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