3.3 Numerical Analysis of Seismic Response of SDOF 3.3 Numerical Analysis of Seismic Response of SDOF ,When t—tlet x),x=x),=)=)口t=4r Let, 主+片式AN=B, Then x+元+rP=4 0=）+-[6-6门 3 1g-1 Then,we can get, 或=或+w1+伐-或) =8,+线当 (3.20) =w++我 移之 (3.21) x=u+w1+号0r+ 6 裤具，裤大学土*红服学接 闺 德典其，同济大学士木工塑学接 国 201510w13 49 模dun 2015/1013 50 3.3 Numerical Analysis of Seismic Response of SDOF 3.3 Numerical Analysis of Seismic Response of SDOF .satisfy the dynamic equilibrium equation, [Example 3.2]Consider a SDOF system with a mass of 200t. a lateral stiffness k of 7200 kN/m,and a damping ratio =0.05. 线+2wB+2xw二+r4+ 6= The record of ground motion acceleration is shown below. Please calculate the displacement,velocity,acceleration, Let, 51+at+ 64 maximum absolute acceleration,and maximum horizontal seismic action of the mass using linear acceleration method then (Take the time step as 0.1s and ends at 1.2s). =民4+25oB+aoA) Table 3-1 s 1a1020.30.405 060.7080.91.01.112 We can obtain,x()use same way to solve all results cms 108 207 300 200 150-198-120 ✉1 00 20151013 2015/1013 3.3 Numerical Analysis of Seismic Response of SDOF Please [Solution]Natrual frequency Review Dynamics o=√/m=√7200/200=6 Conduct iterative calculation according to Eqn.(3.20) (3.21)and (3.22)(Please refer to the textbook for detail to obtain the displacement,velocity,acceleration in Table 3.2) 5a1+58A1+gA12.1+005x6x01+gx01'=109 6 6 S,-求+，l.-十54.03+0昨54.03cm/6 Fm=S.·m=544.03x102×200=1088kN 牌风，济大学土*工表学精 2015/1013 54 99  When let, k t t ＝ = = = = = -         k k k-1 k-1 k k k k k k-1 x x t x x t x x t x x t t t t ， ， ， ， ， k k-1 k-1 k k-1   k k-1 k-1 k 1 x x + x t + x - x t 2 1 1 x x + x t + x t 2 2     ＝ ＝  Then 2 2 k k-1 k-1 k-1 k 1 1 x x + x t + x t + x t 3 6 ＝            k-1 k-1 k k-1 k k-1 t - t x t x t + x t - x t t - t   ＝   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 49 Let, k-1 k-1 K-1 1 x + x t B 2   ＝ 2 k-1 k-1 k-1 k-1 1 x + x t + x t A 3   ＝ Then, we can get, k k-1 k t x B + x 2  ＝  （3.20） 2 k k-1 k t x A + x 6  ＝  （3.21） 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 50  satisfy the dynamic equilibrium equation, 2 2 2 k k-1 k k-1 k g k t t x +2x w B +2x w x +w A +w x - x 2 6   ＝ 2 2 t 6 s 1 t    Let, ＝  then  B A k k  s 2 k g k 1 1 1 x x 2     ＝   We can obtain, , use same way to solve all （3.22） results ( ), ( ) k k x t x t 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 51 [Example 3.2] Consider a SDOF system with a mass of 200t, a lateral stiffness k of 7200 kN/m, and a damping ratio ζ=0.05. The record of ground motion acceleration is shown below. Please calculate the displacement, velocity, acceleration, maximum absolute acceleration, and maximum horizontal seismic action of the mass using linear acceleration method (Take the time step as 0.1s and ends at 1.2s). Table 3-1   2 cm/s g x t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 108 207 300 200 95 0 -150 -198 -120 -18 0 0 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 52 [Solution] Natrual frequency   k m＝ 7200 200＝6 0 1 1 09 6 6 1 0 05 6 0 1 6 1 2 2 2 2 s    t   t   .   .   .  .    Conduct iterative calculation according to Eqn. (3.20), (3.21) and (3.22) (Please refer to the textbook for detail to obtain the displacement, velocity, acceleration in Table 3.2). 3.3 Numerical Analysis of Seismic Response of SDOF 2 F S m 544.03 10 200 1088 kN max a  ＝    ＝ ＝ Please  Review Dynamics 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 54