C2T (Mechanics) Topic Oscillations (Part 1)...2021/01/22 · oscillator including friction, a...

Dr. Avradip Pradhan, Assistant Professor,

Department of Physics, Narajole Raj College, Narajole.

PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)

C2T (Mechanics)

Topic – Oscillations (Part – 1)

Introduction:

The simple harmonic motion (or SHM, in short) plays a vital role in Physics

than one might guess from its humble origin, which is a mass bouncing at the

end of a spring vertically or on a frictionless horizontal surface. The mass

executing SHM is usually called as a harmonic oscillator. It underlies the

creation of sound by musical instruments, the propagation of waves in media,

the analysis and control of vibrations in heavy machinery (e.g. airplanes) and

the time-keeping crystals in digital watches etc. Furthermore, the harmonic

oscillator arises in numerous atomic and optical quantum situations, in quantum

systems such as lasers and it is a recurrent topic in advanced quantum field

theories.

We mostly encounter simple harmonic motion in the periodic motion of a mass

attached to a spring. The treatment there is highly idealized because it neglects

friction and the possibility of a time-dependent driving force. It turns out that

friction is essential for the analysis to be physically meaningful and that the

most interesting applications of the harmonic oscillator generally involve its

response to a driving force. In this e-report we will look at the harmonic

oscillator including friction, a system known as the damped harmonic

oscillator, and then investigate how the system behaves when driven by a

periodic applied force, a system called the driven or forced harmonic oscillator.

Periodic Motion and Oscillatory Motion:

When a mass is found to describe the same path repeatedly in some fixed

interval of time, the motion is said to be periodic. The time taken by the mass to

complete its path once is called its time period. Examples of periodic motion

can be easily found. The motion of the earth round the sun is an example of a

periodic motion, where the time period is one year. The rotation of the earth




about its own axis is also a periodic one, the time period being 24 hours. The

motion of a grandfather clock pendulum is another periodic motion.

The periodic motion of a mass is said to be oscillatory when the motion gets

reversed in direction after a definite interval of time. The motion of a pendulum

is an example of this type of motion. It is important to note that an oscillatory

motion is always a periodic one, but the converse statement is not true. A

periodic motion need not have to be necessarily an oscillatory motion every

time.

Simple Harmonic Motion (SHM):

It is the simplest version of oscillatory motion of a body, with the following two

characteristics.

(a) The magnitude of restoring force (the force which pulls the body back)

acting on the body is always directly proportional to its displacement from a

fixed point on its path, known as the equilibrium position.

(b) The restoring force is always directed towards its equilibrium position.

Differential Equation of SHM. In order to setup the differential equation for

SHM, we need to follow the above two characteristics. If being the

displacement from the equilibrium position, with as the restoring force

acting on the body of mass , then we write

or with

The minus sign indicates that the restoring force is directed towards the centre.

The proportionality constant is often called as the spring constant of the

SHM. According to Newton’s laws of motion,

, where

is the

acceleration of the body. Therefore we get




or

, with

or

This is the differential equation of a body executing SHM. The general solution

of the above equation is

where and are arbitrary constants that can be chosen to make the general

solution meet any two given independent initial conditions. Typically these are

the position and velocity at a time taken to be . The solution can be cast in

a different form by using the trigonometric identity

Applying this to the previous equation takes the solution into the form

where and . Both the expressions are equivalent

and therefore simultaneously used.

Fig. 1

Nomenclature. In the expression , is the amplitude of

the motion (the distance from equilibrium position to a maximum) while is

called the natural frequency (more precisely, the natural angular frequency) of




the oscillator. Angular frequency is generally written in units of s−1. The

circular frequency is the frequency expressed as revolutions per second or

cycles per second. So, we obtain

in hertz, where one hertz (1 Hz) = 1

cycle per second. The quantity is the phase angle of the oscillation at

time and is known as the phase constant or initial phase. The time period of

the motion is given by

. Fig. 1 shows a typical waveform of an ideal

(or frictionless) simple harmonic motion.

Velocity. The velocity can be calculated from the expression of the

displacement as,

.

Therefore, .

At the equilibrium position, , the velocity magnitude becomes its

maximum value at . On the other hand, at the extreme points we get

, and velocity becomes zero, as expected.

Fig. 2

Example of SHM: Simple Pendulum:

A simple pendulum consists of a heavy point mass, known as a bob, suspended

from a rigid support by an inextensible, massless and perfectly flexible string.

We will see that a simple pendulum displays simple harmonic motion to good

approximation if the amplitude of swing is a small angle. Fig. 2 shows a simple




pendulum of length , with mass of the bob , and corresponding weight

.

The bob moves in a circular arc in a vertical plane. Denoting the angle from the

vertical by , we see that the velocity is

and the acceleration is

. The

tangential force is . Thus the equation of motion can be written as

or

.

This is not the equation for SHM because of the sine function, and it cannot be

solved in terms of familiar functions. However, if the pendulum never swings

far from the vertical so that , we can make the approximation ,

giving

.

This is the equation for simple harmonic motion. To put it in standard form, we

obtain

. The motion is therefore periodic, which means it occurs

identically over and over again. The time period is given by or

.

Total Energy of a Harmonic Oscillator:

A harmonic oscillator possesses kinetic energy from its translational motion,

and potential energy from its spring action. The kinetic energy ( ) is given by

The potential energy, which is taken as zero for the unstretched configuration, is

given by




or

The total energy of the harmonic oscillator is therefore written as

constant

The total energy is constant, a familiar feature of motion in systems where the

forces are conservative.

Time Averaged Values. It is also important to know the time averaged values

of both kinetic energy and potential energy. The time average is generally

carried over one time period of oscillation.

The time averaged value of kinetic energy is given by

or

or

.

Similarly the time averaged value of potential energy is given by

or

or

.

Therefore, we find that the time averaged values of and are same and

equal to

.

It is also important to note that

, as expected. In

the averaged form the total energy gets equally distributed among kinetic and

potential energies.




Damped Harmonic Oscillator:

The ideal harmonic oscillator is frictionless. It turns out that friction is often

essential and ignoring it can lead to absurd predictions. Let us therefore

examine the effect of a viscous frictional force which is proportional to the

body’s instantaneous velocity. Mathematically it is written as . This

type of friction is most often encountered, so our analysis will therefore be

widely applicable. For example, in the case of oscillations in electromagnetic or

circuits, the electrical resistance of the circuit precisely plays the role of

viscous retarding force.

The total force acting on the mass is therefore a sum of the restoring force

and the viscous force, . The equation of motion is therefore,

or

This equation can be written in standard form, given below

where

and

This is known as the differential equation for a damped harmonic oscillator. We

will attempt to guess the solution from the Physics of the situation because this

can yield insights that the formal solution may hide. If friction were negligible

the motion would be given by . On the other hand, if the

restoring force were negligible, then the mass would move according to the

equation . We might therefore guess that the trial solution to the

previous equation is of the form

The constants and can be chosen to make this trial solution satisfy the

given equation. and are arbitrary constants for satisfying the initial

conditions. Substituting the trial solution in the equation of motion, we find that

the equation is satisfied provided that




and

This solution is valid when

. This situation is called as

underdamped. The other case (

, known as overdamped) is

discussed later.

Fig. 3

The motion described by the trial solution is known as damped harmonic

motion. Several examples are shown in Fig. 3 for increasing values of

. The

motion is reminiscent of the undamped frictionless harmonic motion described

in the last section. To emphasize this, we can rewrite the equation as

where

. The motion is similar to the undamped case except that

the amplitude decreases exponentially in time and the frequency of oscillation




is less than the undamped frequency . The motion is periodic because the

zero crossings of

are separated by equal time intervals

, but the peaks do not lie exactly halfway between them.

Underdamped, Overdamped and Critically damped Motion. The essential

features of the motion depend on the ratio

. If

, the amplitude

decreases only slightly during the time the cosine function makes many zero

crossings. In this regime, the motion is called lightly damped. If

is

comparatively larger, tends rapidly to zero while the cosine function

makes only a few oscillations. This type of motion is called heavily damped.

For light damping, , but for heavy damping can be significantly

smaller than . If

, the trial solution fails, and the motion is not

oscillatory. The system is now described as overdamped. This solution has no

oscillatory behaviour and can be expressed as

where and are two decay constants, given as

and

Critically damped situation arises at the border line of overdamped and

underdamped motions, when

. The general solution for critical damping

is obtained as

. Here the solution doesn’t have the

oscillatory nature either. The motion is dead-beat as the displacement falls from

maximum to zero very rapidly.




Energy Dissipation in Damped Harmonic Oscillator:

As we know that friction dissipates mechanical energy, so the energy of a

damped oscillator must decay in time. To evaluate the kinetic energy we first

find the velocity by differentiating the trial solution to obtain

We will be most interested in systems with light damping, where

, so

that . This allows us to make an approximation that simplifies the

arithmetic and reveals some universal features

With our approximation that

, the second term in the bracket of the

previous equation can be neglected, giving

where we have used .

In this case, the kinetic energy is expressed as

and the potential energy is written as

.

The total energy is therefore,

or

The see that the total energy is decaying with time and the decay of the total

energy is described by a simple differential equation




which has the simple exponential solution , where is the energy

at time , as shown in Fig. 4.

Fig. 4

The energy’s decay is characterized by the time scale

in which the total

energy decreases from its initial value ( ) by a factor of . is

therefore often called the damping time of the system. In the limit of zero

damping, , so that and is constant. The system behaves like an

undamped oscillator.

It is important to note that we could have found the same result directly from

the work-energy theorem. The rate at which work is done on the system by

friction is

. Using the expression for velocity derived

before and making the approximation for lightly damping case, we can

write from work-energy theorem as

or




Here factor has been taken out of the average since this value will be

essentially constant because of the fact . The time average value of

will be half. Using this value we finally obtain

.

This result is as per our early expectation.

This concludes part 1 of this e-report.

The discussion will be continuing in the part 2 of this e-report.

Reference(s):

An Introduction to Mechanics, Kleppner & Kolenkow, Cambridge

University Press

A Treatise on General Properties of Matter, Chatterjee & Sengupta,

New Central Book Agency

(All the figures have been collected from the above mentioned references)

C2T (Mechanics) Topic Oscillations (Part 1)...2021/01/22 · oscillator including friction, a...

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