Vibratory Motion. Prof. Zheng-yi Feng NCHU SWC. National CHung Hsing University, Department of Soil and Water Conservation

Transcription

1 Vibratory Motio Prof. Zheg-yi Feg NCHU SWC 1

2 Types of vibratory motio Periodic motio Noperiodic motio See Fig. A1, p.58 Harmoic motio Periodic motio Trasiet motio impact Trasiet motio earthquake

3 A powerful tool for dyamic aalysis of liear systems Fig. A.1b ca be expressed as the sum of a series of simple harmoic motios. Fig. A.1c ad A.1d ca be represeted as periodic motios by assumig quiet zoe. See Fig. A, p.58 liear system; superpositio Respose to trasiet loadig ca be expressed as the sum of the resposes to a series of simple harmoic loads. 3

4 Simple harmoic motio, S.H.M. Ca be defied by 3 quatities: Amplitude Frequecy phase Cab be described i: Trigoometric otatio Complex otatio 4

5 Trigoometric otatio u(t) = A si (t+) A: amplitude : circular frequecy : phase agle Thedescribes the amout of time by which the peaks (ad zeros) are shifted from those of a pure sie fuctio. See Fig. A.4 for positive (lead) ad egative (lag) 5

6 Rotatig vector represetatio See Fig. A.5 Period of vibratio, T : time for oe cycle of motio T = / f = 1 / T =usually hertz (Hz), cycle per secod = f 6

7 S.H.M. ca also be described as: u(t) = a cost + b sit See Fig. A.6 Amplitude is ot the simple sum Peaks do ot occurs at the same time as those of the sie or cosie fuctios. 7

8 Cos= si (+90) a 90deg ahead of b See Fig. A.7a Legth of resultat will be sqrt(a +b ) ad it leads b by a agle= ta -1 (a/b) u(t) = A si (t+) 8

9 Complex otatio Much simpler descriptio Euler s law : e i = cos+ i si u( t) aib e See Fig. A.8 clockwise it a ib i t e it is represeted by a vector of uit legth rotatig clockwise at a agular speed,. e couterclockwise 9

10 Other measures of motio Displacemet Velocity u( t) A si( t ) ut ( ) Acos( t ) Acceleratio u( t) Asi( t ) u Frequecy, amplitudes of displacemet, velocity, ad acceleratio are related. Tripartite plot:a harmoic motio ca be described by a sigle poit 10

11 Tripartite plot See Fig. A.9 commoly used to described earthquake motio applied oly to harmoic motio; for other types of motio, must be obtaied by differetiatio ad/or itegratio 11

12 Out of phase with each other betwee displacemet, velocity, ad acceleratio u( t) A si( t ) u ( t) Asi( t / ) u ( t) Asi( t ) u( t) u( t) Ae u() t i Ae Velocity leads displacemet 90 degrees. Acceleratio leads velocity 90 degrees. Acceleratio leads displacemet 180 degrees. it it i it Ae Ae it 1

13 Phase leads & amplitudes See Fig. A10 See Fig. A.11 Leads 90 degree Leads 180 degree 13

14 Fourier Series The Frech mathematicia J.B.J Fourier A periodic fuctio ca be expressed as the sum of a series of siusoids of differet amplitude, frequecy, ad phase. Fourier series: a extraordiarily useful tool 14

15 Process to produce total respose See Figure A.1 Time history of loadig Sum of series of harmoic loads Calculatig resposes of each harmoic load Summig the resposes 15

16 Trigoometric form See E.Q. A.11 for the geeral trigoometric form of the Fourier series for a fuctio of period, T f, ad the Fourier coefficiets, a 0, a, b ; = / T f. a 0 is the average values of x(t) i t=0~t f Usually a 0 = 0 i may geotechical earthquake egieerig applicatios is ot arbitrary; icremet =/ T f. 16

17 Fourier amplitude spectrum & Fourier phase spectrum From EQ A.5 ad EQ A.11 x( t) c 0 c 0 c 1 a 0 ; c a si( t b ta c ad = the amplitude ad phase of the th harmoic. c versus : a Fourier amplitude spectrum; very useful to describes the frequecy cotet of a EQ versus : a Fourier phase spectrum, ) ad 1 ( a / b 17 )

18 See Example A.1 example a 0 =0; sice the average of x(t) is zero. eve fuctio; sie terms are zero; f(t)=f(-t) odd fuctio; cos terms are zero; f(t)=-f(-t) See example A. for c 0, c ad amplitude & phase spetra See Figure EA. for the plots of spectra 18

19 19 Expoetial Form See Fig.EA.3 for oe- ad two-sided Fourier spectra f T 0 t i f t i dt e t x T 1 c e c t x ) ( ) ( * *

20 Discrete Fourier Trasform, DFT For fiite umber of data poits Fourier coefficiets are obtaied by summatio rather tha itegratio Fourier coefficiets of DFT have uits of the origial variable multiplied by time The DFT ca be iverted by usig Iverse DFT (IDFT) The time required for computatio of DFT/IDFT is proportioal to N. 0

21 Fast Fourier Trasform, FFT Cooley ad Tukey(1965) developed a computatioal algorithm for the case where N is a power of kow as FFT. The algorithm: by performig repeated operatios o groups that start with a sigle umber ad icrease i size by a factor of at each of j stages, where N= j. The time is proportioal to N log N. For example, at N=048, the FFT is more tha 180 times faster tha the DFT. 1

22 Power Spectrum Power spectrum: power vs frequecy plot Power of a sigal x(t): Total power: Power spectrum are ofte used to describe earthquake-iduced groud motio. (Fourier amplitude spectrum illustrates how the stregth of a quatity varies with frequecy. ) c 1 b a 1 P ) ( ) ( d c 1 dt t x P f 0 T 0 1 )] ( [ ) (

23 The ed 3