Normal Modes in Coupled Oscillators
By: George Levesque IV
Applied Mathematics
MWF 11-12
Introduction: An interesting problem in physics stems from the extension of block-spring
systems in classical Newtonian mechanics. Previously, in elementary physics, we have
considered the motion of a single block and a single spring and we have derived the associated
equations of motion. But later as we develop, we come across more and more difficult
extrapolations from the same problem. Eventually, we became developed enough to ponder the
problem of what if we had two blocks that were connected by springs to each other, as well as to
two opposing surfaces. The resulting interconnection of springs and blocks can be referred to as
coupled (or linked) oscillation.
Many studies have led to conclusions about what have been called normal modes.
Normal modes can be defined as “the motion that can be decomposed into that of a set of N
independent harmonic oscillators” (Bernstein, 343) where N can be equivalent to the number of
objects (in our case blocks) interconnected by springs. (More generally, the number of normal
modes is equal to the number N of degrees of freedom, which is equivalent to the “N unique
patterns of vibration in which all masses oscillate ate the same frequency with fixed amplitudes
(Givens et al., 87).” For example, if we consider the two block-spring system as before we can
define any of its motions to be some combination of two separate oscillations.
As a result of experimentation, it turns out that that we can represent each of the separate
normal modes by their respective frequencies in the driving force (or the force introduced to
keep the oscillation consistent). The oscillation to the right is that of the lowest frequency and it
is called the symmetrical mode (as is always the case when all of the particles move in the same
direction). The oscillation to the left is that of the higher frequency and it is called the
asymmetrical mode (as all of the particles move in opposing directions). We can also consider
the process for three blocks (as well as the resulting three normal modes).
As a matter of fact we can extrapolate this idea to just about any number of objects
(blocks), though not without difficulty. Even baseball bats have normal modes (though
considering the N normal modes would be difficult since N is approximately equal to something
on the order of 10^23 for the number of atoms within the bat). The following is a consideration
of normal modes on an ordinary Louisville slugger.
Normal bat
f1  179Hz
f 2  582Hz
f 3  1181Hz
Derivations for amplitudes, phase shifts, and equations of motion for who coupled masses:
First we must start from basic principles and complete a free body diagram.
Now we can label these separate forces and place them into the equation  F  ma
And if we utilize the dot notation, as we will continue to, we can rewrite this as  F  mx
Now we must define each of our forces. There is the force, as defined by Hooke’s law, F= -kx.
There is also a force that results from the internal resistence of the spring, we will call it
F  ci xi for each mass. There is also a driving force on the lower block; we will define it to be
kD sin( t ) . It is also important to note that, since the blocks are connected the distance that one
block is away from another adds another form of Hooke’s law in each block’s equation of
motion: F  k 2 ( x2  x1 ) . In the previous equation the positive/negative symbols are used
since for each block it affects the direction of the motion in either the positive or negative
direction. We can sum these forces to get:
 Fmass1  m1x1  k1x1  k 2x2  x1  c1x1
 Fmass2  m2 x2  k 2 x 2  k 2x2  x1  c 2 x 2  k 3 D sin t 
The experiment is defined to use similar masses and springs (with corresponding similar
constants and so would be the resulting resistance). So we can write.
m1  m2  m
c1  c2  c
k1  k 2  k
We can substitute to get:
mx1  kx1  k x2  x1  cx1
mx2  kx2  k x2  x1  cx 2  kDsin t 
We can distribute and simplify:
(2a)
mx1  2kx1  kx2  cx1
mx2  2kx2  kx1  cx 2  kDsin t 
We can define F 0  k 3D and substitute to get
(2b)


mx2  2kx2  kx1  cx 2  F 0 sin t 
If we add 2a and 2b we get:
mx1  2kx1  kx2  cx1
+ mx2  2kx2  kx1  cx 2  F 0 sin t 
m(x1  x2)  2k ( x1  x2)  k ( x1  x2 )  c( x1  x 2)  F 0 sin t 
It is an advantageous to consider the irrelevancy of the coordinate axis. While it is
necessary for one to be chosen, it is irrelevant which position it is chosen to be placed at x = 0.
For example, we ordinarily consider this position to be at the center of the block in its
equilibrium position. However, we can choose different coordinate axis that will eventually
become advantageous to us later as we try to decouple the positions of each mass from being in
the equation of motion of another. Such an axis is chosen (one that is new and different) but at
the same time we choose such a pair of coordinate axis that they move with respect to the blocks
position. As unusual as it may seem, it is useful in decoupling these equations of motion. In the
diagram below we consider the normal coordinate axis (x sub I) and the new ones (q where they
are defined to be q1  x1  x2 and q 2  x 2  x1 ). Notice how q moves while x remains still.
Now in recognition of our ability to shift the coordinate axis we can make the substitutions
involving q1  x1  x2 and q 2  x 2  x1 .
mq1  2kq1  kq1  cq1  F 0 sin( t )
Combine


mq1  kq1  cq1  F 0 sin( t )
(3a)
Rearrange


mq1  cq1  kq1  F 0 sin( t )
If we subtract 2a and 2b we get:
mx1  2kx1  kx2  cx1
- mx2  2kx2  kx1  cx 2  F 0 sin t 
 m(x2  x1)  2k ( x2  x1)  k ( x2  x1)  c( x 2  x1)  F 0 sin t 
Once again remembering our ability to shift the coordinate axis we can make the substitutions
involving q1  x1  x2 and q 2  x 2  x1 .
 mq2  2kq2  kq2  cq 2  F 0 sin( t )
Combine
 mq2  3kq2  cq 2  F 0 sin( t )
Rearrange
(3b)
mq2  cq 2  3kq2  F 0 sin( t )
Now we realize that the complementary solution for equations 3a and 3b are transient (as they
die out quickly and do not result in a compounding effect). Let us consider this notion with
graphical sketches.
The above is what the particular solution looks like. A motion that is something of a simple
harmonic oscillator that is consistent. This is the motion of an object without resistance or
consistent a driving force that would compensate for it (here our system points is that of the
prior).
The above sketch is one of the transient solution (as, it can plainly be seen, that it quickly dies
out). The transient solution is the result of the dampening ability of the springs, which, without a
driving force, would not continuously oscillate. Instead, they would quickly come to rest at its
point of equilibrium.
This sketch is a representation of when both of the above sketches occur at the same time (as
they would since they are both solutions of travel of the same system over time). This shows us
that since the form is close to that of the particular solution we can ignore the transient solution
for simplicity when finding equations of motion for each of the blocks. Instead, let us investigate
the general solution. We assume that the particular solution is of the form qi  Aisim (t   )
Before we find this solution let us manipulate equation 3b. mq1  cq1  kq1  F 0 sin( t ) where we
c
k
F0
  22 and also
D
make the substitutions 2  
and
m
m
m
So we get
q1  2 q1   02 q1  D sin( t )
Now we substitute our general form of the particular solution.
 A 2 sin( t   )  2 A cos(t   )   02 A sin( t   )  D sin(  )  0
Now we can expand this equation using the substitutions
sin( t   )  sin( t ) cos( )  cos(t ) sin(  ) and
cos(t   )  cos(t ) cos( )  sin( t ) sin(  )
 A 2 [sin( t ) cos( )  cos(t ) sin(  )]  2 A[cos(t ) cos( )  sin( t ) sin(  )]  A 02 [sin( t ) cos( )
 cos(t ) sin(  )]  D sin( t )  0
Now let us expand to get:
 A 2 sin( t ) cos( )  A 2 cos(t ) sin(  )  2 A cos(t ) cos( )  2 A sin( t ) sin(  )  A 02 sin( t ) cos( )
 A 02 cos(t ) sin(  )  D sin( t )  0
Now gather like terms cosine (time dependent) terms first followed by sine
(time dependent) terms.
A 2 cos(t ) sin(  )  2 A cos(t ) cos( )  A 02 cos(t ) sin(  )
 A 2 sin( t ) cos( )  2 A sin( t ) sin(  )  A 02 sin( t ) cos( )  D sin( t )  0
Now let’s factor out these like terms
cos(t )[ A 2 sin(  )  2 A cos( )  A 02 sin(  )] 
sin( t )[  A 2 cos( )  2 A sin(  )  A 02 cos( )  D]  0
Now factor A constants
A cos(t )[ 2 sin(  )  2  cos( )   02 sin(  )] 
sin( t ){ A[ 2 cos( )  2  sin(  )   02 cos( )]  D}  0
Factor the cosine terms of the second term.
A cos(t )[ 2 sin(  )  2  cos( )   02 sin(  )] 
sin( t ){ A[( 02   2 ) cos( )  2  sin(  )]  D}  0
Here we note the linear independence of both terms (after all a cosine and sine term with the
same functional constants would never cause their sum to be zero since they would require a
shift of one half pi in order to have the symmetry necessary to equal zero). Here we notice that
the graphed functions are always off by a factor of pi.
While we should remember that similar functions like sin and –sin
Can graphically result in:
which is the graph zero that our sum would require in the last equation.
We notice that, as a result of their linear dependence, that each term must be equal to zero.
So let us consider the sine term as an equation on its own.
A[( 02   2 ) cos( )  2  sin(  )]  D  0
We can rearrange to get
A
D
(   ) cos( )  2  sin(  )
2
0
2
We perform the null operation on the denominator of squaring its contents and taking the square
root at the same time.
A
D
(   ) cos ( )  4  2 2 sin 2 ( )
2
0
2 2
2
We apply the trigonometric identity sin 2 ( x)  cos 2 ( x)  1 to give
A
D
(   )  4  2 2
2
0
2 2
We can even substitute
F0
 D to get
m
F0
m
A
2
2 2
( 0   )  4  2 2
If we consider the cosine term we get:
 2 sin(  )  2  cos( )   02 sin(  )  0
Combine sine terms
(   02 ) sin(  )  2  cos( )  0
2
Now rearrange
(   02 ) sin(  )  2  cos( )
2
sin(  )
 2 
 2
cos( ) (   02 )
Now apply a trigonometric identity to get:
tan(  ) 
 2 
( 2   02 )
Solve
  tan 1
 2 
( 2   02 )
It is also useful to consider the manipulation of equation 3b:
mq2  cq 2  3kq2  F 0 sin( t )
Here we make the substitutions 2  
c
3k
F0
  22 and also
D
and
m
m
m
So we get
q1  2 q1   02 q1  D sin( t )
NOTE THIS IS OF THE SAME FORM AS WHEN WE GOT TO THIS STEP FOR FINDING
THE SOLUTION FOR EQUATION 3A. THE FOLLOWING SOLUTION IS, THEREFORE
THE SAME.
Now we substitute our general form of the particular solution.
 A 2 sin( t   )  2 A cos(t   )   02 A sin( t   )  D sin(  )  0
Now we can expand this equation using the substitutions
sin( t   )  sin( t ) cos( )  cos(t ) sin(  ) and
cos(t   )  cos(t ) cos( )  sin( t ) sin(  )
 A 2 [sin( t ) cos( )  cos(t ) sin(  )]  2 A[cos(t ) cos( )  sin( t ) sin(  )]  A 02 [sin( t ) cos( )
 cos(t ) sin(  )]  D sin( t )  0
Now let us expand to get:
 A sin( t ) cos( )  A cos(t ) sin(  )  2 A cos(t ) cos( )  2 A sin( t ) sin(  )  A 02 sin( t ) cos( )
 A 02 cos(t ) sin(  )  D sin( t )  0
2
2
Now gather like terms cosine (time dependent) terms first followed by sine
(time dependent) terms.
A 2 cos(t ) sin(  )  2 A cos(t ) cos( )  A 02 cos(t ) sin(  )
 A 2 sin( t ) cos( )  2 A sin( t ) sin(  )  A 02 sin( t ) cos( )  D sin( t )  0
Now let’s factor out these like terms
cos(t )[ A 2 sin(  )  2 A cos( )  A 02 sin(  )] 
sin( t )[  A 2 cos( )  2 A sin(  )  A 02 cos( )  D]  0
Now factor A constants
A cos(t )[ 2 sin(  )  2  cos( )   02 sin(  )] 
sin( t ){ A[ 2 cos( )  2  sin(  )   02 cos( )]  D}  0
Factor the cosine terms of the second term.
A cos(t )[ 2 sin(  )  2  cos( )   02 sin(  )] 
sin( t ){ A[( 02   2 ) cos( )  2  sin(  )]  D}  0
Here we note the linear independence of both terms (after all a cosine and sine term with the
same functional constants would never cause their sum to be zero). We notice that, as a result of
their linear dependence, that each term must be equal to zero.
So let us consider the sine term as an equation on its own.
A[( 02   2 ) cos( )  2  sin(  )]  D  0
We can rearrange to get
A
D
(   ) cos( )  2  sin(  )
2
0
2
We perform the null operation on the denominator of squaring its contents and taking the square
root at the same time.
A
D
( 02   2 ) 2 cos 2 ( )  4  2 2 sin 2 ( )
We apply the trigonometric identity sin 2 ( x)  cos 2 ( x)  1 to give
A
D
( 02   2 ) 2  4  2 2
We can even substitute
F0
 D to get
m
F0
m
A
2
2 2
( 0   )  4  2 2
If we consider the cosine term we get:
 2 sin(  )  2  cos( )   02 sin(  )  0
Combine sine terms
( 2   02 ) sin(  )  2  cos( )  0
Now rearrange
( 2   02 ) sin(  )  2  cos( )
sin(  )
 2 
 2
cos( ) (   02 )
Now apply a trigonometric identity to get:
tan(  ) 
 2 
( 2   02 )
Solve
  tan 1
 2 
( 2   02 )
As a result of having resolved answers for two separate equations and ending in a similar form
we can generalize the answers to get
Ais 
D
( i2   2 ) 2  4  2 2
And
 i  tan 1
 2 
( i2   2 )
The time has come to develop the equations of motion for both of the masses in the system. We
must recall qi  Ai sin( t   ) and also the relations q1  x1  x2 and q 2  x 2  x1
Now it is convenient for us to write
(4a)
(4b)
x1  q1  x 2
x2  q2  x1
If we substitute 4b into 4a we get
x1  q1  q 2  x1
Rearrange
2 x1  q1  q 2
Divide
x1  (q1  q 2) / 2
If we put this into our general solution qi  Aisim (t   ) and Ais  A1 / 2 to get
(5a)
x1(t )  A1 sin( t   )  A2 sin( t   )
If we substitute 4a into 4b we get:
x2  q2  q1  x2
Rearrange
2 x 2  q1  q 2
Divide
x 2  (q1  q 2) / 2
If we put this into our general solution qi  Aisim (t   ) and Ais  Ais / 2 to get
(5b)
x 2(t )  A1 sin( t   )  A2 sin( t   )
Experimental Procedure: So, to help with our investigation of the mathematical properties of
such a system of normal modes in coupled oscillation, I, with the kind help and consideration of
Bridgewater State College’s own Dr. Edward F. Deveney (B.S., Ph.D. from The University of
Connecticut) set out to consider the experiment “Direct Observation of Normal Modes in
Coupled Oscillators” by Ryan Givens, O.F. de Alcantara Bonfirm, and Robert B. Ormond (Am.
J. Phys. 71 (1), January 2003).
In this experiment, the experimenters created the two block-three spring system and then
attached the lower spring to something that would create a driving force (in this case they used a
speaker). This driving device was then hooked up to a function generator that would be allow us
to adjust the frequency of the driving device and allow us to maintain such frequencies for
extended periods of time.
With the equipment set we only had to adjust the frequency until the amplitude is at a
maximum, then we check for either the asymmetric or the symmetric normal modes. Once we
find one of the normal modes we only have to record the resulting frequency. The following
data is from the aforementioned study (Givens, et al.) with the suggested springs (k=16.5+/-0.1
N/m) and weights (m=50+/-1 g). The calculations are from the equations f i 
1 
i
,
2
k
3k
  2 , and  2 
2 .
m
m
Frequency (Hz)
Calculated
Measured
Symmetric
2.89+/-0.04
2.8+/-0.1
Asymmetric
5.01+/-0.1
5.0+/-0.1
As the data clearly shows, all our calculations paid off and we were able to be fairly
accurate in determining the frequencies of the normal modes.
Conclusions: It turns out that our derivations worked out very well. Not only do such
mathematics work out very nicely in the real world, but also we have empirical data to support it.
As a result we learn that the experiment and the mathematics are both accurate reflections of
each other and both can model the real world effectively.
Items of Further Investigation: Such an analysis of a single question brings about many
separate tangents and extrapolations of the same topic. Here we took the time to give an in depth
analysis of two objects in coupled oscillation. We also did give brief analysis of the normal
modes of a three-object (and four spring) system. It should be noted that there are other fine
works that give in depth mathematical analysis and/or experimental to this system (Givens et al.
89). In these works we learn that the mathematics is slightly more cumbersome as the result of
having to uncouple the equations involving the interdependence of three objects eventually
resulting in three equations of motion of similar forms. There is also a given empirical analysis


k
 1 
 1  2k
that gives a comparison to the calculations from f1    2  2
, f2   
, and
m
 2 
 2  m



k
 1 
resulting from f i  i for the three normal modes.
f1    2  2
m
2
 2 
Frequency (Hz)
Calculated
Measured
f1
2.21+/-0.03
2.1+/-0.1
f2
4.09+/-0.06
4.1+/-0.1
f3
5.34+/-0.07
5.3+/-0.1
Once again we see support for the calculation of such normal modes empirically.
With an expansion of such a system of coupled oscillation we could study systems that
look like
Or we might also desire to consider a system like the above but extended into the third
dimension like:
Here we have a cubic arrangement of interconnected objects. They are each connected to three
of the other objects (including one in front or behind) by springs. There are also driven by
speakers (six are shown here) as before. The reasoning to the above extrapolations point to our
analysis of molecules undergoing thermic vibrations. Their formation can be arranged in such
ways as above to better represent the lattices of these atoms in molecules (or interconnected
molecules).
We can also take a moment to consider what is going to be involved in finding these
equations of motion. In the 2-D, four-object model, we can see that the initial equations are
going to have each of the objects dependent on the motion of two of the other, nearby, objects.
As a result, there will probably be a more involved substitution to change the coordinate axis (at
least one new axis for each object). The result would be something that is more involved but
would result in similar equations of motion (consisting of amplitudes and sinusoidal equations).
In the 3-D, eight-object model, we can see that we will have the forces on each object dependent
upon three of the surrounding objects. Once again, we have slightly more complex equations but
of the similar form. It seems that the general trend would be that if there are 2^n objects than
there will be 2^n equations where there will be n objects that are affecting each individual object
(resulting in trying to get at least n coordinate axis equations for substitution). Here we should
also note that the types of normal modes that can be shown are more numerous and more
intricate than discussed in our linear two- or three- object system.
We notice that as there are more and more objects in our analysis the more
(exponentially) difficult it becomes to evaluate such equations of motion. This points us in a
direction of considering other mathematical methods to analyze these objects in coupled
oscillation. It may be advantageous to consider the Euler-Lagrange equations of motion. Here
we can consider energy instead of Newton’s forces but we must also take up difficult methods
for considering the nonconservative nature of a real spring-object system (see Goldstein). Also,
there are more advanced methods utilizing eigenvalues and eigenvectors along with Fourier
analysis (see Hathorn). There are also a variety of other experiments that can be carried out in
order to analyze the normal modes of the system (see bibliography).
It should be noted that adding more and more atoms (or objects) to our interconnected
system of coupled oscillation is not always useful. After all, macroscopic objects contain a
nearly incomprehensible number of atoms (for small objects we can consider something on the
order of 10^23 or possibly larger as an extrapolation from Avogadro’s number; a number that
would have twenty three digits like10,000,000,000,000,000,000,000). Here we would be much
better off considering their collective behavior rather than considering the contribution of each
atom whose individual motion consist of the whole object’s motion. However, as methods of
analysis become more advanced it might become useful (or at least entertaining to some) to
investigate such tasks that currently seem quite difficult.
Acknowledgements: As special thanks is necessary for Dr. Edward F. Deveney and his
willingness to contribute his vast body of knowledge, which proved indispensable.
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