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

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