3. Method of Solution
The analysis of the dynamic response to a uniformly distributed moving mass of isotropic rectangular plates resting on a Winkler foundation and subjected to arbitrary boundary conditions is carried out in this section. The generalized two-dimensional integral transform is defined as
(7)
W
¯
(
j
,
k
,
t
)
=
∫
0
L
y
∫
0
L
x
W
(
x
,
y
,
t
)
W
j
(
x
)
W
k
(
y
)
d
x
d
y
with the inverse
(8)
W
(
x
,
y
,
t
)
=
∑
j
=
1
∞
∑
k
=
1
∞
μ
V
j
μ
V
k
W
-
(
j
,
k
,
t
)
W
j
(
x
)
W
k
(
y
)
,
where
(9)
V
j
=
∫
0
L
x
μ
W
j
2
(
x
)
d
x
,
V
k
=
∫
0
L
y
μ
W
k
2
(
y
)
d
y
.
W
j
(
x
)
and
W
k
(
y
)
are, respectively, the beam mode functions in the
x
and
y
directions defined, respectively, as
(10)
W
j
(
x
)
=
Sin
λ
j
x
L
x
+
A
j
Cos
λ
j
x
L
x
+
B
j
Sin
h
λ
j
x
L
x
+
C
j
Cos
h
λ
j
x
L
x
,
W
k
(
y
)
=
Sin
λ
k
y
L
y
+
A
k
Cos
λ
k
y
L
y
+
B
k
Sin
h
λ
k
y
L
y
+
C
k
Cos
h
λ
k
y
L
y
,
where,
A
j
,
B
j
,
C
j
,
A
k
,
B
k
, and
C
k
are constants and
λ
j
,
λ
k
are mode frequencies which are determined using the appropriate boundary conditions.
Applying the generalized two-dimensional integral transform (7), (5) takes the form
(11)
Z
A
F
1
(
0
,
L
x
,
L
y
,
t
)
+
Z
A
F
2
(
0
,
L
x
,
L
y
,
t
)
+
W
¯
t
t
(
j
,
k
,
t
)
-
Z
B
T
B
1
(
t
)
-
Z
B
T
B
2
(
t
)
+
Z
C
W
¯
(
j
,
k
,
t
)
+
T
B
3
(
t
)
+
T
B
4
(
t
)
+
T
B
5
(
t
)
=
M
g
μ
∫
0
L
y
∫
0
L
x
H
(
x
-
c
t
)
H
(
y
-
y
0
)
W
j
(
x
)
W
k
(
y
)
d
x
d
y
,
where
(12)
Z
A
=
D
μ
,
Z
B
=
R
0
,
Z
C
=
K
μ
,
(13)
F
1
(
0
,
L
x
,
L
y
,
t
)
=
∫
0
L
y
(
[
W
j
(
x
)
∂
3
W
(
x
,
y
,
t
)
∂
x
3
-
W
j
′
(
x
)
∂
2
W
(
x
,
y
,
t
)
∂
x
2
m
m
n
m
m
m
+
W
j
′
′
(
x
)
∂
W
(
x
,
y
,
t
)
∂
x
-
W
j
′
′
′
(
x
)
W
(
x
,
y
,
t
)
]
0
L
x
m
m
m
n
m
n
×
W
k
(
y
)
d
y
m
m
m
m
m
+
2
[
W
j
(
x
)
∂
W
(
x
,
y
,
t
)
∂
x
-
W
j
′
(
x
)
W
(
x
,
y
,
t
)
]
m
m
m
m
m
×
W
k
′
′
(
y
)
d
y
[
W
j
(
x
)
∂
W
(
x
,
y
,
t
)
∂
x
]
)
+
∫
0
L
x
(
2
[
W
k
(
y
)
∂
W
(
x
,
y
,
t
)
∂
y
-
W
k
′
(
y
)
W
(
x
,
y
,
t
)
]
0
L
y
m
m
m
m
×
W
j
′
′
(
x
)
d
x
m
m
m
m
+
[
W
k
(
y
)
∂
3
W
(
x
,
y
,
t
)
∂
y
3
h
h
h
h
h
h
-
W
k
′
(
y
)
∂
2
W
(
x
,
y
,
t
)
∂
y
2
+
W
k
′
′
(
y
)
∂
W
(
x
,
y
,
t
)
∂
y
m
m
m
m
m
n
∂
W
(
x
,
y
,
t
)
∂
y
-
W
k
′
′
′
(
y
)
W
(
x
,
y
,
t
)
]
0
L
y
W
j
(
x
)
d
x
∂
W
(
x
,
y
,
t
)
∂
y
]
0
L
y
)
,
(14)
F
2
(
0
,
L
x
,
L
y
,
t
)
=
∫
0
L
y
∫
0
L
x
(
N
^
W
(
x
,
y
,
t
)
W
j
′
υ
(
x
)
W
k
(
y
)
m
m
m
m
i
m
m
i
m
+
2
W
(
x
,
y
,
t
)
W
j
′
′
(
x
)
W
k
′
′
(
y
)
N
^
)
d
x
d
y
+
∫
0
L
y
∫
0
L
x
W
(
x
,
y
,
t
)
W
j
(
x
)
W
k
′
υ
(
y
)
d
x
d
y
,
(15)
T
B
1
(
t
)
=
∫
0
L
y
∫
0
L
x
∂
4
W
(
x
,
y
,
t
)
∂
t
2
∂
x
2
W
j
(
x
)
W
k
(
y
)
d
x
d
y
,
(16)
T
B
2
(
t
)
=
∫
0
L
y
∫
0
L
x
∂
4
W
(
x
,
y
,
t
)
∂
t
2
∂
y
2
W
j
(
x
)
W
k
(
y
)
d
x
d
y
,
(17)
T
B
3
(
t
)
=
M
μ
∫
0
L
y
∫
0
L
x
H
(
x
-
c
t
)
H
(
y
-
y
0
)
m
m
m
m
m
m
m
×
∂
2
W
(
x
,
y
,
t
)
∂
t
2
W
j
(
x
)
W
k
(
y
)
d
x
d
y
,
(18)
T
B
4
(
t
)
=
2
M
c
μ
∫
0
L
y
∫
0
L
x
H
(
x
-
c
t
)
H
(
y
-
y
0
)
m
m
m
m
m
m
m
×
∂
2
W
(
x
,
y
,
t
)
∂
t
∂
x
W
j
(
x
)
W
k
(
y
)
d
x
d
y
,
(19)
T
B
5
(
t
)
=
M
c
2
μ
∫
0
L
y
∫
0
L
x
H
(
x
-
c
t
)
H
(
y
-
y
0
)
m
m
m
m
m
m
m
×
∂
2
W
(
x
,
y
,
t
)
∂
x
2
W
j
(
x
)
W
k
(
y
)
d
x
d
y
.
It is recalled that the equation of the free vibration of a rectangular plate is given by
(20)
D
[
∂
4
W
(
x
,
y
,
t
)
∂
x
4
+
2
∂
4
W
(
x
,
y
,
t
)
∂
x
2
∂
y
2
+
∂
4
W
(
x
,
y
,
t
)
∂
y
4
]
+
μ
∂
2
W
(
x
,
y
,
t
)
∂
t
2
=
0
.
If the free vibration solution of the problem is set as
(21)
W
(
x
,
y
,
t
)
=
W
j
(
x
)
W
k
(
y
)
Sin
Ω
j
,
k
t
,
where
Ω
j
,
k
is the natural circular frequency of a rectangular plate, then substituting (21) into (20) yields
(22)
D
[
W
j
′
υ
(
x
)
W
k
(
y
)
+
2
W
j
′′
(
x
)
W
k
′′
(
y
)
+
W
j
(
x
)
W
k
′
υ
(
y
)
]
-
μ
Ω
j
,
k
2
W
j
(
x
)
W
k
(
y
)
=
0
.
It is well known that for a simply supported rectangular plate,
Ω
j
,
k
2
is given by
(23)
Ω
j
,
k
2
=
D
[
j
4
π
4
L
x
4
+
2
j
2
k
2
π
4
L
x
2
L
y
2
+
k
4
π
4
L
y
4
]
.
Multiplying (22) by
W
(
x
,
y
,
t
)
and integrating with respect to
x
and
y
between the limits 0,
L
x
and 0,
L
y
, respectively, we get
(24)
∫
0
L
y
∫
0
L
x
W
(
x
,
y
,
t
)
W
j
′
υ
(
x
)
W
k
(
y
)
d
x
d
y
+
2
∫
0
L
y
∫
0
L
x
W
(
x
,
y
,
t
)
W
j
′
′
(
x
)
W
k
′
′
(
y
)
d
x
d
y
+
∫
0
L
y
∫
0
L
x
W
(
x
,
y
,
t
)
W
j
(
x
)
W
k
′
υ
(
y
)
d
x
d
y
=
μ
D
Ω
j
,
k
2
∫
0
L
y
∫
0
L
x
W
(
x
,
y
,
t
)
W
j
(
x
)
W
k
(
y
)
d
x
d
y
.
In view of (14) we have
(25)
F
2
(
0
,
L
x
,
L
y
)
=
μ
D
Ω
j
,
k
2
W
¯
(
j
,
k
,
t
)
.
Since
W
¯
(
p
,
q
,
t
)
is just the coefficient of the generalized two-dimensional finite integral transform
(26)
W
(
x
,
y
,
t
)
=
∑
p
=
1
∞
∑
q
=
1
∞
μ
V
p
μ
V
q
W
¯
(
p
,
q
,
t
)
W
p
(
x
)
W
q
(
y
)
,
it follows that
(27)
W
′′
(
x
,
y
,
t
)
=
∑
p
=
1
∞
∑
q
=
1
∞
μ
V
p
μ
V
q
W
¯
(
p
,
q
,
t
)
W
p
′′
(
x
)
W
q
(
y
)
.
Therefore, the integrals (15) and (16) can be rewritten as
(28)
T
B
1
(
t
)
=
∑
p
=
1
∞
∑
q
=
1
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
H
2
(
p
,
j
)
F
(
q
,
k
)
,
where
(29)
H
2
(
p
,
j
)
=
∫
0
L
x
W
p
′′
(
x
)
W
j
(
x
)
d
x
,
F
(
q
,
k
)
=
∫
0
L
y
W
q
(
y
)
W
k
(
y
)
d
y
,
(30)
T
B
2
(
t
)
=
∑
p
=
1
∞
∑
q
=
1
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
H
(
p
,
j
)
F
2
(
q
,
k
)
,
where
(31)
H
(
p
,
j
)
=
∫
0
L
x
W
p
(
x
)
W
j
(
x
)
d
x
,
F
2
(
q
,
k
)
=
∫
0
L
y
W
q
′′
(
y
)
W
k
(
y
)
d
y
.
In order to evaluate the integrals (17), (18), and (19), use is made of the Fourier series representation of the Heaviside function; namely,
(32)
H
(
x
-
c
t
)
=
1
4
+
1
π
∑
n
=
0
∞
Sin
(
(
2
n
+
1
)
π
(
x
-
c
t
)
)
2
n
+
1
.
Similarly,
(33)
H
(
y
-
y
0
)
=
1
4
+
1
π
∑
m
=
0
∞
Sin
(
(
2
m
+
1
)
π
(
y
-
y
0
)
)
2
m
+
1
.
Using (30) and (31), one obtains
(34)
T
B
3
(
t
)
=
M
16
μ
∑
p
=
1
∞
∑
q
=
1
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
F
(
q
,
k
)
H
(
p
,
j
)
+
M
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
×
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
s
(
n
,
p
,
j
)
F
(
q
,
k
)
-
M
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
×
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
c
(
n
,
p
,
j
)
F
(
q
,
k
)
+
M
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
×
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
F
s
(
m
,
q
,
k
)
H
(
p
,
j
)
-
M
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
×
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
F
c
(
m
,
q
,
k
)
H
(
p
,
j
)
+
M
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
m
m
×
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
×
H
s
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
-
M
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
m
m
×
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
×
H
s
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
-
M
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
i
m
m
m
m
m
×
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
i
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
i
m
m
m
m
m
×
H
c
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
+
M
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
n
m
m
m
m
m
×
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
n
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
n
m
m
m
×
H
c
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
,
where
(35)
H
s
(
n
,
p
,
j
)
=
∫
0
L
x
Sin
(
2
n
+
1
)
π
x
W
p
(
x
)
W
j
(
x
)
d
x
,
H
c
(
n
,
p
,
j
)
=
∫
0
L
x
Cos
(
2
n
+
1
)
π
x
W
p
(
x
)
W
j
(
x
)
d
x
,
F
s
(
m
,
q
,
k
)
=
∫
0
L
y
Sin
(
2
m
+
1
)
π
y
W
q
(
y
)
W
k
(
y
)
d
y
,
F
c
(
m
,
q
,
k
)
=
∫
0
L
y
Cos
(
2
m
+
1
)
π
y
W
q
(
y
)
W
k
(
y
)
d
y
.
Similarly,
(36)
T
B
4
(
t
)
=
2
M
c
16
μ
∑
p
=
1
∞
∑
q
=
1
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
H
1
(
p
,
j
)
F
(
q
,
k
)
+
2
M
c
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
×
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
1
s
(
n
,
p
,
j
)
F
(
q
,
k
)
-
2
M
c
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
m
×
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
1
c
(
n
,
p
,
j
)
F
(
q
,
k
)
+
2
M
c
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
×
F
s
(
m
,
q
,
k
)
H
1
(
p
,
j
)
-
2
M
c
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
×
F
c
(
m
,
q
,
k
)
H
1
(
p
,
j
)
+
2
M
c
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
×
H
1
s
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
-
2
M
c
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
×
H
1
s
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
-
2
M
c
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
×
H
1
c
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
+
2
M
c
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
×
H
1
c
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
,
where
(37)
H
1
(
p
,
j
)
=
∫
0
L
x
W
p
′
(
x
)
W
j
(
x
)
d
x
,
H
1
s
(
n
,
p
,
j
)
=
∫
0
L
x
Sin
(
2
n
+
1
)
π
x
W
p
′
(
x
)
W
j
(
x
)
d
x
,
H
1
c
(
n
,
p
,
j
)
=
∫
0
L
x
Cos
(
2
n
+
1
)
π
x
W
p
′
(
x
)
W
j
(
x
)
d
x
,
T
B
5
(
t
)
=
M
c
2
16
μ
∑
p
=
1
∞
∑
q
=
1
∞
μ
2
V
p
V
q
W
-
(
p
,
q
,
t
)
H
2
(
p
,
j
)
F
(
q
,
k
)
+
M
c
2
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
×
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
2
s
(
n
,
p
,
j
)
F
(
q
,
k
)
-
M
c
2
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
m
×
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
2
c
(
n
,
p
,
j
)
F
(
q
,
k
)
+
M
c
2
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
m
m
m
m
m
m
n
m
m
m
×
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
n
×
F
s
(
m
,
q
,
k
)
H
2
(
p
,
j
)
-
M
c
2
4
μ
π
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
i
m
×
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
i
×
F
c
(
m
,
q
,
k
)
H
2
(
p
,
j
)
+
M
c
2
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
×
H
2
s
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
-
M
c
2
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
×
H
2
s
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
-
M
c
2
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
×
H
2
c
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
+
M
c
2
μ
π
2
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
μ
2
V
p
V
q
W
¯
t
(
p
,
q
,
t
)
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
n
i
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
mmmmmmnmmmm
×
H
2
c
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
,
where
(38)
H
2
s
(
n
,
p
,
j
)
=
∫
0
L
x
Sin
(
2
n
+
1
)
π
x
W
p
′′
(
x
)
W
j
(
x
)
d
x
,
H
2
c
(
n
,
p
,
j
)
=
∫
0
L
x
Cos
(
2
n
+
1
)
π
x
W
p
′′
(
x
)
W
j
(
x
)
d
x
,
M
g
μ
∫
0
L
y
∫
0
L
x
H
(
x
-
c
t
)
H
(
y
-
y
0
)
W
j
(
x
)
W
k
(
y
)
d
x
d
y
=
M
g
L
x
L
y
μ
λ
j
λ
k
[
B
f
(
λ
,
j
)
+
Cos
λ
j
c
t
L
x
-
A
j
Sin
λ
j
c
t
L
x
m
m
m
m
m
m
n
m
n
-
B
j
Cos
h
λ
j
c
t
L
x
-
C
j
Sin
h
λ
j
c
t
L
x
]
·
[
B
f
(
λ
,
k
)
+
Cos
λ
k
y
0
L
y
-
A
k
Sin
λ
k
y
0
L
y
m
i
m
m
n
-
B
k
Cos
h
λ
k
y
0
L
y
-
C
k
Sin
h
λ
k
y
0
L
y
]
,
where
(39)
B
f
(
λ
,
j
)
=
-
Cos
λ
j
+
A
j
Sin
λ
j
+
B
j
Cos
h
λ
j
+
C
j
Sin
h
λ
j
,
B
f
(
λ
,
k
)
=
-
Cos
λ
k
+
A
k
Sin
λ
k
+
B
k
Cos
h
λ
k
+
C
k
Sin
h
λ
k
.
Substituting the above expressions in (11) and simplifying, we obtain
(40)
W
¯
t
t
(
j
,
k
,
t
)
+
Ω
j
,
k
2
W
¯
(
j
,
k
,
t
)
+
K
μ
W
¯
(
j
,
k
,
t
)
-
R
0
[
∑
p
=
1
∞
∑
q
=
1
∞
W
-
t
t
(
p
,
q
,
t
)
H
2
(
p
,
j
)
F
(
q
,
k
)
m
m
m
n
m
+
∑
p
=
1
∞
∑
q
=
1
∞
W
¯
t
t
(
p
,
q
,
t
)
H
(
p
,
j
)
F
2
(
q
,
k
)
]
+
Γ
1
L
x
L
y
{
{
1
16
[
∑
p
=
1
∞
∑
q
=
1
∞
F
(
q
,
k
)
H
(
p
,
j
)
]
m
m
m
m
n
m
n
m
m
+
1
4
π
[
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
s
(
n
,
p
,
j
)
F
(
q
,
k
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
c
(
n
,
p
,
j
)
F
(
q
,
k
)
C
o
s
(
2
n
+
1
)
π
c
t
2
n
+
1
)
m
m
m
m
m
m
m
+
∑
p
=
1
∞
∑
q
=
1
∞
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
F
s
(
m
,
q
,
k
)
H
(
p
,
j
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
n
m
m
m
×
F
c
(
m
,
q
,
k
)
H
(
p
,
j
)
C
o
s
(
2
m
+
1
)
π
y
0
2
m
+
1
)
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
(
C
o
s
(
2
n
+
1
)
π
c
t
2
n
+
1
]
m
m
m
m
m
m
m
+
1
π
2
[
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
n
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
n
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
s
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
m
m
m
m
m
m
m
m
m
m
m
-
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
n
×
H
s
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
m
m
m
m
m
m
m
m
m
m
m
-
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
c
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
m
m
m
m
m
m
m
m
m
m
m
+
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
c
(
n
,
p
,
j
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
F
c
(
m
,
q
,
k
)
[
∑
p
=
1
∞
∑
q
=
1
∞
F
(
q
,
k
)
H
(
p
,
j
)
]
]
}
m
m
m
m
m
m
×
W
-
t
t
(
p
,
q
,
t
)
m
m
m
m
m
m
+
2
c
{
1
16
[
∑
p
=
1
∞
∑
q
=
1
∞
H
1
(
p
,
j
)
F
(
q
,
k
)
]
+
1
4
π
m
m
m
m
m
m
m
i
m
m
×
[
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
i
m
m
m
n
m
m
m
m
×
H
1
s
(
n
,
p
,
j
)
F
(
q
,
k
)
m
m
m
m
m
m
m
m
m
m
i
m
n
m
m
m
m
m
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
n
i
m
m
m
m
m
m
×
H
1
c
(
n
,
p
,
j
)
F
(
q
,
k
)
C
o
s
(
2
n
+
1
)
π
c
t
2
n
+
1
)
m
m
m
m
m
m
m
m
m
m
m
+
∑
p
=
1
∞
∑
q
=
1
∞
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
F
s
(
m
,
q
,
k
)
H
1
(
p
,
j
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
F
c
(
m
,
q
,
k
)
H
1
(
p
,
j
)
C
o
s
(
2
m
+
1
)
π
y
0
2
m
+
1
)
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
(
C
o
s
(
2
n
+
1
)
π
c
t
2
n
+
1
]
m
m
m
m
m
m
+
1
π
2
[
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
1
s
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
m
m
m
m
m
m
m
m
m
-
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
1
s
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
m
m
m
m
m
m
m
m
m
-
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
1
c
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
m
m
m
m
m
m
m
m
m
+
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
n
m
n
m
n
m
n
m
n
m
m
m
i
m
m
×
H
1
c
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
[
∑
p
=
1
∞
∑
q
=
1
∞
H
2
(
p
,
j
)
F
(
q
,
k
)
]
]
}
m
m
m
m
m
×
W
-
t
(
p
,
q
,
t
)
m
m
m
m
m
+
c
2
{
1
16
[
∑
p
=
1
∞
∑
q
=
1
∞
H
2
(
p
,
j
)
F
(
q
,
k
)
]
m
m
m
m
m
m
m
m
+
1
4
π
[
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
m
×
H
2
s
(
n
,
p
,
j
)
F
(
q
,
k
)
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
i
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
2
c
(
n
,
p
,
j
)
F
(
q
,
k
)
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
)
m
m
m
m
m
m
m
m
m
m
+
∑
p
=
1
∞
∑
q
=
1
∞
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
i
m
m
m
m
×
F
s
(
m
,
q
,
k
)
H
2
(
p
,
j
)
m
m
m
m
m
m
m
m
m
m
m
m
i
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
n
i
m
m
×
F
c
(
m
,
q
,
k
)
H
2
(
p
,
j
)
C
o
s
(
2
m
+
1
)
π
y
0
2
m
+
1
)
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
(
C
o
s
(
2
n
+
1
)
π
c
t
2
n
+
1
]
m
m
m
m
m
+
1
π
2
[
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
2
s
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
m
m
m
n
m
m
m
m
n
-
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
×
H
2
s
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
m
m
m
m
n
m
m
m
n
-
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
n
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
n
×
H
2
c
(
n
,
p
,
j
)
F
s
(
m
,
q
,
k
)
m
m
m
m
n
m
m
m
n
+
∑
p
=
1
∞
∑
q
=
1
∞
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
m
m
m
m
m
m
m
m
m
m
n
m
m
m
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
n
m
m
m
×
H
2
c
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
[
∑
p
=
1
∞
∑
q
=
1
∞
H
2
(
p
,
j
)
F
(
q
,
k
)
]
]
}
}
×
W
-
(
p
,
q
,
t
)
=
M
g
L
x
L
y
μ
λ
j
λ
k
×
[
B
f
(
λ
,
j
)
+
Cos
λ
j
c
t
L
x
-
A
j
Sin
λ
j
c
t
L
x
m
m
m
m
-
B
j
Cos
h
λ
j
c
t
L
x
-
C
j
Sin
h
λ
j
c
t
L
x
]
·
[
B
f
(
λ
,
k
)
+
Cos
λ
k
y
0
L
y
-
A
k
Sin
λ
k
y
0
L
y
m
m
m
m
-
B
k
Cos
h
λ
k
y
0
L
y
-
C
k
Sin
h
λ
k
y
0
L
y
]
,
where
(41)
Γ
1
=
M
μ
L
x
L
y
.
Equation (40) is the generalized transformed coupled nonhomogeneous second order ordinary differential equation describing the flexural vibration of the rectangular plate under the action of uniformly distributed loads travelling at constant velocity. It is now the fundamental equation of our dynamical problem and holds for all arbitrary boundary conditions. The initial conditions are given by
(42)
W
¯
(
j
,
k
,
0
)
=
0
,
W
¯
t
(
j
,
k
,
0
)
=
0
.
In what follows, two special cases of (40) are discussed.
4. Solution of the Transformed Equation
4.1. Isotropic Rectangular Plate Traversed by Uniformly Distributed Moving Force
An approximate model which assumes the inertia effect of the uniformly distributed moving mass
M
as negligible is obtained when the mass ratio
Γ
1
is set to zero in (40). Thus, setting
Γ
1
=
0
in (40), one obtains
(43)
W
¯
t
t
(
j
,
k
,
t
)
+
Ω
j
,
k
2
W
¯
(
j
,
k
,
t
)
-
R
0
[
∑
p
=
1
∞
∑
q
=
1
∞
H
2
(
p
,
j
)
F
(
q
,
k
)
+
H
(
p
,
j
)
F
2
(
q
,
k
)
]
×
W
¯
t
t
(
p
,
q
,
t
)
+
K
μ
W
¯
(
j
,
k
,
t
)
=
M
g
L
x
L
y
μ
λ
j
λ
k
[
B
f
(
λ
,
j
)
+
Cos
λ
j
c
t
L
x
-
A
j
Sin
λ
j
c
t
L
x
m
m
m
m
m
n
m
m
m
-
B
j
Cos
h
λ
j
c
t
L
x
-
C
j
Sin
h
λ
j
c
t
L
x
]
·
[
B
f
(
λ
,
k
)
+
Cos
λ
k
y
0
L
y
-
A
k
Sin
λ
k
y
0
L
y
m
m
m
m
-
B
k
Cos
h
λ
k
y
0
L
y
-
C
k
Sin
h
λ
k
y
0
L
y
]
.
This represents the classical case of a uniformly distributed moving force problem associated with our dynamical system. Evidently, an analytical solution to (43) is not possible. Consequently, we resort to the modified asymptotic technique due to Struble discussed in Nayfeh [24]. By this technique, we seek the modified frequency corresponding to the frequency of the free system due to the presence of the effect of the rotatory inertia. An equivalent free system operator defined by the modified frequency then replaces (43). To this end, (43) is rearranged to take the form
(44)
W
¯
t
t
(
j
,
k
,
t
)
+
(
Ω
m
f
*
2
(
1
-
ε
1
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
)
)
×
W
¯
(
j
,
k
,
t
)
-
ε
1
(
1
-
ε
1
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
)
×
∑
p
=
1
p
≠
j
∞
∑
q
=
1
q
≠
k
∞
L
x
L
y
{
H
2
(
p
,
j
)
F
(
q
,
k
)
+
H
(
p
,
j
)
F
2
(
q
,
k
)
}
·
W
¯
t
t
(
p
,
q
,
t
)
=
P
1
(
1
-
ε
1
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
)
·
[
B
f
(
λ
,
j
)
+
Cos
λ
j
c
t
L
x
-
A
j
Sin
λ
j
c
t
L
x
m
m
m
m
-
B
j
Cos
h
λ
j
c
t
L
x
-
C
j
Sin
h
λ
j
c
t
L
x
]
,
where
(45)
Ω
m
f
*
2
=
(
Ω
j
,
k
2
+
K
μ
)
,
ε
1
=
R
0
L
x
L
y
,
P
1
=
M
g
L
x
L
y
μ
λ
j
λ
k
W
(
λ
k
,
y
0
)
,
W
(
λ
k
,
y
0
)
=
B
f
(
λ
,
k
)
+
Cos
λ
k
y
0
L
y
-
A
k
Sin
λ
k
y
0
L
y
-
B
k
Cos
h
λ
k
y
0
L
y
-
C
k
Sin
h
λ
k
y
0
L
y
.
We now set the right hand side of (44) to zero and consider a parameter
η
<
1
, for any arbitrary ratio
ε
1
, defined as
(46)
η
=
ε
1
1
+
ε
1
.
Then,
(47)
ε
1
=
η
+
o
(
η
2
)
.
And, consequently, we have that
(48)
1
(
1
-
η
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
)
=
1
+
η
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
+
o
(
η
2
)
,
where
(49)
|
η
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
|
<
1
.
Substituting (47) and (48) into the homogeneous part of (44) one obtains
(50)
W
¯
t
t
(
j
,
k
,
t
)
+
(
Ω
m
f
*
2
+
η
Ω
m
f
*
2
L
x
L
y
m
m
m
×
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
Ω
m
f
*
2
)
W
¯
(
j
,
k
,
t
)
-
η
{
∑
p
=
1
p
≠
j
∞
∑
q
=
1
q
≠
k
∞
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
}
×
W
¯
t
t
(
p
,
q
,
t
)
=
0
.
When
η
is set to zero in (50) one obtains a situation corresponding to the case in which the rotatory inertia effect on the vibrations of the plate is regarded as negligible. In such a case, the solution of (50) can be obtained as
(51)
W
¯
(
j
,
k
,
t
)
=
C
m
f
Cos
[
Ω
m
f
*
t
-
ϕ
m
f
]
,
where
C
m
f
and
ϕ
m
f
are constants.
Since
η
<
1
, Struble’s technique requires that the solution to (50) be of the form [24]
(52)
W
¯
(
j
,
k
,
t
)
=
A
(
j
,
k
,
t
)
Cos
[
Ω
m
f
*
t
-
ϕ
(
j
,
k
,
t
)
]
+
η
W
¯
1
(
j
,
k
,
t
)
+
o
(
η
2
)
,
where
A
(
j
,
k
,
t
)
and
ϕ
(
j
,
k
,
t
)
are slowly varying functions of time. To obtain the modified frequency, (52) and its derivatives are substituted into (50). After some simplifications and rearrangements, one obtains
(53)
[
2
Ω
m
f
*
ϕ
˙
(
j
,
k
,
t
)
+
η
Ω
m
f
*
2
L
x
L
y
×
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
ϕ
˙
(
j
,
k
,
t
)
+
η
Ω
m
f
*
2
]
×
A
(
j
,
k
,
t
)
Cos
[
Ω
m
f
*
t
-
ϕ
(
j
,
k
,
t
)
]
-
2
A
˙
(
j
,
k
,
t
)
Ω
m
f
*
Sin
[
Ω
m
f
*
t
-
ϕ
(
j
,
k
,
t
)
]
-
η
∑
p
=
1
p
≠
j
∞
∑
q
=
1
q
≠
k
∞
{
[
Ω
m
f
*
t
-
ϕ
(
p
,
q
,
t
)
]
L
x
L
y
{
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
}
m
m
m
m
m
m
m
·
{
2
Ω
m
f
*
ϕ
˙
(
p
,
q
,
t
)
-
Ω
m
f
*
2
}
A
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
×
Cos
[
Ω
m
f
*
t
-
ϕ
(
j
,
k
,
t
)
]
m
m
m
m
m
m
m
m
-
2
A
˙
(
p
,
q
,
t
)
Ω
m
f
*
Sin
[
Ω
m
f
*
t
-
ϕ
(
p
,
q
,
t
)
]
}
neglecting terms to
o
(
η
2
)
.
The variational equations are obtained by equating the coefficients of
Sin
[
Ω
m
f
*
t
-
ϕ
(
j
,
k
,
t
)
]
and
Cos
[
Ω
m
f
*
t
-
ϕ
(
j
,
k
,
t
)
]
on both sides of (53) to zero; thus, one obtains
(54)
-
2
A
˙
(
j
,
k
,
t
)
Ω
m
f
*
=
0
,
[
2
Ω
m
f
*
ϕ
˙
(
j
,
k
,
t
)
+
η
Ω
m
f
*
2
L
x
L
y
×
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
Ω
m
f
*
]
=
0
.
Evaluating (54) one obtains
(55)
A
(
j
,
k
,
t
)
=
A
j
k
,
ϕ
(
j
,
k
,
t
)
=
-
η
Ω
m
f
*
L
x
L
y
2
×
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
t
+
ϕ
f
,
where
A
j
k
and
ϕ
f
are constants.
Therefore, when the effect of the rotatory inertia correction factor is considered, the first approximation to the homogeneous system is given by
(56)
V
(
j
,
k
,
t
)
=
A
j
k
Cos
[
γ
p
m
f
t
-
ϕ
f
]
,
where
(57)
γ
p
m
f
=
Ω
m
f
*
×
{
1
+
η
2
(
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
)
}
represents the modified natural frequency due to the presence of rotatory inertia correction factor. It is observed that when
η
=
0
, we recover the frequency of the moving force problem when the rotatory inertia effect of the plate is neglected. Thus, to solve the nonhomogeneous equation (44), the differential operator which acts on
W
¯
(
j
,
k
,
t
)
and
W
¯
(
p
,
q
,
t
)
is now replaced by the equivalent free system operator defined by the modified frequency
γ
p
m
f
; that is,
(58)
W
¯
t
t
(
j
,
k
,
t
)
+
γ
p
m
f
2
W
¯
(
j
,
k
,
t
)
=
P
p
m
f
[
B
f
(
λ
,
j
)
+
Cos
λ
j
c
t
L
x
-
A
j
Sin
λ
j
c
t
L
x
m
m
m
m
m
m
-
B
j
Cos
h
λ
j
c
t
L
x
-
C
j
Sin
h
λ
j
c
t
L
x
]
,
where
(59)
P
p
m
f
=
P
1
(
1
-
η
L
x
L
y
[
H
2
(
j
,
j
)
F
(
k
,
k
)
+
H
(
j
,
j
)
F
2
(
k
,
k
)
]
)
.
To obtain the solution to (58), it is subjected to a Laplace transform. Thus, solving (58) in conjunction with the initial condition, the solution is given by
(60)
V
-
(
j
,
k
,
t
)
=
P
p
m
f
[
C
j
(
γ
p
m
f
Sin
h
α
j
t
-
α
j
Sin
γ
p
m
f
t
)
γ
p
m
f
(
α
c
2
+
γ
p
m
f
2
)
B
f
(
λ
,
j
)
(
1
-
Cos
γ
p
m
f
t
)
γ
p
m
f
+
Cos
α
j
t
-
Cos
γ
p
m
f
t
γ
p
m
f
2
-
α
j
2
m
m
m
m
m
m
-
A
j
(
γ
p
m
f
Sin
α
j
t
-
α
j
Sin
γ
p
m
f
t
)
γ
p
m
f
(
γ
p
m
f
2
-
α
j
2
)
m
m
m
m
m
m
-
B
j
(
Cos
h
α
j
t
-
Cos
γ
p
m
f
t
)
α
j
2
+
γ
p
m
f
2
m
m
m
m
m
m
-
C
j
(
γ
p
m
f
Sin
h
α
j
t
-
α
j
Sin
γ
p
m
f
t
)
γ
p
m
f
(
α
c
2
+
γ
p
m
f
2
)
]
,
where
(61)
α
j
=
λ
j
c
L
x
.
Substituting (60) into (8), we have
(62)
W
(
x
,
y
,
t
)
=
∑
j
=
1
∞
∑
k
=
1
∞
1
V
j
V
k
P
p
m
f
[
C
j
(
γ
p
m
f
Sin
h
α
j
t
-
α
j
Sin
γ
p
m
f
t
)
γ
p
m
f
(
α
c
2
+
γ
p
m
f
2
)
B
f
(
λ
,
j
)
(
1
-
Cos
γ
p
m
f
t
)
γ
p
m
f
m
m
m
m
m
m
m
m
m
i
m
+
Cos
α
j
t
-
Cos
γ
p
m
f
t
γ
p
m
f
2
-
α
j
2
m
m
m
m
m
m
m
m
i
m
m
-
A
j
(
γ
p
m
f
Sin
α
j
t
-
α
j
Sin
γ
p
m
f
t
)
γ
p
m
f
(
γ
p
m
f
2
-
α
j
2
)
m
m
m
m
m
m
m
i
m
m
m
-
B
j
(
Cos
h
α
j
t
-
Cos
γ
p
m
f
t
)
α
j
2
+
γ
p
m
f
2
m
m
m
m
m
m
m
m
i
m
m
-
C
j
(
γ
p
m
f
Sin
h
α
j
t
-
α
j
Sin
γ
p
m
f
t
)
γ
p
m
f
(
α
j
2
+
γ
p
m
f
2
)
]
·
(
Sin
λ
j
x
L
x
+
A
j
Cos
λ
j
x
L
x
+
B
j
Sin
h
λ
j
x
L
x
+
C
j
Cos
h
λ
j
x
L
x
)
·
(
Sin
λ
k
y
L
y
+
A
k
Cos
λ
k
y
L
y
+
B
k
Sin
h
λ
k
y
L
y
+
C
k
Cos
h
λ
k
y
L
y
)
.
Equation (62) represents the transverse displacement response to a uniformly distributed force moving at constant velocity for arbitrary boundary conditions of a rectangular plate incorporating the effects of rotatory inertia correction factor and resting on elastic foundation.
4.2. Isotropic Rectangular Plate Traversed by Uniformly Distributed Moving Mass
In this section, we consider the case in which the mass of the structure and that of the load are of comparable magnitude. Under such condition, the inertia effect of the uniformly distributed moving mass is not negligible. Thus,
Γ
1
≠
0
, and the solution to the entire equation (40) is required. This gives the moving mass problem. An exact analytical solution to this dynamical problem is not possible. Again, we resort to the modified asymptotic technique due to Struble discussed in Nayfeh [24]. Evidently, the homogenous part of (40) can be replaced by a free system operator defined by the modified frequency due to the presence of rotatory inertia correction factor
R
0
. To this end, (40) can now be rearranged to take the form
(63)
W
¯
t
t
(
j
,
k
,
t
)
+
2
c
Γ
1
L
x
L
y
G
2
(
j
,
k
,
t
)
1
+
Γ
1
L
x
L
y
G
1
(
j
,
k
,
t
)
W
¯
t
(
j
,
k
,
t
)
+
c
2
Γ
1
L
x
L
y
G
3
(
j
,
k
,
t
)
+
γ
p
m
f
2
1
+
Γ
1
L
x
L
y
G
1
(
j
,
k
,
t
)
W
¯
(
j
,
k
,
t
)
+
Γ
1
L
x
L
y
1
+
Γ
1
L
x
L
y
G
1
(
j
,
k
,
t
)
×
∑
p
=
1
p
≠
j
∞
∑
q
=
1
q
≠
k
∞
{
G
a
(
p
,
q
,
t
)
W
¯
t
t
(
p
,
q
,
t
)
+
2
c
G
b
(
p
,
q
,
t
)
m
m
m
m
m
m
m
m
×
W
¯
t
(
p
,
q
,
t
)
+
c
2
G
c
(
p
,
q
,
t
)
W
¯
(
p
,
q
,
t
)
}
=
M
g
L
x
L
y
W
(
λ
k
,
y
0
)
μ
λ
j
λ
k
[
1
+
Γ
1
L
x
L
y
G
1
(
j
,
k
,
t
)
]
×
[
B
f
(
λ
,
j
)
+
Cos
λ
j
c
t
L
x
-
A
j
Sin
λ
j
c
t
L
x
m
m
m
n
m
-
B
j
Cos
h
λ
j
c
t
L
x
-
C
j
Sin
h
λ
j
c
t
L
x
]
,
where
(64)
G
1
(
j
,
k
,
t
)
=
1
16
F
(
k
,
k
)
H
(
j
,
j
)
+
1
4
π
[
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
s
(
n
,
j
,
j
)
F
(
k
,
k
)
m
m
m
m
m
m
m
m
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
c
(
n
,
j
,
j
)
F
(
k
,
k
)
)
m
m
m
m
m
n
+
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
F
s
(
m
,
k
,
k
)
H
(
j
,
j
)
m
m
m
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
F
c
(
m
,
k
,
k
)
H
(
j
,
j
)
)
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
s
(
n
,
j
,
j
)
]
-
1
π
2
[
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
×
H
s
(
n
,
j
,
j
)
F
c
(
m
,
k
,
k
)
-
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
×
H
c
(
n
,
j
,
j
)
F
s
(
m
,
k
,
k
)
+
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
×
H
c
(
n
,
p
,
j
)
F
c
(
m
,
q
,
k
)
[
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
]
G
2
(
j
,
k
,
t
)
=
1
16
F
(
k
,
k
)
H
1
(
j
,
j
)
+
1
4
π
[
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
1
s
(
n
,
j
,
j
)
F
(
k
,
k
)
m
m
m
m
m
n
n
m
m
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
1
c
(
n
,
j
,
j
)
F
(
k
,
k
)
)
m
i
m
m
m
m
+
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
F
s
(
m
,
k
,
k
)
H
1
(
j
,
j
)
m
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
F
c
(
m
,
k
,
k
)
H
1
(
j
,
j
)
)
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
1
s
(
n
,
j
,
j
)
F
(
k
,
k
)
]
+
1
π
2
[
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
×
H
1
s
(
n
,
j
,
j
)
F
s
(
m
,
k
,
k
)
m
m
m
i
m
m
-
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
×
H
1
s
(
n
,
j
,
j
)
F
c
(
m
,
k
,
k
)
m
m
i
m
m
m
-
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
×
H
1
c
(
n
,
j
,
j
)
F
s
(
m
,
k
,
k
)
m
m
i
m
m
m
+
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
×
H
1
c
(
n
,
j
,
j
)
F
c
(
m
,
k
,
k
)
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
]
G
3
(
j
,
k
,
t
)
=
1
16
F
(
k
,
k
)
H
2
(
j
,
j
)
+
1
4
π
[
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
2
s
(
n
,
j
,
j
)
F
(
k
,
k
)
m
m
m
m
m
m
m
i
m
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
2
c
(
n
,
j
,
j
)
F
(
k
,
k
)
)
m
m
m
m
+
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
F
s
(
m
,
k
,
k
)
H
2
(
j
,
j
)
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
F
c
(
m
,
k
,
k
)
H
2
(
j
,
j
)
)
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
2
s
(
n
,
j
,
j
)
F
(
k
,
k
)
]
+
1
π
2
[
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
×
H
2
s
(
n
,
j
,
j
)
F
s
(
m
,
k
,
k
)
m
m
m
m
i
i
m
-
∑
n
=
0
∞
∑
m
=
0
∞
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
×
H
2
s
(
n
,
j
,
j
)
F
c
(
m
,
k
,
k
)
m
m
m
i
i
m
m
-
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
m
m
m
m
m
m
m
×
H
2
c
(
n
,
j
,
j
)
F
s
(
m
,
k
,
k
)
m
m
i
m
i
m
m
+
∑
n
=
0
∞
∑
m
=
0
∞
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
·
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
m
m
m
i
i
m
m
×
H
2
c
(
n
,
j
,
j
)
F
c
(
m
,
k
,
k
)
∑
m
=
0
∞
sin
(
2
n
+
1
)
π
c
t
2
n
+
1
]
,
G
a
(
p
,
q
,
t
)
=
G
1
(
j
,
k
,
t
)
|
j
=
p
,
k
=
q
,
G
b
(
p
,
q
,
t
)
=
G
2
(
j
,
k
,
t
)
|
j
=
p
,
k
=
q
,
G
c
(
p
,
q
,
t
)
=
G
3
(
j
,
k
,
t
)
|
j
=
p
,
k
=
q
.
Going through similar argument as in the previous section, we obtain the first approximation to the homogenous system when the effect of the mass of the uniformly distributed load is considered as
(65)
W
¯
(
j
,
k
,
t
)
=
Δ
j
k
e
-
β
(
j
,
k
)
t
Cos
[
γ
m
m
t
-
ϕ
m
]
,
where
ϕ
m
and
Δ
j
k
are constants. Consider
(66)
β
(
j
,
k
)
t
=
c
Γ
a
L
x
L
y
H
c
(
j
,
k
)
,
Γ
a
=
Γ
1
1
+
Γ
1
,
where
(67)
γ
m
m
=
γ
p
m
f
{
1
-
Γ
a
L
x
L
y
2
[
H
b
(
j
,
k
)
-
c
2
H
a
(
j
,
k
)
γ
p
m
f
2
]
}
is called the modified natural frequency representing the frequency of the free system due to the presence of the uniformly distributed moving mass and
(68)
H
a
(
j
,
k
)
=
1
16
F
(
k
,
k
)
H
2
(
j
,
j
)
+
1
4
π
[
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
2
s
(
n
,
j
,
j
)
F
(
k
,
k
)
m
m
m
m
m
m
m
m
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
2
c
(
n
,
j
,
j
)
F
(
k
,
k
)
)
m
m
m
m
m
+
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
F
s
(
m
,
k
,
k
)
H
2
(
j
,
j
)
m
m
m
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
F
c
(
m
,
k
,
k
)
H
2
(
j
,
j
)
)
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
]
,
H
b
(
j
,
k
)
=
1
16
F
(
k
,
k
)
H
(
j
,
j
)
+
1
4
π
[
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
s
(
n
,
j
,
j
)
F
(
k
,
k
)
m
m
m
m
m
m
m
n
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
c
(
n
,
j
,
j
)
F
(
k
,
k
)
)
m
m
m
m
i
n
m
+
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
F
s
(
m
,
k
,
k
)
H
(
j
,
j
)
m
m
m
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
F
c
(
m
,
k
,
k
)
H
(
j
,
j
)
)
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
s
(
n
,
j
,
j
)
]
,
H
c
(
j
,
k
)
=
1
16
F
(
k
,
k
)
H
1
(
j
,
j
)
+
1
4
π
[
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
H
1
s
(
n
,
j
,
j
)
F
(
k
,
k
)
m
m
m
m
m
m
m
m
-
Sin
(
2
n
+
1
)
π
c
t
2
n
+
1
H
1
c
(
n
,
j
,
j
)
F
(
k
,
k
)
)
m
m
m
m
m
+
∑
m
=
0
∞
(
Cos
(
2
m
+
1
)
π
y
0
2
m
+
1
F
s
(
m
,
k
,
k
)
H
1
(
j
,
j
)
m
m
m
m
m
m
m
m
m
-
Sin
(
2
m
+
1
)
π
y
0
2
m
+
1
F
c
(
m
,
k
,
k
)
H
1
(
j
,
j
)
)
∑
n
=
0
∞
(
Cos
(
2
n
+
1
)
π
c
t
2
n
+
1
]
.
To solve the nonhomogeneous equation (63), the differential operator which acts on
W
¯
(
j
,
k
,
t
)
and
W
¯
(
p
,
q
,
t
)
is replaced by the equivalent free system operator defined by the modified frequency
γ
m
m
; that is,
(69)
W
¯
t
t
(
j
,
k
,
t
)
+
γ
m
m
2
W
¯
(
j
,
k
,
t
)
=
Γ
a
g
L
x
2
L
y
2
W
(
λ
k
,
y
0
)
λ
j
λ
k
×
[
B
f
(
λ
,
j
)
+
Cos
λ
j
c
t
L
x
m
m
m
i
m
-
A
j
Sin
λ
j
c
t
L
x
-
B
j
Cos
h
λ
j
c
t
L
x
-
C
j
Sin
h
λ
j
c
t
L
x
]
.
Evidently, (69) is analogous to (58). Thus, using similar argument as in the previous section, solution to (69) can be obtained as
(70)
W
(
x
,
y
,
t
)
=
∑
j
=
1
∞
∑
k
=
1
∞
1
V
j
V
k
Γ
a
g
L
x
2
L
y
2
W
(
λ
k
,
y
0
)
λ
j
λ
k
m
m
m
m
m
n
×
[
C
j
(
γ
m
m
Sin
h
α
j
t
-
α
j
Sin
γ
m
m
t
)
γ
m
m
(
α
j
2
+
γ
m
m
2
)
B
f
(
λ
,
j
)
(
1
-
Cos
γ
m
m
t
)
γ
m
m
m
m
m
m
m
m
m
m
+
Cos
α
j
t
-
Cos
γ
m
m
t
γ
m
m
2
-
α
j
2
m
m
m
m
m
m
m
m
-
A
j
(
γ
m
m
Sin
α
j
t
-
α
j
Sin
γ
m
m
t
)
γ
m
m
(
γ
m
m
2
-
α
j
2
)
m
m
m
m
m
m
m
m
-
B
j
(
Cos
h
α
j
t
-
Cos
γ
m
m
t
)
α
j
2
+
γ
m
m
2
m
m
m
m
m
m
m
m
-
C
j
(
γ
m
m
Sin
h
α
j
t
-
α
j
Sin
γ
m
m
t
)
γ
m
m
(
α
j
2
+
γ
m
m
2
)
]
·
(
Sin
λ
j
x
L
x
+
A
j
Cos
λ
j
x
L
x
+
B
j
Sin
h
λ
j
x
L
x
+
C
j
Cos
h
λ
j
x
L
x
)
·
(
Sin
λ
k
y
L
y
+
A
k
Cos
λ
k
y
L
y
+
B
k
Sin
h
λ
k
y
L
y
+
C
k
Cos
h
λ
k
y
L
y
)
.
Equation (70) represents the transverse displacement response to uniformly distributed masses moving at constant velocity of an isotropic rectangular plate resting on elastic foundation for various end conditions.