3.2.3 Arbitrary excitation 3.2.3 Arbitrary excitation +,+/=p0) Displacement Response under Arbitrary excitatior Duhamel Intogral(:哈查积分) c 0= mom)r f■elastic restoring force■kx time-dependent appled force rim.ic.h- 3.2.4 Base excitation 3.2.4 Base excitation Base excitation(Earthquakes) Base excitation (Earthquakes) +c欧+=- And dividing by m R+2Em+o2x=-式 Then +c+=-m 是大生林杠情 3.3 Numerical Analysis of Seismic Response 3.3 Numerical Analysis of Seismic Response of SDOF Basic Principles From then 0,e,0)A 0=)+2[)6】 Derivate the moment,k 【04,0e x),) 。年 0小取e小-取- 0=ta小60-w小rl-e- 移具大华持红精8 Where =inertia force= = damping force = = elastic restoring force = k x = time-dependent applied force is the total acceleration of the mass, and , x are the velocity and displacement of the mass relative to the base. f f f pt i  d  s  cx d f s f pt x x i f mx No. 43 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 43 3.2.3 Arbitrary excitation Displacement Response under Arbitrary excitation Duhamel Integral (杜哈密积分)                  F e sin t d m x t t t        2 0 2 1 1 1 d  t F Ft() p( )  d  p(t) dv( ) t  t'=t-  2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 44 3.2.3 Arbitrary excitation Base excitation (Earthquakes) x y ground mass k c y x ground mass k c g x g m  x   cx   kx  m  x  2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 45 3.2.4 Base excitation And dividing by m g m  x   cx   kx  m  x  Then g  x   x   x   x  2 2                   x t x e sin t d t t g         2 0 2 1 1 1  2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 46 Base excitation (Earthquakes) x ground mass k c g x 3.2.4 Base excitation  Basic Principles  From the moment, (k-1) Derivate the moment, k principles：linear acceleration increase Method: linear acceleration method, Newmark-β method, and Wilson-θ method. -1 -1 -1 ( ), ( ), ( ) kkk x t x t x t ( ), ( ), ( ) kkk x t x t x t x t x t x t ( ), ( ), ( ) -1 -1 -1 ( ), ( ), ( ) kkk x t x t x t( ), ( ), ( ) kkk x t x t x t t k -1 t kt 3.3 Numerical Analysis of Seismic Response 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 47         k-1 k-1 k k-1 k k-1 t - t x t x t + x t - x t t - t   ＝             k-1 k-1 k-1 t t t k k-1 k-1 k-1 t t t k k-1 x t - x t x t dt x t dt + t - t dt t - t    ＝              2 k-1 k-1 k-1 k-1 k k-1 k k-1 1 t - t x t x t + x t t - t + x t - x t 2 t - t   ＝                   2 3 k k-1 k-1 k-1 k-1 k-1 k-1 k-1 k k-1 1 1 x t - x t x t x t + x t t - t + x t t - t + t - t 2 6 t - t ＝ t 区段[ tk-1 , t ]积分 3.3 Numerical Analysis of Seismic Response of SDOF 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 48