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Wize University Physics Textbook (Master) > Periodic Motion: Oscillations

Coupled Springs

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Coupled Simple Harmonic Oscillators


In some physical systems we have a set of harmonic oscillators that are coupled. For instance, we have have a set of masses and a set of springs that connects these masses to each other. In general, this system of coupled oscillators could be very challenging to analyze analytically. However, here we consider a simple case when two identical masses which are connected to each other and the walls around them with three identical springs as shown in the picture below:

We define the equilibrium of each mass as a point at which the net force acting on them is zero and any displacement respect to these equilibrium points are shown by x1x_1 and x2x_2 for left and right masses in the picture, respectively.

Using Hooke's law and Newton's second law, we can write down the equation of motion for each of the masses separately:


md2x1dt2=−kx1+k(x2−x1)m\dfrac{d^2x_1}{dt^2}=-kx_1+k(x_2-x_1)

md2x2dt2=−kx2+k(x1−x2)m\dfrac{d^2x_2}{dt^2}=-kx_2+k(x_1-x_2)


Depending on the condition at which we start oscillation in this system, we would see different type of behavior. The two most important ones are known as normal modes.


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Symmetric Mode (In Phase Mode)

The first normal mode exists when we move both masses to the same direction with the same amount and release them to oscillate. In this mode, the middle spring is never stretched or compressed. The frequency of the oscillation in this case is equal to:


ω=km\omega = \sqrt{\dfrac{k}{m}}


Anti-Symmetric Mode (Out of Phase Mode)

The second normal mode exists when we move masses to opposite directions but still with the same amount and release them to oscillate. In this mode, the middle spring is also changing in size. The frequency of the oscillation in this case is equal to:



ω=3km\omega = \sqrt{\dfrac{3k}{m}}



Practice: Coupled Oscillators


a) Consider two masses of 900900 g each, connected between themselves and the walls around them by three identical springs, all arranged in a straight line. They are oscillating in the symmetric mode. If it takes them 33 s to compete 55 oscillations, what is their spring constant? Leave the answer in exact form.

b) If the same springs oscillate in the anti-symmetric mode, by what factor should the masses be increased so that the period of oscillation stays the same?

c) Still in the anti-symmetric mode, by what factor should the spring constant change so that the period of oscillation stays the same, if the masses are kept the same as in part a)?