Spring 2016
OSCILLATION & WAVES
Free
Forced
OSCILLATION
OSCILLATION
Damped / Un-damped
Coupled
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OSCILLATORS
x(t )  bx(t )   x(t )  0
2
0
 i) Underdamped
Small friction
s
2

0
m 4m 2
x(t )  A cos( ' t   ) e
 t
2m
s
b2
' 

m 4m 2
 ii) critically damped
s
2

0
m 4m 2
e


2m
t
{C1  C2t}
 iii) overdamped
s
2

0
2
m 4m
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e


2m
t
2
{C1e
k

t
4 m2 m
 C2 e

2
4m

2
k
t
m
}
Underdamped SHOs
x(t )  A cos( ' t   ) e
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t
2m
s
b2
' 

m 4m 2
Critical Damping
s
2

0
2
m 4m
e


2m
t
{C1  C2t}
x ( 0)  0
x (0)  V
x  Vte
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

2m
t
Under vs. Critical Damping
s
2
' 

0
m 4m 2
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Heavy vs. Critical Damping
e


2m
t
2
{C1e
k

t
2
4m m
 C2 e

2
4m

2
k
t
m
}
x(0)  0
x (0)  2
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Heavy vs. Critical Damping
x(0)  0
x (0)  2
s
2



0
m 4m 2
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Minimum Relaxation Time from Perturbation
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Linear 2nd Order Differential Eqn
d 2 x(t )
dx(t )
a2
 a1
 a0 x(t )  f (t )
dt
dt
For complementary solution:
Take trial solution : x=emt, m is constant
m_1, m_2,……….will be the roots.
If all roots are real and distinct, then solution x=c1em_1*t+c2em_2*t+……
If some roots are complex, if a+ib then a-ib will be root and
solution will be eat(c1 cos(bt) +c2 sin(bx)) +……
5. If some roots are repeated, say m_1 repeated k times, then
solution will be (c1 + c2t+ …..cktk-1)e m_1* t
1.
2.
3.
4.
For Particular solution:
Trial solution to be assumed depending on the form of f(t)
1/22/2016solution = Complimentary + Particular Solution
General
Underdamped SHOs
x(t )  A cos( ' t   ) e
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t
2m
s
b2
' 

m 4m 2
How to Describe the Damping
 What is the rate of amplitude decaying?
Logarithmic decrement
What is the time taken by amplitude to decay
to 1/e (=0.368) times of its original value ?
Relaxation time
How many cycles does the rate of energy decay
to 1/e (=0.368) of its original value ?
Quality Factor
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Logarithmic Decrement
x(t )  A cos( ' t   ) e
 t
2m
s
b2
' 

m 4m 2
Measurement of t‘ and logarithmic decrement  gives a
very convenient method of determining the damping.
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Relaxation time t
Modulus of Decay
A  A0  e
t
2m
Amplitude at t=0, A=A0
 t
1
A0  A0  e 2 m
e
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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The Meaning of Quality Factor
Q-value (Q)
 '2
 '  0
E ( N )  E0  e
2 N
Q
The number of cycles (or complete oscillations)
through which the system moves in decaying to 1/e
energy
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The Quality Factor of a Damped SHO
Q-value (Q)
 t
1 2
E  sA  E0  e m
2
(1/e)E0 = E0e-r/m(Δt) ; Δt = m/r
Q = ω´Δt = ω´m/r = 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.
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The Meaning of Quality Factor
Q-value (Q)
2π (Energy stored in system/Energy lost per cycle)
e.g., An electron in an atom has a Q of 5*107
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Quiz-Jan-21, your name
Three traces below show the temporal response of
three SHOs that occur after a external perturbation.
Which one undergo under-damping process?
A. Blue
B. Red
C. Black
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Energy Dissipation
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Electrical LC Oscillator with Loss
Find charge on the
capacitor at time t
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Faraday Law
The interplay between B and E
You may know steady current generates magnetic field (or flux B)
B
I
But also: increasing B generates counter-current because of a ‘rotating’ E field
Or in formula form :
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    
  E  dl   B  dS
t S
C
Electrical LC Oscillator with Loss
Find charge on the
capacitor at time t
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Electrical LC Oscillator with Loss
Find charge on the
capacitor at time t
V0
 qt 
V0
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qt   V0C