Damped Oscillations 11/29/2018 Damped Oscillations
(Free) Damped Oscillations The equation of motion is Let us now find out the solution 11/29/2018 Damped Oscillations
Try a solution In the equation Substitution yields 11/29/2018 Damped Oscillations
The equation has the roots and 11/29/2018 Damped Oscillations
then the general solution Situation-1:Underdamped or let us call then the roots are then the general solution 11/29/2018 Damped Oscillations
General solution: Underdamped 11/29/2018 Damped Oscillations
Different Initial Conditions Case-1.Released from extremity 11/29/2018 Damped Oscillations
Underdamped Oscillations 11/29/2018 Damped Oscillations
an example : 11/29/2018 Damped Oscillations
Phase Comparison 11/29/2018 Damped Oscillations
Logarithmic Decrement 11/29/2018 Damped Oscillations
What is the rate of amplitude dying ? How to describe the damping of an Oscillator 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. 11/29/2018 Damped Vibration
Amplitude of nth Oscillation: An = A0e-βnT Logarithmic Decrement (δ) Amplitude of nth Oscillation: An = A0e-βnT This measures the rate at which the oscillation dies away 11/29/2018 Damped Vibration
(1/e)E0 = E0e-2β(Δt) ; Δt = 1/2β 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) 11/29/2018 Damped Vibration
Example: LCR in series Find charge on the capacitor at time t. 11/29/2018 Damped Vibration
Example: LCR in series Find charge on the capacitor at time t. 11/29/2018 Damped Vibration
Example: Conductor Torsion constant Uniform magnetic field B Mass Square coil Side = a Resistance 11/29/2018 Damped Vibration
E.M.F. Flux change: 11/29/2018 Damped Vibration
Current: Force: Torque: 11/29/2018 Damped Vibration
11/29/2018 Damped Vibration
Relaxation time: Moment of inertia: 11/29/2018 Damped Vibration
a problem 11/29/2018 Damped Oscillations
Different Initial Conditions Case-2. Impulsed at equilibrium General solution: Underdamped 11/29/2018 Damped Oscillations
Situation-2: Overdamped 11/29/2018 Damped Oscillations
General solution: Overdamped Case-1. Released from extremity 11/29/2018 Damped Oscillations
General solution: Overdamped Case-2. Impulsed at equilibrium 11/29/2018 Damped Oscillations
General solution: Overdamped Case-3. position xo : velocity vo 11/29/2018 Damped Oscillations
High damping 11/29/2018 Damped Oscillations
High damping 11/29/2018 Damped Oscillations
Situation-3: Critically damped Identical roots - General solution 11/29/2018 Damped Oscillations
General solution: Critically damped Case-1. Released from extremity 11/29/2018 Damped Oscillations
General solution: Critically damped Case-2. Impulsed at equilibrium 11/29/2018 Damped Oscillations
Critically damped 11/29/2018 Damped Oscillations
Comparison 11/29/2018 Damped Oscillations
Comparison 11/29/2018 Damped Oscillations
Comparison 11/29/2018 Damped Oscillations
Summary 11/29/2018 Damped Oscillations