Hello wjs-ga,
The mass and stiffness matrices in your problem describe a two
degree-of-freedom system.
A good reference for solving for the natural frequencies and mode
shapes for a two degree-of-freedom system can be found at:
University of Saskatchewan, College of Engineering
http://www.engr.usask.ca/classes/CE/804/notes/mdof_dis.pdf
As described in this reference, for this type of system, the mass
matrix is given by
[m1 0
0 m2]
The stiffness matrix is given by
[k11 k12
k21 k22]
(See Eq. 8.2a in the reference)
k11 = k1 + k2; k12 = -k2
k21 = -k2; k22 = k2
(See Eq. 8.3)
Based on the information that you have provided in your problem,
m1 = 1; m2 = 1
k11 = 2; k12 = -1; k21 = -1; k22 = 2
With this information, you can use Eq. 8.8 to calculate the natural
frequencies:
(w1)^2=1/2[(2/1)+(2/1)]-SQRT[1/4(((2/1)-(2/1))^2)+((-1)^2)/(1)(1))
(w2)^2=1/2[(2/1)+(2/1)]+SQRT[1/4(((2/1)-(2/1))^2)+((-1)^2)/(1)(1))
(I hope that’s not too confusing. It’s a little difficult to make the
equation readable.)
This gives you w1^2 = 1 and w2^2 = 3
Therefore the natural frequencies are w1 = 1 and w2 = SQRT(3) = 1.732.
Eq. 8.9 gives you the mode shapes:
First mode shape:
[Q1 K12 -1 -1
-- = ------------- = ------------ = -----
Q2]1 m1(w1)^2 –k11 (1)(1)^2 – 2 -1
As stated in the reference, Q1 can be arbitrarily set to one.
Therefore if Q1 = 1, then Q2 = 1. The masses are moving in phase.
I’ll try to plot it, but I don’t know how this is going to look after
it is posted.
m1 m2
.______.
/ \
/ \
Second mode shape:
[Q1 K12 -1 -1
-- = ------------- = -------------- = ----
Q2]2 m1(w2)^2 –k11 (1)(SQRT(3))^2 – 2 1
Therefore if Q1 = 1, then Q2 = -1. The masses are moving
out-of-phase.
M1
/\
/--\--/
\/
m2
The modal matrix is
[1 1
1 –1]
Other references:
Kettering University
http://www.gmi.edu/~drussell/Demos/multi-dof/multi-dof.html
Structural Dynamics Research Laboratory
University of Cincinnati
http://www.sdrl.uc.edu/ucme662/lectures.pdf
Mechanical Engineering Department
United States Naval Academy
http://web.usna.navy.mil/~ratcliff/EM423/MDOF/Twodof.pdf
Mechanical Vibrations, Second Edition, Singiresu S. Rao
Addison-Wesley Publishing Company, 1990
I hope this has been helpful. If you have questions, please request
clarification prior to rating the answer.
Googlenut
Google Search Terms:
spring mass mode shapes "two degree of freedom"
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natural frequencies mode shapes
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