Types of Oscillators: Simple Harmonic, Damped and Forced Oscillators
Simple harmonic oscillators exhibit periodic motion with constant amplitude, while damped and forced types introduce energy loss or external driving forces.
Simple Harmonic Motion
Simple harmonic motion (SHM) is a type of oscillating motion. A pendulum or an object hanging from a spring will oscillate harmonically when there is a small deviation from its equilibrium or rest state. A restoring force proportionate to x is created when slight displacement x occurs from the equilibrium position. This force always acts in the opposite direction of the displacement, towards the equilibrium position.
where k is the spring constant (force constant).
This motion is used to model many physical systems in the real world where a mass oscillates about an equilibrium position.
Simple Pendulum
A simple pendulum consists of a point mass (bob) suspended from a fixed support by a light, inextensible string of length L, free to oscillate in a vertical plane.
Mass Attached to a Spring
A mass attached to a spring fixed to a rigid wall executes SHM when displaced from equilibrium.
Oscillations of a Liquid Column in a U-Tube
The motion of an oscillating liquid column in a U-tube is SHM with time period:
where h is the height of the liquid column in one arm at the equilibrium position of the liquid and g is the acceleration due to gravity.
Therefore, the time period T is independent of the density of the liquid.
LC Oscillator
Another example is that of an LC electrical circuit — an inductance L connected across a capacitance C carrying charge q performs electrical oscillations analogous to SHM.
The general governing equation for simple harmonic motion is:
where:
- m : Mass of the ball (kg)
- F₀ : Applied force (N)
- b : Damping factor (Ns/m)
- k : Spring constant (N/m)
- ω : Angular frequency (rad/s)
- t : Time (s)
- ϕ : Phase angle
Free Oscillators
Free oscillations, such as the swinging motion of a pendulum or the vibration of an elastic rod, occur when the motion is sustained by gravitational or elastic restoring forces. Undamped free vibration is the simplest form of vibrating motion.
Example: Simple pendulum oscillations, short-range tuning fork sound, sounds of musical instruments (e.g., organ pipes), etc. are a few instances of free oscillations.
Nature of Free Oscillators
In free oscillations, the amplitude and frequency of a freely vibrating body remain constant. During this motion, the system neither gains nor loses energy.
In this type of motion, both the damping factor (b) and the forcing frequency (F₀) are equal to zero.
Thus, equation (3) becomes:
where the solution of the equation is:
which represents simple harmonic motion. Here ω0 is the natural angular frequency of the system:
Damped Oscillators
When a moving particle interacts with resistive forces like air or friction, it gradually loses kinetic energy in damped oscillations. A particle's displacement gradually decreases over time and eventually reaches its state of rest as a result of the resistance provided by external forces.
Example: Shock Absorbers
If the damping force is proportional to the velocity v with a damping constant b, then:
The equation of motion (3) becomes:
The solution of equation (7) is:
where γ = b / (2m), and:
- b : Damping factor (Ns/m)
- m : Mass of the ball (kg)
The amplitude is given by:
ω0 is the natural angular frequency of the system and ω is the angular frequency of the damped oscillator:
Damping is of three types:
1. Under Damping
When damping is small, the system vibrates at first approximately as if there were no damping, but the amplitude of the oscillations decreases exponentially.
Example: A swinging pendulum decreasing in amplitude until it comes to a stop.
Under damped condition: b < 2√2
where γ = b / (2m), b = Damping factor, m = Mass of the ball (kg).
The amplitude is given by:
2. Critical Damping
When a critically damped oscillator is displaced from the equilibrium, it returns to its equilibrium position in the shortest possible time without oscillating. Critical damping occurs when the position of the mass asymptotically approaches zero. Depending on the initial velocity, the system either moves monotonically towards zero or reaches a maximum displacement before reversing direction and returning to equilibrium.
Example: Car suspension systems prevent the car from oscillating after travelling over a bump in the road.
Critically damped condition: b = 2√2 (or ≈ 2.8)
3. Over Damping
When a heavily damped oscillator is displaced from the equilibrium, it takes a long time to return to its equilibrium position without oscillating. When damping is large, the system does not oscillate at all.
Example: Car suspension systems prevent the car from oscillating after travelling over a bump in the road.
Over damped condition: b > 2.8
where γ = b / (2m)
Forced Oscillators
Forced oscillations occur when an oscillating system is driven by a periodic external force.
Example: Applying a periodic force onto a swinging pendulum or turning off a radio.
The governing equation is:
where:
- ω : Angular frequency for forced oscillations
- F₀ : Applied force for the forced oscillator
Resonance
If the forcing frequency is close to the natural frequency of the system and the system is lightly damped, huge vibration amplitudes may occur. This phenomenon is known as resonance.
Some examples are:
- Musical instruments (acoustic resonance).
- Most clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal.
- Tidal resonance.
- Orbital resonance of some moons in the solar system.
- Resonance of the basilar membrane in the ear.
- Increasing the height of a child's swing by pushing it at each swing.
- A wineglass breaking when a loud sound is produced at its resonant frequency.
Additional Information
Some of the other types of oscillators are:
Coupled Oscillators
Coupled oscillations occur when two or more oscillating systems are connected in such a way that energy can be exchanged between them. Coupled oscillators occur in nature, for example, in the Earth–Moon system, and in biological systems such as natural pacemakers in the heart.