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 Hookes 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 Hookes law, and such a system is called a simple harmonic oscillator.

If the net force can be described by Hookes 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 lifePendulum

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 clockPendulums 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 OscillationsSimple 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.

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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 motionPeriodic 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 Motionperiodic 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).DisplacementThe 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 waveformDamping

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.AmplitudeWhen 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.PhaseWhen 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 ResonanceWhen 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.htmlhttp://www.acoustics.salford.ac.uk/feschools/waves/shm4.phphttp://en.wikipedia.org/wiki/Simple_harmonic_motionhttp://en.wikipedia.org/wiki/Harmonic_oscillatorhttp://en.wikipedia.org/wiki/Harmonic_motion