matlab programming – eigenvalue problems and mechanical
TRANSCRIPT
MATLAB Programming –Eigenvalue Problems and Mechanical Vibration
0)( =⋅−=⋅ xIAxxA λλ
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
A Coupled Mass Vibration Problem
EOM:
0)(0)(
122212
122111
=−++=−−+
xxkxkxmxxkxkxm
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Vibration Solutions – harmonic response
)cos(2
1
2
1 ϕω +⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡t
cc
xx
0/)(/
//)(
2
1
212
2
2212
=⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡
++−−−++−
cc
mkkmkmkmkk
ωω
Trial solution:
Matrix representation of EOM:
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Vibrational Frequencies and Mode Shapes
⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡+=
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡=
112
11
222
12122
112
1121
Acc
mkk
Acc
mk
ω
ω
Characteristic Equation (Determinant = 0 ):
22
21 kmkk ±=−+ ω
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Vibrations as a general class of “Eigenvalue Problems”
0/)(/
//)(
2
1
212
2
2212
=⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡
++−−−++−
cc
mkkmkmkmkk
ωω
Recast EOM:
As:
0)(
1001
/)(///)(
00
/)(///)(
2
12
2
1
212
221
2
12
2
2
1
212
221
=⋅−⋅=⋅
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡+−−+
⎥⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⋅⎥⎦
⎤⎢⎣
⎡+−−+
xIAxIxA
cc
cc
mkkmkmkmkk
cc
cc
mkkmkmkmkk
λλ
ω
ωω
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Eigenvalue equation, Eigenvalues, Eigenvectors
0)( =⋅−=⋅ xIAxxA λλ
Eigenvalue equation:
Eigenvalues (angular frequencies of the vibration):
2ωλ =
Eigenvectors (mode shape of the vibration):
⎥⎦
⎤⎢⎣
⎡=
2
1
cc
x
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Solving Eigenvalue Problem in MATLABLook at the problem numerically:
mNkmNkkgm /2/11 21 ===
m=1;k1=1;k2=2;A=[(k1+k2)/m -k2/m; -k2/m (k1+k2)/m][X,L]=eig(A);XL
Simple m-file:
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
MATLAB output of simple vibration problem
X =
-0.7071 -0.7071-0.7071 0.7071
L =
1.0000 00 5.0000
eigenvector 1 eigenvector 2
eigenvalue 1
eigenvalue 2
Ok, we get the same results as solving the characteristics equation… so what is the big deal?
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
A more complex vibrations problem
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
EOM for a more complex problem
EOM can be gotten from free body diagrams of each mass
0)(0)(0)(
0)(
342646444
442315354333
463312263222
3522152111
=−−++=−−−+++=−−−+++
=−−+++
xkxkxkkxmxkxkxkxkkkxmxkxkxkxkkkxm
xkxkxkkkxm
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Characteristic equation of more complex problem:
0
0
0
4
3
2
1
4
642
4
4
4
6
3
4
3
5422
3
3
3
5
2
6
2
3
2
6322
2
2
1
5
1
2
1
5212
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−−
++−−
++−−
++−−
cccc
mkk
mk
mk
mk
mkkk
mk
mk
mk
mk
mkkk
mk
mk
mk
mkkk
ω
ω
ω
ω
Solving this and find roots is no longer so simple!It is now an eighth order polynomial….
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Look at more complex vibration as eigenvalue problem
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−−
−++
−−
−−++
−
−−++
4
3
2
1
2
4
3
2
1
4
64
4
4
4
6
3
4
3
542
3
3
3
5
2
6
2
3
2
632
2
2
1
5
1
2
1
521
1000010000100001
0
0
cccc
cccc
mkk
mk
mk
mk
mkkk
mk
mk
mk
mk
mkkk
mk
mk
mk
mkkk
ω
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Solving more complex problem by MATLAB
Pick some numerical values:
mNkmNkmNkmNkmNkmNk
kgmkgmkgmkgm
/6/5/4/3/2/1
4321
654
321
4321
======
====
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Solving complex vibration problem by MATLAB
m1=1;m2=2;m3=3;m4=4;k1=1;k2=2;k3=3;k4=4;k5=5;k6=6;A=[(k1+k2+k5)/m1 -k2/m1 -k5/m1 0; -k2/m2 (k2+k3+k6)/m2 -k3/m2 -k6/m2; -k5/m3 -k3/m3 (k3+k4+k5)/m3 -k4/m3; 0 -k6/m4 -k4/m4 (k4+k6)/m4][X,L]=eig(A);XL
Create another m-file:
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Characteristic matrix of the eigenvalue problem
A =
8.0000 -2.0000 -5.0000 0-1.0000 5.5000 -1.5000 -3.0000-1.6667 -1.0000 4.0000 -1.3333
0 -1.5000 -1.0000 2.5000
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Frequencies and mode shapes of complex problem
X =0.4483 0.9456 0.6229 -0.16500.5156 -0.1826 -0.0411 -0.90060.5031 -0.2591 0.5788 0.31640.5292 0.0735 -0.5246 0.2481
L =0.0878 0 0 0
0 9.7562 0 00 0 3.4858 00 0 0 6.6702
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Multiple element vibration problems –finite element simulation
Inspired by Aladdin web site
Consider the vibrationalmodes of a four stories building.The mass are assumed to be concentrated at the floors.The walls constitutes springs.This can be models as a1-D system.
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Write equation of motion for the four floors
4443344
34332233
23221122
12111
2)(2)(2)(2)(2)(2
)(2
xkxxkxmxxkxxkxmxxkxxkxm
xxkxm
−−=−+−=−+−=
−=
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Write down the Mass and Stiffness Matrix
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
4
3
2
1
000000000000
mm
mm
M
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−−+−
−+−−
=
)22(2002)22(2002)22(20022
433
3322
2211
11
kkkkkkk
kkkkkk
K
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Using MATLAB Eig with Mass & Stiffness Matrix Directly
MVDKV =
Eig can also operate on the eigenvalue equationIn this form where:K is the stiffness matrix, V is the matrix containingAll the eigenvectors, M is the mass matrix, andD is a diagonal matrix containing the eigenvalues
[V,D]=eig(K,M)
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Mode shape of building oscillation
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.0151
1.5
2
2.5
3
3.5
4
08.213.159.012.0 22224321==== ωωωω
160012008004004500300030001500
4321
4321
========
kkkkmmmm
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Mode shape of building oscillation 2
83.293.170.0042.0 22224321==== ωωωω
160012008004004500300030001500
4321
4321
========
kkkkmmmm
-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.021
1.5
2
2.5
3
3.5
4
Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].