R
v R n
R R
v R n
R R
v n
R v
n R
v n
R
v n
R
v n
R
v n
R
v n
R
v n
R
R s
R s w
v n
R R
T T K
T K K
T K
T s K K s
T K s
s
T s s T K
s s s K
K T
s T K
s K K
T s T K
s K K s
s
s T
T s K s
T T s
K s s
s
T T s
K s s
s
T T s
K s s
T T s
K s s
s
T T s
K s
T T s
K s s s
T T s
K s s s
F F
F F F
T T s
K s F
