short version : 13. oscillatory motion wilberforce pendulum
TRANSCRIPT
Short Version : 13. Oscillatory Motion
Wilberforce Pendulum
Disturbing a system from equilibrium results in oscillatory motion.
Absent friction, oscillation continues forever.
Oscillation
13.1. Describing Oscillatory Motion
Characteristics of oscillatory motion:
• Amplitude A = max displacement from
equilibrium.
• Period T = time for the motion to repeat itself.
• Frequency f = # of oscillations per unit time.
1fT
[ f ] = hertz (Hz) = 1 cycle / ssame period T same amplitude A
A, T, f do not specify an oscillation completely.
Oscillation
13.2. Simple Harmonic Motion
Simple Harmonic Motion (SHM): F k x
2
2
d xm k xd t
cos sinx t A t B t Ansatz:
sin cosd x
A t B td t
22 2
2cos sin
d xA t B t
d t 2 x
k
m
angular frequency
2T x t T x t 2
mT
k
1
2fT
2nd order diff. eq 2 integration const.
cos sinx t A t B t
sin cosd x
v t A t B td t
A, B determined by initial conditions
0 1
0 0
x
v
1A
0B cosx t t
( t ) 2
x 2A
Amplitude & Phase
cos sinx t A t B t cosC t
cos cos sin sinC t t cos
sin
A C
B C
C = amplitude
= phase
Note: is independent of amplitude only for SHM.
Curve moves to the right for < 0.
2 2C A B
1tanB
A
Oscillation
Velocity & Acceleration in SHM
cosx t A t
sind x
v t A tdt
2
22
cosd x
a t A tdt
2x t
|x| = max at v = 0
|v| = max at a = 0
cos2
A t
2 cosA t
Application: Swaying skyscraper
Tuned mass damper :
Damper highly damped.
Overall oscillation overdamped.
Taipei 101 TMD:
41 steel plates,
730 ton, d = 550 cm,
87th-92nd floor.
Also used in:
• Tall smokestacks
• Airport control towers.
• Power-plant cooling towers.
• Bridges.
• Ski lifts.
Movie
Tuned Mass Damper
Example 13.2. Tuned Mass Damper
The tuned mass damper in NY’s Citicorp Tower consists of a 373-Mg (vs 101’s 3500
Mg) concrete block that completes one cycle of oscillation in 6.80 s.
The oscillation amplitude in a high wind is 110 cm.
Determine the spring constant & the maximum speed & acceleration of the block.
2
3 2 3.1416373 10
6.80kg
s
2
T
53.18 10 /N m
2 3.1416
6.80 s
10.924 s
2
2k m
T
2m
Tk
maxv A 10.924 1.10s m 1.02 /m s
2maxa A 210.924 1.10s m 20.939 /m s
The Vertical Mass-Spring System
k
m
Spring stretched by x1 when loaded.
mass m oscillates about the new equil. pos. with freq
The Torsional Oscillator
= torsional constant
I
I
2
2
dIdt
Used in timepieces
The Pendulum
sinm g L g
2
2
dIdt
Small angles oscillation: sin
2
2
dI m g Ldt
m g L
I
Simple pendulum (point mass m):
2I m Lg
L
LT
g
Tτ 0
sin
Conceptual Example 13.1. Nonlinear Pendulum
A pendulum becomes nonlinear if its amplitude becomes too large.
(a)As the amplitude increases, how will its period changes?
(b)If you start the pendulum by striking it when it’s hanging vertically,will it undergo oscillatory motion no matter how hard it’s hit?
(a) sin increases slower than smaller longer period
(b) If it’s hit hard enough, motion becomes rotational.
The Physical Pendulum
Physical Pendulum = any object that’s free to swing
Small angular displacement SHM
m g L
I
13.4. Circular & Harmonic Motion
Circular motion: cosx t r t
siny t r t2 SHO with same A &
but = 90
x = Rx = Rx = 0
Lissajous Curves
GOT IT? 13.3.
The figure shows paths traced out by two pendulums swinging with
different frequencies in the x- & y- directions.
What are the ratios x : y ?
1 : 2 3: 2
Lissajous Curves
13.5. Energy in Simple Harmonic Motion
cosx t A tSHM: sinv t A t
21
2K m v
21
2U k x 2 21
cos2k A t
2 2 21sin
2m A t 2 21
sin2k A t
21
2E K U k A
= constant
Energy in SHM
Potential Energy Curves & SHM
F k xLinear force:
U F d x
parabolic potential energy:
21
2k x
Taylor expansion near local minimum:
min
22
min min2
1
2x x
d UU x U x x x
d x
2min
1
2const k x x
min
0x x
dU
d x
Small disturbances near equilibrium points SHM
13.6. Damped Harmonic Motion
Damping (frictional) force:
dF b vd x
bd t
Damped mass-spring:
2
2
d x d xm k x bd t d t
Ansatz:
i tx t A e
i tv t i A e
2 i ta t A e
2m k i b
2
2 2
b k bim m m
where
2
b
m 2 2
0 0
k
m
sinusoidal oscillation
Amplitude exponential decay
set
i
costx t A e t
Real part 實數部份 :
costx t A e t 2
b
m 2 2
0
At t = 2m / b, amplitude drops to 1/e of max value.
(a) For 0 is real, motion is oscillatory ( underdamped )
(c) For is imaginary, motion is exponential ( overdamped )
(b) For 0 = 0, motion is exponential ( critically damped )
0
k
m
0
i i
i
i
i tx t A e
Damped & Driven Harmonic Motion
13.7. Driven Oscillations & Resonance
External force Driven oscillator
0 cosext dF F tLet d = driving frequency
2
02cos d
d x d xm k x b F td t d t
Prob 75: cos dx A t
0
222 2
0d
d
FA
bm
m
0
k
m = natural frequency
Resonance: 0d
( long time )
Damped & Driven Harmonic Motion
Buildings, bridges, etc have natural freq.
If Earth quake, wind, etc sets up resonance, disasters result.
Resonance in microscopic system:
• electrons in magnetron microwave oven
• Tokamak (toroidal magnetic field) fusion
• CO2 vibration: resonance at IR freq Green house effect
• Nuclear magnetic resonance (NMR) NMI for medical use.
Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation.
Tacoma Bridge