94
V. Ionica et al.
P 2 (s) = T 2 I 1 s
2
+ R 1 c(T 1 R 2 + T 2 R 1 )s + R 1 k(T 1 R 2 + T 2 R 1 );
(12)
k ≈ 1
N
m
(13)
Applying the Laplace transform function to the functions (9), we obtain the desired
time functions in the form below:
θ i (t) =
1
2 f 3
A i (t) −
B i (t)
C
, i = 1, 2
( 1 4 )
where
A i (t) =
2c i e
2
− 2c i d f − 2b i e f + 2a i f
2
− 2c i e f + 2b i f
2 t + c i f
2 t
, i = 1, 2
(15)
B i (t) = 2
cosh
et
2d
− sinh
et
2d
· {C
f
−b i e + d
2
+ c i
e
2
− d f
· cosh
C
2d
t
+
f
−b i e
2
+ 2b i d f + a i e f
+ c i
e
3
− 3d f
· sinh
C
2d
t
;
(16)
C =
e 2 − 4d f ;
(17)
d = I 1 I 2 ; e = c
R 1 I 2 − I 1 R
2
2
; f = R
2
1 I 2 − I 1 R 2 ;
c 1 = −R 2 (T 1 R 2 + T 2 R 1 ); c 2 = R 1 (T 1 R 2 + T 2 R 1 );
a 1 = T 1 I 1 ; a 1 = T 2 I 1 ; b 1 = −R 2 c(T 1 R 2 + T 2 R 1 ); b 2 = R 1 c(T 1 R 2 + T 2 R 1 ) (18)
The application we proposed at the beginning of the paper led us to the graphical
representations in Fig. 2.
θ 1 [deg]
θ 2 [deg]
time [sec]
time [sec]
Fig. 2 Graphical representation of angular displacements θ 1 = θ 1 (t) and θ 2 = θ 2 (t)