Term Project
Analysis of the Vibration Characteristics of a Simple Structure
by:
Michael C. Ramsey
December 9, 1999
NAE
Fall 1999
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Table of Contents
Introduction
2
Problem Development
3
Numerical Background
8
Computational Application
9
Results
10
Comparison
13
Conclusion
13
1
Introduction
The general subject of interest is the behavior of a "high" degree of freedom
(henceforth referred to as DOF) system. The behavior of such a system is a function of
the physical properties of that system. Such a system can also be described in terms of its
natural frequencies and mode shapes. This report demonstrates a method and application
of finding the natural frequencies and mode shapes for a multi DOF system
The properties that allow us to mathematically model and predict the behavior of
vibrating systems are of particular interest to the engineer. There are historical instances
of catastrophic failures that were the result of systems being excited at or near a natural
frequency. One such instance is the Tacoma Narrows Bridge [1, page 267].
The Tacoma Narrows Bridge fell prey to aerodynamically induced demons. The
winds that bombarded the bridge excited destructive torsional modes, and the negative
damping provided by the aerodynamic excitation induced a catastrophic failure. Tacoma
Narrows is one example of a structure suffering damage due to vibration.
Today in earthquake prone parts of the world, engineers are developing
"earthquake proof" buildings and "active" sky-risers that use counter weights to offset the
induced motion of the Buildings. Earthquakes have claimed the lives of tens of thousands
of people in the last decade in places like Turkey, Mexico, Taiwan and Japan. To better
understand the dynamic nature of these traditionally static structures, we must be able to
model their behavior. The first step in understanding the dynamics is to find two defining
system parameters, the natural frequencies and mode shapes.
2
Problem Development
Photo of a building rocking in a recent Earthquake in Taiwan.
In order to solve the stated problem, we must first be able to describe the system
in terms we can readily analyze. Therefore, we will begin the mathematical background
for the analysis.
Consider the following 2 DOF system:
M1
K1
M2
K2
We write the equations of motion for this system and we obtain:
m1 x1  k1 x1  k 2 ( x2  x1 )  0
m 2 x2  k 2 ( x1  x2 )  0
We assume oscillatory motion where:
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x (t )  X sin( t )
x  X cos(t )
x   2 X sin( t )
When we substitute into the equations of motion we can develop the following
dynamic matrix:
( 2 m1  (k1  k 2 ))  k 2   X 1  0

   
2
k 2  ( m 2  k1 ))
  X 2  0
When we solve for 2 we find the eigenvalues for this matrix, which equate
physically to the square of the natural frequencies. When we substitute the eigenvalues
back into the matrix and find a non-zero solution for X1 and X2, we find the
eigenvectors, which correspond to the mode shape of the system.
Now we can develop a mass and stiffness matrix for a simplified structure. The
following schematic represents the simplified structure we will analyze:
5th Story
4th Story
3rd Story
2nd Story
1st Story
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Each level of the building has been simplified to be a concentrated block
of equal mass. Also, we will assume that the stiffness provided by the vertical beams is
uniform throughout the structure. We will also assume that the motion of the floors is
strictly lateral and the floors do not rotate about any axis.
To develop a set of equations in matrix form, we will follow a procedure outlined
by Thomson [1, page 168]. Using this method we assume a unit displacement of one
floor, and hold the others constant at zero. The forces required to maintain this
configuration give us the first column of the stiffness matrix in terms of its stiffness
coefficients.
K11
5th Story
K21
4th Story
K21
3rd Story
K21
2nd Story
K21
1st Story
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These values give us the first column of our stiffness matrix. Carrying the
procedure out, we develop the following stiffness matrix:
 k11
k
 21
 k31

k 41
 k51
k12
k13
k14
k 22
k 23
k 24
k32
k33
k34
k 42
k 43
k 44
k52
k53
k54
k15 
k 25 
k35 

k 45 
k55 
We find that this matrix has several traits that make it conducive to analysis. First,
The matrix is tridiagonal, with all other terms equal to zero. Furthermore it is symmetric
with K12=K21= K23=K32 etc. Also, K11=K22=K33=K44=2*K55.
We can determine the value of each stiffness element using standard beam theory.
If we treat the supporting structure as a single beam with an Area Moment of Inertia of
"I", and length "L", we can compute the lateral stiffness of each floor as the stiffness of a
Fixed-Fixed beam of length 2L.
K11 
192  E  I
2  L 3
Where E is the Modulus of Elasticity.
The off diagonal terms are then equal to –1/2 multiplied by K11.
In this case, we are assuming the floors are of equal mass, "M". So we can
develop a mass matrix as such:
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M
0

0

0
 0
0
0
0
M
0
0
0
M
0
0
0
M
0
0
0
0
0 
0

0
M 
The equations of motion in Matrix form then appear as:
M
0

0

0
 0
0
0
0
M
0
0
0
M
0
0
0
M
0
0
0
0   X 1   k11
0   X 2  k 21
 
0   X 3   k 31


0   X 4  k 41
 
M   X 5  k 51
k12
k13
k14
k 22
k 23
k 24
k 32
k 33
k 34
k 42
k 43
k 44
k 52
k 53
k 54
k15   X 1  0
k 25   X 2  0
   
k 35   X 3   0

k 45   X 4  0
   
k 55   X 5  0
Or for convenience as:
M X   K X   0
We can now combien these matrices into a single dynamic matrix "A", where:
A  invM  K 
Computing the inverse of a matrix can often be a tedious process, but since the
mass matrix is only along a single diagonal, inv(M11)=1/M11, making the computation
of the inverse of M trivial.
With the dynamic matix complete, we are ready to solve for the eigenvalues and
corresponding eigenvectors and interperet their physical signifigance.
7
Numerical Background
One powerful method for finding the eigenvalues is the "QR Algorithm" [2, page
573]. This method has the advantage of not incurring high roundoff error like other
deflation methods. It is also particularly suited to our situation since we are finding the
eigenvalues of a symmetric, tridiagonal matrix. This is advantageous in that we do not
have to perform a Householders [2, page 565] or similar method to achieve the desired
matrix condition. The "QR Algorithm" is employed to find the eigenvalues of our
dynamic matrix.
Once the eigenvalues are determined, we still need to determine the associated
eigenvectors of the matrix. Again, the tridiagonal, symmetric nature of the dynamic
matrix is advantageous. The following method is employed to determine the non-zero
solution of the eigenvectors:
1. Subtract the value of the eigenvalue from each term along the main diagonal
of the dynamic matrix.
2. Set X1 equal to one.
3. Solve for X2 using row one of the dynamic matrix.
4. Now that X1 and X2 are known, solve for X3 using the second row of the
dynamic matrix.
5. Repeat this procedure until the eigenvector is defined.
This is the general method used to find the eigenvalues and eigenvectors for our system.
8
Computational Application
The code developed to analyze this system is a Fortran 90 program built
using MS Developer Studio. The code first reads the parameters of the building (the
system of interest) from a text file. The values entered into the text file are:
1. Height of the building in inches.
2. Total mass of the building in lb*s2/in.
3. Modulus of the beam material in psi.
4. Area Moment of Inertia in inches4.
5. Number of floors above ground.
Given this data and the assumptions mentioned, the mass and stiffness matrices
are defined. The Dynamic matrix is then developed and a subroutine to find the
eigenvalues and corresponding eigenvectors is invoked. The three largest and most
dominant eigenvalues are returned along with the associated eigenvectors. The data is
printed to the screen.
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Results
The following data is analyzed using the program "BeamVibration.for".
1. Height = 600 inches.
2. Mass=25.87 lb*s2/in.
3. E=30e6 psi.
4. I=25.13 inches4.
5. Floors=5
After running the program, we find the following results:
EigenValues:
1. 357.369
2. 589.756
3. 767.182
EigenVectors:
1. 1 0.28 -.92 -.55 0.76
2. 1 -.83 -.39 1.09 -.59
3. 1 -1.68 1.83 -1.40 0.52
These Eigenvalues correspond to the following frequencies:
1. 3.01 Hz.
2. 3.87 Hz.
3. 4.41 Hz.
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The mode shapes are represented by the following graphs:
Mode Shape For Frequency=3.01 Hz
Floor
6
5
4
3
2
1
0
-5
0
5
Unit Displacement
Mode Shape For Frequency=3.87 Hz
Floor
6
5
4
3
2
1
0
-5
0
5
Unit Displacement
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Mode Shape For Frequency=4.41 Hz
Floor
6
5
4
3
2
1
0
-5
-3
-1
1
3
5
Unit Displacement
These Graphs display an expected characteristic of the system. As the Frequency
increases, the distortion of the building becomes more severe.
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Comparison
The accuracy of the code can be checked using other software packages. The dynamic
matrix is analyzed using Maple and Algorithm 9.1 and the following results are found:
Fortran EigenValue
Maple EigenValue
Percent Difference
767.18
777.51
1.3%
This result indicates that the method used is of comparable accuracy to the packaged
method in Algorithm 9.1.
Conclusions
This report lays out the groundwork for a means of analyzing a vertical structure
undergoing lateral vibration. This method can be useful in analyzing a structure and the
way it could react during an earthquake or in a windstorm. For the data given, we found
the first three natural frequencies and mode shapes. This was accomplished by analyzing
the structure, and developing a dynamic matrix. Next, we found the eigenvalues and
eigenvectors that give us insight into the system. Upon comparison with other numerical
methods, we found the values determined to be in close agreement. This method could
be further developed to track the center of mass of the structure and the absolute
displacement.
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Symbols Used
=Natural Frequency
F=Frequency (Hz)
L=Length
E=Modulus of Elasticity
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Sources
1) Theory of Vibrations with Applications, by William T. Thomson, Fifth Edition,
Prentice Hall.
2) Numerical Analysis, by Richard L. Burden, Sixth Edition, Brooks/Cole Publishing
Company.
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