《结构动力学》课程教学课件（讲稿）04 Response to periodic loading

Wuhan University of TechnologyChapter 4Responsetoperiodicloading4.1 Fourier series expressions of periodic loading4.2 Response to the Fourier series loading5-1

5-1 Wuhan University of Technology 4.1 Fourier series expressions of periodic loading 4.2 Response to the Fourier series loading Chapter 4 Response to periodic loading

Wuhan University of Technology4.1 Fourier series expressions of periodic loadingtp(t)Fig.4-1Arbitraryperiodic loadingBecause any periodic loading can be expressed as a series ofharmonic loading terms, the response analysis procedures presentedin Chapter 3 have a wide range of applicability.To treat the case of an arbitrary periodic loading of period Tp, asindicated in Fig.41, it is convenient to express it in a Fourier seriesformwith harmonicloading components atdiscrete values offrequency.5-2

5-2 Wuhan University of Technology 4.1 Fourier series expressions of periodic loading Fig. 4-1 Arbitrary periodic loading  Because any periodic loading can be expressed as a series of harmonic loading terms, the response analysis procedures presented in Chapter 3 have a wide range of applicability.  To treat the case of an arbitrary periodic loading of period Tp, as indicated in Fig. 41, it is convenient to express it in a Fourier series form with harmonic loading components at discrete values of frequency

WuhanUniversityof Technology4.1 Fourier series expressions of periodic loadingThe wellknown trigonometric form of the Fourier series is given byp(t)=ao+Za,cosa,t+Eb,sina,2元,=n,=TThe harmonic amplitude coefcients can be evaluated using theexpressions="p(odf p(t)cos 0,tdt n=1,2,3... p(t)sina,tdt n=1,2,3,5-3

5-3 Wuhan University of Technology 4.1 Fourier series expressions of periodic loading 0 1 1 ( ) cos sin n n nn n n pt a a t b t           The wellknown trigonometric form of the Fourier series is given by 1 2 n p n n T      0 0 1 ( ) Tp p a p t dt T   0 2 ( )cos n=1,2,3,. Tp n n p a p t tdt T    0 2 ( )sin n=1,2,3,. TP n n P b p t tdt T    The harmonic amplitude coefcients can be evaluated using the expressions

Wuhan University of Technology4.1 Fourier series expressions of periodic loadingWhen p(t) is of arbitrary periodic form, the integrals in Egs. (43) mustbe evaluated numerically.This can be done by dividing the period TpintoN equal intervals△t (Tp =N△t),evaluating the ordinates of theintegrand in each integral at discrete values of t = tm = m△t (m =O,1,2,..,N) denoted by qo, qi, q2..,qn, and then applying thetrapezoidalruleofintegrationinaccordancewithN-IgNqoq(t)dt=tQ2m='q()dt=Ai2qm7773ip(m)aN=l2Atqmaqm=p(tm)cos,(mAt)p(tm)sina,(mAt)5-4

5-4 Wuhan University of Technology 4.1 Fourier series expressions of periodic loading 1 0 0 1 () ( ) 2 2 N Tp m m q qN q t dt t q            ￾  1 0 1 ( ) p N T m q t dt t qm       0 1 1 2 N n p m n a t a qm T b           1 ( ) 2 ( )cos ( ) ( )sin ( ) m m n m n p t qm p t m t p t mt                    When p(t) is of arbitrary periodic form, the integrals in Eqs. (43) must be evaluated numerically. This can be done by dividing the period Tp into N equal intervalsΔt (Tp = N Δt), evaluating the ordinates of the integrand in each integral at discrete values of t = t m = m Δt (m = 0,1,2,.,N) denoted by q 0, q 1, q 2,.,q N, and then applying the trapezoidal rule of integration in accordance with

Wuhan Universityof Technology4.1 Fourier series expressions of periodic loadingExponential Formcos ,t =-[exp(i,t)+exp(-i,)sin ,t=[exp(io,1)-exp(-i,0)]p(t)= Z p, xp(ia,)p(t)exp(-i,t)dtn=0.±1.±2....2元nmp(tm)exp(-n=0.1.2...(N-1)N5-5

5-5 Wuhan University of Technology 4.1 Fourier series expressions of periodic loading     1 cos exp( ) exp( ) 2 sin exp( ) exp( ) 2 n nn n nn t it it i t it it         ( ) exp( ) n n n p t p it      0 1 ( )exp( ) Tp n n p P p t i t dt T     n  0. 1. 2    1 1 1 2 ( )exp( ) N n m nm P p tm i N N       n N  0.1.2 ( 1)   Exponential Form

Wuhan University of Technology4.2 Response to the Fourier series loadingThe response of a linear system to this loading may be obtained bysimplyaddinguptheresponsestotheindividualharmonicloadings.InChapter3[Eg.(310)l,itwasshownthatthesteadystateresponseproducedinanundampedSDOFsystembythenthsinewaveharmonicof Eq.(41) (afteromitting thetransient responseterm)isgiven bybv.(t)sinw,tk1-β2β,=a,/oTheharmonicamplitudethesteadystateresponseproducedbythenthcosinewaveharmonicinEg.(41)is0v.(0)=o=ao/kcoOSW,tB-5-6

5-6 Wuhan University of Technology 4.2 Response to the Fourier series loading   2 1 sin 1 n n n n b vt t k           / n n     2 1 cos 1 n n n n a vt t k           0 0 v ak  / The response of a linear system to this loading may be obtained by simply adding up the responses to the individual harmonic loadings. In Chapter 3 [Eq. (310)], it was shown that the steadystate response produced in an undamped SDOF system by the nth sinewave harmonic of Eq. (41) (after omitting the transient response term) is given by The harmonic amplitude the steadystate response produced by the nth cosinewave harmonic in Eq. (41) is

Wuhan Universityof Technology4.2Response to the Fourier series loadingThetotalperiodicresponseoftheundampedstructurethencanbeexpressedasthesumoftheindividualresponsestotheloadingtermsinEq.(41)asfollows:(ancosa,t+b,sina,t)The total steadystate response for damped system is given by6+2(1-β)+(25B)Sx([25a,B,+b,(1-B.)]sina,+[a,(1-B,)-25b,B,Jcosa,025-7

5-7 Wuhan University of Technology 4.2 Response to the Fourier series loading   0 2 1 1 1 ( cos sin ) 1 n nn n n n vt a a t b t k                              2 2 0 22 2 1 1 1 2 1 sin 1 2 cos (1 ) (2 ) nn n n n n n nn n n n n vt a a b t a b t k                                         The total periodic response of the undamped structure then can be expressed as the sum of the individual responses to the loading terms in Eq. (41) as follows: The total steadystate response for damped system is given by

Wuhan University of Technology4.2Response to theFourier series loadingExampleE41.Asanexampleoftheresponseanalysisofaperiodicallyloaded structure,considerthesystemandloadingshowninFig.E41.Theloadinginthiscaseconsistsofthepositiveportionofasimplesinefunction.TheFouriercoefficientsofEq.(41)arefoundby using Eqs. (42) and (43) to obtaint P(t)27PosirFPop(t)m00000kOTpTpTpTp2225(a)(b)FIGURE E4-1 Example analysis of response to periodic loading:(a)SDOF system; (b)periodicloading.5-8

5-8 Wuhan University of Technology p(t) 2 p0 sin  t Tp p0 t Tp Tp Tp Tp    22 2 2 (b) p(t) 2 p0 sin  t Tp p0 t Tp Tp Tp Tp    22 2 2 (b) 4.2 Response to the Fourier series loading Example E41. As an example of the response analysis of a periodically loaded structure, consider the system and loading shown in Fig. E41. The loading in this case consists of the positive portion of a simple sine function. The Fourier coefficients of Eq. (41) are found by using Eqs. (42) and (43) to obtain FIGURE E4-1 Example analysis of response to periodic loading: (a) SDOF system; (b) periodic loading

Wuhan University of Technology4.2 Response to the Fourier series loading2元t-PoPosin2元0oddnTh22元t2元ntdtPsin二2a.cOSPoTJoT,Tneven元PoTh2n=12元t2元ntdt =2P,sinsinTPZTJo0n>1222元Pep(t)cos2a,tcos4可,sinw,tcos6可,t-+235315T5-9

5-9 Wuhan University of Technology 2 0 0 1 2 sin Tp o o p pt p a p dt T T     2 0 0 2 0 2 22 sin cos 2 1 Tp n o p pp n odd t nt a p dt p T TT n even n                  0 2 0 2 22 1 sin sin 2 0 1 Tp n o p pp p t nt n b p dt T TT n            0 11 1 1 22 2 1 sin cos 2 cos 4 cos 6 2 3 15 35 p pt t t t t                 4.2 Response to the Fourier series loading

Wuhan University of Technology4.2 Response to the Fourier series loadingIfitisnowassumedthatthestructureofFig.E41isundamped,andiffor example, the period of loading is taken as fourthirds the period ofvibration of the structure, i.e.,34na,B.4300ThesteadystateresponsegivenbyEq.(415)becomes88元Pov(t)=cos2a,tsino,t+cos40,t+715k元605-10

5-10 Wuhan University of Technology 4.2 Response to the Fourier series loading 1 1 4 3 3 4 p n T n n T           11 1 88 1 1 sin cos 2 cos 4 7 15 60 o p vt t t t k               If it is now assumed that the structure of Fig. E41 is undamped, and if, for example, the period of loading is taken as fourthirds the period of vibration of the structure, i.e., The steadystate response given by Eq. (415) becomes