16 Small Oscillations III

Ashok Goyal

1. Introduction

Problem of the motion of elastic string or a spring on which identical masses are placed at regular intervals was attempted by Newton and was eventually solved by Johann Bernoulli. It gave rise to the principle of superposition. In this unit we will visit the problem and will indicate how a solution leading to free vibrations can be obtained . In the earlier unit we discussed how a system, specifically a chain of molecules when displaced from equilibrium would oscillate freely. The general oscillations are complicated but a great simplification is achieved by considering the motion in normal mode coordinates. In its normal mode, a system oscillates with one of its natural frequencies called Resonant Frequency. In order to obtain knowledge of intermolecular forces, one would need to study the vibrations of the molecular chain This would require setting the chain to oscillatory motion with the help of external forces. If the natural/resonant frequencies of the system lie in the ‘Acoustical’ range, the system can be set into oscillations by second waves falling on the system. In case the resonant frequencies lie in the ‘optical’ range, the external driving force can be provided by shining light/laser beam on the system. In addition the system would have dissipative (frictional) forces present which would result in the damping of the motion.

In this unit we will study the vibrations of a loaded string. We will the study the oscillations of such a complex system i.e. vibrations of a linear molecular chain in the presence of external force.

2. Vibrations of a Loaded String

Consider an elastic string of length L on which are placed n identical masses at regular intervals with spacing a between them. The two ends of the string are fixed and the string is in equilibrium. The string is now pulled up by a small amount and let go setting the string into vibratory motion.

We will try to find the conditions on  and  such that boundary conditions are satisfied. Substituting  from (16.11) into the equation (16.8).We get

(In deriving the expressions for eigen-frequencies we have followed the method given in ‘Classical Dynamics’ by Marion for the solution of the characteristic equation (16.10). For an alternate derivation see ‘Classical Mechanics’ by S.N. Biswas).

3. Forced and Damped Vibrations

In the presence of an external driving force which is taken to vary sinosoidally with time with a driving frequency , the system is forced to oscillate with the frequency of driving force and not by the resonant frequencies (eigen-frequencies). If  is the generalized force corresponding to the coordinate , then the generalized force  for the normal coordinate  is given by

The equation of motion in normal coordinates now is given by

These equations are a set of n inhomogeneous second-order differential equations which can be solved once the generalized force  is specified. For the force  we take

A complete solution of the differential equation (16.24) consists of a general solution of the homogenous equation plus a particular solution of the inhomogeneous equation. The general solutions of the homogeneous equation are the free modes of vibration with characteristic frequencies of the system. The particular solution of the equation (16.24) with  given in (16.25) can be written as

Dissipate Force

If a dissipative force exists in the system which is proportional to the velocity of the particle, then just as in the case of restoring force proportional to deviations from the equilibrium position can be derived from a potential energy function.

In physical situations where the kinetic and potential energy functions can be diagonalised by transformation to normal coordinates, it frequently happens that the dissipation function too is diagonalized. We will consider this situation to write the equations of motion in normal coordinates as :

• In the presence of driving force, the normal coordinate oscillates with the driving frequency.
• In the presence of a dissipative force proportional to the velocity of the particle, the amplitude of oscillations is damped.