11/30/2015Damped Oscillations1

11/30/2015Damped Oscillations2 Let us now find out the solution The equation of motion is (Free) Damped Oscillations

11/30/2015Damped Oscillations3 Try a solution In the equation Substitution yields

11/30/2015Damped Oscillations4 The equation has the roots and

11/30/2015Damped Oscillations5 Situation-1 : Underdamped or then the roots are let us call then the general solution

11/30/2015Damped Oscillations6 General solution: Underdamped

11/30/2015Damped Oscillations7 Case-1.Released from extremity Different Initial Conditions

11/30/2015Damped Oscillations8 Underdamped Oscillations

11/30/2015Damped Oscillations9 an example :

11/30/2015Damped Oscillations10 Phase Comparison

11/30/2015Damped Oscillations11 Logarithmic Decrement

11/30/2015Damped Vibration  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 time  What 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

11/30/2015Damped Vibration Logarithmic Decrement (δ) Amplitude of n th Oscillation: A n = A 0 e -βnT This measures the rate at which the oscillation dies away

11/30/2015Damped Vibration Relaxation time (τ) Amplitude : A = A 0 e -βt ; at t=0, A=A 0 (1/e)A 0 = A 0 e -βτ Quality factor (Q) Energy : ½k(Amplitude) 2 ; E=E 0 e -2βt (1/e)E 0 = E 0 e -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)

11/30/2015Damped Vibration Example: LCR in series Find charge on the capacitor at time t.

11/30/2015Damped Vibration Example: LCR in series Find charge on the capacitor at time t.

11/30/2015Damped Vibration Example: Mass Resistance Conductor Square coil Side = a Uniform magnetic field B Torsion constant

11/30/2015Damped Vibration Flux change: E.M.F.

11/30/2015Damped Vibration Current: Force: Torque:

11/30/2015Damped Vibration

11/30/2015Damped Vibration Relaxation time: Moment of inertia:

11/30/2015Damped Oscillations22 a problem

11/30/2015Damped Oscillations23 General solution: Underdamped Case-2. Impulsed at equilibrium Different Initial Conditions

11/30/2015Damped Oscillations24 Situation-2 : Overdamped

11/30/2015Damped Oscillations25 General solution: Overdamped Case-1. Released from extremity

11/30/2015Damped Oscillations26 General solution: Overdamped Case-2. Impulsed at equilibrium

11/30/2015Damped Oscillations27 General solution: Overdamped Case-3. position x o : velocity v o

11/30/2015Damped Oscillations28 High damping

11/30/2015Damped Oscillations29 High damping

11/30/2015Damped Oscillations30 Situation-3 : Critically damped General solution Identical roots -

11/30/2015Damped Oscillations31 General solution: Critically damped Case-1. Released from extremity

11/30/2015Damped Oscillations32 General solution: Critically damped Case-2. Impulsed at equilibrium

11/30/2015Damped Oscillations33 Critically damped

11/30/2015Damped Oscillations34 Comparison

11/30/2015Damped Oscillations35 Comparison

11/30/2015Damped Oscillations36 Comparison