FEM

1
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2
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3
FEM
?????
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4
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????FEM???
FEM
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5
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6
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7
Fig. 2.1 Model of the analysis.
8
??
???-?????
?????
9
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10
0
11
?????
?????
??????????????????
J???? ? ?
V?j1,fa????????????????????
12
Newton-Raphson?
NLFEM
( fj1 )
Newton-Raphson?
Modal FEM0
( fj1fa )
Newton-Raphson?
Modal FEMa
??NLFEM???Modal FEMa
Modal FEMb
fj1fb
13
?????(??????, fj1 )
?????(??????, fj1jj )
?????(??????, f yij1 )
14
???????
15
Table 2.1 Specification of the beam.
Table 2.2 Specification of the external force.
16
fajj (j2,3,,8)FRFs , displacements and
strains
17
??????- beam 1 -
18
Fig. 2.2
FRFs obtained by FEM, single-mode(fj1,
j0) and two-mode(fj1jjj2,3,,8)
analyses for beam 1.
19
Fig. 2.3
Displacements and normal strains obtained by FEM,
single-mode(fj1, j0) and
two-mode(fj1jjj2,3,,8) analyses for
beam 1 at 50Hz.
20
fayi (i2,3,,8) FRFs , displacements and
strains
21
??????- beam 1 -
22
Fig. 2.4
FRFs obtained by FEM, single-mode(fj1,
i0) and two-mode(fyij1i1,2,,8)
analyses for beam 1.
23
Fig. 2.5
Displacements and normal strains obtained by FEM,
single-mode(fj1, i0) and
two-mode(fyij1i1,2,,8) analyses for
beam 1 at 50Hz.
24
??????(??????????)- beam 2 -
25
Fig. 2.6
FRFs obtained by FEM, single-mode(fj1,
i0) and two-mode(fyij1i1,2,,8)
analyses for beam 2.
26
Fig. 2.7
Displacements and normal strains obtained by FEM,
single-mode(fj1, i0) and
two-mode(fyij1i1,2,,8) analyses for
beam 2 at 75Hz.
27

but
?
?
28
NLFEM????FRFs , modes , strains
29
Fig. 2.8
FRFs obtained by NLFEM and numerical integration
for beam 1 10 elements, with damping term
2MdV/dt(N.I.).
30
NLFEM ??????????????
31
Fig. 2.9
Modes and normal strains obtained by NLFEM and
numerical integration(NI) at 20 , 30 , 40 , 50
Hz for beam 1.
32
NLFEM???
??
???????
33
Fig. 2.10
Modes and normal strains obtained by FEM at 20 ,
30 , 40 , 50 Hz for beam 1, at 45 , 55 , 65 ,
75 Hz for beam 2.
34
?????????????????? ????
Vq qHz????NLFEM???
y q??????????
j q??????????
f yqjq and yqj1
35
beam 1f yqjq and yqj1differen
ce of FRF , sqrt(k/m)
36
Fig. 2.11
Differences between NLFEM and Modal FEM, and
sqrt(k/m) / w1 by using fyqjq and
yqj1 (q20,21,,50) for beam 1.
37
f yw1/2pj1
G.E.(??????)
38
?????????- beam 1 -
39
beam 1w1 , m , k , b ???
40
Table 2.4 Convergence characteristics of linear
natural frequency for beam 1.
41
Table 2.5 Convergence characteristics of
coefficients of G.E. for beam 1.
42
??????????
43
Fig. 2.12
Comparison between the backbone curve and the
numerical integration for beam 1 10 elements,
with damping term 2MdV/dt(N.I.).
44
Yamaki?Mori?????????
45
Fig. 2.13
Comparison between the backbone curve obtained by
the present method and that by Yamaki and Mori
for beam 1.
46
??????????????????????????????
47
Table 3.1 Specifications of the plates.
Table 3.2 Specifications of the external forces.
48
Fig. 3.1 Finite element model of a rectangular
plate.
49
???????????
50
?????????
51
j1 , w1
( fj1 )
Newton-Raphson? at ww1
h1w1
V w1
Newton-Raphson? at ww1
with Vinit j1 h1w1
y 1???????1????
V w1
52
plate 1?????????
53
Fig. 3.2
Comparison between the backbone curve obtained by
the present method and the frequency ratios in
other papers for plate 1.
54
plate 2?????????
55
Fig. 3.3
Comparison between the backbone curve obtained by
the present method and the frequency ratios in
other papers for plate 2.
56
????????????- plate 3 -
57
?????????
58
y2y1y5 M , K1????????????????
im , ik1 ?????
0
59
plate 3Rebeiro ? Petyt ???????
60
?????(??????, )
???????
????
??
61
Fig. 3.4
(a/2,b/2)
(a/4,b/4)
1st
3rd
Amplitude of 1st and 3rd harmonic of plate 3 at
(x,y) (a/2,b/2) and (a/4,b/4).
62
???????,???????????
63
??????
64
Table 3.3 Specification of the beam.
Table 3.4 Specification of the external force.
65
?????? ?????????
66
Fig. 3.5
Comparison between the backbone curve obtained by
the present method and that by Yamaki and Mori.
67
2????????????L???
68
Fig. 3.6 The geometry and support conditions of
L-shaped plate.
69
L???Shi ? Mei???????
70
Fig. 3.7
Comparison between the backbone curve obtained by
the present method and frequency ratios obtained
by Shi and Mei.
71
? ?
72
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FEM???
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