11/30/2015damped oscillations1. 11/30/2015damped oscillations2 let us now find out the solution the...
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04/18/23 Damped Oscillations 2
Let us now find Let us now find out the solutionout the solution
The equation The equation of motion is of motion is
(Free) Damped (Free) Damped OscillationsOscillations
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Try a Try a solutionsolution
In the In the equationequation
SubstitutiSubstitution yieldson yields
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Situation-Situation-
11::UnderdampedUnderdampedoror
then the then the roots areroots are
let us let us callcall
then the then the general general solutionsolution
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Case-1Case-1.Released .Released from extremityfrom extremity
0:0At xa, xt
Different Initial Different Initial ConditionsConditions
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What is the rate of amplitude dying ? Logarithmic decrement What is the time taken by amplitude to decay to 1/e (=0.368) times of its original value ? Relaxation timeWhat is the rate of energy decaying to 1/e (=0.368) times of its original value ? Quality Factor
The time for a natural decay process to reach zero is theoretically infinite. Measurement in terms of the fraction e-1 of the original value is a very common procedure in Physics.
How to describe the damping of an Oscillator
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Logarithmic Decrement (δ)Amplitude of nth Oscillation: An = A0e-βnT
This measures the rate at which the oscillation dies away
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Relaxation time (τ)Amplitude : A = A0e-βt ; at t=0, A=A0
(1/e)A0 = A0e-βτ
Quality factor (Q)Energy : ½k(Amplitude)2 ; E=E0e-2βt
(1/e)E0 = E0e-2β(Δt) ; Δt = 1/2β Q = ω´Δt = ω´/2β = π/δ
Quality factor is defined as the angle in radians through which the damped system oscillates as its energy decays to e-1 of its original energy.
Show that Q = 2π (Energy stored in system/Energy lost per cycle)
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Example:
Mass
Resistance
Conductor
Square coil Side = a
Uniform magnetic field B
Torsion constant
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General solution: General solution: UnderdampedUnderdamped
Case-2.Case-2. Impulsed Impulsed at equilibriumat equilibrium
0 speed 0At vx
Different Initial Different Initial ConditionsConditions
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General solution: General solution: OverdampedOverdamped
Case-1.Case-1. Released Released from extremityfrom extremity 2
02
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General solution: General solution: OverdampedOverdamped
Case-2.Case-2. Impulsed Impulsed at equilibriumat equilibrium
20
2
0 speed 0At vx
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General solution: General solution: OverdampedOverdamped
Case-3.Case-3. position position xxo o
: velocity : velocity vvoo
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Situation-3Situation-3: : CCritically ritically damped damped
General General solutionsolution
21Identical Identical roots -roots -
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General solution: General solution: Critically dampedCritically damped
Case-1.Case-1. Released Released from extremityfrom extremity 0:0At xa, x t
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General solution: General solution: Critically dampedCritically damped
Case-2.Case-2. Impulsed Impulsed at equilibriumat equilibrium 0 speed 0At vx