1. Damped Oscillations
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2. (Free) Damped Oscillations
The equation of motion is
Let us now find out the solution
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3. Try a solution In the equation Substitution yields 11/29/2018
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4. The equation
has the roots
and
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5. then the general solution
Situation-1:Underdamped or
let us call
then the roots are
then the general solution
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Damped Oscillations

6. General solution: Underdamped
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7. Different Initial Conditions
Case-1.Released from extremity
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8. Underdamped Oscillations
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9. an example :
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10. Phase Comparison
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11. Logarithmic Decrement
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12. 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.
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Damped Vibration

13. 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
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Damped Vibration

14. (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)
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15. Example: LCR in series Find charge on the capacitor at time t.
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16. Example: LCR in series Find charge on the capacitor at time t.
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17. Example: Conductor Torsion constant Uniform magnetic field B Mass
Square coil
Side = a
Resistance
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Damped Vibration

18. E.M.F.
Flux change:
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19. Current:
Force:
Torque:
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20. 11/29/2018
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21. Relaxation time:
Moment of inertia:
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22. a problem
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23. Different Initial Conditions
Case-2. Impulsed at equilibrium
General solution: Underdamped
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24. Situation-2: Overdamped
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25. General solution: Overdamped
Case-1. Released from extremity
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26. General solution: Overdamped
Case-2. Impulsed at equilibrium
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27. General solution: Overdamped
Case-3. position xo : velocity vo
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28. High damping
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29. High damping
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30. Situation-3: Critically damped
Identical roots -
General solution
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31. General solution: Critically damped
Case-1. Released from extremity
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32. General solution: Critically damped
Case-2. Impulsed at equilibrium
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33. Critically damped
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34. Comparison
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35. Comparison
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36. Comparison
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37. Summary
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