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**Damped Systems**

- Damping is an effect which causes a Reduction in the amplitude of an oscillation as a Result of energy being drained from the system to overcame frictional/other Resistive forces.
- These are the systems in which kinetic friction is zero. Technical Name of Kinetic friction in any Vibrations system is known as, Damping.

**Damping in Any System**

Damping force ox = (x = velocity)

= c·x

Where, c = coefficient of damping

The equilibrium equation of a damped spring-mass system is given as

It is a 2^{nd} order differential equation.

**Types of Damping Systems**

**Overdamped System(ζ>1)**

No vibrations will be present in an over-damped system.

Critically Damped Systems(ζ = 1):

∴ No Vibrations**Note:**Critical damping response is much faster than overdamping response i.e. in critically-damped systems, the body once displaced comes into its Equilibrium Position much faster than that in overdamped systems.**Under Damped Systems (ζ < 1)**

∴ Damped frequency, ω_{d}= ω_{n}(1 - ξ^{2})^{1/2}

Time period T_{d}= 2π/w_{d}(Second)

Linear frequency, f_{n}= 1/T_{d}(Hz)

Hence, Amplitude of under-Damped Vibrations is not constant and it decreases exponentially w.r.t. time.**Logarithmic Decrement (δ)**

δ = In (Decrement ratio)

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