] D ν [ 1 4 h * [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i , j + 1 k + 1 ψ X , T i + 1 , j k + 1 ψ X , T i , j k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i , j + 1 k ψ X , T i + 1 , j k ψ X , T i , j k ] ] (35)

1 4 r * [ M X Y i + 1 , j + 1 k + 1 + M Y i + 1 , j k + 1 + M Y i , j + 1 k + 1 + M Y i , j k + 1 M Y i + 1 , j + 1 k M Y i + 1 , j k M Y i , j + 1 k M Y i , j k ] = D 1 ν 2 [ 1 4 k * [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i , j + 1 k + 1 ψ X , T i + 1 , j k + 1 ψ X , T i , j k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i , j + 1 k ψ X , T i + 1 , j k ψ X , T i , j k ] ] + D 1 ν 2 [ 1 4 h * [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i , j + 1 k + 1 ψ Y , T i + 1 , j k + 1 ψ Y , T i , j k + 1 + ψ Y , T i + 1 , j + 1 k + ψ Y , T i , j + 1 k ψ Y , T i + 1 , j k ψ Y , T i , j k ] ] (36)

1 4 r * [ Q x i + 1 , j + 1 k + 1 + Q x i + 1 , j k + 1 + Q x i , j k + 1 + Q x i , j + 1 k + 1 Q x i + 1 , j + 1 k Q x i + 1 , j k Q x i , j + 1 k Q x i , j k ] = α h G [ 1 8 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i + 1 , j k + 1 + ψ X , T i , j k + 1 + ψ X , T i , j + 1 k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i + 1 , j k + ψ X , T i , j + 1 k + ψ X , T i , j k ] ] α h G [ 1 4 h * [ D Y i + 1 , j + 1 k + 1 + D Y i + 1 , j k + 1 D Y i , j k + 1 D Y i , j + 1 k + 1 + D Y i + 1 , j + 1 k + D Y i + 1 , j k D Y i , j + 1 k D Y i , j k ] ] (37)

1 4 r * [ Q Y i + 1 , j + 1 k + 1 + Q Y i + 1 , j k + 1 + Q Y i , j k + 1 + Q Y i , j + 1 k + 1 Q Y i + 1 , j + 1 k Q Y i + 1 , j k Q Y i , j + 1 k Q Y i , j k ] = α h G [ 1 8 [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i + 1 , j k + 1 + ψ Y , T i , j k + 1 + ψ Y , T i , j + 1 k + 1 + ψ Y , T i + 1 , j + 1 k + ψ Y , T i + 1 , j k + ψ Y , T i , j + 1 k + ψ Y , T i , j k ] ] α h G [ 1 4 k * [ D Y i + 1 , j + 1 k + 1 + D Y i , j + 1 k + 1 D Y i + 1 , j k + 1 D Y i , j k + 1 + D Y i + 1 , j + 1 k + D Y i , j + 1 k D Y i + 1 , j k D Y i , j k ] ] (38)

1 8 [ D T i + 1 , j + 1 k + 1 + D T i + 1 , j k + 1 + D T i , j k + 1 + D T i , j + 1 k + 1 D T i + 1 , j + 1 k D T i + 1 , j k D T i , j + 1 k D T i , j k ] = 1 4 r * [ W x i + 1 , j + 1 k + 1 + W x i , j + 1 k + 1 + W x i + 1 , j k + 1 + W x i , j k + 1 W x i + 1 , j + 1 k W x i , j + 1 k W x i + 1 , j k W x i , j k ] (39)

1 8 [ D Y i + 1 , j + 1 k + 1 + D Y i + 1 , j k + 1 + D Y i , j k + 1 + D Y i , j + 1 k + 1 + D Y i + 1 , j + 1 k + D Y i + 1 , j k + D Y i , j + 1 k + D Y i , j k ] = 1 4 h * [ W x i + 1 , j + 1 k + 1 + W x i + 1 , j k + 1 W x i , j k + 1 W x i , j + 1 k + 1 + W x i + 1 , j + 1 k + W x i + 1 , j k W x i , j + 1 k W x i , j k ] (40)

1 8 [ D T i + 1 , j + 1 k + 1 + D T i + 1 , j k + 1 + D T i , j k + 1 + D T i , j + 1 k + 1 + D T i + 1 , j + 1 k + D T i + 1 , j k + D T i , j + 1 k + D T i , j k ] = 1 4 k * [ W x i + 1 , j + 1 k + 1 + W x i , j + 1 k + 1 W x i + 1 , j k + 1 W x i , j k + 1 + W x i + 1 , j + 1 k + W x i , j + 1 k W x i + 1 , j k W x i , j k ] (41)

The set of algebraic equations to be solved may be written in matrix form as:

R i , j + 1 S i , j + 1 / + P i + 1 , j + 1 S i + 1 , j + 1 / = T i , j + 1 S i , j + 1 / Y i + 1 , j S i + 1 , j / + Z k i = 1 , 2 , 3 , , N 1 , j = 1 , 2 , 3 , , M 1 (42)

where N and M are the number of the modal points along x- and y-axes respectively, Zk is a matrix representing the right half of Equations (16)-(26) defined by

Z k = A i , j S i , j 0 + B i , j + 1 S i , j + 1 0 + C i + 1 , j S i + 1 , j 0 + D i + 1 , j + 1 S i + 1 , j + 1 0 + E 1 (43)

The terms of the above Equations ((42) and (43)) can be represented in matrix form as follows:

R i , j + 1 S i , j + 1 / = [ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ] [ M X i , j + 1 M Y i , j + 1 M X Y i , j + 1 Q X i , j + 1 Q Y i , j + 1 ψ X T i , j + 1 ψ Y T i , j + 1 D T i , j + 1 W i , j + 1 D X i , j + 1 D Y i , j + 1 ]

Numerical Simulation

For numerical work the coupled differential Equations (16)-(26) were solved using the central difference formula of finite difference method. The following values of the various parameters were used: h = 1, h1 = 0.2, ρ = 0.8, ρL = 0.5, B = 0.5, U = 5.5, D = 0.63, ν = 0.2, M = 10, ML = 0.05, K = 100, G1 = 10, γ = 0.5, A = 6, g = 9.8, θ = 30, α = 0.01, G = 200, r* = 1, h* = 1, k* = 1, r = 2. Equations (31) to (41) can now be written as follows:

0.125 [ Q x i + 1 , j + 1 k + 1 + Q x i + 1 , j k + 1 + Q x i , j + 1 k + 1 + Q x i , j k + 1 + Q x i + 1 , j + 1 k + Q x i + 1 , j k + Q x i , j + 1 k + Q x i , j k ] 0.25 [ M X Y i + 1 , j + 1 k + 1 + M X Y i , j + 1 k + 1 M X Y i + 1 , j k + 1 M X Y i , j k + 1 + M X Y i + 1 , j + 1 k + M X Y i , j + 1 k M X Y i + 1 , j k M X Y i , j k ] 0.25 [ M X i + 1 , j + 1 k + 1 + M X i + 1 , j k + 1 M X i , j k + 1 M X i , j + 1 k + 1 + M X i + 1 , j + 1 k + M X i + 1 , j k M X i , j + 1 k M X i , j k ]

= { 0.00002 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i + 1 , j k + 1 + ψ X , T i , j + 1 k + 1 + ψ X , T i , j k + 1 ψ X , T i + 1 , j + 1 k ψ X , T i + 1 , j k ψ X , T i , j + 1 k ψ X , T i , j k ] } + { 0.00075 [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i + 1 , j k + 1 + ψ Y , T i , j + 1 k + 1 + ψ Y , T i , j k + 1 ψ Y , T i + 1 , j + 1 k ψ Y , T i + 1 , j k ψ Y , T i , j + 1 k ψ Y , T i , j k ] } { 0.00121 [ M X i + 1 , j + 1 k + 1 + M X i + 1 , j k + 1 + M X i , j + 1 k + 1 + M X i , j k + 1 M X i + 1 , j + 1 k M X i + 1 , j k M X i , j + 1 k M X i , j k ] }

{ 0.0008 [ M X i + 1 , j + 1 k + 1 + M X i + 1 , j k + 1 M X i , j k + 1 M X i , j + 1 k + 1 + M X i + 1 , j + 1 k + M X i + 1 , j k M X i , j + 1 k M X i , j k ] } + { 0.00121 [ M Y i + 1 , j + 1 k + 1 + M Y i + 1 , j k + 1 + M Y i , j + 1 k + 1 + M Y i , j k + 1 M Y i + 1 , j + 1 k M Y i + 1 , j k M Y i , j + 1 k M Y i , j k ] } + { 0.00667 [ M Y i + 1 , j + 1 k + 1 + M Y i + 1 , j k + 1 M Y i , j k + 1 M Y i , j + 1 k + 1 + M Y i + 1 , j + 1 k + M Y i + 1 , j k M Y i , j + 1 k M Y i , j k ] } (44)

0.125 [ Q Y i + 1 , j + 1 k + 1 + Q Y i + 1 , j k + 1 + Q Y i , j + 1 k + 1 + Q Y i , j k + 1 + Q Y i + 1 , j + 1 k + Q Y i + 1 , j k + Q Y i , j + 1 k + Q Y i , j k ] 0.25 [ M X Y i + 1 , j + 1 k + 1 + M X Y i , j + 1 k + 1 M X Y i + 1 , j k + 1 M X Y i , j k + 1 + M X Y i + 1 , j + 1 k + M X Y i , j + 1 k M X Y i + 1 , j k M X Y i , j k ] 0.25 [ M X i + 1 , j + 1 k + 1 + M X i + 1 , j k + 1 M X i , j k + 1 M X i , j + 1 k + 1 + M X i + 1 , j + 1 k + M X i + 1 , j k M X i , j + 1 k M X i , j k ]

= 0.0667 { 1 4 r [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i + 1 , j k + 1 + ψ Y , T i , j + 1 k + 1 + ψ Y , T i , j k + 1 ψ Y , T i + 1 , j + 1 k ψ Y , T i + 1 , j k ψ Y , T i , j + 1 k ψ Y , T i , j k ] } + 0.00002 { 1 4 r [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i + 1 , j k + 1 + ψ Y , T i , j + 1 k + 1 + ψ Y , T i , j k + 1 ψ Y , T i + 1 , j + 1 k ψ Y , T i + 1 , j k ψ Y , T i , j + 1 k ψ Y , T i , j k ] } + 0.00046 [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i + 1 , j k + 1 ψ Y , T i , j k + 1 ψ Y , T i , j + 1 k + 1 + ψ Y , T i + 1 , j + 1 k + ψ Y , T i + 1 , j k ψ Y , T i , j + 1 k ψ Y , T i , j k ] 0.30313 { 1 4 r [ M X i + 1 , j + 1 k + 1

+ M X i + 1 , j k + 1 + M X i , j + 1 k + 1 + M X i , j k + 1 M X i + 1 , j + 1 k M X i + 1 , j k M X i , j + 1 k M X i , j k ] } 1.66722 { 1 4 k [ M X i + 1 , j + 1 k + 1 + M X i + 1 , j k + 1 M X i , j k + 1 M X i , j + 1 k + 1 + M X i + 1 , j + 1 k + M X i + 1 , j k M X i , j + 1 k M X i , j k ] } + 0.30313 { 1 4 r [ M X i + 1 , j + 1 k + 1 + M X i + 1 , j k + 1 + M X i , j + 1 k + 1 + M X i , j k + 1 M X i + 1 , j + 1 k M X i + 1 , j k M X i , j + 1 k M X i , j k ] } + 1.66722 { 1 4 h [ M X i + 1 , j + 1 k + 1 + M X i + 1 , j k + 1 M X i , j k + 1 M X i , j + 1 k + 1 + M X i + 1 , j + 1 k + M X i + 1 , j k M X i , j + 1 k M X i , j k ] } (45)

0.25 [ Q x i + 1 , j + 1 k + 1 + Q x i + 1 , j k + 1 Q x i , j k + 1 Q x i , j + 1 k + 1 + Q x i + 1 , j + 1 k + Q x i + 1 , j k Q x i , j + 1 k Q x i , j k ] + 0.25 [ Q x i + 1 , j + 1 k + 1 + Q x i , j + 1 k + 1 Q x i + 1 , j k + 1 Q x i , j k + 1 + Q x i + 1 , j + 1 k + Q x i , j + 1 k Q x i + 1 , j k Q x i , j k ] + 12.5 [ W x i + 1 , j + 1 k + 1 + W x i , j + 1 k + 1 + W x i + 1 , j k + 1 + W x i , j k + 1 + W x i + 1 , j + 1 k + W x i , j + 1 k + W x i + 1 , j k + W x i , j k ] + 1.25 [ D T i + 1 , j + 1 k + 1 + D T i + 1 , j k + 1 + D T i , j k + 1 + D T i , j + 1 k + 1 D T i + 1 , j + 1 k D T i + 1 , j k D T i , j + 1 k D T i , j k ] + 2.5 [ D x i + 1 , j + 1 k + 1 + D x i + 1 , j k + 1 D x i , j k + 1 D x i , j + 1 k + 1 + D x i + 1 , j + 1 k + D x i + 1 , j k D x i , j + 1 k D x i , j k ]

+ 2.5 [ D Y i + 1 , j + 1 k + 1 + D Y i , j + 1 k + 1 D Y i + 1 , j k + 1 D Y i , j k + 1 + D Y i + 1 , j + 1 k + D Y i , j + 1 k D Y i + 1 , j k D Y i , j k ] + 0.008335 [ g sin θ + 1 4 [ D T i + 1 , j + 1 k + 1 + D T i + 1 , j k + 1 + D T i , j k + 1 + D T i , j + 1 k + 1 D T i + 1 , j + 1 k D T i + 1 , j k D T i , j + 1 k D T i , j k ] ] + 0.011459 [ D T i + 1 , j + 1 k + 1 + D T i + 1 , j k + 1 + D T i , j k + 1 + D T i , j + 1 k + 1 D T i + 1 , j + 1 k D T i + 1 , j k D T i , j + 1 k D T i , j k ] + 0.005729 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i , j + 1 k + 1 + ψ X , T i + 1 , j k + 1 + ψ X , T i , j k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i , j + 1 k + ψ X , T i + 1 , j k + ψ X , T i , j k ] 0.052101 [ M x i + 1 , j + 1 k + 1

+ M x i , j + 1 k + 1 + M x i + 1 , j k + 1 + M x i , j k + 1 + M x i + 1 , j + 1 k + M x i , j + 1 k + M x i + 1 , j k + M x i , j k ] + 0.01042 [ M Y i + 1 , j + 1 k + 1 + M Y i , j + 1 k + 1 + M Y i + 1 , j k + 1 + M Y i , j k + 1 + M Y i + 1 , j + 1 k + M Y i , j + 1 k + M Y i + 1 , j k + M Y i , j k ] 0.005729 [ Q x i + 1 , j + 1 k + 1 + Q x i + 1 , j k + 1 + Q x i , j + 1 k + 1 + Q x i , j k + 1 Q x i + 1 , j + 1 k Q x i + 1 , j k Q x i , j + 1 k Q x i , j k ] 0.0315105 [ Q Y i + 1 , j + 1 k + 1 + Q Y i + 1 , j k + 1 Q Y i , j + 1 k + 1 Q Y i , j k + 1 + Q Y i + 1 , j + 1 k + Q Y i + 1 , j k Q Y i , j + 1 k Q Y i , j k ] = 0 (46)

0.25 [ M x i + 1 , j + 1 k + 1 + M x i + 1 , j k + 1 + M x i , j + 1 k + 1 + M x i , j k + 1 M x i + 1 , j + 1 k M x i + 1 , j k M x i , j + 1 k M x i , j k ] = 0.63 [ 1 4 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i + 1 , j k + 1 ψ X , T i , j k + 1 ψ X , T i , j + 1 k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i + 1 , j k ψ X , T i , j + 1 k ψ X , T i , j k ] ] 0.126 [ 1 4 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i , j + 1 k + 1 ψ X , T i + 1 , j k + 1 ψ X , T i , j k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i , j + 1 k ψ X , T i + 1 , j k ψ X , T i , j k ] ] (47)

0.25 [ M Y i + 1 , j + 1 k + 1 + M Y i + 1 , j k + 1 + M Y i , j + 1 k + 1 + M Y i , j k + 1 M Y i + 1 , j + 1 k M Y i + 1 , j k M Y i , j + 1 k M Y i , j k ] = 0.63 [ 1 4 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i , j + 1 k + 1 ψ X , T i + 1 , j k + 1 ψ X , T i , j k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i , j + 1 k ψ X , T i + 1 , j k ψ X , T i , j k ] ] 0.126 [ 1 4 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i , j + 1 k + 1 ψ X , T i + 1 , j k + 1 ψ X , T i , j k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i , j + 1 k ψ X , T i + 1 , j k ψ X , T i , j k ] ] (48)

0.25 [ M X Y i + 1 , j + 1 k + 1 + M Y i + 1 , j k + 1 + M Y i , j + 1 k + 1 + M Y i , j k + 1 M Y i + 1 , j + 1 k M Y i + 1 , j k M Y i , j + 1 k M Y i , j k ] = 0.252 [ 1 4 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i , j + 1 k + 1 ψ X , T i + 1 , j k + 1 ψ X , T i , j k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i , j + 1 k ψ X , T i + 1 , j k ψ X , T i , j k ] ] + 0.252 [ 1 4 [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i , j + 1 k + 1 ψ Y , T i + 1 , j k + 1 ψ Y , T i , j k + 1 + ψ Y , T i + 1 , j + 1 k + ψ Y , T i , j + 1 k ψ Y , T i + 1 , j k ψ Y , T i , j k ] ] (49)

0.25 [ Q x i + 1 , j + 1 k + 1 + Q x i + 1 , j k + 1 + Q x i , j k + 1 + Q x i , j + 1 k + 1 Q x i + 1 , j + 1 k Q x i + 1 , j k Q x i , j + 1 k Q x i , j k ] = 2 [ 1 8 [ ψ X , T i + 1 , j + 1 k + 1 + ψ X , T i + 1 , j k + 1 + ψ X , T i , j k + 1 + ψ X , T i , j + 1 k + 1 + ψ X , T i + 1 , j + 1 k + ψ X , T i + 1 , j k + ψ X , T i , j + 1 k + ψ X , T i , j k ] ] 2 [ 1 4 [ D Y i + 1 , j + 1 k + 1 + D Y i + 1 , j k + 1 D Y i , j k + 1 D Y i , j + 1 k + 1 + D Y i + 1 , j + 1 k + D Y i + 1 , j k D Y i , j + 1 k D Y i , j k ] ] (50)

0.25 [ Q Y i + 1 , j + 1 k + 1 + Q Y i + 1 , j k + 1 + Q Y i , j k + 1 + Q Y i , j + 1 k + 1 Q Y i + 1 , j + 1 k Q Y i + 1 , j k Q Y i , j + 1 k Q Y i , j k ] = 2 [ 1 8 [ ψ Y , T i + 1 , j + 1 k + 1 + ψ Y , T i + 1 , j k + 1 + ψ Y , T i , j k + 1 + ψ Y , T i , j + 1 k + 1 + ψ Y , T i + 1 , j + 1 k + ψ Y , T i + 1 , j k + ψ Y , T i , j + 1 k + ψ Y , T i , j k ] ] 2 [ 1 4 [ D Y i + 1 , j + 1 k + 1 + D Y i , j + 1 k + 1 D Y i + 1 , j k + 1 D Y i , j k + 1 + D Y i + 1 , j + 1 k + D Y i , j + 1 k D Y i + 1 , j k D Y i , j k ] ] (51)

0.125 [ D T i + 1 , j + 1 k + 1 + D T i + 1 , j k + 1 + D T i , j k + 1 + D T i , j + 1 k + 1 D T i + 1 , j + 1 k D T i + 1 , j k D T i , j + 1 k D T i , j k ] = 0.25 [ W x i + 1 , j + 1 k + 1 + W x i , j + 1 k + 1 + W x i + 1 , j k + 1 + W x i , j k + 1 W x i + 1 , j + 1 k W x i , j + 1 k W x i + 1 , j k W x i , j k ] (52)

0.125 [ D Y i + 1 , j + 1 k + 1 + D Y i + 1 , j k + 1 + D Y i , j k + 1 + D Y i , j + 1 k + 1 + D Y i + 1 , j + 1 k + D Y i + 1 , j k + D Y i , j + 1 k + D Y i , j k ] = 0.25 [ W x i + 1 , j + 1 k + 1 + W x i + 1 , j k + 1 W x i , j k + 1 W x i , j + 1 k + 1 + W x i + 1 , j + 1 k + W x i + 1 , j k W x i , j + 1 k W x i , j k ] (53)

0.125 [ D T i + 1 , j + 1 k + 1 + D T i + 1 , j k + 1 + D T i , j k + 1 + D T i , j + 1 k + 1 + D T i + 1 , j + 1 k + D T i + 1 , j k + D T i , j + 1 k + D T i , j k ] = 0.25 [ W x i + 1 , j + 1 k + 1 + W x i , j + 1 k + 1 W x i + 1 , j k + 1 W x i , j k + 1 + W x i + 1 , j + 1 k + W x i , j + 1 k W x i + 1 , j k W x i , j k ] (54)

The matrices now appear as follows:

R i , j + 1 S i , j + 1 / = [ 0.3 0 0.3 0.1 0 0 0 0 0 0 0 0.3 0 0.3 0 0.1 0 0.1 0 0 0 0 0.1 0 0 0 0 0 0 0 1.3 2.5 2.5 1.3 0 0 0 0 0.1 0 0 0 0 0 0 0.3 0 0 0 0.2 0 0 0 0 0 0 0 0.3 0.1 0.1 0 0 0 0 0 0 0 0 0 0.3 0 0.3 0 0.5 0 0 0 0 0 0 0 0.3 0 0.3 0.5 0 0 0 0 0 0 0 0 0 0 0.1 0.3 0 0 0 0 0 0 0 0 0 0.1 0.3 0 0 0 0 0 0 0 0 0 0.1 0.3 0 0 ] [ M X i , j + 1 M Y i , j + 1 M X Y i , j + 1 Q X i , j + 1 Q Y i , j + 1 ψ X T i , j + 1 ψ Y T i , j + 1 D T i , j + 1 W i , j + 1 D X i , j + 1 D Y i , j + 1 ]

The above is at a particular node (i, j + 1). Similar matrices can be shown for the other nodes, but for brevity sake.

4. Results Discussion

The paper set out to analyse, numerically, the vibration of rectangular elastic orthotropic damped inclined Mindlin plate, because of applied force, using finite difference method. The plate was supported by a Pasternak foundation. Deflection of the plate was calculated for specific values of foundation parameter and contact area of the plate. It was observed that Mindlin plate has highest maximum amplitude when compared with Non-Mindlin plate. The response maximum amplitude decreases with an increase in the value of the subgrade’s shear modulus for fixed value of foundation stiffness, contact area and velocity. It was noticed that the response amplitude of the plate continuously supported by a Pasternak foundation is less than that of the plate not resting on any elastic subgrade. As the foundation stiffness and shear modulus increase the response amplitude decreases. Also, it was observed that as the contact area increases the response maximum amplitude increases with fixed values of the foundation stiffness and the subgrade’s shear modulus. Finally it was observed that the maximum amplitude increases as the velocity increases.

5. Conclusion

The structure of interest was an inclined Mindlin rectangular plate on Pasternak elastic foundation, under the influence of a uniform partially distributed moving load. The problem was to use finite difference technique to solve the governing equation of a moving load problem. The dynamic response of the whole system was determined by solving the resulting first order coupled partial differential equations obtained from governing equations for the simply supported Mindlin plate. The study has contributed to scientific knowledge by showing that Pasternak foundation, on which the inclined Mindlin plate rests, has a significance effect on the dynamic response of the plate to a partially distributed moving load. The effect of rotating inertia and shear deformation on the dynamic response of the inclined Mindlin plate to the moving load gives more realistic results for practical application, especially when such inclined plate is supported by a Pasternak type of subgrade foundation.

Cite this paper
Agarana, M. , Ehigbochie, A. (2018) Forced Vibration Numerical Analysis of Rectangular Elastic Orthotropic Damped Inclined Mindlin Plate Using Finite Difference Algorithm. Applied Mathematics, 9, 618-632. doi: 10.4236/am.2018.96043.
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