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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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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.
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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.
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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
Dr. Avradip Pradhan, Assistant Professor,
Department of Physics, Narajole Raj College, Narajole.
PAPER: C2T (Mechanics) TOPIC(s): Oscillations (Part – 1)
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)