The document discusses damped harmonic oscillators. It defines damping as the mechanism that results in dissipation of an oscillator's energy, and gives equations to model damped simple harmonic motion. There are three cases of damping: overdamped where motion is non-oscillatory, critically damped where displacement decays exponentially in minimum time, and underdamped where displacement decays exponentially within the time of critical damping. Damping can be measured via relaxation time, the time for amplitude to decay to 1/e of its initial value, or logarithmic decrement, the ratio of amplitudes between oscillations.

1. Simple Harmonic Motion (SHM)

2. Presentation Outline
 Damped Harmonic Oscillator
 Forced Harmonic Oscillator
5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences
2

3. 5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences
Damped vs Undamped Oscillations

4. Damping
The mechanism that results in dissipation of the
energy of an oscillator is called damping.
In mechanical oscillator energy damping may due to
viscous drag, friction and structure.
In case of electric oscillator energy may dissipated
due to the presence of resistance in the circuit or
due to emission of electromagnetic waves.
The general equation of damping is given as
Fd = r(dx/dt)
Where r is called damping constant.
5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences

5. For mechanical oscillator
Fd represents resistive force and dx/dt is velocity.
and r is defined as resistive force per unit velocity of the oscillating
part of the system.
For electrical oscillator
Fd represents voltage and dx/dt = dq/dt is the current and thus, r is
defined as voltage per unit current. It is equivalent to resistance.
Damping factor depends upon the dx/dt which for the simple
harmonic oscillations varies sinusoidally, therefore the damping factor
also varies sinusoidally, that is it varies periodically with time.
5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences
Damping

6. Equations of damped SHM
and its solutions
5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences
02 2
2
2
 x
dt
dx
b
dt
xd
o
Mechanical Oscillator
Where, 2b= r/m and ω2
o = S/m
Where, r= damping constant S is the spring constant and
m is mass
E1 E2O

7. 5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences
Equations of damped SHM
and its solutions











 



 
 tbtb
bt oo
eAeAex
2222
21

Solutions
A1 and A2 are constants whose values depends
upon the initial conditions of the oscillator

8. The behaviour of the oscillator is mainly determined by the term
depending upon the relative value of b and ωo
three cases are arise as discussed below
(i) b2 > ω2o called heavily/over damped oscillations. In this case
motion is non oscillatory and system takes longer time to return
to the equilibrium state.
(ii) b2 = ω2o called critical damped oscillations. In this case
displacement decays to zero exponentially and the system
returns to the initial state in the minimum possible time.
(iii) b2 < ω2o called light damped oscillations. In this case
displacement decays to zero exponentially almost during same
time in which the critically damped oscillation returns to the
intial state.
5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences
22
ob 

9. 5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences

10. Damping can measured in term of relaxation time,
logarithmic decrement
Relaxation time(τ)
It is the time for the `amplitude' to decay to 1/e times of
its initial value.
For Mechanical oscillator
τ = 1/b=2m/r,
For electrical oscillator
τ = 1/b=2L/R, L is inductance and R is resistance
5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences

11. It is ratio of the amplitude of oscillation at any instant and
that one
time period after it.
Let A be the amplitude at time t and AT is amplitude at
time t+T
δ= Loge (A/AT )
For Mechanical oscillator
δ= rT/2m
For electrical oscillator
δ= RT/2L
5/9/2020
Dr Arun Upmanyu, Associate Professor
Department of Applied Sciences
Logarithmic Decrement(δ)