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MatLab Programming

Eigenvalue Problems and Mechanical Vibration

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

k5

k1 k2 k3 k4

m3 m4

x1 x2 x3 x4

m2m1

k6

Figure by MIT OCW.

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A Coupled Mass Vibration Problem

EOM:

0)(

0)(

122212

122111

=++

=+

xxkxkxm

xxkxkxm

&&

&&

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

k1

x1 x2

k1k2

m m

Figure by MIT OCW.

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Vibration Solutions harmonic response

)cos(2

1

2

1 +

=

t

c

c

x

x

0/)(/

//)(

2

1

212

2

221

2

=

++

++

c

c

mkkmk

mkmkk

Trial solution:

Matrix representation of EOM:

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Vibrational Frequencies and Mode Shapes

=

+=

=

=

112

11

2

22

1212

2

1

12

1121

Acc

mkk

Acc

mk

Characteristic Equation (Determinant = 0 ):

2

2

21kmkk =+

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

Vib ti l l f Ei l P bl

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Vibrations as a general class of Eigenvalue Problems

0/)(/

//)(

2

1

21

2

2

2212

=

++

++c

c

mkkmk

mkmkk

Recast EOM:

As:

0)(

10

01

/)(/

//)(

0

0

/)(/

//)(

2

12

2

1

212

221

2

1

2

2

2

1

212

221

=

=

=

+

+

=

+

+

xIA

xIxA

c

c

c

c

mkkmk

mkmkk

c

c

c

c

mkkmk

mkmkk

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Eigenvalue equation, Eigenvalues, Eigenvectors

0)( == xIAxxA

Eigenvalue equation:

Eigenvalues (angular frequencies of the vibration):

2=

Eigenvectors (mode shape of the vibration):

=

2

1

c

cx

ite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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Solving Eigenvalue Problem in MatLab

Look at the problem numerically:

mNkmNkkgm /2/1121===

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, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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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 solvingthe characteristics equation so what is the big deal?

Cite as: Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007.MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].