15 Small Oscillations II

Ashok Goyal

1. Introduction

In the preceding unit we considered the motion of a complex conservative system with large number of degrees of freedom in the limit of small departure of the system from its stable equilibrium state. We saw that a transformation from the Cartesian coordinates to Normal coordinates allows the description of the system in these coordinates which oscillate with single specified frequencies. Determination of Normal coordinates in general could be quite complicated. In this unit with the help of some representative problems with two and three degrees of freedom, we would illustrate how this can be accomplished in practice.

2. Two Coupled Harmonic Oscillators

Let us consider a simple example of the motion of two identical harmonic oscillators connected by a spring. Each spring connected to the oscillators has a force constant . We consider the motion of the oscillator in the x-direction as shown. The system has two degrees of freedom represented by the coordinates  and  measured from the equilibrium state.

The kinetic energy of the

For a non-trivial solution to exist, the determinant of the coefficients  must vanish. Thus the characteristic equantion is obtained by putting

we thus see that if initial condition is given by  at

The particles would oscillate with the frequency and oscillations are out of phase. This is called the ‘antisymmetric’ mode.

On the other hand, if at 0,  the particles would oscillate with frequency  and are in phase.This is the ‘symmetric’ mode.

The two distinct modes we considered are the particular solutions. The general solution is a linear combination of these two solutions. We can choose coordinates so as to decouple the motion. An examination of equation (15.4) shows that if we define

The equations of motion is terms of these coordinates become

The coordinates  and  are now to independent uncoupled coordinates called the normal coordinates. In terms of normal coordinates the solutions of the system are

Thus  represents the antisymmetric and   the symmetric modes of the problem.

3. Vibrations of Triatomic Molecule

We will illustrate in this example, the general method for obtaining the resonant frequencies of vibration and the normal modes. We consider a linear tri-atomic molecule like CO2. In the equilibrium configuration of the molecule, two oxygen atoms of mass m are on the two sides of the carbon atom of mass M. All the three atoms are located linearly with equal equilibrium distance b between the central carbon and side oxygen atoms. The inter-atomic forces responsible for keeping the oxygen atoms bounds to the carbon atom arise from the Coulomb forces between the electrons of these atoms. However, in the limit of small oscillations, the interaction forces can be replaced by simple harmonic forces and we have a simple models of linear tri-atomic molecules as shown below:

We thus have two masses m bound to a central mass M by springs with force constant k each.  The molecule has three degrees of freedom. We choose the rectangular coordinates  and  measured from the equilibrium position. In these coordinates, the potential energy is given by

(The equations of motion in normal coordinates are given by  which implies a uniform transnational motion in the coordinate)

This translational motion can be eliminated by imposing the constraint that the centre of mass given by

remains at rest. With this constraint the original problem reduces to a problem with two degrees of freedom. The second resonant frequency  appears to be the frequency of atoms of mass  on a spring of constant  and as if the central atom is at rest. In the third mode with frequency  all the particles appear to participate.The eigenvectors corresponding to a particular eigen value  are to be determined from the eigen value equation.

We will now obtain the components of eigenvectors for each frequency.

Hence the normal modes are as depicted below:

4. Summary

•  In some simple problems, it is possible to obtain the normal modes by inspection of the equations of motion.