Several types of motion repeat themselves over and over: vibration of the quartz crystal in the watch, swinging pendulum of the grandfather clock, sound vibrations generated by the organ pipe and back-and-forth motion of pistons in a car engine. This type of motion is known as periodic motion or oscillation.
The body which undergoes periodic motion always has the stable equilibrium position. When it is moved away from this position and released, a force or torque comes into play to pull it back toward equilibrium. There are several oscillatory systems; vibrations of molecules and interaction between atoms, oscillations of the electrical circuit and springs. There is reciprocal nature of correspondence between mechanical compliance and electrical capacitance.
Simple Harmonic Motion (SHM):
If one displaces the system from the position of stable equilibrium, system will move back and forth, i.e., it will oscillate about equilibrium position. Maximum displacement is known as amplitude, A. Time, T, to go through one complete cycle is known as period of oscillation and its inverse is known as the frequency, f.
f = 1/t
For several systems, if amplitude is small enough, restoring force F is directly proportional to displacement from equilibrium x and satisfies Hook's law, provided by:
F = - kx
Where k is the positive constant called as force constant and has units of N/m (or kg/s2). Motion of such system is known as simple harmonic motion (SHM). We can calculate motion using Newton's second law ( F = ma) to have
-kx = md2x/dt2 = mx
The solution of this equation provides displacement, x as the function of time, t. General solution is of form
x = Acos(wt + Φ) Where Φ is known as phase and it states initial displacement x = Acos Φ.
Equation of the simple harmonic motion is therefore x + kx/m = 0 or x + ω2x = 0
By definition, after a period T later the motion repeats itself, therefore:
x = A cos ωt = A cos(ωt + ωT) = A cos ωt cos ωt - A sin ωt sin ωt
This equation can be solved if we set
ωT = 2π = ω = 2π/T = 2πf = Angular frequency
Simple harmonic motion along the straight line can be represented by projection of uniform circular motion along the diameter of reference circle.
Projection of motion of the particle along y-axis signifies that particle also shows simple harmonic motion. Hence, uniform circular motion can be considered as the combination of two simple harmonic motions, one along x-axis and other along y-axis, with two differing in phase 90o.
Conservation of Energy in SHM:
The number of energy conserving physical systems which show simple harmonic oscillation about a stable equilibrium state exist. One of the major features of such oscillation is that, once excited, it never dies away. However, the majority of the oscillatory systems that we usually encounter in everyday life suffer some kind of irreversible energy loss due; for example, to frictional or viscous heat generation whilst they are oscillating.
Energy of Simple Harmonic Motion:
Energies in the simple harmonic motion are:
Kinetic energy, K.E = 1/2 mv2 = 1/2mx2 = 1/2mω2A2sin2(ωt + Φ) and
Potential energy, P.E = 1/2kx2 = 1/2kx2 = 1/2kA2cos2(ωt + Φ)
So total energy is
E = K.E + P.E = 1/2 mv2 + kx2 = 1/2mω2A2sin2(ωt + Φ) + 1/2 kA2cos2(ωt + Φ)
= 1/2kA2 (For a spring mass: mω2 = k)
Time average kinetic energy and time average potential energy are stated as:
K.E = 1/T∫t t+T 1/2mx2dt and
P.E = 1/T∫t t+T 1/2kx2dt
Where T is the period of oscillation.
Damped Oscillatory Motion:
Many simple harmonic oscillators in real world are damped - mechanical oscillators, electrical oscillators, etc. Suppose that damping force linear in velocity is applied to harmonic oscillator. For the mechanical oscillator, this could be the frictional force. For the electrical oscillator, this could be a resistive element. We investigate effect of damping on SHO. In real systems, dissipative forces (non- conservative forces) retard th oscillatory motion by causing amplitude to decrease. As a result, mechanical energy of system diminishes with time. Therefore, oscillatory motion of system is damped. Non - conservative force ( known as damping force) is about equal
-rv = -rx
r is the constant giving damping strength and v is velocity. Equation of the damped harmonic oscillatory motion is
md2x/dt2 + rdx/dt + kx = 0
Solution of differential equation is of the form
x(t) = Ae-t/τ cos(ωt + Φ)
For simplicity, let us take x = A at t = 0, then Φ = 0
If we plug solution
x(t) = Ae-t/τ cos(ωt) in Newton's second law, damping time, τ will be:
τ = 2m/r
And angular frequency ω as ω = ω0√(1-1/(ωoτ)2)
Where ω0 = √k/m is the un-damped angular frequency
Underdamping, system oscillates with steady decreasing amplitude when it is displaced and released. In critical damping, system no longer oscillates but returns to the equilibrium position without oscillation when it is displaced and released. For overdamping, there is no oscillation but system returns to equilibrium more slowly than with critical damping.
Forced Oscillations and Resonance:
When the oscillatory system is applied on by the external force we say that system is driven (or forced).To compensate energy loss in system in the damped system because of retarding forces, the external force is applied. This force acts in direction of motion of oscillator and does the positive work on system. Consequently of which, amplitude of motion remains constant when energy input per cycle exactly equals energy lost because of damping. System that oscillates in this way goes through forced oscillation.
Let the external oscillatory force F = Fo cos(ωdt)
Where F is varying force with time t, F0 driving force and ωd driving angular frequency. Newtos's second law for the system becomes
m dx2/dt+ r dx/dt + kx = Fo cosωdt
Again, if we try a solution of the form
x(t) = Acos(ωdt)
And plug in Newton's second law, we get amplitude which has the resonance form
A(ωd) = Fo/(√(ωd - ωo)2 + r2ωd2/m2)
When system executing forced oscillation behaves in such a way that natural frequency becomes equal to frequency of oscillation, system is said to be in resonance. At resonance, applied force remains in phase with the velocity so that power transferred to system is of maximum value.
Normal Frequencies and Normal Mode of Vibration: Two Body Oscillations
A vibration involving only one independent variable, say x (or y), is called a normal mode of vibration and has its own normal frequency. Consider the coupled oscillations, when two masses are connected with each other by strings and oscillating together,
The masses are placed on a frictionless track and joined up by ideal strings. There are two kinds of motion which distinguish themselves by being very simple and the two differential equations of motion are:
k(x2 - 2x1)i = md2x1/dx1i
k(x1 - 2x2)i = md2x2/dx2i
If we solve two equations we will get two frequencies each corresponding to type of motion.
In one type of motion x1 and x2 remain equal and whole system oscillates back and forth without stretching of the middle string. In this type of motion the frequency of oscillation is provided by,
ωo = √k/m
In the other type of motion, x1 and x2 remain exactly opposite and motion is like "in and out" kind. In this situation, net restoring force on body is three times as compared to previous case and therefore frequency of oscillation of system is provided by, ωo = √3k/m = √3ωo
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