Free, Forced, and Damped Oscillations: Calculation & Examples

An oscillation is simply the periodic back-and-forth motion between two positions or states. We have seen many real-life scenarios of such motion in daily life, such as the side-to-side swing of a pendulum or the up-and-down motion of spring with a weight show oscillation.

Due to the absence of 'eternal motion' in physical experiments, we encounter various types of oscillations, including free, forced, and damped oscillations. The derived mathematical expressions of such motion become very useful in tailoring the efficiency of a mechanical system.

• Periodic motion repeats itself in a regular cycle, like a sine wave—a wave with eternal motion.
• Oscillation depends upon the period, frequency and amplitude of oscillation.
• An oscillating movement occurs around an equilibrium point. 

Here, we will learn more about free, forced, and damped oscillations and solve some related questions.

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What is Oscillation?

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Oscillation is the periodic fluctuation of an object about its mean value or between two fixed states. The point at which the body starts moving is called the mean or equilibrium position.

• It is a time-dependent quantity that is calculated in terms of a condition of equilibrium.
• The motion produces a continuous, repeated, alternating waveform without any input.
• The unit of oscillation is Hertz.
• We can also induce electrical oscillations in a circuit through a device called an oscillator.
• The oscillograph and oscilloscope are two more instruments based on oscillation principles.
• Oscillators might be mechanical or electrical, but they all function in the same way.
• It occurs in everything from tides and the pendulum of a clock to our decision-making process.

Damped, free and forced oscillation depends upon three quantities, which are as follows:

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Damped, Free and Forced Oscillation

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Oscillation can be classified into three categories: Damped, free and forced oscillation.

Damped Oscillation

Damped oscillation is a type of oscillation in which the amplitude decreases with respect to time. The reduction in amplitude is due to external variables like friction or air resistance, which cause damping, resulting in energy loss from the system. 

An object is called damped when there is a difference between the applied restoring force and the restraining force acting on the object. The fading oscillations of a pendulum are an example. Underdamped and overdamped oscilllation are two different types of damped oscillation. 

• The situation when the amplitude of oscillation never becomes zero even after significant deceleration is called underdamped oscillation.
• The situation when the amplitude of oscillation reaches zero due to a significant reduction in amplitude is called overdamped oscillation.

Free Oscillation

A free oscillation occurs when a body vibrates at its own frequency. Without any external force to set the oscillation, this oscillation has a constant amplitude and time. 

• Free oscillation does not experience any form of damping.
• It is a type of oscillation that exhibits natural frequency, which corresponds to the constant amplitude, energy, and time period.
• The vibrations in a tuning fork are an example.

Forced Oscillation

A forced oscillation occurs when any object oscillates as a result of an external periodic force. Since other external energy is supplied to it, the amplitude of the oscillation is damped but remains constant. It is also known as a driven or forced harmonic oscillator.

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Calculation of Damped, Free and Forced Oscillation

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When a body is in the oscillatory motion then three terms are used in the calculation namely time period, frequency and amplitude. Each succeeding string vibration takes the same amount of time as the preceding one.

• So, we get ‘period’ (T), the amount of time it takes to complete one oscillation. 
• Consider l l be the length of the string, then its time period is given by

T=2π√l/g

Where,
T : time period of oscillation
L : length of string 
g : acceleration due to gravity

• The relation between time period and frequency is given by

  frequency (η) = 1/ time period (T)

η = 1/ T

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Examples of Oscillations

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Oscillatory motion is prominent in-

• Pendulum Clock.
• Tuning Fork.
• Swing.
• Flapping of Wings.
• A freely hanging Bob.
• String Musical Instruments.
• Spring Toy.
• Alternating Current.

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Simple Harmonic Motion

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Simple harmonic motion is defined as a repeated back-and-forth movement about a central position with the maximum displacement on one side equal to the maximum displacement on the other. 

• The time interval between each full vibration is the same. 
• It is characterized by a variable acceleration that is proportionate to the displacement from the equilibrium point.
• It is a special case of oscillatory motion.
• Linear Simple Harmonic Motion and Angular Simple Harmonic Motion are two types of Simple Harmonic Motion.
• Furthermore, the time period between complete vibrations is constant and independent of the greatest displacement size. 
• Mathematically, it can be represented as:

F = -kx

• where F stands for force
• x stands for displacement
• k stands for a constant or restoring force constant.

An example of Simple Harmonic Oscillator 

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Things to Remember

• Oscillation refers to the periodic back-and-forth movement of something between two positions or states.
• Resonance is the phenomenon of driving a system with a frequency equal to its natural frequency.
• The forced oscillation occurs when a body oscillates as a result of an external periodic force.
• A free oscillation occurs when a body vibrates at its frequency.
• Damped oscillation is a type of oscillation that reduces with time.