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5 Variational Problems of Conditional Extrema
Example 5.3.5 There is a separately excited direct current motor controlled by the
armature, the equation of motion is
J
dω
dt
+ M f = C m I a
where, M f is the constant load torque, J is the moment of inertia, C m is a constant.
In the time t 1 , the motor starts from the stationary state, after turning a certain angle,
it stops, namely
ω(0) = ω(t 1 ) = 0,
t 1
0
ωdt = a (constant)
in the time interval [0, t 1 ], find the armature current I a which makes the loss of the
armature winding
Q =
t 1
0
I
2
a dt
be a minimum.
Solution Constructing the auxiliary functional
Q
∗
=
t 1
0
(I
2
a + λω)dt =
t 1
0
1
C 2
m
J
dω
dt
+ M f
2
+ λω
dt
(5.1)
The Euler equation of the functional is
λ −
2J
2
C 2
m
d
2
ω
dt 2 = 0
( 5 . 2 )
Managing Eq. (2) and integrating twice, we get
ω =
C
2
m λ
4J 2 t
2
+ c 1 t + c 2
(5.3)
From the boundary condition ω(0) = ω(t 1 ) = 0, we give c 2 = 0, c 1 = −
C
2
m λ
4J 2 t 1 ,
substituting them into the expression (3), we obtain
ω =
C
2
m λ
4J 2 t (t − t 1 )
(5.4)
Substituting ω into the isoperimetric condition, there is
t 1
0
ωdt =
C
2
m λ
4J 2
t 1
0
t (t − t 1 )dt = −
C
2
m λt
3
1
24J 2 = a
(5.5)