Project Proposal
Natural Frequencies and Mode Shapes of an Undamped Linear
Vibrating System
Michael C. Ramsey
12 October 1999
Numerical Analysis for Engineers
I propose to find the natural frequencies and mode shapes of an undamped linear
vibrating system of multiple degrees of freedom under free vibration. This is a fairly
straight forward process as long as the system has only 2 degrees of freedom. Once the
degrees of freedom begin to increase, the manual computational effort becomes
inhibitive. The results of the code generated to solve these types of problems can be
easily checked using Matlab and other available software packages.
The general system I will analyze can be represented by the following 2 degree of
freedom system:
M1
K1
M2
K2
K3
To find the natural frequencies of this system, the sytem must first be modeled
mathematically. The equations of motion that describe this system are:
m1 x1  k1 x1  k 2 ( x2  x1 )  0
m 2 x2  k 3 x2  k 2 ( x1  x2 )  0
Now we assume oscillatory motion where:
x (t )  X sin( t )
x  X cos(t )
x   2 X sin( t )
We substitute into the equations of motion and divide out the sin term to end up with:
  2 m1 X 1  k1 X 1  k2 ( X 2  X 1 )  0
  2 m 2 X 2  k3 X 2  k 2 ( X 1  X 2 )  0
We now represent this set of equations in matrix format:
(  2 m1  ( k1  k 2 ))  k 2   X 1  0

    
2
k

(


m

(
k

k
))
 X 2   0
2
3
2 
 2
To solve for the natural frequencies (eigen-values) we find the determinate of the first
matrix in 2 and then take the square root. This can be performed using a root finding
method like Newton-Raphson. Now we find the mode shapes (eigen-vectors) and the
problem is complete. This method will be generalized to solve problems of n degrees of
freedom.
This data by itself ,while interesting, is of little use. Perhaps we could use the result to
determine if a natural frequency existed in the operating range of the machine. I would
propose further effort in this project to finding the deflections and stress associated with
this system under forced vibration.