ExpPhys I Lect22

8/13/2019 ExpPhys I Lect22

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Dr. M arei ke Zi nk /

P ro f . Dr. Jo sef A. Ks

Experimental Physics IWinter 2013/14


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Resonance disaster

in mechanics and construction a resonance disaster

describes the destruction of a building or a technicalmechanism by induced vibrations at a system's resonancefrequency, which causes it to oscillate

periodic excitation optimally transfers to the system the

energy of the vibration and stores it there

because of this repeated storage and additional energyinput the system swings ever more strongly, until its load

limit is exceeded

http://www.youtube.com/watch?feature=player_detailpage&v=moUfbNwHDTs


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The 1940 Tacoma Narrows Bridgewas the first TacomaNarrows Bridge, a suspension bridge in the U.S. state of

Washington that spanned the Tacoma Narrows strait of PugetSound between Tacoma and the Kitsap Peninsula.


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http://upload.wikimedia.org/wikipedia/commons/transcoded/1/19/Tacoma_N

arrows_Bridge_destruction.ogg/Tacoma_Narrows_Bridge_destruction.ogg.480p.webm

Broughton Suspension Bridge

On 12 April 1831, the bridge collapsed, reportedly due to mechanical resonanceinduced by troops marching in step and as a result of the incident, the British

Army issued an order that troops should "break step" when crossing a bridge.

Millennium Bridge, London


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The bridge's movements were caused by a 'positive feedback'

phenomenon, known as synchronous lateral excitation. Thenatural sway motion of people walking caused small sidewaysoscillations in the bridge, which in turn caused people on the

bridge to sway in step, increasing the amplitude of the bridge

oscillations and continually reinforcing the effect.Resonant vibrational modes due to vertical loads (such aspedestrians) are well understood in bridge design. In the caseof the Millennium Bridge, because the lateral motion causedthe pedestrians loading the bridge to directly participate withthe bridge, the vibrational modes had not been anticipated bythe designers. The crucial point is that when the bridge

lurches to one side, the pedestrians must adjust to keep fromfalling over, and they all do this at exactly the sametime. Hence the situation is similar to soldiers marching inlockstep, but horizontal instead of vertical.


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The lateral vibration problems of the Millennium Bridge are

very unusual, but not entirely unique. Any bridge with lateralfrequency modes of less than 1.3 Hz, and sufficiently lowmass, could witness the same phenomenon with sufficientpedestrian loading.


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Coupled oscillators


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Normal modes:

The harmonic oscillator and the systems it models have asingle degree of freedom. More complicated systems have

more degrees of freedom


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Normal modes in a crystal


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N linear coupled oscillators


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Any other possible oscillation can be expressed as a linearcombination of these normal modes.


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Superposition of oscillations

All shapes can be decomposed in a linear decomposition of

harmonic waves.

One dimensional superposition:


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Beat (acoustics)

interference between two sounds of slightly different

frequencies, perceived as periodic variations in volumewhose rate is the difference between the two frequencies

when the two tones are close in pitch but not identical, the

difference in frequency generates the beating

the volume varies like in a tremolo as the sounds alternatelyinterfere constructively and destructively


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Fourier series

A Fourier series decomposes

periodic functions or periodicsignals into the sum of a (possiblyinfinite) set of sines and cosines (orcomplex exponentials)


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s(x) denotes a function of the real variable x, and s isintegrable on an interval [x

0, x

0+ P], for real numbers x

0andP


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Fourier coefficients


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frequency spectrum

,


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Fourier transform

mathematical transformation employed to transform signalsbetween time (or spatial) domain and frequency domain

for defining the Fourier transform of an integrable functionf(x):

inverse transform:

for any real number .

,

for any real numberx.


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Fourier analysisis the study of the way general functionsmay be represented or approximated by sums of simplertrigonometric functions.


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Lissajous curve

the graph of a system of parametric equations:


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The appearance of the figure is highly sensitive to the ratioa/b. For a ratio of 1, the figure is an ellipse, with special cases

including circles (A= B, = /2 radians) and lines (= 0).Another simple Lissajous figure is the parabola (a/b= 2,=/2). Other ratios produce more complicated curves, which

are closed only if a/b is rational. The visual form of thesecurves is often suggestive of a three-dimensional knot, andindeed many kinds of knots, including those known asLissajous knots, project to the plane as Lissajous figures.


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The animation below shows the curve adaptation with

continuously increasing fraction from 0 to 1 in steps of 0.01.(=0)


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Below are examples of Lissajous figures with =/2, an oddnatural numbera, an even natural numberb, and |a b| = 1.

a= 2, b= 1 (2:1) a= 3, b= 2 (3:2) a= 5, b= 4 (5:4)


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Noise (signal processing)

noise is a general term for unwanted (and, in general,unknown) modifications that a signal may suffer duringcapture, storage, transmission, processing, or conversion


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Signal processing noise can be classified by its statistical properties (sometimescalled the "color" of the noise) and by how it modifies the intended signal:

Additive noise, gets added to the intended signal White noise

Additive white Gaussian noise Pink noise Black noise

Gaussian noise Flicker noise, with 1/fpower spectrum Brown noise or Brownian noise, with 1/f2 power spectrum Contaminated Gaussian noise, whose PDF is a linear mixture of Gaussian

PDFs

Power-law noise Cauchy noise

Multiplicative noise, multiplies or modulates the intended signal Quantization error, due to conversion from continuous to discrete values Poisson noise, typical of signals that are rates of discrete events Shot noise, e.g. caused by static electricity discharge Transient noise, a short pulse followed by decaying oscillations Burst noise, powerful but only during short intervals Phase noise, random time shifts in a signal


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White noise

a random signal with a flat (constant) power spectral density a signal that contains equal power within any frequency band

with a fixed width

under most types of discrete Fourier transform the transformwill be a Gaussian white noise, too; that is, its n Fouriercoefficients will be independent Gaussian variables with zeromean and the same variance

ExpPhys I Lect22

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