Physics 106 Lecture 12
Oscillations – II
SJ 7thEd.: Chap 15.4, Read only 15.6 & 15.7
• Recap: SHM using phasors (uniform circular motion)
Ph i l d l l
• Physical pendulum example • Damped harmonic oscillations • Forced oscillations and resonance. • Resonance examples and discussion
– music
– structural and mechanical engineering – waves
• Sample problems • Sample problems
• Oscillations summary chart
Damped Oscillations
• Non-conservative forces may be present – Friction is a common nonconservative force – No longer an ideal system (such as those dealt with
so far)
• The mechanical energy of the system diminishes in
neglect gravity
• The mechanical energy of the system diminishes in time, motion is said to be damped
• The motion of the system can be decaying oscillations if the damping is “weak”.
• If damping is “strong”, motion may die away without oscillating. • Still no driving force, once system has been started
Add Damping: E
mechnot constant, oscillations not simple
neglect gravity
bv
F
•Spring oscillator as before, but with dissipative force Fdamp
Fdampviscous drag force, proportional to velocity
such as the system in the figure, with vane moving in fluid.
bv
F
damp=
−
•
Previous force equation gets a new, damping force term
)
t
(
dx
( )
)
t
(
x
d
2dt
)
t
(
dx
b
x(t)
k
dt
)
t
(
x
d
m
F
net=
2=
−
−
x(t)
m
k
dt
)
t
(
dx
m
b
dt
)
t
(
x
d
−
=
+
2 2 new termSolution for Damped oscillator equation
x(t)
m
k
dt
)
t
(
dx
m
b
dt
)
t
(
x
d
−
=
+
2
2
new term Solution: modified oscillations 4 2 2 m b m k'
≡
−ω
)
t
'
cos(
e
x
)
t
(
x
m bt mω
+
φ
=
−2 exponentiallydecaying envelope frequencyaltered ωor imaginary’ can be real
0
k
ω
0=
: natural frequency
m
ω
: natural frequency
2 2 0'
( / 2 )
b
m
ω
≡
ω
−
Damped physical systems can be of three types
Solution: damped oscillations 2 4 2 m b m k'
≡
−ω
)
t
'
cos(
e
x
)
t
(
x
m bt mω
+
φ
=
−2U d
d
d
ll
b
2
k
Underdamped: small
22 , for which is positive.
4 b k m <m
ω
Critically damped:
0 4 2 0 2 2 ≈ ω ω ≡ ≈ ' m k m b which for2
b
<
km
2
b
=
km
4m mOverdamped:
for which ' isimaginarym k m b > ≡ω2 ω 0 2 2 4
cos( )
cosh( )
(
) / 2
sin( )
sinh( )
(
) / 2
x x x xix
x
e
e
ix
x
e
e
− −=
=
+
=
=
−
Math Review:
cos(
ix
+
y
)
=
cos( ) cos( ) sin( ) sin( )
ix
y
−
ix
y
Types of Damping, cont (
Link to Active Fig.)
a) an underdamped oscillator
b) a critically damped oscillator
c) an overdamped oscillator
For critically damped and overdamped oscillators there is no
periodic motion and the angular frequency
ω
has a different
meaning
Weakly damped oscillator :
m k m 4 b 2 0 2 2 ω ≡ <<( )
2 bt-X
2 0 2 4'
k b m mω
≡
−≈
ω
2 0( )
cos(
)
bt m mx t
=
x e
−ω
t
+
ϕ
slow decay
of amplitude envelope
e
x
(t)
x
2m m m≈
mX =
small fractional change in amplitude during one complete cycle ) t os( c ω0 +φ ≈Weakly damped oscillator :
small fractional m k m 4 b 2 0 2 2 ω ≡ <<
e
x
(t)
x
2m bt -m m≈
A X : Amplitude m = 2 0 2 4'
k b m mω
≡
−≈
ω
2 0( )
cos(
)
bt m mx t
=
x e
−ω
t
+
ϕ
small fractional change in amplitude during one completecycle slow decay of amplitude envelope ) t os( c ω0 +φ ≈
Velocity with weak damping: find derivative ) t ' sin( e v ) t ( x dt d ) t ( v m bt m ω +φ ≈ = −2
exponentially frequency ~ altered ω
m 0 m velocity maximum
x
v
=
−
ω
(t)
kx
(t)
mv
U(t)
K(t)
E
2 2 1 2 2 1 mech
=
+
=
+
Mechanical energy decays exponentially in an
“weakly damped” oscillator (small b)
2 0
( )
cos(
)
bt m mx t
=
x e
−ω
t
+
ϕ
Velocity with weak damping: find derivative ) t ' sin( e v ) t ( x dt d ) t ( v m bt m ω +φ ≈ = −2 altered m 0 m velocity maximum
x
v
=
−
ω
exponentially decaying envelope altered frequency ~ ω0 2 0cos(
)
2
bt m mb
x
e
t
m
ω
ϕ
−⎛
−
⎞
+
⎜
⎟
⎝
⎠
term is negligible, because
b
is small..
Substitute previous solutions:
(t)
kx
(t)
mv
U(t)
K(t)
E
2 2 1 2 2 1 mech
=
+
=
+
Mechanical energy decays exponentially in an
“weakly damped” oscillator (small b)
bt ) t ' sin( e x ) t ( v m bt m ω +φ ω − ≈ −2 0 ) t ' cos( e x ) t ( x m bt m ω +φ = −2
)
t
'
(
os
c
e
kx
)
t
'
(
sin
e
x
m
E
2 m / bt 2 m 2 1 2 m / bt 2 m 2 1 mech
φ
+
ω
+
φ
+
ω
ω
=
− − 2 0As always: cos2(x) + sin2(x) = 1
exponential decay at twice the rate of amplitude decay
m k ≡ ω2 0 Also:
e
kx
)
t
(
E
2m bt/m 2 1 mech
=
−∴
Damped physical systems can be of three types
Solution: damped oscillations 2 4 2 m b m k'
≡
−ω
)
t
'
cos(
e
x
)
t
(
x
m bt mω
+
φ
=
−2 exponentiallydecaying envelope frequencyaltered ωor imaginary’ can be real
Underdamped:
0 2 0 2 2 4 << m≡ω ω'≈ω k m b which forThe restoring force is large compared to the damping force. The system oscillates with decaying amplitude
Critically damped:
0 4 2 0 2 2 ≈ ω ω ≡ ≈ ' m k m b which forThe restoring force and damping force are comparable in effect.
The system can not oscillate; the amplitude dies away exponentially
Overdamped:
imaginary is which for ' m k m b > ≡ω2 ω 0 2 2 4The damping force is much stronger than the restoring force.
The amplitude dies away as a modified exponential
Note: Cos( ix ) = Cosh( x )
Forced (Driven) Oscillations and Resonance
An external driving force starts oscillations in a stationary system
The amplitude remains constant (or grows) if the energy input per cycle exactly equals (or exceeds) the energy loss from damping
Eventually, Edriving = Elostand a steady-state condition is reached
Oscillations then continue with constant amplitude
Oscillations are at the driving frequency ωD
F
D(t)
)
'
t
cos(
F
)
t
(
F
D
=
0
ω
D
+
φ
Oscillating driving force applied to
d
d
ill t
Equation for Forced (Driven) Oscillations
ω
0= natural frequency
ω
D= driving frequency of external force
External driving force function:
0
k
m
ω
=
F
D(t)
)
'
t
cos(
F
)
t
(
F
D=
0ω
D+
φ
2 2( )
( )
( ) -b
- x(t)
m
net D
dx t
d x t
F
F t
k
dt
dt
=
=
Solution for Forced (Driven) Oscillations
)
'
t
cos(
F
)
t
(
F
D=
0ω
D+
φ
2 2( )
( )
( ) -b
- x(t)
m
net D
dx t
d x t
F
F t
k
dt
dt
=
=
Solution (steady state solution):
F
D(t)
Th t
l
ill t
t
th
)
t
cos(
A
)
t
(
x
=
ω
D+
φ
Solution (steady state solution):
2 2 2 0 2 0 ) m b ( ) ( m / F A D D ω + ω − ω =
where
The system always oscillates at the
driving frequency ω
Din steady-state
The amplitude A depends on how
close ω
Dis to natural frequency ω
0 0k
m
ω
=
Amplitude of the driven oscillations:
resonance
The largest amplitude
oscillations occur at or
2 2 2 0 2 0 ) m b ( ) ( m / F A D D ω + ω − ω =near RESONANCE (
ω
D~
ω
0)
As damping becomes
weaker
Æ
resonance sharpens
resonance sharpens
&
amplitude at
resonance increases.
Resonance
At resonance, the applied force is in phase with the velocity and the power Fov transferred to the oscillator is a
maximum.
Th lit d f t
The amplitude of resonant oscillations can become enormous when the damping is weak, storing enormous amounts of energy
Applications:
• buildings driven by earthquakes
• buildings driven by earthquakes
• bridges under wind load
• all kinds of radio devices, microwave
