Determining the Normal Modes of Oscillation for a Spring
System with Two Masses
H. Potter
(Completed 23 March 2006)
The frequencies of the normal modes of oscillation of a low-friction spring system
with two masses were determined experimentally and compared to the theoretical
values. The experimental low-energy frequency was .548 Hz, .3% higher than the
theoretical value of .546 Hz. The experimental high-energy frequency was .940
Hz, .7% lower than the theoretical value of .946 Hz. Data was then taken twice
for an arbitrary motion of the system, and the presence of both the high-energy
and low-energy frequency peaks was observed in both cases.
I. Introduction
Arbitrary periodic motion can be difficult to analyze because it is rarely a function
of a simple sine or cosine; however, there are certain situations in which the arbitrary
periodic motion of a system can be separated into linear combinations of just a few
simple sine and cosine waves. This, however, requires that the system have a few
linearly independent normal modes of oscillation. When this is the case, any arbitrary
motion of the system can be expressed in terms of these base sinusoidal functions.
II. Experiment
One system that can be characterized using normal mode analysis is a spring
system with two masses. Each mass is allowed to oscillate back and forth along a
straight low-friction railing. The oscillations are caused by three springs, one that
connects the masses to each other, and one more for each mass that attaches it to a
stationary object. A motion sensor is placed at each end of the railing so that each one
points in towards the carts on the railing in order to detect the motion of the cart closest to
it.
In this experiment three springs of the same make and model were used so that
the spring constant of each spring could be assumed to be identical. In order to determine
the value of k for the springs, a motion sensor was set up below a spring with a hanger
attached to it and an initial height was recorded. A small known mass was then added to
the hanger and a new equilibrium position was determined. Another mass was then
added, and a new equilibrium position was determined. This was done once more in
order to obtain three data sets. For each data set the total mass added M was known, as
was the change in the length of the spring from its equilibrium position L. The spring
constant for each data set was then calculated as
k
Mg
,
L
(1)
where g is the acceleration due to gravity. These three values for k were then averaged to
determine a best experimental value for k.
The mass of each cart was also measured using a tri-beam balance. The masses
were close enough that it was reasonable to take them to be equally massive, with mass
equal to their average mass. In combination with the value for the spring constant
determined previously, this measurement enabled the theoretical calculation of the
frequencies of the normal modes.
Two data sets of each type of oscillatory motion were taken after these
measurements were made. The low-energy normal mode was analyzed first. This mode
occurs when both masses are oscillating in sync with one another. This motion was
produced by pushing the masses in the same direction by the same amount shortly after
data recording was initiated. The high-energy mode was analyzed second. This mode
occurs when the masses are oscillating exactly opposite each other: when one mass is as
far left as it will get, the other mass is as far right as it will get. This motion was
produced by pushing the masses in opposite directions by the same amount shortly after
data recording was initiated. Lastly, arbitrary oscillatory motion was produced by
pushing the masses in an asymmetric manner. In each case, the data was recorded and
analyzed on a computer using a Fast Fourier Transform in order to determine the peak
frequencies at which the carts were oscillating. The specific value for the peak frequency
in each case was taken to be the mean of the upper and lower bounds on the peak
frequency that could easily be read off of the graph as specific data points. The final
values for the high energy and low energy frequencies were taken to be the means of all
relevant peak data from all data sets.
III. Theory
The theory behind determining the normal mode frequencies for the oscillating
spring system is significant enough to handle separately.
Let x1 be the distance to the left mass as measured by the left motion detector, and
x1e be the corresponding equilibrium position. Let x2 be the distance to the right mass as
measured by the right motion detector, and x2e be the corresponding equilibrium position.
The mean mass m of the oscillating masses was measured, and the mean spring constant
k of the springs was also determined experimentally. It is assumed that the mass of the
springs is negligible in comparison to that of the massive carts.
According to Hooke’s Law,
Fspring  k springx .
(2)
Using this law the force on the left mass m1 and the right mass m2 can be expressed in
terms of k, x1, x1e, x2, and x2e, as follows. Let D be the total distance from one motion
sensor’s zero position to the other motion sensor’s zero position. Then, taking right to be
the positive direction, it follows that
F1  k x1  x1e   k D  x1  x2   D  x1e  x2e   2k x1  x1e   k x2  x2e  , (3)
and that
2
F2  k x2  x2e   k D  x1  x2   D  x1e  x2e   k x1  x1e   2k x2  x2e  .
(4)
Due to the direction in which x2 is measured, however, this leads to the system of
differential equations
d 2 x1
k
  2x1  x1e   x 2  x 2e 
2
m
dt
(5)
2
d x2
k
  x1  x1e   2x 2  x 2e 
m
dt 2
because
F2  m
d 2 x2
.
dt 2
(6)
Now define
y1  x1  x1e ,
(7)
y 2  x2  x2e ,
(8)
y 
y   1,
 y2 
(9)
2 1 
A

1 2 
(10)
and
so that the system of differential equations (5) becomes just
d2y
k
  Ay .
2
m
dt
(11)
This is a much more compact form, but it does not yet yield a solution. In order to
complete the solution, it must further be assumed that y1 and y2 are of the form
y1  A1 cost  1 
(12)
y2  A2 cos(t  2 ) ,
(13)
and
3
where ω is a common normal angular frequency of vibration, and φ1 and φ2 are just phase
shifts that can be used to fit the cosines to a specific curve. Assuming solutions of this
form enables the second derivative of y to expressed as
y
d 2 y    2 A1 cost  1  
2 1
2

       y .
2
2
dt
 y2 
  A2 cost  2 
(14)
Thus Equation 11 can now be further simplified to
Ay 
2
k
 
m
y  y ,
(15)
where λ is just the common variable used to represent an eigenvalue, because this
equation is precisely the definition of the eigenvalue of a matrix: a constant λ is an
eigenvalue of a matrix A if there exists a nonzero vector y for which Equation (15) holds.
The eigenvalue is determined by the condition that
det( A  I )  0 .
(16)
In this case this means that
2   2     1  2  4  3    3  1  0 ,
(17)
so the eigenvalues for this system are 1 and 3. Relating this back to the value of λ in this
case, this means that

k
m
(18)

3k
m
(19)
and
are the angular frequencies of the high and low energy normal modes of oscillation for
this system. The frequencies of the high and low energy normal modes of oscillation can
then be found using the relationship
f 

.
2
(20)
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It is worth noting that by adjusting the parameters A1, A2, φ1, and φ2, Equations
(7), (8), (12), and (13) can be used to determine the functional form in both x1 and x2 of
any arbitrary motion of the system.
IV. Results
Finding k:
mc (kg)
md (kg)
mr (kg)
0.0499
0.0501
0.0497
x1 (m)
x2 (m)
x3 (m)
1.018
0.852
0.688
k1 (N/m) k2 (N/m) k3 (N/m)
2.95
2.97
2.93
Carts:
m1 (kg)
9.8 0.25040
x4 (m)
m2 (kg)
0.518 0.25030
k (N/m)
m (kg)
g (m/s2)
2.95
0.25035
Table 1: Preliminary data and calculated values.
Low-Energy Mode:
Run
Side
1
Left
1
Right
2
Left
2
Right
f (Hz)
0.5480
0.5480
0.5480
0.5490
High-Energy Mode:
Run
Side
1
Left
1
Right
2
Left
2
Right
f (Hz)
0.9390
0.9390
0.9405
0.9415
Mixed Mode:
Run
Side
1
Left
1
Right
2
Left
2
Right
flow (Hz)
fhigh (Hz)
0.5480
0.9395
0.5485
0.9400
0.5480
0.9395
0.5480
0.9390
Table 2: Frequency data for the two distinct normal modes and two mixed mode runs.
V. Data Analysis
Means:
flow (Hz)
0.548
Theoretical Values:
flow (Hz)
0.546
fhigh (Hz)
0.940
fhigh (Hz)
0.946
Percentage Discrepancy:
flow (Hz)
0.3%
fhigh (Hz) -0.7%
Table 3: Final data for mean experimental frequencies, and theoretical frequencies.
VI. Conclusion
The agreement between experimentally obtained peak frequencies and
theoretically predicted peak frequencies was incredible. Either method would thus be
reliable enough to use in order to determine the peak frequencies of a different but similar
system. By determining the normal modes of a system, all modes of the system are
understood because any arbitrary motion of such a system can be expressed as linear
combinations of its normal modes.
In this experiment the fact that the two normal modes of oscillation, namely the
high-energy and the low-energy modes, spanned all possible motions of the system was
verified. This was done by determining the peak frequency for each of the two normal
modes independently. Arbitrary motion of the system was then analyzed and two peak
frequencies, one corresponding to each normal mode frequency, were observed. This
indicated that the arbitrary motion was indeed a linear combination of the two normal
modes. This verification of the physical reality of normal modes of oscillation for a
specific system is of far more significance than the determination of the specific normal
frequencies of oscillation for this system.
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