1

Forced oscillator with damping
resistance force is proportional to speed :
c dx dt
Equation of motion
m
2
d x dt
2 
c dx dt

kx
 2

For convenience:
c

m
 Also,
k=m

0 2
In ‘complex’ method 2
x
ˆ   
x
ˆ   0 2 
F
ˆ
m e
3

Define Solution of damped harmonic motion
x
ˆ 
m
  0 2
F
ˆ   2 
i
 
R

m
  0 2 1   2 
i
  4

x
ˆ   ˆ
R

i
  
F e
0
i
 
F
 Re of
F e
0
i
(  )
x
 
F
0 cos(    ) 5

x

m
(  0 2
F
0   2 2 )  2 6

Maximum amplitude at    0 2   2 2 7

or 1
R
 1 
e i
     
i
 
m
  0 2   2 
i
  1  cos  tan    
i
 sin     0 2    2 
m
  0 2   2  
im


  2  2
f

F
0
m
 Somnath Bharadwaj and Pratik Khastgir, Department of Physics and Meteorology, IIT Kharagpur, 721 302 India http://www.cts.iitkgp.ernet.in/~phy1/ SB 8

Average energy of the oscillator
E

k
2 * 9

 2 
m
2   0 2   2  1
i
 0 2   2 
i
  
m
2     2 1   0 2  2  2  10

Plot of  2 vs   near resonance   2   0  11

Work done by the force per second  Power
P

F dx dt

m
  
dx dt
  2
d x dt
2     0 2
x dx dt
    
m dx dt
2 
d dt
   1 2
m dx dt
2  1 2
m
 0 2
x
2    
m dx dt
2 12

P

d dt
 k.e of motion + p.e of motion   
m dx dt
2 Stored Energy 13

P
 
m
  
dx dt
   2  1 2 2
x
0 2 14

E
 1 2
m dx dt
2  1 2
m
 0 2
x
2  1 2
m
  2   0 2  1 2
x
0 2 15

Q
 2  Stored energy in oscillator Work done by force per cycle  2  1 2
m
  2 2   0 2 
x
2
x
2 2      2   0 2 16

Another measure of response Q factor
Q
   0

Forced and Damped Oscillation 17

1. Stimulation motor 2. Rotating pendulum 3. Bearing block 4. Electromagnetic break 5. Spiral spring
18

Eddy current brake
If a large conductive metal plate is moved through a magnetic field which intersects perpendicularly to the sheet, the magnetic field will induce small "rings" of current which will actually create internal magnetic fields opposing the change. This is why a large sheet of metal swung through a strong magnetic field will stop as it starts to move through the field. 19

Eddy current 20

Pohl’s Pendulum: Compare model SHO and Experiment
1. Find frequency of oscillations with different damping current Check: the frequency of oscillation should decrease with increase in damping current. 2. In a semi log plot Check whether amplitude decreases logarithmically with time
3. Check whether the maximum of resonance curve shifts
towards lower frequency with increasing in damping.
4. Check whether the width of the resonance curve is more with
higher damping current
5. Check the phase difference between the exciter and pendulum
--- much above resonance and much below the resonance.
21

Damped forced oscillation
x
ˆ 
m
  0 2
F
ˆ   2 
i
  • Effect of damping is negligible at low and high frequency regime • Effect of damping is large near resonance Increase in width and shift in resonance frequency with an increase in damping

FEYNMAN LECTURES ON PHYSICS VOL I Author : RICHARD P FEYNMAN, IIT KGP Central Library : Class no. 530.4