PR Kerr Solution HKB Rev#1.Nb

8/16/2019 PR Kerr Solution HKB Rev#1.Nb

1/7

k1 .

k2 .

Gs .

.

.

p .

q .

a .

b .

h .

Dx .

Dy .

B .

.

w Sin p Pi x a Sinq Pi y b Sin t

Sinp x

a Sin

q y

b Sint

f1 1 k2 k1 Dx D w, x, 4 2 B D w, x, 2, y, 2 Dy D w, y, 4 h D w, t, 2

1 k2

k1

Dx p4 4 Sin p xa

Sin q yb

Sint a4

2 B p2 4 q2 Sin p xa

Sin q yb

Sint

a2 b2

Dy 4 q4 Sin p xa

Sin q yb

Sint b4

h 2 Sinp x

a Sin

q y

b Sint

f2 Dx D w, x, 4 2 B D w, x, 2, y, 2 Dy D w, y, 4 h D w, t, 2Dx p4 4 Sin

p x

a Sin

q y

b Sin

t

a4

2 B p2 4 q2 Sin

p x

a Sin

q y

b Sin

t

a2 b2

Dy 4 q4 Sin p xa

Sin q yb

Sint b4

h 2 Sinp x

a Sin

q y

b Sint


2/7

f3 Gs k1 Df2, x, 2 Df2, y, 2

1

k1Gs

Dx p6 6 Sin p xa

Sin q yb

Sint

a6

2 B p4 6 q2 Sin p xa

Sin q yb

Sint

a4 b2

Dx p4 6 q2 Sin

p x

a Sin

q y

b Sin

t

a4 b2

2 B p2 6 q4 Sin

p x

a Sin

q y

b Sin

t

a2 b4

Dy p2 6 q4 Sin p xa

Sin q yb

Sint

a2 b4

Dy 6 q6 Sin p xa

Sin q yb

Sint

b6

h p2 2 2 Sin p xa

Sin q yb

Sint

a2

h 2 q2 2 Sin p xa

Sin q yb

Sint

b2

f4 k2 w Gs D w, x, 2 D w, y, 2

k2 Sinp x

a Sin

q y

b Sint

Gs

p2 2 Sin p xa

Sin q yb

Sint a2

2 q2 Sin p xa

Sin q yb

Sint b2

ftot Simplifyf1 f3 f4 w

1

a6 b6 k1b6 Dx Gs p6 6 a2 b4 p4 4 b2 Dx k1 k2 2 B Dx Gs 2 q2

a4 b2 p2 2 2 b2 B k1 k2 2 q2 2 B Dy Gs 4 q4 b4 Gs k1 h 2 a6 b2 Dy k1 k2 4 q4 Dy Gs 6 q6

b4 Gs 2 q2 k1 h 2 b6 k1 k2 h k1 2 h k2 2

ftota Apartftot

k2 Gs p2 2

a2

Dx Gs p6 6

a6 k1

Gs 2 q2

b2

2 B k1 k2 p2 4 q2

a2 b2 k1

Dy k1 k2 4 q4

b4 k1

2 B Dy Gs p2 6 q4

a2 b4 k1

Dy Gs 6 q6

b6 k1

p4 4 b2 Dx k1 b2 Dx k2 2 B Gs 2 q2 Dx Gs 2 q2a4 b2 k1

h a2 b2 k1 a2 b2 k2 b2 Gs p2 2 a2 Gs 2 q2 2

a2 b2 k1

2

PR Kerr Solution HKB rev#1.nb


3/7

omega2 Simplify

k2 Gs p2 2

a2

Dx Gs p6 6

a6 k1

Gs 2 q 2

b2

2 B k1 k2 p2 4 q 2

a2 b2 k1

Dy k1 k2 4 q 4 b4 k1

2 B Dy

Gs p2 6 q 4

a2 b4 k1

Dy Gs 6 q 6

b6 k1

p4 4 b2 Dx k1 b2 Dx k2 2 B Gs 2 q 2 Dx Gs 2 q 2a4 b2 k1

a2 b2 k1 h a2 b2 k1 a2 b2 k2 b2 Gs p2 2 a2 Gs 2 q 2

b6 Dx Gs p6 6 a2 b4 p4 4 b2 Dx k1 k2 2 B Dx Gs 2 q2 a4 b2 p2 2 b4 Gs k1 2 b2 B k1 k2 2 q2 2 B Dy Gs 4 q4 a6 b6 k1 k2 b4 Gs k1 2 q2 b2 Dy k1 k2 4 q4 Dy Gs 6 q6

a4

b4

h b2

Gs p2

2

a2

b2

k1 k2 Gs 2

q2

Auxilary I

Persamaan arah X

px A1 Cos p Pi x a A2 Sin p Pi x A3 1 Pi x a Cosh Pi x a A4 1 Pi x a Sinh Pi x a

A1 Cosp x

a A3 1

x

aCosh

x

a A2 Sinp x A4 1

x

aSinh

x

a

Boundary ConditionsBC1 : x = 0 ; w = 0

bc1 px . x 0A1 A3

BC2 : x = 0; Dxd 2 w dx2

+ vy d

2w

dy2 = k1. dw

dx

W2x D

D

px, x

, x

;

W2y DD px, y, y;

Wx1 D px, x;

bc2 Dx W2x vy W2y k1 Wx1 . x 0

k1 A2 p A3

a

A4

a Dx

A1 p2 2

a2

A3 2 2

a2

2 A4 2 2

a2

BC3 : x = a; w = 0

bc3 px . x aA1 Cosp A3 1 Cosh A2 Sina p A4 1 Sinh

PR Kerr Solution HKB rev#1.nb 3


4/7

BC4 : x a; Dx d 2 w dx2

vy d

2w

dy2 k1. dw

dx

bc4 Dx W2x vy W2y k1 Wx1 . x a

k1 A2 p Cosa p A3 Cosh

a A4

1

Cosh

a A1 p Sinp

a

A4 Sinh a

A3 1 Sinh

a

Dx A1 p2 2 Cosp

a2

2 A4 2 2 Cosh a2

A3 2 2 1 Cosh

a2

A2 p2 2 Sina p 2 A3 2 2 Sinh

a2

A4 2 2 1 Sinh a2

Collect bc1, A1, A2, A3, A4A1 A3

Collect bc2, A1, A2, A3, A4

A2 k1 p A1 Dx p2 2

a2 A4

k1

a

2 Dx 2 2

a2 A3

k1

a

Dx 2 2

a2

Collect bc3, A1, A2, A3, A4A1 Cosp A3 1 Cosh A2 Sina p A4 1 Sinh

Collect bc4, A1, A2, A3, A4

A1

Dx p2 2 Cosp a2

k1 p Sinp a

A2 k1 p Cosa p Dx p2 2 Sina p A3 k1 Cosh

a

Dx 2 2 1 Cosh a2

2 Dx 2 2 Sinh

a2

k1 1 Sinh a

A4 2 Dx 2 2 Cosh

a2

k1 1 Cosh a

k1 Sinh a

Dx 2 2 1 Sinh

a2

Memasukkan nilai A1, A2, A3, A4 ke dalam

Matrixbc11 bc12 bc13 bc14

bc21 bc22 bc23 bc24

bc31 bc32 bc33 bc34

bc41 bc42 bc43 bc44

A1

A2

A3

A4

=

0

0

0

0

4



5/7

mx

1 0

Dx p 2 2

a2 k1 p

Cos p Sina p

Dx p 2 2 Cos

p

a2 k1 p Sin

p

a k1 p Cosa p Dx p2 2 Sina p k1

Cosh

a

;

Detmx Det mx Simplify1

a42 p Dx p Cosp a k1 Sinp

a k1

2 Dx

Sina p

a

2

k1 p 1

Sinh

a2 p k1 Cosa p Dx p Sina p a k1 2 Dx Cosp Dx p2 1 Sinh

a2 p 1 k1 Cosa p Dx p Sina p a k1 2 Dx Cosh a k1 Dx Sinh

p a2 k1 Cosp Dx p Sina p 2 Dx a k1 k1 Cosh a k1 Dx 1 Sinh

a2 k1 p 1 Cosh a k1 Dx Sina p 2 Dx a k1 k1 Cosh a k1 Dx 1 Sinh

a k1 2 Dx Sina p a2 k1 p 1 Sinh a k1 Dx 1 Cosh 2 Dx a k1 k1 Sinh

Auxilary IIAuxilary II

Persamaan arah Yarah Persamaan Y

py A1 Cosq Pi y b A2 Sinq Pi y A3 1 Pi y b Cosh Pi y b A4

1

Pi y

b

Sinh

Pi y

b

A1 Cos

q y

b A3 1

y

bCosh

y

b A2 Sin q y A4 1

y

bSinh

y

b

Boundary ConditionsBC1y : y = 0 ; w = 0

bc1y py . y 0A1 A3

BC2y : y = 0;

Dyd

2w

dx2 + vx

d 2w

dy2 = k2.dw

dx



6/7

W2x1 DD py, x, x;

W2y1 DD py, y, y;

Wy1 D py, y;

bc2y

Dy

W2y1

vx

W2x1

k2

Wy1 . y

0 k2 A2 q

A3

b

A4

b Dy

A1 2 q2

b2

A3 2 2

b2

2 A4 2 2

b2

BC3y : y = b; w = 0

bc3y py . y bA1 Cos q A3 1 Cosh A2 Sinb q A4 1 Sinh

BC4 : y b; Dyd 2 w dy2

vx d

2w

dx2 k2. dw

dy

bc4y Dy W2y1 vx W2x1 k2 Wy1 . y b

k2 A2 q Cosb q A3 Cosh

b

A4 1 Cosh b

A1 q Sin qb

A4 Sinh

b

A3 1 Sinh b

Dy A1 2 q2 Cos q

b2

2 A4 2 2 Cosh b2

A3 2 2 1 Cosh

b2

A2 2 q2 Sinb q 2 A3 2 2 Sinh

b2

A4 2 2 1 Sinh b2

Collect bc1y, A1, A2, A3, A4A1 A3

Collect bc2y, A1, A2, A3, A4

A2 k2 q A1 Dy 2 q2

b2 A4

k2

b

2 Dy 2 2

b2 A3

k2

b

Dy 2 2

b2

Collect bc3y, A1, A2, A3, A4A1 Cos q A3 1 Cosh A2 Sinb q A4 1 Sinh

6



7/7

Collect bc4y, A1, A2, A3, A4

A1Dy 2 q2 Cos q

b2

k2 q Sin qb

A2

k2 q Cos

b q

Dy 2 q2 Sin

b q

A3

k2 Cosh b

Dy 2 2 1 Cosh b2

2 Dy 2 2 Sinh

b2

k2 1 Sinh b

A4 2 Dy 2 2 Cosh

b2

k2 1 Cosh b

k2 Sinh b

Dy 2 2 1 Sinh

b2

Memasukkan nilai A1, A2, A3, A4 ke dalam

Matrixbc11 bc12 bc13 bc14

bc21 bc22 bc23 bc24

bc31 bc32 bc33 bc34

bc41 bc42 bc43 bc44

A1

A2

A3

A4

=

0

0

0

0

my

1 0

Dy 2 q 2

b2 k2 q

Cos

q

Sin

b q

Dy 2 q 2 Cos q b2

k2 q Sin q

b k2 q Cos b q Dy 2 q 2 Sin b q k2 Cosh

b

;

Detmy Det my Simplify1

b42 q Dy q Cos q b k2 Sin q

b k2 2 Dy Sinb q b2 k2 q 1 Sinh b2 q k2 Cosb q Dy q Sinb q b k2 2 Dy Cos q Dy q2 1 Sinh

b2 q 1 k2 Cosb q Dy q Sinb qb k2 2 Dy Cosh b k2 Dy Sinh

q b2 k2 Cos q Dy q Sinb q2 Dy b k2 k2 Cosh b k2 Dy 1 Sinh

b2 k2 q 1 Cosh b k2 Dy Sinb q2 Dy b k2 k2 Cosh b k2 Dy 1 Sinh

b k2 2 Dy Sinb q b2 k2 q 1 Sinh

b k2 Dy 1 Cosh 2 Dy b k2 k2 Sinh


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PR Kerr Solution HKB Rev#1.Nb