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Two SpringCoupled Masses
Consider a mechanical system consisting of two identical masses
which are free to slide over a frictionless horizontal surface. Suppose that
the masses are attached to one another, and to two immovable
walls, by means of three identical light horizontal springs of spring constant , as
shown in Figure 14. The instantaneous state of the system
is conveniently specified by the displacements of the left and
right masses, and , respectively. The extensions
of the left, middle, and right springs are , , and ,
respectively, assuming that corresponds to the equilibrium configuration in which the springs are all
unextended. The equations of motion of the two masses
are thus
Here, we have made use of the fact that a mass attached to the left end of a
spring of extension and spring constant experiences a horizontal force ,
whereas a mass attached to the right end of the same spring experiences an
equal and opposite force .
Figure 14:
Two degree of freedom massspring system.

Equations (142)(143) can be rewritten in the form
where
. Let us search for a solution in which the two
masses oscillate in phase at the same angular frequency, . In other words,
where , , and are constants. Equations (144) and
(145) yield
or
where
. Note that by searching for a solution
of the form (146)(147) we have effectively converted the system of two coupled
linear differential equations (144)(145) into the much simpler system of two coupled linear algebraic
equations (150)(151). The latter equations have the trivial solutions
, but also yield

(152) 
Hence, the condition for a nontrivial solution is

(153) 
In fact, if we write Equations (150)(151) in the form of a homogenous (i.e., with a null righthand side) matrix equation, so that

(154) 
then it becomes apparent that the criterion (153) can also be obtained by
setting the determinant of the associated matrix to zero.
Equation (153) can be rewritten

(155) 
It follows that

(156) 
Here, we have neglected
the two negative frequency roots of (155)i.e.,
and
since a negative frequency oscillation
is equivalent to an oscillation with an equal and opposite positive frequency, and
an equal and opposite phase: i.e.,
.
It is thus apparent that
the dynamical system pictured in Figure 142 has two unique frequencies of oscillation: i.e.,
and
. These are called the normal frequencies of the
system.
Since the system possesses two degrees of
freedom (i.e., two independent coordinates are needed to specify its
instantaneous configuration) it is not entirely surprising that it possesses two
normal frequencies. In fact, it is a general rule that a dynamical system
with degrees of freedom possesses normal frequencies.
The patterns of motion associated with the two normal frequencies
can easily be deduced from Equation (152). Thus, for
(i.e.,
), we
get
, so that
where and are constants. This first pattern of motion corresponds to the two masses
executing simple harmonic oscillation with the same amplitude and phase.
Note that such an oscillation does not stretch the middle spring.
On the other hand, for
(i.e.,
), we get
, so that
where and are constants. This second pattern of motion
corresponds to the two masses executing simple harmonic oscillation with the
same amplitude but in antiphase: i.e., with a phase shift of radians. Such oscillations do stretch the
middle spring, implying that the restoring force associated with
similar amplitude displacements is greater for the second
pattern of motion than for the first. This accounts for the higher
oscillation frequency in the second case. (The inertia is the same in both cases, so the
oscillation frequency is proportional to the square root of the restoring force
associated with similar amplitude displacements.) The two distinctive
patterns of motion which we have found are called the normal modes of
oscillation of the system. Incidentally, it is a general rule that a dynamical system
possessing degrees of freedom has unique normal modes of oscillation.
Now, the most general motion of the system is a
linear combination of the two normal modes. This immediately follows because
Equations (142) and (143) are linear equations. [In other
words, if and are solutions then so are and ,
where is an arbitrary constant.] Thus, we can write
Note that we can be sure that this represents the most general solution
to Equations (142) and (143) because it contains four
arbitrary constants: i.e., , , , and . (In general,
we expect the solution of a secondorder ordinary differential equation to
contain two arbitrary constants. It, thus, follows that the solution of a system of two
coupled ordinary differential equations should contain four arbitrary constants.)
Of course, these constants are determined by the initial conditions.
For instance, suppose that , , , and
at . It follows, from (161) and (162), that
which implies that
and
. Thus, the
system evolves in time as
where
, and
use has been made of the trigonometric identities
and
. This evolution is
illustrated in Figure 15. [Here,
. The solid curve
corresponds to , and the dashed curve to .]
Figure 15:
Coupled oscillations in a two degree of freedom massspring system.

Finally, let us define the socalled normal coordinates,
It follows from (161) and (162) that, in the presence of both normal
modes,
Thus, in general, the two normal coordinates oscillate sinusoidally with
unique frequencies, unlike the regular coordinates, and see Figure 15.
This suggests that the equations of motion of the system should look particularly simple when expressed in terms of the normal coordinates. In fact, it is easily seen that the
sum of Equations (144) and (145) reduces to

(173) 
whereas the difference gives

(174) 
Thus, when expressed in terms of the normal coordinates, the equations of motion
of the system reduce to two uncoupled simple harmonic oscillator
equations.
Of course, the most general solution to Equation (173) is (171),
whereas the most general solution to Equation (174) is (172).
Hence, if we can guess the normal coordinates of a coupled oscillatory
system then the determination of the normal modes of oscillation is considerably simplified.
Next: Two Coupled Circuits
Up: Coupled Oscillations
Previous: Coupled Oscillations
Richard Fitzpatrick
20101011