Free, Forced, Damped Oscillations and Resonance - Basic

definition

Damped vibrations

The periodic vibrations of a body of decreasing amplitude in presence of a resistive force are called damped vibrations. Some examples of damped vibrations are oscillations of branch of a tree, sound produced by tuning fork over longer distances, etc. In fact all vibrations on earth's surface in the absence of an external force are damped vibrations.

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Nature of damped vibrations

Usually in damped vibrations, frictional force is proportional to the velocity of the vibrating object and has tendency to resist the motion. The rate of energy lost to the medium depends on the nature of the medium.

definition

Forced vibrations

Vibrations of a body under the constant influence of an external periodic force acting on it are called the forced vibrations. The external applied force is called the driving force. Amplitude of body oscillating under forced oscillations can be decreasing, constant or increasing depending on various factors like difference in amount of driving force and resistive force, difference in frequency of driving force and actual vibrations and difference in phase of driving force and actual vibrations.

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Free vs forced vibrations

Free vibrations	Forced vibrations
Vibrations of body in absence of any external force.	Vibrations of body in presence of external driving force.
Frequency of vibrations depends on the source of vibrations.	Frequency of vibrations depends on the source of vibrations and the frequency of the driving force.
Frequency of vibrations remains constant.	Frequency of vibrations can be changed by changing frequency of driving force.
Amplitude of vibration is constant.	Amplitude of vibration can be decreasing, increasing or constant.

definition

Resonance

The resonance is a special case of forced vibrations. The frequency and phase of driving force is same as that of the free vibrations. Amplitude of the vibrations keep increasing in magnitude continuously.

definition

Damped Oscillations

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems.

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Demonstration of resonance with tuning forks

Resonance can be demonstrated by the help of two tuning forks of same frequency. Mount the two tuning forks upon two separate sound boxes such that their open ends face each other. Strike one of the tuning forks. It is observed that the other tuning fork starts vibrating and a loud sound is heard. This happens because the sound waves from first tuning fork forces a resonant vibrations in the second tuning fork.

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Write total force and write differential equation of motion for damped oscillations

Total force in damped oscillations is:

c \frac{d x}{d t} + k x

(Due to damper and spring.)

Final differential equation for the damper is:

m \frac{d ^{2} x}{d t ^{2}} + c \frac{d x}{d t} + k x = 0

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Displacement as a function of time in damped SHM

x (t) = A c o s (w^{'} t + ϕ) e^{\frac{- b t}{2 m}}

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Examples of resonance

Examples of resonance are sympathetic vibrations of pendulums, machine parts, instruments, air column and tuning fork, oscillations of bridge, etc.

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Resonance Definition

Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others. Frequencies at which the response amplitude is a relative maximum are known as the system's resonant frequencies, or resonance frequencies.

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Experimental demonstration of resonance in air column

The apparatus for the experiment consists of a long cylindrical plastic tube attached to a water reservoir. The length of the water column may be changed by raising or lowering the water level while the tuning fork is held over the open end of the tube. Resonance is indicated by the sudden increase in the intensity of the sound when the column is adjusted to the proper length. The resonance is a standing wave phenomenon in the air column and occurs when the column length is:

λ / 4, 3 λ / 4, 5 λ / 4