A simple system of coupled harmonic oscillators

Imagine two mass-spring systems whose masses interact through a third spring (which produces an elastic force), as in the figure alongside. A system of this kind is a very simple example of coupled oscillators.

In studying this system we will assume that:

• the masses are point-like, that is, we will study their motion without worrying about their shape and size; we do not consider, for example, their rotations or the space they occupy;
• the forces are elastic (also called harmonic forces), that is, restoring forces proportional to the elongation (or compression) of the spring with respect to its rest position, through a negative proportionality constant (called the spring constant);
• the oscillations are longitudinal: the masses can only oscillate "horizontally".

Why we study this system

The physical system described, despite the many assumptions we have introduced and will introduce to make it easier to study, shows a surprising variety of behaviours compared with the simple harmonic oscillator. Indeed, the coupling between the oscillators allows them to exchange energy, but the dynamics of this exchange depends on all the parameters of the system, that is, on all the masses and elastic forces involved. Even a relatively weak coupling can produce large energy transfers. The figure alongside, for example, shows the motion of two oscillators when the force coupling them is 10 times weaker than the force making each of them oscillate. The oscillations behave exactly like those that in acoustics cause the phenomenon of beats. In turn, each pendulum damps its own oscillation while the other oscillates more and more widely, but after a certain time the roles are reversed. Each of the two pendulums oscillates at a frequency very close to the one at which it would oscillate if it were free, while the beat frequency is much slower.

Using Lissajous figures for the analysis, the behaviour of the motion can be visualised in another way. In fact, the figure and animation below simulate only a short period of time. As we shall see below, the ratio between the periods of the two oscillators is not rational, which means that the Lissajous curves would end up covering the whole area bounded by the quadrilateral if the system were left to evolve for an infinite time. The animation shows a fast motion along the curve corresponding to the oscillation of the two pendulums, and a slower alternation that moves the major axis from horizontal to vertical and back. This alternation corresponds to the beat frequency. Without friction the system keeps its total energy and goes on oscillating like this forever.

Lissajous figure for the first 10 periods of oscillation

Let us look at some remarkable properties of this physical system.

1. Compared with the single oscillator, it has more than one resonance frequency.
2. Its overall motion can be broken down, in a sense we will clarify, into the "sum" of several harmonic oscillations involving the whole system. We will call these oscillations the normal modes of the system.
3. It is the first step towards studying a chain of oscillators, in which the oscillations propagate to form a wave.

non-normal mode	normal mode 1 (ω1)	normal mode 2 (ω2)
The animations alongside represent periodic motions, and can therefore be shown with a finite number of frames. However, as explained in the text, their frequencies are not in a rational ratio. Consequently any superposition of the two normal modes produces a non-periodic mode of vibration, which cannot be shown with a finite number of frames.

Analysis of the system

First of all we must write the expression of the force acting on each mass M. At any given instant t it equals the sum of the forces exerted by the two "springs" with which it interacts. From the reference frame in the figure it is easy to obtain

• for the mass on the left

${\displaystyle F_1=-k{x}_1+k(x_2-x_1)}$

• for the mass on the right

${\displaystyle F_2=-k(x_2-x_1)-k{x}_2}$

Calling ${\displaystyle a_1,a_2}$ the accelerations of mass 1 and mass 2 respectively, and applying Newton's second law to the two masses, we obtain:

${\displaystyle \{\begin{array}{ll}M{a}_1=-k{x}_1+k[x_2-x_1]& \\ M{a}_2=F_2=-k(x_2-x_1)-k{x}_2\end{array}}$

(1)

Note that the proportionality between acceleration and displacement x(t) has been lost: the two masses therefore do not move in harmonic motion. The harmonicity of the motion can be partly recovered as follows: adding and subtracting the two equations of system (1) we obtain the system:

${\displaystyle \{\begin{array}{ll}M(a_1+a_2)=-k(x_1+x_2)& \\ M(a_1-a_2)=-3k(x_1-x_2)\end{array}}$

(2)

Now introducing the new coordinates

${\displaystyle \{\begin{array}{ll}{X}_1=x_1+x_2& \\ X_2=x_1-x_2\end{array}}$

(3)

remembering that acceleration is the second derivative of position, and calling ${\displaystyle a_{n1},a_{n2}}$ the accelerations of the new coordinates, we obtain

${\displaystyle \{\begin{array}{ll}{a}_{n1}={\ddot{X}}_1=a_1+a_2& \\ a_{n2}={\ddot{X}}_2=a_1-a_2\end{array}}$

(4)

System (2) then becomes

${\displaystyle \{\begin{array}{ll}{a}_{n1}=-\frac{k}{M}{X}_1& \\ a_{n2}=-\frac{3k}{M}{X}_2\end{array}}$

(5)

Note how the equations have now become decoupled. Indeed, the first equation contains only variables with index 1, and the second only variables with index 2. This means they can be solved independently, and there is no longer any need to solve a system.

For this reason the new coordinates ${\displaystyle X_1,X_2}$ are called normal coordinates. Now note that the new "normal" equations define the well-known harmonic motion, whose solution we already know. We then obtain

${\displaystyle X_1(t)=A_1\sin ({\omega }_{1}t+{\varphi }_1)}$

(6)

${\displaystyle X_2(t)=A_2\sin ({\omega }_{2}t+{\varphi }_2)}$

(6')

where we have set

${\displaystyle {\omega }_1=\sqrt{\frac{k}{M}}\text{ and }{\omega }_2=\sqrt{\frac{3k}{M}}}$

(7)

Of course, going back to the coordinates ${\displaystyle x_1(t)\text{ and }{x}_2(t)}$, from system (3) we obtain:

${\displaystyle x_1=\frac{1}{2}(X_1+X_2)}$,

(8)

${\displaystyle x_2=\frac{1}{2}(X_1-X_2)}$

(8')

from which, substituting (6), we finally obtain

${\displaystyle x_1(t)=\frac{1}{2}{A}_1\cdot \sin ({\omega }_{1}t+{\varphi }_1)+\frac{1}{2}{A}_2\cdot \sin ({\omega }_{2}t+{\varphi }_2)}$

(9)

${\displaystyle x_2(t)=\frac{1}{2}{A}_1\cdot \sin ({\omega }_{1}t+{\varphi }_1)-\frac{1}{2}{A}_2\cdot \sin ({\omega }_{2}t+{\varphi }_2)}$

(9')

Equations (9) and (9') represent the infinitely many time laws according to which the two masses can move: the system of the two masses will select just one of them so as to satisfy the particular initial conditions it is in at time t=0. Fixing the initial conditions of this dynamical system means specifying:

• the initial position of the two masses (in this case the values of ${\displaystyle x_1(0)}$ and ${\displaystyle x_2(0)}$)

• the initial velocity of the two masses (in this case the values of ${\displaystyle v_1(0)}$ and ${\displaystyle v_2(0)}$)

	Normal mode 1 is obtained by imposing ${\displaystyle x_1(0)=x_2(0)=0}$ ${\displaystyle v_1(0)=v_2(0)=v_0}$ which is equivalent to pushing the two masses, starting from the rest position, in the same direction with the same initial velocity.
	Normal mode 2 is obtained by imposing ${\displaystyle x_1(0)=x_2(0)=0}$ ${\displaystyle v_1(0)=-v_2(0)}$ which is equivalent to pushing the two masses, starting from the rest position, in opposite directions with the same initial velocity (in magnitude)

It is interesting to note that any initial condition other than the two above generates a motion that is a "combination" of the two normal motions; the surprising aspect is that this motion is not periodic: however long we wait, the system of the two masses will never pass through a configuration it has already visited. This incredible property, all the more incredible given the simplicity of the system examined, comes from the incommensurability of the frequencies of the two normal modes (that is, from the impossibility of expressing their ratio as a rational number). Indeed:

${\displaystyle \frac{{\omega }_2}{{\omega }_1}=\sqrt{3}}$

(10)

Further reading and links

• On the page on normal modes you will discover that the possibility of expressing complicated oscillations as a superposition of simpler modes is not limited to the simple system described on this page, but is a general procedure applicable to every vibrating system;

• On the page on resonance you will discover that the values of the frequencies of the normal modes decisively determine the ways in which a vibrating system can receive energy from or give energy to the outside;

• Despite its simplicity, the system of two coupled oscillators described on this page can serve as a paradigm for more complicated systems of oscillators. They can be, for example,
  • a building and the foundation it rests on, which in an earthquake is set oscillating and transfers part of its motion to the building itself (to find out more, visit the page resonance in buildings).
  • a violin string and the sound box of the violin itself (to find out more, visit the sections violin and sound and resonance).