MATLAB Programming – Eigenvalue Problems and Mechanical ...

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


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:


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


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


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


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:


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?


A more complex vibrations problem


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


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….


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


Solving more complex problem by MATLAB

Pick some numerical values:

mNkmNkmNkmNkmNkmNk

kgmkgmkgmkgm

/6/5/4/3/2/1

4321

654

321

4321

======

====


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:


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


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


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.


Write equation of motion for the four floors

4443344

34332233

23221122

12111

2)(2)(2)(2)(2)(2

)(2

xkxxkxmxxkxxkxmxxkxxkxm

xxkxm

−−=−+−=−+−=

−=


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


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)


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


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


MATLAB Programming – Eigenvalue Problems and Mechanical ...

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