Transcript Damped and Forces Oscillations

-Damped and Forces Oscillations
-Resonance
AP Physics C
Mrs. Coyle
Objectives
• Identify and analyze forced and damped
oscillations.
• Develop a qualitative understanding of
resonance
• Identify situations in which a system will
resonate in response to a sinusoidal external
force.
Damped Oscillations
• Non conservative forces are
present (ex:friction, resistive
forces, damping forces by a
“dashpot” device).
• The amplitude and thus the
mechanical energy is
reduced over time.
Damped Oscillation- Ex 1
Fdamping  bv
SFx = -k x – bv = ma
dx
 kx  b
 ma
dt
Equation of Motion:
d2x
dx
m 2  b  kx  0
dt
dt
2
k
b
 

m 4m 2
Solution:
x  Ae

b
t
2m
cos  t   
Damped Oscillation
x  Ae

b
t
2m
cos  t   
k
b2
 

m 4m 2
• The amplitude is reduced
over time and eventually
the oscillation stops.
b

2m
Damped Oscillation- Ex 2
• Resistive (retarding force), R
R=-bv
where b is a constant
called the damping coefficient
viscous
liquid
• SFx = -k x – bvx = max
• Equation of Motion:
d2x
dx
m 2  b  kx  0
dt
dt
• Solution b
x  Ae

2m
t
cos(t   )

k  b 


m  2m 
2
Types of Damping
A. Underdamped
• If Rmax = bvmax < kA
k
b2
 

m 4m 2
B. Critically damped
• When b reaches a critical value bc such that
bc / 2 m = k/m=02 , the system will not oscillate (quick
return to equilibrium).
•   k is called the natural frequency
0
m
C. Overdamped
• If Rmax = bvmax > kA and b/2m > 0 (return to equilibrium
without oscillation).
Video of dampers
• https://www.youtube.com/watch?v=xp2pGxF
zrzI
Forced vibrations: when an external force
causes a system to oscillate
External Force
Fext  F0 cos t
dx
SFx = kx  b  F0 cos t  ma
dt
2
dx
dx
m 2  b  kx  F0 cos t
dt
dt
x  A0 sin  t  0 
Fext  F0 cos t
• Amplitude of a forced (driven) oscillation:
F0
A

2

m

2 2
0
 b 


m


2
– 0 is the natural frequency of the undamped
oscillator
How can a damped system have an undamped motion
(no decrease in amplitude)?
• To compensate for the loss of mechanical
energy due to the resistive force, apply a forced
vibration of equal energy.
Resonance: increase in
amplitude due to addition of an
external force.
• When the frequency of the
driving force is near the
natural frequency ( 0)
an increase in amplitude
F0
occurs
m
A

2


2 2
0
 b 


m


2
• The natural frequency 0
is also called the
resonance frequency of
the system
• At resonance the applied force and v are both
proportional to sin (t + ) , so force and velocity are
in phase.
• The power transferred to the oscillator (P=F . v) is a
maximum at resonance.
Takoma Narrows Bridge: collapses in November, 1940, under
42mph winds (opened in July 1940)
• https://www.youtube.com/watch?v=xox9BVS
u7Ok