同济大学：《结构动力学》课程教学资源（教案讲义）Lecture 07 Frequency Domain Method

Structural Dynamics Lecture 7 闺 同桥大学 土本2程学收 COLLEGE OF CIVIL

Structural Dynamics Lecture 7

Frequency Domain Method ·Time-domain method The time domain analysis procedure can be determine the response of any linear SDOF system to any arbitrary loading Frequency Domain Method Sometime,it is more convenient to perform the analysis in the frequency domain.Especially when the structural parameter are frequency- dependent.The FD is much superior to TD

Frequency Domain Method • Time-domain method The time domain analysis procedure can be determine the response of any linear SDOF system to any arbitrary loading • Frequency Domain Method Sometime, it is more convenient to perform the analysis in the frequency domain. Especially when the structural parameter are frequency￾dependent. The FD is much superior to TD

Frequency Domain Method Response to period excitation Fourier series in real form Fourier series in complex form Response to arbitrary excitation Fourier integral/Fourier transform Discrete Fourier transform Fast Fourier transform

Frequency Domain Method • Response to period excitation Fourier series in real form Fourier series in complex form • Response to arbitrary excitation Fourier integral/ Fourier transform Discrete Fourier transform Fast Fourier transform

Review:Response to Periodic Excitation Response of SDOF to harmonic excitation Using Fourier series expansion to expand the periodic excitation into summation of harmonic excitations Linear elastic system,superposition principle is applicable 闺 土本2程季院

• Response of SDOF to harmonic excitation • Using Fourier series expansion to expand the periodic excitation into summation of harmonic excitations • Linear elastic system, superposition principle is applicable Review: Response to Periodic Excitation

For any periodic excitation p(t),Fourier series representation p0=a+2a,os@4+2,smo 0,=j0=j元 2π a=p()cos(o) tn=1,2,3… b,=是p0sin(o4）n=l2,3 闺 土本鞋李悦

            0 1 1 1 0 0 0 0 cos sin 2 1 2 cos 1, 2,3,... 2 sin 1, 2,3,... p p p j j j j j j j p T p T j j p T j j p p t a a t b t j j T a p t dt T a p t t dt n T b p t t dt n T                           • For any periodic excitation p(t), Fourier series representation

Other possible definition of Fourier series coefficients--orthonormal a=/)边 a-IS,f()d a-2f0esu灿k0&=7,Wosh ()sino.dr b,=7J,f④)sinw,d a=元p0 p()coste)d1.2. ()sin(.3 国 士系2鞋学院

6 / 2 / 2 2 ( )cos , 0 T k k T a f t tdt k T      / 2 / 2 2 ( )sin T k k T b f t tdt T       / 2 0 / 2 1 T T a f t dt T    • Other possible definition of Fourier series coefficients -- orthonormal 0   1 T T a f t dt T    1 ( )cos k k T T a f t tdt T     1 ( )sin k k T T b f t tdt T               0 0 0 0 1 2 cos 1,2,3,... 2 sin 1,2,3,... p p p T p T j j p T j j p a p t dt T a p t t dt n T b p t t dt n T          

Steady state response to each harmonic component Frequency ratio B=n ao uo=ao /k a,coso4268,sino,+(1-Bcos (1-P,2+(2B,) smo冬下gmo1-2 (1-B,2)+(2B,) 闺 土本程李悦

0 a 0 0 u a k  / /    j j n  cos j j a t        2 2 2 2 2 sin 1 cos 1 2 j j j j c j j j j a t t u k            sin j j b t        2 2 2 2 1 sin 2 cos 1 2 j j j j s j j j j b t t u k            ： ， ： • Steady state response to each harmonic component Frequency ratio

The total steady state response to the periodic excitation u(0)=4+∑(g,+%,） k a248,)+b1-B1mo 合k(1-B2+(2B,)j 分1a1-B)-b(23, cos@t 合k(（1-B,)+(2β，) 闺 土本之程季院

                    0 1 0 2 2 2 2 1 2 2 2 2 1 1 2 1 sin 1 2 1 1 2 cos 1 2 c s j j j j j j j j j j j j j j j j j j j u t u u u a k a b t k a b t k                                  • The total steady state response to the periodic excitation

Response to periodic excitation: complex Fourier series Let the periodic function p(t)be separated into its harmonic components by means of complex Fourier-series expansion p()=∑PeaW 2 00= 7p0em-01*2 T 闺 土本程李悦

Response to periodic excitation: complex Fourier series • Let the periodic function p(t) be separated into its harmonic components by means of complex Fourier-series expansion   i j t  0  j j p t P e      0 0 2 T        0 0 0 0 1 0, 1, 2, T i j t P p t e dt j j T       

Steady-state response Response to jth term in Fourier series 4,()）=U,eUa U;=H(j)P, Adding responses to all excitation terms leading to ()=∑H(Uo)PeUa 闺 土本2程学院

Steady-state response • Response to jth term in Fourier series • Adding responses to all excitation terms leading to   i j t  0  j j u t U e   U H j P j j   0       0  0 i j t j j u t H j P e      