Examples of oscillations & waves : Earthquake – Tsunami Electric guitar – Sound wave Watch – quartz crystal Radar speed-trap Radio telescope. Part Two: Oscillations, Waves, & Fluids. Examples of fluid mechanics : Flow speed vs river width Plane flight.
Describing Oscillatory Motion
Simple Harmonic Motion
Applications of Simple Harmonic Motion
Circular & Harmonic Motion
Energy in Simple Harmonic Motion
Damped Harmonic Motion
Driven Oscillations & Resonance
Dancers from the Bandaloop Project perform on vertical surfaces, executing graceful slow-motion jumps.
What determines the duration of these jumps?
pendulum motion: rope length & g
Absent friction, oscillation continues forever.
Examples of oscillatory motion:
Microwave oven: Heats food by oscillating H2O molecules in it.
CO2 molecules in atmosphere absorb heat by vibrating global warming.
Watch keeps time thru oscillation ( pendulum, spring-wheel, quartz crystal, …)
Earth quake induces vibrations collapse of buildings & bridges .
Characteristics of oscillatory motion:
same period T
same amplitude A
[ f ] = hertz (Hz) = 1 cycle / s
A, T, f do not specify an oscillation completely.
An oscillating ruler completes 28 cycles in 10 s & moves a total distance of 8.0 cm.
What are the amplitude, period, & frequency of this oscillatory motion?
Amplitude = 8.0 cm / 2 = 4.0 cm.
Simple Harmonic Motion (SHM):
2nd order diff. eq 2 integration const.
( t ) 2
C = amplitude
Note: is independent of amplitude only for SHM.
Curve moves to the right for < 0.
|x| = max at v = 0
|v| = max at a = 0
Two identical mass-springs are displaced different amounts from equilibrium & then released at different times.
Of the amplitudes, frequencies, periods, & phases of the subsequent motions, which are the same for both systems & which are different?
Same: frequencies, periods
amplitudes ( different displacement )
phases ( different release time )
Tuned mass damper :
Damper highly damped ,
Overall oscillation overdamped.
Taipei 101 TMD:
41 steel plates,
730 ton, d = 550 cm,
Also used in:
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.
Spring stretched by x1 when loaded.
mass m oscillates about the new equil. pos.
= torsional constant
Used in timepieces
Small angles oscillation:
Simple pendulum (point mass m):
Tarzan stands on a branch as a leopard threatens.
Jane is on a nearby branch of the same height, holding a 25-m-long vine attached to a point midway between her & Tarzan.
She grasps the vine & steps off with negligible velocity.
How soon can she reach Tarzan?
(a) sin increases slower than
Physical Pendulum = any object that’s free to swing
Small angular displacement SHM
When walking, the leg not in contact of the ground swings forward,
acting like a physical pendulum.
Approximating the leg as a uniform rod, find the period for a leg 90 cm long.
Forward stride = T/2 = 0.8 s
2 SHO with same A & but = 90
x = R
x = R
x = 0
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
Energy in SHM
parabolic potential energy:
Taylor expansion near local minimum:
Small disturbances near equilibrium points SHM
Damping (frictional) force:
Amplitude exponential decay
is real, motion is oscillatory ( underdamped )
is imaginary, motion is exponential ( overdamped )
= 0, motion is exponential ( critically damped )
Damped & Driven Harmonic Motion
A car’s suspension has m = 1200 kg & k = 58 kN / m.
Its worn-out shock absorbers provide a damping constant b = 230 kg / s.
After the car hit a pothole, how many oscillations will it make before the amplitude drops to half its initial value?
Time required is
# of oscillations:
bad shock !
External force Driven oscillator
d= driving frequency
( long time )
= natural frequency
Damped & Driven Harmonic Motion
If Earth quake, wind, etc sets up resonance, disasters result.
Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation.
Resonance in microscopic system: