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  Simple Harmonic Motion III, Physics tutorial

Damped Oscillations:

The amplitude of oscillations of, for instance, a simple pendulum, slowly decreases to zero over time because of resistive force arising from surrounding air in this case. In other forms of s. h. m. it will occur from surrounding medium (like liquid or gas). Motion for such oscillations is not thus a perfect s. h. m. It is said to be damped by air resistance, i.e., there is stable loss of energy as energy is converted to other forms. Generally it will be internal energy through friction but energy may also be radiated away. For instance, a vibrating turning fork loses energy by sound radiation.

The behavior of the mechanical system depends on extent of damping. For instance, mass hanging from the coiled spring and immersed in the liquid when set to vibrate, goes through more damping than when it is in air. Know that undamped oscillations are said to be free. When vibrating system is greatly damped, no oscillations take place. System just slowly returns to its equilibrium position. Now, when time taken for displacement to be zero is extremely small, vibrating system is said to be significantly damped.

When damping forces are proportional to velocity, v, period remain constant as amplitude diminishes and oscillator is said to be isochronous. It will interest you to know that motion of some devices is decisively damped on reason to get the certain desired objective. For instance, shock absorbers on the car vitally damp suspension of vehicle and hence resist the setting up of vibration that could make control difficult or cause damage. In shock absorber the motion of suspension up or down is opposed by viscous forces when liquid passes through transfer tube from one side of piston to the other. Instruments like balances and electrical meters are vitally damped so that pointer moves quickly to the correct position without oscillating. Damping is frequently produced by electro-magnetic forces.

Forced Oscillation and Resonance:

Barton's Pendulums:

A number of paper coned pendulums of length varying from 1/4 m to 3/4 m, each loaded with the plastic curtain ring are suspended from same string as the driver pendulum that has heavy bob and a length of 1/2 m.

348_Barton Pendulums.jpg

When driver pendulum is pulled well aside and then released, it oscillates in the plane perpendicular to plane of the diagram. After a short time, motion settles down and all the other pendulums oscillate with extremely almost same frequency as that of the driver although with different amplitudes. This is the example of forced oscillation. Out of the set of pendulums, one whose length equals that of driver pendulum has greatest amplitude of vibration. Therefore, its natural frequency of oscillation is same as the frequency of driving pendulum. This is the example of resonance and driving oscillator passes on its energy most easily to other system, i.e., proper cone pendulum of the same length. Therefore, when rings on paper cone pendulums are removed, their masses decrease and so damping increases. All amplitudes are then found to be reduced and that of resonance frequency being less pronounced.

Energy Considerations:

Whether or not a body is at or close to resonance, oscillator settles down in the steady state where energy supplied from driver per cycle is equal to energy dissipated per cycle. Sharpness of resonance, known as Q-factor is equal to:

= energy lost per cycle/energy at the start of the cycle

It is also provided by

Q = f0/Δf

Where Δf is the width of the resonance curve

When

x = xmax/√2

Xmax being maximum value of displacement x and where fo is the resonant frequency.

Phase:

At resonance, an oscillator lags behind the driver by 90o i.e. it is 90o out of phase with driver. When driver is at a much lower frequency than oscillator's natural frequency (fd < fN) oscillator is in step with driver. When driver frequency is much higher than natural frequency (fd > fN), driver and oscillator are 180o out of phase.

524_Phase.jpg

S. H. M. - A Mathematical Model:

S. h. m. is entirely an idealized situation which doesn't exist in nature or in practical world. Real oscillators like a motor cycle on its suspension, tall chimney swaying in wind, atoms or ions vibrating in the crystal etc only estimated to ideal kind of motion we call s.h.m. Simple harmonic motion is the mathematical model, helpful because it represents several real oscillations because of its simplicity. It doesn't have complications like damping, variable mass and stiffer (elastic modulus). Only condition it (s. h. m.) has to satisfy is that restoring force must be directed towards centre of motion and be proportional to displacement.

The more complex model might, for instance, take damping into consideration and therefore may be the better explanation of the particular oscillator. Such may perhaps not be extensively appropriate. Conversely, if the model is very simple, it may be of little use for dealing with actual systems. Therefore, a model should have just right degree of complexity. Mathematical s. h. m. has this and so is useful in practice.

Physical Pendulum:

It is not always that pendulum comprises of the massless string with the pointlike mass at the end of it. Sometimes a pendulum can comprise of the suspended swinging object of some form. We call this physical pendulum. Any object can be suspended from any point on object and serve as physical pendulum. This shows fact that s. h. m. is the general feature of motion about the stable equilibrium. You can even set up the physical pendulum, with measuring ruler in the room.

Therefore, the so-called physical pendulum is any real pendulum in which all mass is taken to be concentrated at the point. If a body with irregular shape pivoted about the horizontal frictionless axis O and displaced from vertical by an angle Θ. Distance from pivot to centre of gravity is h, the moment of inertia of pendulum about the axis through pivot is I and the mass of pendulum is m. Weight mg causes the restoring torque τ of value provided by

τ = - mgh sinΘ

When released, body oscillates about the equilibrium position. Note that, unlike s.h.m., motion of the physical pendulum is not simple harmonic as the torque τ is proportional not to Θ but to sin Θ. Though, if Θ is small, we can again estimate sin Θ by Θ so that motion becomes approximately harmonic.

Suppose this approximation then,

τ = (mgh)Θ

Effective torque constant is

K1 = -τ/Θ = mgh

Therefore, period of physical pendulum is

T = 2Π√I/K1 = 2Π√I/mgh

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