Embed Size (px)
Citation preview
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
2( ) ( )u u u + + =
18
19
20
i ij ij
u fx
− =
0i
i
qT s
x
= −
: stresses
: body force
: density
: displacements
: equilibrium temperature
: entropy rate per unit volume
: thermal flux
ij
i
i
i
f
u
T
s
q
s
r
0
: elasticity tensor
: strain tensor
: thermomechanical coupling tensor
: specific heat capacity
: temperature
k : thermalconductivities
ijkl
ij
ij
E
ij
C
C
e
b
q
ij ijkl kl ijC T = −
0ij ij E
Ts c
T = +
1( )
2
jiij
j i
uu
x x
= +
i ijj
Tq k
x
= −
ij ijkl kl ij ij i i ij j i i iV V V VC dV TdV u u dV n u dA f u dV
− + = +
21
0 , ,ij ij E i ij j i iV V VT TdV c TdV k T dV q n TdA
+ + = −
= uu H U = H Θ u =B U ,i = B Θ
: displacement vector
: temperature
: strain vector
: nodal displacementvector
: nodal temperature vector
: displacement interpolation matrix
: temperature interpolation matrix
: strain-displecement matrix
: temper
q
e
Q
u
q
u
q
u
U
H
H
B
B ature-gradient interpolation matrix
: nodal force vectorF
+ + =u uuqMU K K U FQ
+ + =uq qq qqC U C K QQ Q
22
23
24
Navier＇s equation Ordinary differential equation
Transformed responseResponse
Analytical solution
Inverse integral transformation
Finite Element Analysis
Integral transformation
25
26
27
28
29
50 μm
60 μm
70 μm
80 μm
90 μm
100 μm1
2
3
4
5
6
1
2
3
4
5
6
30
+ + =
+ + =
u uu
u
q
q qq qq
MU K K U F
C U C K Q
Q
Q Q
+ + =MU CU KU F
31
32
33
34
35
36
37
38
2
2( 2 ) ru
r z t
+ + =
2
2
( )( 2 ) zur
z r r t
+ − =
:1st Lame parameter
: 2nd Lame parameter
:
:
r r z
r z
u u u
r r z
u u
z r
+ +
−
1
0
ˆ ( , , ) ( , , ) ( ) j tr ru z k u z r t rJ kr e drdt
+ +
−
=
0
0
ˆ ( , , ) ( , , ) ( ) j tz zu z k u z r t rJ kr e drdt
+ +
−
=
0
1
( ) : Bessel functions of the zeroth order
( ) : Bessel functions of the first order
J kr
J kr
39
0
0
ˆ ( , , ) ( , , ) ( )j tz zu z k e dt u z r t rJ kr dr
+ +
−
=
1
0
ˆ ( , , ) ( , , ) ( )j tr ru z k e dt u z r t rJ kr dr
+ +
−
=
0
0
1ˆ( , , ) ( , , ) ( )
2
j tz zu z k e dk u z k kJ kr d
+ +−
−
=
1
0
1ˆ( , , ) ( , , ) ( )
2
j tr ru z k e dk u z k kJ kr d
+ +−
−
=
40
0
1
( ) : Bessel functions of the zeroth order
( ) : Bessel functions of the first order
J kr
J kr
( , )( , )
0zz
f r tr t
=
at z h
at z h
= +
= −
( , ) 0rz r t = at z h=
[ ( ) ( / 2)] ( )H r H r d t− −( ) Heaviside step function
beam diamter
H x
d
=
=
41
42
ˆ ( , , ) cosh( ) cosh( ) sinh( ) sinh( )r s s a au z k A k z D z B k z C z = − + −
ˆ ( , , ) sinh( ) sinh( ) cosh( ) cosh( )z s s a au z k A z D k z B z C k z = − − − −
0 0ˆ ˆ( , , ) ( , ) ( , ) ( ) ( , ) ( , ) ( )
s a
s j t a j tz z z
k k
u h x t H h f k kJ kr e d H h f k kJ kr e d + +
− −
− −
= +
1 1ˆ ˆ( , , ) ( , ) ( , ) ( ) ( , ) ( , ) ( )
s a
s j t a j tr r r
k k
u h x t H h f k kJ kr e d H h f k kJ kr e d + +
− −
− −
= +
43
44
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
45
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
46
0 10 20 30 40 50 60 70 80 90 100-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
47
0 10 20 30 40 50 60 70-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
0 10 20 30 40 50 60 70-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
48
0 10 20 30 40 50 60 70-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
Experiment
Simulation 1
49
0 10 20 30 40 50 60 70-1.00
-0.50
0.00
0.50
1.00
Time (s)
No
rma
l D
isp
lace
me
nt
(no
rma
lize
d)
Theory
Simulation 2
50
51
52
53
S
T
A
A =
: the total area of the frequency range.TA
: the area of the particular frequency range.SA
54
S
T
A
A =
: the total area of the frequency range.TA
: the area of the particular frequency range.SA
55
56
57
58
59
60
61
62
ˆ ( , , ) cosh( ) cosh( ) sinh( ) sinh( )r s s a au z k A k z D z B k z C z = − + −
ˆ ( , , ) sinh( ) sinh( ) cosh( ) cosh( )z s s a au z k A z D k z B z C k z = − − − −
0 0ˆ ˆ( , , ) ( , ) ( , ) ( ) ( , ) ( , ) ( )
s a
s j t a j tz z z
k k
u h x t H h f k kJ kr e d H h f k kJ kr e d + +
− −
− −
= +
1 1ˆ ˆ( , , ) ( , ) ( , ) ( ) ( , ) ( , ) ( )
s a
s j t a j tr r r
k k
u h x t H h f k kJ kr e d H h f k kJ kr e d + +
− −
− −
= +
4 4( )cosh( )sinh( )( , )
8
s sr
s s
j k h hH h
k
−=
4 4( )sinh( )cosh( )( , )
8
a ar
s a
j k h hH h
k
−=
2 2( )sinh( )sinh( )( , )
4
s sz
s
j k h hH h
− −=
2 2( )cosh( )cosh( )( , )
4
a az
a
j k h hH h
− −=
