An Introduction to Free Forced Damped Oscillations
In this section, we will define oscillating motion which comes under the type of periodic movements. You will also come across detailed explanations on concepts like how to find periods of oscillation, simple harmonic motion, and different types of oscillatory motions, which will help you gain a better understanding of the subject.
In our daily lives, we come across several such incidents that involve repetitive movements of objects. Be it the movement of your bicycle wheels, a football’s motion as you kick it, or strings of a playing guitar. If you observe closely, a lot of these movements take place in a rhythm while others do not. The former are grouped as periodic motions while the latter are categorized as non-periodic motions.
What is Oscillation?
Oscillation is a periodic variation in time of a matter about its mean value or between two fixed states.
One cycle of periodic to and fro motion of the body about its central position is one oscillation. Mechanical oscillations are called vibrations. A particle is vibrating means it oscillates between two points between its central point.
The height or the maximum distance that the oscillation takes place at is called the amplitude. It is measured in metres. While the time taken to complete one complete oscillation cycle is called its time period, the number of complete cycles that occur in one second is called the frequency. Frequency basically reciprocates the time period. The time period is calculated in seconds and the frequency is measured in hertz.
Properties of Oscillation
Here are the properties of oscillation:
The frequency is defined as the number of complete oscillations per unit time.
The amplitude is the maximum displacement of an oscillator from its equilibrium position.
The time period is the time taken for one complete oscillation, in seconds.
The relationship of frequency and period is f = 1/T.
What are Some Examples of Oscillation?
There are some of the observed motions that serve as examples of oscillation.
Here are some of them.
The motion of a pendulum bob in a clock. It is a repetitive motion with the bob rising to its right, coming back to its centre position, again rising to its left, and finally going back to its centre again. This total cycle constitutes 1 oscillation.
Movement of springs - the repetitive downward and upward motion.
Tides in the sea.
The vibration of strings in a guitar and other string instruments.
Elastic media.
A sine wave.
Calculation of Oscillation
Several measurements define oscillation. Here, we have explained each parameter with their respective unit of measure and the formulae to calculate their value and also understand how to define oscillating motion.
Period of Oscillation
The time taken by an oscillating body to complete one cycle of motion is termed as its oscillation period. It is generally measured in second and is denoted by T.
Here’s how to calculate period of oscillation.
\[T = 2 \pi (\sqrt{L}{g}) \]
Where L represents the length of a pendulum and g is the acceleration due to gravity.
Frequency of Oscillation
The number of oscillations a body can complete in one second is known as its frequency of oscillation. It is primarily expressed in the SI unit of Hertz and is denoted by the letter f.
You can calculate the value of f with its relation to period T of oscillation as follows:
\[f = \frac{1}{T}\]
Amplitude of Oscillation
The maximum amount of displacement of an oscillating body from its central position is known as its amplitude. Its value is measured in meter and it is denoted by A.
With a known value of A, displacement x is calculated as -
x = A cos 2πft
Where t denotes the time for which oscillation is taking place.
Simple Harmonic Motion
Simple harmonic motion (SMH) is defined as a periodic oscillation of an unaltering frequency and a constant amplitude where the magnitude of restoring force on the object is always acting towards its equilibrium position and is directly proportional to its displacement.
Simple harmonic motion is the simplest form of oscillatory motion meaning. It is an ideal condition where the maximum displacement on one side of the equilibrium position is exactly equal to that on the other side. This aspect is also required to define oscillating motion.
SHM is denoted by y, and its mathematical expression is -
y = A sin ωT or A cos ωT
Where A stands for amplitude, ω stands for angular frequency, and T stands for time.
A perfect example of the simple harmonic motion is again the motion of a simple pendulum because, on its displacement in one direction, a proportional restoring force acts on it in the opposite direction.
What are the Types of Oscillation?
Based on the external forces acting on them, simple harmonic motions are classified into three significant categories. Now we will define oscillating motion and its types with a brief description of each.
Free Oscillation
A free oscillation is an ideal condition where a particle’s motion is not under the influence of any external resistance. It is a motion with a natural frequency of the particle and constant amplitude, energy, and period.
It is an ideal condition because, in reality, every oscillating object undergoes some form of interaction with external conditions resulting in loss of energy.
An example of free oscillations is the motion of a simple pendulum in a vacuum.
Damped Oscillation
A damping oscillation is one in which the moving particle gradually loses its kinetic energy on interaction with resistive forces like air or friction. Due to this resistance offered by external forces, the displacement of a particle slowly reduces with time and ultimately reaches its state of rest.
Damped oscillations are classified based on the difference in energy between the applied restoring force and the restraining force acting on it. Basically, it is an oscillation that fades away with time, which means the oscillations will reduce in their magnitude with time.
The damping can be natural or forced.
The best example of damped oscillations is a simple pendulum oscillating under natural conditions. Shock absorbers in vehicles are an example of damping devices that reduce the vehicle’s vibrations.
The different types of damped oscillation are:
Under damped oscillations: Damping constant < 1
Critically damped oscillations: Damping constant = 1
Over damped oscillations: Damping constant > 1
What is Electromagnetic Damping?
Damping plays a crucial role in controlling the object’s motion. Electromagnetic damping uses electromagnetically induced current to control, regulate or slow down the motion of an object without the use of any physical contact. The damping force is basically electromagnetically applied.
The damping technique depends on Eddy current and Electromagnetic induction. A damping force is generated when the Eddy current and magnetic field interact with each other and create a resistive force. And this force opposes the conductor or object’s motion.
So, the electromagnetic damping force is directly proportional to the following three factors: the induced Eddy Current, the object’s speed and magnetic field’s strength. This means that the faster the object’s motion, the greater the damping. And the slower the object’s motion, the lower the damping, resulting in stopping the object smoothly and effectively.
The concept of Free Forced Damped Oscillations constitutes a significant portion of Class 11 Physics. It makes up almost 14% of the syllabus with some key concepts essential for competitive exams.
If you are considering the answer to define oscillating motion, you need a tutor or solutions for self-study. Download the Vedantu app for detailed information on oscillating motion definition.
FAQs on Free Forced Damped Oscillations
1. What are free, damped, and forced oscillations?
These are three types of oscillatory motion, distinguished by the forces acting on the system.
- Free Oscillation: An ideal oscillation where a body vibrates at its own natural frequency without any external resistive or driving forces. The amplitude remains constant over time.
- Damped Oscillation: A real-world oscillation where resistive forces, like air resistance or friction, cause the amplitude to decrease over time until the motion stops.
- Forced Oscillation: An oscillation where an external periodic force is continuously applied to the body, making it oscillate at the frequency of the external force, not its natural frequency.
2. Can you provide real-world examples for each type of oscillation: free, damped, and forced?
Yes, here are some common examples that illustrate the differences:
- Free Oscillation: A pendulum swinging in a perfect vacuum is a theoretical example. In reality, this is an idealisation as some damping is always present.
- Damped Oscillation: A child on a swing after being pushed once. The swing's height (amplitude) gradually decreases and it eventually stops due to air resistance and friction. The shock absorbers in a car also use damping to prevent excessive bouncing.
- Forced Oscillation: A person continuously pushing a child on a swing at regular intervals. The swing is forced to oscillate at the frequency of the pushes. Another example is the vibration of a washing machine drum during the spin cycle, driven by the motor.
3. What are the basic conditions for an object to be in Simple Harmonic Motion (SHM)?
Simple Harmonic Motion (SHM) is the simplest form of oscillation. For an object to perform SHM, it must satisfy two key conditions as per the CBSE/NCERT syllabus for 2025-26:
- There must be a restoring force that always acts to bring the object back to its equilibrium (mean) position.
- This restoring force must be directly proportional to the object's displacement from the equilibrium position and act in the opposite direction (F ∝ -x).
4. Why do most oscillations in the real world, like a plucked guitar string, eventually stop?
Most real-world oscillations stop because of damping. When a guitar string is plucked, it vibrates in the air and is connected to the guitar's body. Energy from the vibrating string is gradually lost to the surroundings in two main ways:
- Air Resistance: The string has to push air molecules out of the way, which dissipates energy.
- Internal Friction: Energy is also lost as heat within the material of the string and the guitar's structure.
5. What is resonance, and how is it related to forced oscillations?
Resonance is a specific phenomenon that occurs during forced oscillations. It happens when the frequency of the external driving force exactly matches the natural frequency of the oscillating system. When this alignment occurs, the system absorbs maximum energy from the driver, causing the amplitude of the oscillation to become very large. A famous example is a singer shattering a glass by singing at its precise natural resonant frequency.
6. How is a person pushing a swing a good example of both forced oscillation and resonance?
A person pushing a swing perfectly illustrates these concepts. The person provides the external periodic force, making it a forced oscillation. If the person gives small, random pushes, the swing moves erratically. However, to make the swing go high, the person must time their pushes to match the swing's natural back-and-forth rhythm. This act of pushing in sync with the swing's natural frequency is an example of achieving resonance. At resonance, each push adds maximum energy, causing the swing's amplitude (height) to increase dramatically.
7. What is the governing principle behind the equation for a damped oscillator?
The motion of a damped oscillator is governed by two opposing forces. The equation of motion is based on Newton's second law, considering these forces:
- Restoring Force (F = -kx): This force, as seen in SHM, always tries to pull the object back to its equilibrium position. It is proportional to the displacement 'x'.
- Damping Force (F = -bv): This is a resistive force, like friction or air drag, that opposes the motion. It is proportional to the velocity 'v' of the object.
8. In a forced oscillation, why does the object oscillate at the driver's frequency and not its own natural frequency?
This is a key aspect of forced oscillations. When the external driver starts, the object initially tries to oscillate at a combination of its own natural frequency and the driver's frequency. This initial phase is called the transient state. However, damping forces are always present in the system. These damping forces quickly remove the energy from the natural oscillation component. Eventually, the natural oscillation dies out, and the system is left with no choice but to follow the continuous energy input from the external driver. This is the steady state, where the system oscillates at the exact frequency of the driver.
9. What would happen if there were no damping in a system undergoing forced oscillation at its resonant frequency?
In a purely theoretical system with zero damping, forcing it to oscillate at its resonant frequency would lead to a phenomenon called resonance catastrophe. At resonance, the system absorbs energy from the driver most efficiently. Without any damping to dissipate this energy, the amplitude of the oscillation would increase with every cycle, theoretically growing infinitely large. In the real world, this would cause the system to break or fail, which is why damping is a critical safety feature in engineering structures like bridges and buildings to prevent resonance from external forces like wind or foot traffic.
