Lektion 1
Lektion 1
Lektion 1
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FEM<br />
M5<br />
Introduktion til FEM<br />
Søren Heide Lambertsen
Hvad anvender man FEA til.
Elementtyper<br />
1D<br />
1 frihedsgrad
Elementtyper<br />
2D<br />
2-3 frihedsgrader
Elementtyper<br />
2D
Elementtyper<br />
3D<br />
6 frihedsgrader(Bjælke) 3 frihedsgrader for Solid
Elementtyper<br />
3D
Eksempel 1<br />
K 1<br />
D 1<br />
D 2
Kx=F<br />
[k]{d}={R}<br />
K<br />
[Stivhedsmatrice] {Flytninger}= {kraftvektor}<br />
F<br />
x
Eksempel 1<br />
R 1 R 2<br />
K 1<br />
[k]{d}={R}<br />
D 1<br />
D 2
Løsning eksempel 1
Eksempel 2<br />
k 1 k 2<br />
U 1<br />
U 2<br />
U 3
Matrice eksempel 2<br />
R 1 R 2 R 3<br />
k 1 k 2<br />
U 1<br />
L 1 L<br />
U 2<br />
2<br />
U 3
• Løsning eksempel 2
• Løsning eksempel 2
Pause<br />
18
Diskretisering af model<br />
Diskretisere<br />
19
Elementtyper<br />
20
Mesh<br />
21
Convergense<br />
22
Bjælkeelement.<br />
DOF = <br />
23
Stivhedsmatricen<br />
6X6 matrice<br />
∙<br />
<br />
<br />
<br />
<br />
<br />
<br />
=<br />
<br />
<br />
<br />
<br />
<br />
<br />
24
Stivhedsmatricen<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
25
Stivhedsmatricen<br />
<br />
<br />
<br />
<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
26
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
<br />
<br />
<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
27
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
<br />
<br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
28
Stivhedsmatricen<br />
http://people.civil.aau.dk/~lda/Notes/<br />
Deformationsmetoden for Rammekonstruktioner<br />
Side 72<br />
29
Stivhedsmatricen<br />
30
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
<br />
<br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
31
Stivhedsmatricen<br />
32
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
0 12EI/L 3 <br />
<br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
33
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
0 12EI/L 3 <br />
<br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
34
Stivhedsmatricen<br />
35
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
0 12EI/L 3 6EA/L<br />
<br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
36
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
0 12EI/L 3 6EA/L<br />
0 <br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
37
Stivhedsmatricen<br />
38
Stivhedsmatricen<br />
q 1 q 2<br />
u 1 u 2<br />
v 1<br />
v 2<br />
EA/L 0 0<br />
0 12EI/L 3 6EA/L<br />
0 −6EA/L 2 <br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
39
Stivhedsmatricen<br />
40
Stivhedsmatricen<br />
q 1 q 2<br />
N 1 N 2<br />
V 1<br />
V 2<br />
EA/L 0 0<br />
0 12EI/L 3 6EA/L<br />
0 6EA/L 2 4EA/L<br />
−EA/L 0 0<br />
∙<br />
u 1<br />
v 1<br />
θ 1<br />
u = 2<br />
v 2<br />
θ 2<br />
R u1<br />
R v1<br />
R θ1<br />
R u2<br />
R v2<br />
R θ2<br />
41
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
Frihedsgrader og matrice<br />
42
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
3 frihedsgrader i hvert punkt<br />
15X15 matrice<br />
43
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
44
Stivhedsmatricen<br />
u 1<br />
q 1 q 2<br />
u q<br />
u 3<br />
2 3<br />
q 4<br />
u 4<br />
q 5<br />
u 5<br />
V 1<br />
V 2<br />
V 3<br />
V 4<br />
V 5<br />
X<br />
45
Stivhedsmatricen<br />
u 1<br />
q 1 q 2<br />
u q<br />
u 3<br />
2 3<br />
q 4<br />
u 4<br />
q 5<br />
u 5<br />
V 1<br />
V 2<br />
V 3<br />
V 4<br />
V 5<br />
X X<br />
X XX X<br />
X XX X<br />
X XX X<br />
X X<br />
46
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
Nyt nummersystem.<br />
47
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X<br />
X<br />
X<br />
X<br />
48
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X<br />
X<br />
X<br />
X<br />
49
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X X<br />
X X<br />
50
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X<br />
X<br />
X<br />
X<br />
51
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
Nu 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X<br />
X<br />
X<br />
X<br />
52
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
XX<br />
53
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X<br />
X<br />
X<br />
X X<br />
X XX X<br />
X XX<br />
54
Stivhedsmatricen<br />
u 1<br />
q 1 q 4<br />
u q<br />
u 3<br />
4 3<br />
q 2<br />
u 2<br />
q 5<br />
u 5<br />
V 1<br />
V 4<br />
V 3<br />
V 2<br />
V 5<br />
X<br />
X<br />
X<br />
XX X X<br />
X XX X<br />
X XX<br />
X<br />
X<br />
55
Eksempel<br />
P<br />
1 2 3<br />
P<br />
56
Eksempel<br />
1 2 3<br />
P<br />
0<br />
D =<br />
0<br />
<br />
<br />
<br />
<br />
0<br />
0<br />
<br />
57
Eksempel<br />
1 2 3<br />
P<br />
R =<br />
R 1N<br />
R 1V<br />
0<br />
0<br />
−P<br />
0<br />
R 3N<br />
R 3V<br />
0<br />
58
Eksempel<br />
59<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
0<br />
0<br />
X<br />
X<br />
X<br />
XX<br />
XX<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
XX<br />
XX<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
XX<br />
XX<br />
XX<br />
X<br />
X<br />
X<br />
0<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
∙<br />
0<br />
0<br />
<br />
<br />
<br />
<br />
0<br />
0<br />
<br />
=<br />
R 1N<br />
R 1V<br />
0<br />
0<br />
−P<br />
0<br />
R 3N<br />
R 3V<br />
0
Eksempel<br />
60<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
X<br />
0<br />
0<br />
XX<br />
XX<br />
XX<br />
X<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
XX<br />
XX<br />
XX<br />
X<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
XX<br />
XX<br />
XX<br />
X<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
0<br />
0<br />
X<br />
X<br />
∙<br />
<br />
<br />
<br />
<br />
<br />
0<br />
0<br />
0<br />
0<br />
=<br />
0<br />
−P<br />
0 0<br />
0<br />
R 1N<br />
R 1V<br />
R 3N<br />
R 3V
Eksempel<br />
61<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
X<br />
0<br />
0<br />
XX<br />
XX<br />
XX<br />
X<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
XX<br />
XX<br />
XX<br />
X<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
XX<br />
XX<br />
XX<br />
X<br />
XX<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
0<br />
0<br />
X<br />
X<br />
X<br />
X<br />
X<br />
0<br />
X<br />
0<br />
0<br />
X<br />
X<br />
∙<br />
<br />
<br />
<br />
<br />
<br />
0<br />
0<br />
0<br />
0<br />
=<br />
0<br />
−P<br />
0 0<br />
0<br />
R 1N<br />
R 1V<br />
R 3N<br />
R 3V<br />
k reduceret<br />
k 12
Eksempel<br />
k reduceret D = R<br />
D = R k<br />
−1<br />
reduceret<br />
Reaktioner bliver da:<br />
R reaktioner = k 12 D<br />
62
FEM<br />
BM5 kursusgang<br />
Introduktion til FEM (Modal analyse)<br />
Søren Heide Lambertsen
I dag<br />
1. Hvad er modal analyse<br />
2. Egenfrekvens,Dæmpning<br />
3. Beregning´s grundlaget for modal analyse<br />
4. Eksempel 1<br />
5. Opsætning af modal analyse i Ansys workbench<br />
6. Opsætning af Prestress modal analyse i Ansys Workbench
Modal analyse<br />
Beregningsmetoden giver mulighed for at bestemme egenfrekvenser<br />
for en struktur.
Fjeder/masse system<br />
u = u t<br />
K<br />
m<br />
u<br />
F
Fjeder/masse system<br />
u = u t<br />
K<br />
u = du<br />
dt<br />
m<br />
u<br />
F
Fjeder/masse system<br />
u = u t<br />
K<br />
u = du<br />
dt<br />
m<br />
u<br />
a = u = du2<br />
d 2 t<br />
F
Fjeder/masse system<br />
u = u sin (ωt)<br />
K<br />
m<br />
u<br />
F
Fjeder/masse system<br />
u = u sin (ωt)<br />
K<br />
u = uω cos(ωt)<br />
m<br />
u<br />
F
Fjeder/masse system<br />
u = u sin (ωt)<br />
K<br />
u = uω cos(ωt)<br />
a = u = −uω 2 sin (ωt)<br />
m<br />
u<br />
F
Fjeder/masse system<br />
f = ma<br />
f = mu<br />
K<br />
m<br />
u<br />
F
Fjeder/masse system<br />
f = ma<br />
f = mu<br />
K<br />
f = mu<br />
f − ku − cu = mu<br />
m<br />
u<br />
F
Fjeder/masse system<br />
f = ma<br />
f = mu<br />
K<br />
f = mu<br />
f − ku − cu = mu<br />
f = mu + cu + ku<br />
m<br />
u<br />
F
Fjeder/masse system<br />
f = ma<br />
f = mu<br />
K<br />
f = mu<br />
f − ku − cu = mu<br />
f = mu + cu + ku<br />
m<br />
u<br />
F
Fjeder/masse system<br />
c = 0 f = 0<br />
K<br />
m<br />
u<br />
F
Fjeder/masse system<br />
c = 0 f = 0<br />
K<br />
f = mu + cu + ku<br />
m<br />
u<br />
F
Fjeder/masse system<br />
c = 0 f = 0<br />
K<br />
f = mu + cu + ku<br />
0 = mu + ku<br />
m<br />
u<br />
F
Fjeder/masse system<br />
0 = mu + ku<br />
K<br />
m<br />
u<br />
F
Fjeder/masse system<br />
0 = mu + ku<br />
K<br />
u = u sin (ωt)<br />
m<br />
u<br />
F
Fjeder/masse system<br />
0 = mu + ku<br />
K<br />
u = u sin (ωt)<br />
u = −uω 2 sin (ωt)<br />
m<br />
u<br />
F
Fjeder/masse system<br />
0 = mu + ku<br />
K<br />
u = u sin (ωt)<br />
u = −uω 2 sin (ωt)<br />
m<br />
u<br />
0 = −muω 2 sin (ωt) + ku sin (ωt)<br />
F
Fjeder/masse system<br />
0 = −muω 2 sin (ωt) + ku sin (ωt)<br />
K<br />
m<br />
u<br />
F
Fjeder/masse system<br />
0 = −muω 2 sin (ωt) + ku sin (ωt)<br />
K<br />
0 = −mω 2 + k<br />
m<br />
u<br />
F
Fjeder/masse system<br />
0 = −muω 2 sin (ωt) + ku sin (ωt)<br />
K<br />
0 = −mω 2 + k<br />
ω =<br />
k m<br />
m<br />
u<br />
F
Dæmpning
Dæmpning<br />
c = 0<br />
C cr = 2m<br />
k m<br />
c < c cr
Dæmpning<br />
c = 0<br />
C cr = 2m<br />
k m<br />
c < c cr<br />
ω d = ω 1 − ξ 2 ξ = C<br />
C cr
Dæmpning<br />
c = 0<br />
ξ < 0,15
Dæmpning<br />
c = 0<br />
ξ < 0,15<br />
ω d = ω 1 − ξ 2
Dæmpning<br />
c = 0<br />
ξ < 0,15<br />
ω d = ω 1 − ξ 2<br />
ω d = 30<br />
1 − 0,15 2 = 29,66 rad/s
Dæmpning<br />
c = 0<br />
ξ < 0,15<br />
ω d = 30<br />
ω d = ω 1 − ξ 2<br />
1 − 0,15 2 = 29,66 rad/s<br />
ω d = 30<br />
2π = 4,77Hz<br />
ω = 29,66<br />
2π<br />
= 4,72Hz
Dæmpning<br />
c = 0<br />
ξ < 0,15<br />
ω d = 30<br />
ω d = ω 1 − ξ 2<br />
1 − 0,15 2 = 29,66 rad/s<br />
ω d = 30<br />
2π = 4,77Hz<br />
ω = 29,66<br />
2π<br />
ω d ≈ ω<br />
= 4,72Hz
Matrise system<br />
D = D sin (ωt)
Matrise system<br />
D = D sin (ωt)<br />
D<br />
= − D ω 2 sin (ωt)
Matrise system<br />
D = D sin (ωt)<br />
D<br />
= − D ω 2 sin (ωt)<br />
0 = [m]{D} + [k]{D}
Matrise system<br />
D = D sin (ωt)<br />
D<br />
= − D ω 2 sin (ωt)<br />
0 = [m]{D} + [k]{D}<br />
0 = − m D ω 2 sin ωt + [k] D sin (ωt)
Matrise system<br />
D = D sin (ωt)<br />
D<br />
= − D ω 2 sin (ωt)<br />
0 = [m]{D} + [k]{D}<br />
0 = − m D ω 2 sin ωt + [k] D sin (ωt)
Matrise system<br />
D = D sin (ωt)<br />
D<br />
= − D ω 2 sin (ωt)<br />
0 = [m]{D} + [k]{D}<br />
0 = − m D ω 2 sin ωt + [k] D sin (ωt)<br />
0 = − m D ω 2 + [k] D 0 = (−[m]ω 2 + [k]) D
Mass<br />
Mass lumping<br />
m = m 2<br />
1 0<br />
0 1
Mass<br />
Consistent mass metrix<br />
0<br />
L<br />
[N] T N ρA dx =<br />
Beam:<br />
m<br />
420<br />
156 22L 54 −13L<br />
22L 4L 2 13L −3L 2<br />
54 13L 156 −22L<br />
−13L −3L 2 −22L 4L 2
Eksempel<br />
EI 12 −6L m<br />
L 3 −6L −4L2 − ω2<br />
420<br />
156 −22L<br />
−22L −4L 2 v 2<br />
θ 2<br />
= 0 0<br />
ω 1 = 3,533<br />
ω 2 = 34,81<br />
1/2<br />
EI<br />
mL 3<br />
1/2<br />
EI<br />
mL 3<br />
102
Eksempel<br />
ω 1 = 3,533<br />
ω 2 = 34,81<br />
1/2<br />
EI<br />
mL 3<br />
1/2<br />
EI<br />
mL 3<br />
103