Example: Prove that the angular frequency of a vertical spring
with a spring constant k and a hanging mass m is still given by
k
m
ω=
Physics 106 Lecture 12
Oscillations – II
SJ 7th Ed.: Chap 15.4, Read only 15.6 & 15.7
•
•
•
•
•
Recap: SHM using phasors (uniform circular motion)
Ph i l pendulum
Physical
d l
example
l
Damped harmonic oscillations
Forced oscillations and resonance.
Resonance examples and discussion
– music
– structural and mechanical engineering
– waves
• Sample problems
• Oscillations summary chart
1
Goals
Oscillations in the presence of damping
Oscillations in the presence of oscillating external force
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
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
2
Add Damping: Emech not constant, oscillations not simple
• Spring oscillator as before, but with dissipative force
Fdamp
neglect
gravity
such as the system in the figure, with vane moving in fluid.
Fdamp viscous drag force, proportional to velocity
Fdamp = −bv
• Previous force equation gets a new, damping force term
Fnet = m
d2x(t)
dt
2
= − kx(t)
()− b
dx(t)
dt
new term
2
d x(t)
+
dt2
b dx(t)
k
= − x(t)
m dt
m
Solution for Damped oscillator equation
new term
2
d x(t)
2
dt
Solution:
modified
oscillations
ω0 =
+
b dx(t)
k
= − x(t)
m dt
m
x(t ) = x me
exponentially
decaying envelope
k
m
−
bt
2m
cos(ω' t + φ)
altered
frequency
ω’ can be real
or imaginary
ω' ≡
k b2
−
m 4m2
: natural frequency
ω ' ≡ ω02 − (b / 2m) 2
• Recover undamped solution for b Æ 0
3
Damped physical systems can be of three types
x( t ) = x me
Solution:
damped
oscillations
U d d
Underdamped:
d small
ll
−
bt
2m
cos(ω' t + φ)
ω' ≡
k b2
−
m 4m2
b < 2 km
k
2
b
k
<
, for which ω is positive.
4m 2 m
Critically damped:
b2
4m
Overdamped:
2
≈
b = 2 km
k
≡ ω02
m
b2
4m2
>
for which
k
≡ ω02
m
ω' ≈ 0
for which
ω'
is imaginary
Math Review: cos(ix) = cosh( x) = (e x + e − x ) / 2
sin(ix) = sinh( x) = (e x − e − x ) / 2
cos(ix + y ) = cos(ix) cos( y ) − sin(ix) sin( y )
Types of Damping, cont.
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
4
b2
k
<<
≡ ω20
m
4m2
Weakly damped oscillator :
ω'≡
k b2
−
m 4 m2
≈ ω0
x(t ) = xm e
x m (t)
( ) ≈ x me
Xm =
-
−
bt
2m
cos(ω0t + ϕ )
bt
2m
2
slow decay
of amplitude envelope
≈ cos(ω0 t + φ)
b2
k
<<
≡ ω20
2
m
4m
Weakly damped oscillator :
ω'≡
k b2
−
m 4 m2
≈ ω0
x m (t) ≈ x m
small fractional
change in
amplitude
during one
complete cycle
x(t ) = xm e
−
bt
2m
cos(ω0t + ϕ )
bt
2m
e
slow decay
of amplitude envelope
Amplitude : X m = A
small fractional
change in amplitude
during one complete
cycle
≈ cos(ω0 t + φ)
Velocity with weak damping: find derivative
bt
v( t ) =
−
d
x(t ) ≈ vme 2m sin(ω' t + φ)
dt
exponentially
decaying envelope
maximum velocity
v m = − ω0 x m
altered
frequency ~ ω0
5
Mechanical energy decays exponentially in an
“weakly damped” oscillator (small b)
Emech = K(t) + U(t) =
x(t ) = xm e
−
bt
2m
1
2
mv 2 (t) +
1
kx 2 (t)
2
cos(ω0t + ϕ )
Velocity with weak damping: find derivative
bt
v( t ) =
−
d
x(t ) ≈ vme 2m sin(ω' t + φ)
dt
exponentially
decaying envelope
maximum velocity
v m = − ω0 x m
altered
frequency ~ ω0
bt
⎛ b ⎞ − 2m
xm ⎜ −
e
cos(ω0t + ϕ ) term is negligible, because b is small..
⎟
⎝ 2m ⎠
Mechanical energy decays exponentially in an
“weakly damped” oscillator (small b)
Emech = K(t) + U(t) =
1
2
mv 2 (t) +
1
kx 2 (t)
2
Substitute previous solutions:
x(t ) = x me
−
bt
2m
Emech =
cos(ω' t + φ)
1
2
v( t )
≈ − ω0 x me
−
bt
2m
2 −bt / m
mω02 x m
e
sin2 (ω' t + φ)
1
2 − bt / m
kx m
e
cos2 (ω' t
2
As always: cos2(x) + sin2(x) = 1
Also:
ω02 ≡
sin(ω' t + φ)
+
+ φ)
k
m
∴ Emech (t ) =
1
2
kx m
2
Initial mechanical energy
e −bt / m
exponential decay at twice
the rate of amplitude decay
6
Damped physical systems can be of three types
Solution:
damped
oscillations
x( t ) = x me
exponentially
decaying envelope
Underdamped:
b2
4m
2
<<
−
bt
2m
cos(ω' t + φ)
altered
frequency
k
≡ ω02
m
ω' ≡
ω’ can be real
or imaginary
for which
k b2
−
m 4m2
ω' ≈ ω0
ƒ The restoring force is large compared to the damping force.
ƒ The system oscillates with decaying amplitude
Critically damped:
b2
4m
2
≈
k
≡ ω02
m
for which
ω' ≈ 0
ƒ The restoring force and damping force are comparable in effect.
ƒ The system can not oscillate; the amplitude dies away exponentially
Overdamped:
b2
4m
2
>
k
≡ ω02
m
for which
ω'
is imaginary
ƒ The damping force is much stronger than the restoring force.
ƒ The amplitude dies away as a modified exponential
ƒ Note: Cos( ix ) = Cosh( x )
Forced (Driven) Oscillations and Resonance
ƒ An external driving force starts oscillations in a stationary system
ƒ The amplitude remains constant (or grows) if the energy input per cycle
exactly equals (or exceeds) the energy loss from damping
ƒ Eventually, Edriving = Elost and a steady-state condition is reached
ƒ Oscillations then continue with constant amplitude
ƒ Oscillations are at the driving frequency ωD
FD (t ) = F0 cos(ωD t + φ' )
Oscillating driving force applied to
a damped
d
d oscillator
ill t
FD(t)
7
Equation for Forced (Driven) Oscillations
ω0 = natural frequency
k
m
ω0 =
ωD = driving frequency of external force
External driving force function:
FD (t ) = F0 cos(ωD t + φ' )
Fnet = FD (t ) -b
dx(t )
d 2 x(t )
- k x(t) = m
dt
dt 2
FD(t)
Solution for Forced (Driven) Oscillations
Fnet = FD (t ) -b
dx(t )
d 2 x(t )
- k x(t) = m
dt
dt 2
FD (t ) = F0 cos(ωD t + φ' )
Solution (steady state solution):
x(t ) = A cos(ωD t + φ)
where
A=
F0 / m
(ωD2 − ω02 ) 2 + (
bωD 2
)
m
FD(t)
The system
Th
t
always
l
oscillates
ill t att the
th
driving frequency ωD in steady-state
The amplitude A depends on how
close ωD is to natural frequency ω0
“resonance”
ω0 =
k
m
8
Amplitude of the driven oscillations:
A=
F0 / m
(ωD2 − ω02 ) 2 + (
ƒ The largest amplitude
oscillations occur at or
near RESONANCE (ωD ~
ω0)
bωD 2
)
m
resonance
As damping becomes
weaker
Æ
resonance sharpens
&
amplitude at
resonance increases.
Resonance
ƒ At resonance, the applied force is in
phase with the velocity and the power
Fov transferred to the oscillator is a
maximum.
ƒ The
Th amplitude
lit d off resonantt
oscillations can become enormous
when the damping is weak, storing
enormous amounts of energy
Applications:
• buildings driven by earthquakes
• bridges under wind load
• all kinds of radio devices, microwave
• other numerous applications
9
Forced resonant torsional oscillations due to
wind - Tacoma Narrows Bridge
Roadway collapse - Tacoma Narrows Bridge
10
Twisting bridge at resonance frequency
Breaking glass with voice
Another Breaking glass with voice
Yet another breaking glass with voice
11