同济大学：《结构动力学》课程教学资源（教案讲义）Lecture 06 Numerical Evaluation of Dynamic Response

Structural Dynamics Lecture 6 Numerical Evaluation of Dynamic Response 自 同海大学 土本工相学院

Structural Dynamics Lecture 6 Numerical Evaluation of Dynamic Response

Time-Domain Analysis:methodology given m(t)+cx(t)+kx(t)=p(t) Pi given 'd 3u!peoT p} m++k= time Time stepping m++c++k=+ seek seek time

Time-Domain Analysis:methodology Time-Domain Analysis:methodology mx(t)cx(t)kx(t)p(t) L o a d i n g p time D i s p l a c e m e n t x time p1 x1 t1 p2 x2 t2 given seek i xi kxi pi mx c   i 1 xi 1 kxi 1 pi 1 mx c          pi ti xi given pi+1 tj xi+1 seek Time stepping

m4,+c4;+u,=P2 △ △t △ t计 u:-u(t1)

Duhamel's integral provides a general result for evaluating the response of a linear SDOF system to arbitrary force. When the force is simple,Duhamel's integral method is applicable. If the force is complicated function,evaluation of the integral requires numerical method and usually not efficient The exciting force exists only in digitized form,such as earthquake records. u()=p()(i-)d

• Duhamel’s integral provides a general result for evaluating the response of a linear SDOF system to arbitrary force. • When the force is simple, Duhamel’s integral method is applicable. • If the force is complicated function, evaluation of the integral requires numerical method and usually not efficient • The exciting force exists only in digitized form, such as earthquake records.       0 t u t p d   h t   

Accordingly,two basic approaches: Numerical evaluation of the Duhamel integral; Principle of Superposition->Linear system Direct integration of the equation of motion- step-by-step integration.Valid for both Linear and Non-linear systems->more commonly utilized. Response:at discrete time,usually with equal time internal

• Accordingly, two basic approaches: • Numerical evaluation of the Duhamel integral; Principle of Superposition → Linear system • Direct integration of the equation of motion – step-by-step integration. Valid for both Linear and Non-linear systems → more commonly utilized. • Response: at discrete time, usually with equal time internal

Convergence:As the time step decreases,the numerical solution should approach the exact solution; Stability:the numerical solution should be stable in the presence of numerical round-off errors, Accuracy:the numerical procedure should provide results that are close enough to the eact solution

• Convergence: As the time step decreases, the numerical solution should approach the exact solution; • Stability: the numerical solution should be stable in the presence of numerical round-off errors; • Accuracy: the numerical procedure should provide results that are close enough to the eact solution

Time-Domain Analysis .(1)Interpolation of Excitation Method .(2)Central Difference Method ·(3)Newmark's Method ·(4)Vilson-O Method .(5)State space method

Time-Domain Analysis • (1) Interpolation of Excitation Method • (2) Central Difference Method • (3) Newmark’s Method • (4) Wilson- Method • (5) State space method 

Interpolation of Excitation linear PiAPi mi+cxi+kxi Pi △ti p+1 Pi △pi ti≤t≤ti+l △ti m成t+cx+kxt=Pi+ piT △t mXi+1+cXi+1+kxi+1=Pi+l ti+l

Interpolation of Excitation Interpolation of Excitation  i t τ Δt Δp pkxxcxm i i      ittt  i i 1 t t t        i i i t p linear p i 1 xi 1 kxi 1 pi 1 mx c          pi+1 i1 t i xi kxi pi mx  c   pi i t pi

Interpolation of Excitation m戊+cx,+kX,=p;+ pi元 △t\ constant linear ramp Solution of x(t)consists of 3 parts Step response due to pi A Ramp response due tot Free vibration response

Interpolation of Excitation Interpolation of Excitation           i i i t p   pkxxcxm constant linear ramp Solution of x(t) consists of 3 parts - Step response due to pi - Ramp response due to - Free vibration response    i i t P

Recurrence Equations APLt m+k=p ur)(o.) sino,t ；on△t ↑ Free vibration Step force Ramp force l回-4 isino.+立coso,r+朵sino,r+若oA k 91(1-cos0,) 4+1=A4,+B,+CP,+Dp+l u'i+1=A'u,+B'u+C'p,+D'P+1 For constant time interval,ABCD need to be computed only once

Recurrence Equations Recurrence Equations i i i m p u ku p t          1 cos  si os n c sin i n i i i n n n n i n i p k p k u u u t t                             Free vibration Step force Ramp force   sin co   1 s sin 1 cos i i n i n n n n n i i n p p k k t u u u                       1 1 1 1 ' ' ' ' ' i i i i i i i i i i u Au Bu Cp Dp u A u B u C p D p               For constant time interval, ABCD need to be computed only once