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