Modal Analysis and Resonance
In Multi-Degree-of-Freedom Oscillators
Physics 4304 – Mechanics
By Chris Carr
May 7, 2004
The motion of an oscillator under the influence of an external driving force is
often discussed in physics. For simplicity sake, the feedback of the driver is neglected,
which is often an acceptable approximation. The problem arises when two or more
oscillators are connected, allowing energy to be transferred back and forth. The study of
coupled oscillations is important because it can be used to model several real-life
systems, such as an automobile suspension system, a multi-story building, or the
interaction between atoms on a molecular level. Modal analysis and resonance are two
important concepts to understand when dealing with multi-degree-of-freedom oscillatory
systems.
Oscillating harmonic systems have been studied for hundreds of years. Daniel
Bernoulli was serving as chair of mathematics at St. Petersburg when he worked with
Leonard Euler from 1727 to 1733 studying oscillating systems. It was in this time that
Bernoulli defined the simple nodes and the frequencies of oscillation of a system. His
work with harmonic vibrations was mostly related to its application in stringed musical
instruments. After leaving St. Petersburg, Bernoulli continued to study oscillations as
they occurred in the air in an organ pipe. Although much of the groundwork for the
theory of oscillations was done by Bernoulli, it was Lagrange who generalized this to the
oscillatory motion of a system of particles with a finite number of degrees of freedom.
The simplest example of the motion of a coupled harmonic oscillator can be
modeled by two equal masses attached on either side to fixed ends by springs of stiffness
k0 and attached to each other by a spring of stiffness k1. This configuration is shown in
Figure 1. It is assumed that the motion is restricted to the line connecting the masses, so
this is a two-degree-of-freedom system.
1
Figure 1: Simple Coupled Harmonic Oscillator3
The positions of each mass m from its equilibrium position are measured by the
coordinates x1 and x2. Using Newton’s Equations, the motion of the masses of the
system are described by Equations 1 and 2.
mx1  k 0 x1  k1 x2  x1   0
(1)
mx2  k 0 x2  k1 x2  x1   0
(2)
These differential equations are nontrivial due to the coupling term k1(x2-x1). The
resulting equation of motion of either mass shows that there are two normal modes of
motion of the mass, symmetric and antisymmetric. All oscillations of the masses can be
written as linear combinations of the two modes. The position of the mass at x1 as a
function of time is shown in Equation 3, where the subscript a refers to the antisymmetric
mode and s refers to the symmetric mode.
x1 t   Aa cos a  t  a   As cos s  t   s 
(3)
Through some tedious mathematics, the frequency of the system is found to be
described by Equation 4 (this derivation is shown on page 8). The plus/minus in the
equation shows that there are two characteristic frequencies, or eigenfrequencies, for the
system. The two frequencies are shown in Equations 5 and 6.
2

k 0  k1  k1
m
(4)
1 
k 0  2k1
m
(5)
2 
k0
m
(6)
The frequency ω1 corresponds to the antisymmetrical mode, which occurs when
the system is oscillating out-of-phase. Figure 2 shows the antisymmetrical mode of
oscillation. The two masses will oscillate both outward and both inward at the same
time. The frequency ω2 corresponds to the symmetrical mode of oscillation, where the
masses are moving in-phase. Figure 3 shows the masses in phase. The general motion of
the system is a linear combination of the symmetrical and antisymmetrical modes.
Figure 2: Antisymmetrical Mode
Figure 3: Symmetrical Mode
3
For this example, there are two natural frequencies. Each of these frequencies
c 
will have an associated eigenvector  1  (the process for finding the eigenvectors is
c 2 
explained on page 10). The eigenvectors are important because the solution to the simple
harmonic oscillator can be written in the form shown in Equation 7. The eigenvector
describes the preferred motion of the system. It shows the relationship between the
motions of the two masses. For every amount c1 that the first mass is displaced, the
second mass will be displaced an amount c2.
 x1  c1 
     cos  t   
 x2  c2 
(7)
Modal analysis is important when observing a driven multi-degree-of-freedom
oscillatory system. Structural analysis is an important application of modal analysis. For
example, imagine a three-story structure where each floor is approximately the same
weight. The stiffness of the walls is such that the second floor is twice as stiff as the third
and the first is three times the stiffness of the third. For the sake of obtaining numerical
values, let m=18(103) slugs and k3=18(103) lb/ft. The mode shapes (or eigenvalues) are
defined in Table 1.
4
Table 1: Mode Shapes of 3-DOF Structure
Mode 1
ω1 = 12.17 rad/s
Mode 2
ω2 = 24.35 rad/s
Mode 3
ω3 = 50.95 rad/s
 1 
C1  2.66
4.78


 1 
C2   1.62 
 2.08


C3
 1 


  1.40
 0.20 


From this example, it can be deduced that the higher the natural frequency, the
lower the amplitude of vibration. Also, there are more “turning points” where the
displacement goes from positive to negative as the frequency is increased. Analyzing the
eigenvectors is a quick and easy way to understand and visualize the motion of the
system. This becomes increasingly valuable when dealing with systems that have several
degrees of freedom.
The natural frequencies and mode shapes are fundamental properties of a freely
vibrating, undamped multi-degree-of-freedom system. Regardless of the choice of
coordinates, the same set of natural frequencies will be obtained. These frequencies are
fundamental properties which are independent of the choice of coordinates. The mode
shapes will be different in the two cases because the mode shapes depend on the choice
of coordinates. However, the motions of the physical system defined by the mode shapes
will be identical.
Resonance is an important phenomenon to understand when dealing with the
forced vibration of a multi-degree-of-freedom system. Resonance will occur when the
driving frequency matches one of the eigenfrequencies of the system. An example of this
5
would be a driven two-mass, three-spring system where the motion can be described by
Equation 8. For this case, {F(t)} will be assumed to be a harmonic excitation of the form
{F0}cosΩt, where {F0} is a known constant vector. Through some tedious algebra, it can
be shown that the particular solution for the displacement x1 and x2 is Equation 9.
mx k x  F t 
(8)

1

F0
1
3
x p t  
cos t 

2
2
2k
 1   2 1   2
1
2






(9)
The steady-state response of the system can be plotted against the excitation
frequency of the input. There are two peaks in this plot, one at ω1 and the other at ω2.
This can clearly be seen by the 1  
2
2
term. When Ω=ω, this term will be zero,
leading to an infinitely large peak. Of course in application there will be some inequality,
but the term will still be very small, leading to very large peaks. This large amplitude
vibrational response to the input frequency is known as resonance. Both masses will
move harmonically at the same frequency Ω as the excitation. The plot of the steadystate response of the system against the excitation frequency is shown in Figure 4, where
the amplitude has been nondimensionalized by the factor
6
F0
2k
.
Figure 4: Response to Excitation Frequencies2
Resonance is important to understand, especially when dealing with an autosuspension system where the tires and suspension can be approximated as springs with
the axles and the car body acting as the masses. It would be preferable for the operating
speed of the vehicle to be such that the mode shape of the vehicle calls for the violent
vibration to be in the axles rather than in the car body.
Resonance has applications in many aspects of life. Most musical instruments
depend on resonance to create sound. Electricity and magnetism is another important
application of resonance. A famous example of the importance of understanding
resonance is the Tacoma Narrows Bridge in Tacoma Washington. The bridge, one of the
longest suspension bridges in the world when it was built, self-destructed due to the
resonance effects caused by wind-induced vibration. Soldiers are now taught to march in
breakstep rather than in lockstep when crossing bridges to avoid a similar catastrophe.
7
Bibliography
1. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Bernoulli_Daniel.html
2. Burton, T.D. Introduction to Dynamic Systems Analysis, McGraw-Hill, Inc., 1994.
3. Bernhard, Jonte. The Coupled Harmonic Oscillator – A Lab using MBL-tools for an
Uppper-Level Mechanics Course.
4. Thornton, Stephen T. and Jerry B. Marion, Classical Dynamics of Particles and
Systems, 5th Edition, Brooks/Cole – Thomson Learning, 2004.
5. http://www.mathphysics.com/calc/eigen.html
6. http://encyclopedia.thefreedictionary.com/Resonance
8
Derivation of System Frequency for First Example
The equations of motion can be written in matrix form as shown in Equation 10.
m 0   x1  k 0  k1
 0 m  x     k

  2 
1
 k1   x1  0
    
k 0  k1   x2  0
( 10 )
The assumed solution to the differential equation is shown in Equation 7. The second
derivative of the solution is shown in Equation 11.
 x1  c1 
     cos  t   
 x2  c2 
(7)
c
 x1 
2 1
      cos  t   
x2 
c2 
( 11 )
When the position and acceleration equations are substituted back into Equation 10, the
result is Equation 12. By factoring out the like term, the simplified result is Equation 13.
k  k1
m 0   c1 
2 
   cos  t      0

 0 m c2 
  k1
  k 0  k1

 k
1

 k1  c1 
0
   cos  t      

k 0  k1  c2 
0
 k1 
m 0    c1 
0
    cos  t      
2 



k 0  k1 
 0 m  c 2 
0
( 12 )
( 13 )
Since the solution must be valid for all time, the cosine term will generally be nonzero, so
Equation 14 must be the case. The mass and stiffness matrices can be combined to yield
the result shown in Equation 15.
  k 0  k1

 k
1

 k1 
m 0    c1  0
2 

   c   0
k 0  k1 
0
m

  2   
k 0  k1  m 2

 k1

  c1  0
 k1

 
2 
k 0  k1  m  c2  0
( 14 )
( 15 )
For the solution to be nontrivial, the determinant of the coefficient matrix must be zero.
By expanding the determinant and simplifying, Equation 16 is obtained.


m 2 4  2mk 0  k1  2  k 02  2k 0 k1  0
( 16 )
9
When the Quadratic Equation is used to solve for ω2 the result is Equation 17. Thus the
natural frequency ω is Equation 4.
2 
k 0  k1  k1
m
( 17 )

k 0  k1  k1
m
(4)
To find the eigenvector associated with either frequency, substitute that frequency into
Equation 15. Substituting the frequency shown in Equation 6 into Equation 15 will result
in two equations (linearly dependent equations) for the constants c1 and c2, shown in
Equations 18 and 19.
2k 0  k1 c1  k1c2  0
( 18 )
 k1c1  2k 0  k1 c2  0
( 19 )
Solving for c1 in terms of c2 will result in the eigenvector. The stiffness ratio ε is the ratio
of k0 to k1.
c  
1 

2  1
( 20 )
10