Harmonic Motion

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Introdction
Simple harmonic motion
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium
point. The oscillations may be periodic such as the motion of a pendulum or random such as the
movement of a tire on a gravel road.
Vibration is occasionally "desirable". For example the motion of a tuning fork, the reed in a
woodwind instrument or harmonica, or mobile phones or the cone of a loudspeaker is desirable
vibration, necessary for the correct functioning of the various devices.
More often, vibration is undesirable, wasting energy and creating unwanted sound – noise. For
example, the vibrational motions of engines, electric motors, or any mechanical device in
operation are typically unwanted. Such vibrations can be caused by imbalances in the rotating
parts, uneven friction, the meshing of gear teeth, etc. Careful designs usually minimize unwanted
vibrations.
The study of sound and vibration are closely related. Sound, or "pressure waves", are generated
by vibrating structures (e.g. vocal cords); these pressure waves can also induce the vibration of
structures (e.g. ear drum). Hence, when trying to reduce noise it is often a problem in trying to
reduce vibration.
The oscillations of a system in which the net force can be described by Hooke’s law are of
special importance, because they are very common. They are also the simplest oscillatory
systems. Simple Harmonic Motion (SHM) is the name given to oscillatory motion for a system
where the net force can be described by Hooke’s law, and such a system is called a simple
harmonic oscillator.
If the net force can be described by Hooke’s law and there is no damping (by friction or other
non-conservative forces), then a simple harmonic oscillator will oscillate with equal
displacement on either side of the equilibrium position, as shown for an object on a spring in
Figure. The maximum displacement from equilibrium is called the amplitude X. The units for
amplitude and displacement are the same, but depend on the type of oscillation. For the object on
the spring, the units of amplitude and displacement are meters; whereas for sound oscillations,
they have units of pressure (and other types of oscillations have yet other units). Because
amplitude is the maximum displacement, it is related to the energy in the oscillation.

Types of vibration in oscillations

Free vibration occurs when a mechanical system is set off with an initial input and then
allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing
and then letting go or hitting a tuning fork and letting it ring. The mechanical system will then
vibrate at one or more of its "natural frequency" and damp down to zero.

Forced vibration is when a time-varying disturbance (load, displacement or velocity) is
applied to a mechanical system. The disturbance can be a periodic, steady-state input, a transient
input, or a random input. The periodic input can be a harmonic or a non-harmonic disturbance.
Examples of these types of vibration include a shaking washing machine due to an imbalance,
transportation vibration (caused by truck engine, springs, road, etc.), or the vibration of a
building during an earthquake. For linear systems, the frequency of the steady-state vibration
response resulting from the application of a periodic, harmonic input is equal to the frequency of
the applied force or motion, with the response magnitude being dependent on the actual
mechanical system.

Example of simple harmonic motion related to our life

Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely.[1] When a
pendulum is displaced sideways from its resting equilibrium position, it is subject to a restoring
force due to gravity that will accelerate it back toward the equilibrium position. When released,
the restoring force combined with the pendulum's mass causes it to oscillate about the
equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and
a right swing, is called the period. A pendulum swings with a specific period which depends
(mainly) on its length.

The simple gravity pendulum is an idealized mathematical model of a pendulum. This is
a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. When
given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are
subject to friction and air drag, so the amplitude of their swings declines.

Clock pendulums

Pendulum and anchor escapement from grandfather clock

Pendulums in clocks are usually made of a weight or bob suspended by a rod of wood or metal.
To reduce air resistance (which accounts for most of the energy loss in clocks) the bob is
traditionally a smooth disk with a lens-shaped cross section, although in antique clocks it often
had carvings or decorations specific to the type of clock. In quality clocks the bob is made as
heavy as the suspension can support and the movement can drive, since this improves the
regulation of the clock (see Accuracy below).

Each time the pendulum swings through its center position, it releases one tooth of the escape
wheel The force of the clock's mainspring or a driving weight hanging from a pulley, transmitted
through the clock's gear train, causes the wheel to turn, and a tooth presses against one of the
pallets, giving the pendulum a short push. The clock's wheels, geared to the escape wheel, move
forward a fixed amount with each pendulum swing, advancing the clock's hands at a steady rate.
The pendulum always has a means of adjusting the period, usually by an adjustment nut under
the bob which moves it up or down on the rod. Moving the bob up decreases the pendulum's
length, causing the pendulum to swing faster and the clock to gain time. Some precision clocks
have a small auxiliary adjustment weight on a threaded shaft on the bob.
Damped and Driven Oscillations

Simple harmonic oscillators that we encounter in the real world do not oscillate forever.
Unlike the Energizer Bunny, they do not keep going and going. There is usually some friction
present and that friction causes the motion to become smaller and smaller or to decay or to die
out or to damp out. "Damped" simply means gradually decreasing.

The
amplitude
of a
mass oscillating under water gets smaller or decays as time goes on. This is a damped harmonic
oscillator.

Springs on an
automobile turn the car into a
harmonic oscillator. Shock absorbers turn it into a damped harmonic oscillator and keep it from
continuing to bounce up and down after every bump or pot hole a tire encounters. Shock
absorbers designed for a lightweight sports car would not provide enough damping for a heavy
van and would allow several oscillations after each bounce. Likewise, shock absorbers designed
for a heavy pickup would provide too much damping for a small sports coupe and would give a
stiff, uncomfortable ride. Shock absorbers must be designed with the mass of the car and the
stiffness of the springs taken into account.

The swing is first pulled back, and then pushed forward. It then oscillates back and forth
'on its own' until it slowly comes to rest at its equilibrium position - the middle position between
the two extremes of displacement. This is usually the place where an object will naturally rest if
no external forces are applied to it.
Once the swing has been pulled away from its equilibrium position, the force of gravity
will act to bring it back. This force will always act in a direction towards the equilibrium
position, and is known as a restoring force. If the swing is pulled higher into the air (larger
displacement amplitude) then there will be a larger restoring force acting on it, and when the
pusher lets go it will travel further. This shows that the restoring force is proportional to the
distance of the swing from the equilibrium position.

\

This is an example of resonance of a driven harmonic oscillator. If forces from the
outside are given at the resonance frequency of the oscillator, the resulting amplitude may be
quite large.

Motion of a swing

If given one strong push, a swing will move back and forth because the earth's gravity is pulling
down while the rope is making the swing move in a partial circle. When the swing has returned
to the starting point, it can't go down any more, but still has a lot of momentum to use up, so it
moves beyond the center and up on the other side until gravity slows it down and pulls it back to
the center, but it still has too much momentum.. and so forth. This whole operation is periodic:
each complete swing takes the same amount of time, regardless of how far the swing moves.
When the arc of swing is large, the swing moves quickly, when the arc is small, the swing moves
slowly. The amount of time it takes the swing to complete its cycle depends on the length of the
rope and nothing else. If you make a graph of the distance the swing travels crossways with time,
the result is a sine wave.

The only thing slowing the swing down is friction with the air and within the rope. If you were to
replace the energy lost to this friction, you could keep the swing going forever. If you try this by
giving the swing extra pushes, you very quickly discover you must push at the right time and in
the right direction, or you waste your push or perhaps slow the swing down. Specifically, the
time between pushes must correspond with the period of the swing, and you must push in the
same direction the swing is already going.

Periodic motion
Periodic motion is motion which repeats itself. The frequency of motion is the number of
times the motion is repeated per second. The period of the motion is the time required for one
cycle or repetition.
Simple harmonic motion is a particular kind of periodic motion and occurs for very diverse
systems when they are disturbed slightly from equilibrium The restoring force which brings a
simple harmonic oscillator back to equilibrium is proportional to how far it has been disturbed
from equilibrium.. The amplitude of a simple harmonic oscillator is the maximum distance that it
moves from equilibrium. The period or frequency of a simple harmonic oscillator is independent
of its amplitude. This makes simple harmonic oscillators very important in keeping accurate
time.

During simple harmonic motion, energy is transferred from one form to another throughout the
cycle but the total energy remains constant. For a horizontal spring and mass, energy changes
from kinetic energy to elastic potential energy and back again. A simple pendulum is another
example of a simple harmonic oscillator as long as its amplitude does not get very large; energy
changes from kinetic energy to gravitational potential energy and back again. The period or
frequency of a mass and spring is determined by the spring constant and the mass. The period or
frequency of a simple pendulum is determined by the length of the pendulum and the
acceleration due to gravity.

A damped oscillator has friction present which causes its amplitude to gradually decrease with
time. An external force which is applied with the resonant frequency of an oscillator can cause
the oscillator to have a very large amplitude.

Periodic Motion
periodic motion is motion which repeats itself. The frequency of motion is the number of times
the motion is repeated per second. The period of the motion is the time required for one cycle or
repetition.
Simple harmonic motion is a particular kind of periodic motion and occurs for very diverse
systems when they are disturbed slightly from equilibrium The restoring force which brings a
simple harmonic oscillator back to equilibrium is proportional to how far it has been disturbed
from equilibrium.. The amplitude of a simple harmonic oscillator is the maximum distance that it
moves from equilibrium. The period or frequency of a simple harmonic oscillator is independent
of its amplitude. This makes simple harmonic oscillators very important in keeping accurate
time.

During simple harmonic motion, energy is transferred from one form to another throughout the
cycle but the total energy remains constant. For a horizontal spring and mass, energy changes
from kinetic energy to elastic potential energy and back again. A simple pendulum is another
example of a simple harmonic oscillator as long as its amplitude does not get very large; energy
changes from kinetic energy to gravitational potential energy and back again. The period or
frequency of a mass and spring is determined by the spring constant and the mass. The period or
frequency of a simple pendulum is determined by the length of the pendulum and the
acceleration due to gravity.

A damped oscillator has friction present which causes its amplitude to gradually decrease with
time. An external force which is applied with the resonant frequency of an oscillator can cause
the oscillator to have a very large amplitude.
When an object moves in a repeated pattern over regular time intervals, it is undergoing periodic
motion.
One complete succession of the pattern is called a cycle of vibration.
The time required to complete one cycle is the period(T).

Displacement

The displacement amplitude tells us how 'big' the oscillations are - we can use the peak
value (the maximum positive displacement from the equilibrium position) or the peak-to-peak
value (the distance between negative-maximum to positive-maximum).
The time period of the oscillation is the time taken for the object to travel through one complete
cycle. We can measure from the equilibrium position, as in the animation, or from any other
point on the cycle, as long as we measure the time taken to return to the same point with the
same direction of travel.
Can you see why the first time the mass returns to the equilibrium position, we've only reached
half a cycle?
Displacement on its own will sometimes do...but often it is also useful to understand the
oscillation in terms of velocity or acceleration. These three things are very similar

Velocity and Acceleration
When our oscillating object reaches maximum displacement (when it is as far from
equilibrium as it can get) it changes direction. This must mean that there is an instant in time
when it is not moving...and so at the point of maximum displacement, its speed is zero. Then it
speeds up in the opposite direction, and travels fast through the equilibrium position before
starting to slow again in preparation for the next change in direction.
Velocity is just another word for speed, with the extra feature that it has direction and therefore
can be negative or positive. (If you walk backwards at 4m.p.h. - difficult without falling over then your velocity = -4m.p.h.)
Inertia
Let's go back to thinking about masses and springs. Imagine a mass held between two springs as
shown in the animation below. If the mass is moved away from the equilibrium position the
restoring forces provided by the springs will make the mass oscillate back and forth in a similar
manner to the playground swing

An increase in mass increases the inertia (reluctance to change velocity...i.e. reluctance to
accelerate / decelerate) of the object. (In this example we have switched off gravity, so the heavy
mass does not 'sag' on the springs. Note - massive objects have inertia even in outer space where
they have no weight!). An increased inertia means that the springs will not be able to make the
mass change direction as quickly. This increases the time period of the oscillation. The greater
the inertia of an oscillating object the greater the time period; this lowers the frequency of its
oscillations.
The oscillating objects we've looked at up to now all vibrate with a rather special 'shape' or
waveform

Damping
When we were talking about playground swings, we mentioned damping - a loss of
energy from, in that example, movement (kinetic) energy to heat. There other examples of
damping we could think about - here's one:
Imagine hitting a cymbal. This causes the cymbal to oscillate. These oscillations cause the air
around the cymbal to vibrate, and these vibrations travel to your ear and you hear this as sound.
The sound of the cymbal will eventually die away, as the air resistance and internal losses within
the cymbal reduce the restoring forces, causing the oscillations to get smaller and smaller.
Placing your hand on the cymbal after hitting it can greatly speed up the damping process, as
your fingers absorb the kinetic energy (being soft and a bit 'pudgy'!) very effectively..
Normally it is hard to see a cymbal vibrating because it is moving too fast. Here we've slowed it
down to 1/ 80th of normal speed. You can see that the oscillations take ages to die away, as
damping is small. For many oscillations (including this one) the damping forces are roughly
proportional to velocity, and this leads to an exponential decay of amplitude over time.

Amplitude
When a simple pendulum or a mass on a spring is not in motion but is allowed to hang freely, the
position it assumes is called the rest or equilibrium position.
When in motion, the distance from the equilibrium position to the maximum displacement is the
amplitude(A) of the vibration.

Frequency
One of the most common terms used to describe periodic motion is frequency(f), which is the
number of cycles completed in a specific time interval.

Phase
When two vibrating objects have the same amplitude and frequency, they may not be at the same
point in their cycles at the same time.
When this occurs we say that there is a phase difference between them

.

Phase
If, at any time, the two objects are moving in opposite directions, they are vibrating out of phase.
If they are always going in opposite directions at the same time then they are 180oout of phase or
in ant phase.

Natural Frequencies and Resonance
When you push a child on a swing you do not need to push very hard to make the child swing
higher and higher. What you do have to do is push at the right times, that is, with a frequency
equal to the natural frequency of the swing and child. As well, the cycle of pushing must be in
phase with the motion of the child and the swing.

Conclusion
I learned how we could find the spring constant of a spring from Hanging a mass and measuring
the displacement of the spring and Having a mass oscillate on the spring.
One factor that could have caused a difference in our values for k is that the oscillating mass
could have lost some potential energy causing the spring to oscillate slower. We can further study
spring constants by studying a spring that compresses and finding its constant.

Reference
http://physics4abalewis.blogspot.com/2012/12/hookes-law-and-simple-harmonic-motion.html
http://www.acoustics.salford.ac.uk/feschools/waves/shm4.php
http://en.wikipedia.org/wiki/Simple_harmonic_motion
http://en.wikipedia.org/wiki/Harmonic_oscillator
http://en.wikipedia.org/wiki/Harmonic_motion

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