Appendix A Vibratory Motion - 國立中興大學web.nchu.edu.tw/pweb/users/feng/lesson/2291.pdf ·...

National CHung Hsing University, Department of Soil and Water Conservation1

Vibratory Motion

Prof. Zheng-yi FengNCHU SWC


Types of vibratory motion

•Periodic motion•Nonperiodic motion•See Fig. A1, p.528

–Harmonic motion–Periodic motion–Transient motion –impact–Transient motion –earthquake


A powerful tool for dynamicanalysis of linear systems

•Fig. A.1b can be expressed as the sum of aseries of simple harmonic motions.

•Fig. A.1c and A.1d can be represented asperiodic motions by assuming “quiet”zone.

•See Fig. A2, p.528•linear system; superposition•Response to transient loading can be

expressed as the sum of the responses to aseries of simple harmonic loads.


Simple harmonic motion, S.H.M.

•Can be defined by 3 quantities:–Amplitude–Frequency–phase

•Cab be described in:–Trigonometric notation–Complex notation


Trigonometric notation

•u(t) = A sin (t+)•A: amplitude•: circular frequency•: phase angle•The describes the amount of time by

which the peaks (and zeros) are shiftedfrom those of a pure sine function.

•See Fig. A.4 for positive (lead) andnegative (lag)


Rotating vector representation

•See Fig. A.5•Period of vibration, T : time for one cycle

of motion•T = 2/ • f = 1 / T = usually hertz (Hz), cycle

per second•= 2f


•S.H.M. can also be described as:•u(t) = a cos t + b sin t•See Fig. A.6

–Amplitude is not the simple sum …–Peaks do not occurs at the same time as

those of the sine or cosine functions.


•Cos = sin (+90) a 90deg ahead of b

•See Fig. A.7a•Length of resultant will be sqrt(a2+b2)

and it leads b by an angle = tan-1(a/b)

•u(t) = A sin (t+)


Complex notation

•Much simpler description•Euler’s law : ei= cos + i sin

•See Fig. A.8•eit is represented by a vector of unit length

rotating clockwise at an angular speed, .

titi e2

ibae

2iba

tu

)(

clockwise counterclockwise


Other measures of motion

•Displacement

•Velocity

•Acceleration

•Frequency, amplitudes of displacement,

velocity, and acceleration are related.

•Tripartite plot：a harmonic motion can be

described by a single point

)cos()( tAtu

utAtu 22 )sin()(

)sin()( tAtu

20070328


Tripartite plot

•See Fig. A.9•commonly used to described earthquake

motion•applied only to harmonic motion; for other

types of motion, must be obtained bydifferentiation and/or integration


Out of phase with each other …•between displacement, velocity, and

acceleration

•Velocity leads displacement 90 degrees.

•Acceleration leads velocity 90 degrees.

•Acceleration leads displacement 180degrees.

)/sin()( 2tAtu

)sin()( tAtu 2

)sin()( tAtu tiAeitu )(

ti2ti22 AeAeitu )(

tiAetu )(


Phase leads & amplitudes

•See Fig. A10•See Fig. A.11•Leads 90 degree•Leads 180 degree


Fourier Series

•The French mathematician J.B.J Fourier•A periodic function can be expressed as

the sum of a series of sinusoids ofdifferent amplitude, frequency, and phase.

•Fourier series: an extraordinarily usefultool


Process to produce total response

•See Figure A.12–Time history of loading–Sum of series of harmonic loads–Calculating responses of each harmonic load–Summing the responses


Trigonometric form

•See E.Q. A.11 for the generaltrigonometric form of the Fourier series fora function of period, Tf , and the Fouriercoefficients, a0, an, bn; n=2n / Tf.

•a0 is the average values of x(t) in t=0~Tf

•Usually a0 = 0 in many geotechnicalearthquake engineering applications

•n is not arbitrary; increment n=2/ Tf.


Fourier amplitude spectrum &Fourier phase spectrum

•From EQ A.5 and EQ A.11

•cn and n= the amplitude and phase of the nthharmonic.

•cn versusn: a Fourier amplitude spectrum;very useful to describes the frequency contentof an EQ

•n versusn: a Fourier phase spectrum

)/(tan,;

)sin()(

nn1

n2n

2nn00

1nnnn0

baandbacac

tcctx


example

•See Example A.1

•a0=0; since the average of x(t) is zero.

•even function; sine terms are zero; f(t)=f(-t)

•odd function; cos terms are zero; f(t)=-f(-t)

•See example A.2 for c0, cn and n–amplitude & phase spetra

•See Figure EA.2 for the plots of spectra


Exponential Form

•See Fig.EA.3 for one- and two-sidedFourier spectra

f

n

n

T

0

ti

fn

ti

nn

dtetxT1

c

ectx

)(

)(

*

*


Discrete Fourier Transform, DFT

•For finite number of data points•Fourier coefficients are obtained by

summation rather than integration•Fourier coefficients of DFT have units of

the original variable multiplied by time•The DFT can be inverted by using Inverse

DFT (IDFT)•The time required for computation of

DFT/IDFT is proportional to N2.


Fast Fourier Transform, FFT•Cooley and Tukey(1965) developed an

computational algorithm for the casewhere N is a power of 2 known as FFT.

•The algorithm: by performing repeatedoperations on groups that start with asingle number and increase in size by afactor of 2 at each of j stages, where N=2j.

•The time is proportional to N log2N.•For example, at N=2048, the FFT is more

than 180 times faster than the DFT.


Power Spectrum•Power spectrum: power vs frequency plot•Power of a signal x(t):

• Total power:

•Power spectrum are often used to describeearthquake-induced ground motion.

•(Fourier amplitude spectrum illustrates how thestrength of a quantity varies with frequency. )

2n

2n

2nn c

21

ba21

P )()(

dc21

dttxPnf

0

2n

T

0

2

1nn )]([)(


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Appendix A Vibratory Motion - 國立中興大學web.nchu.edu.tw/pweb/users/feng/lesson/2291.pdf ·...

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