Transcript High Damping - s3.amazonaws.com

Transient
dx
resistance force is proportional to speed : c
dt
For convenience: c  m
Also,
k=m02
Equation of motion
2
d x
dx
2
m 2  m  m 0 x  0
dt
dt
Underdamped
  20

1    i  
2

2
0
2    i  
2
2
0

2
4

2
4



2

2
 i
 i
x1  Ae
 t 2 i t
x2  Ae
e
 t 2  i t
e
x  Cx1  Dx2  e
 t / 2
 Ee
i t
 Fe
 i t

For x to be a real solution of equation of motion F=E*
xe

t
2
 Ee
i  t
*  i t
E e

Damped cosine oscillation
x
-t/2
e
cos t
t
Logarithmic decrement
x
1.0
A1
A2
0.5
A3
A4
0.0
0
25
50
t
-0.5
-1.0
An
T
  ln

An 1
2
To find E and E* we choose the initial conditions
at t  0
x  x0 and
x0
v0  
x0
2
E  i
2
2
x0
v0  
x0
*
2
E  i
2
2
dx
 v0
dt
x0
v0  
 t / 2
2
xe
[ x0 cos  t 
sin  t ]

Phase difference between x and v is not simply p/2.
High Damping
  20


2
4


2
4
1   
2   
2
2

2
0

2
0
x1
x2

 Ae


 Ae

  2   2 / 4 02 t
  2   2 / 4 02 t
x  Cx1  Dx2  e
 t / 2
 Ee
 2 / 4 02 t
 Fe
  2 / 4 02 t

For very high damping
  20

2
4
 
2
0

2
1
4

2
0
2
2 
 1 

2
 

2
0
2
For very high damping
© SB
Two different time scale involved in damped SHM
T0
TD
1
0
1

Critical Damping
  20
General Solution
t
x( t )  e [ A  Bt ]
x
1.5
low damping
high damping
critical damping
1.0
0.5
0.0
0
-0.5
-1.0
10
20
30
40
50
60
t
Transient : Damped oscillation with no forcing
© Walter Fendt (2006)
http://www.walter-fendt.de/ph14e/springpendulum.htm
Pohl’s Pendulum
Forced and Damped Oscillation
1. Stimulation motor 2. Rotating pendulum 3. Bearing block
4. Electromagnetic break 5. Spiral spring
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.
Eddy current
Transients
•Equation of motion
d 2x
dx
2
m 2  m  m 0 x  0
dt
dt
•For low damping
xe

t
2
Ce
i  t
 De
 i t

  02 
2
4
•For high damping

x  e  t / 2  Ee


2
4
02
)t
 Fe
•For critical damping
t
x( t )  e [ A  Bt ]

2
4
02 ) t




Problem 11
A thin uniform disc of mass m and radius R suspended by an elastic thread
performs small torsional oscillations in a liquid. The moment of elastic forces
emerging in the thread is equal to f, where  is a constant and f is the
angle of rotation from the equilibrium position. The resistance force acting
on a unit area of the disc is equal to F1=hv, where v is the velocity of the
given element of the disc relative to the liquid. Find the frequency of small
oscillations.
Problem 8
A particle is displaced from the equilibrium position by a distance
l=1.0 cm and left alone. What is the distance that the particle covers
till its oscillations die down, if the logarithmic decrement is equal to
0.20.
Problem 23
A mass m=50 gm is suspended by a massless spring with a spring
constant 20 N/m. The mass performs steady state vertical oscillation
of amplitude 1.3 cm. Due to an external harmonic force of frequency
=25 sec-1. The displacement lags behind the force by an angle
f=3p/4. Find
(a) The quality factor of the given oscillator
(b) The work performed by the external force in one oscillation