3.6 Free Vibration of MDOF Systems M3d6IFresibration of MDOF Systems Divide the uncoupled equations by M Displacements: i+2s.o.+o.=P0 Contribution of the nth mode to the M. displacement (is: .Solve the above equation for the modal coordinate u.(0=g.9.) similar to an SDF system.g ( .Combining these modal contributions gives the total displacement: 0=2u0=2a.0 -A/ 海贝。济大学土木工 前廊贝。同济大学土木工塞单璃 20151020 439 2015/10/20 3.6 Free Vibration of MDOF Systems 3.6 Free Vibration of MDOF Systems Modal Analysis:Summary .Define the structural properties: Modal Analysis:Summary (cont'd) .Determine the mass matrix m and the stiffness matrix k. Compute n-modal displacements ut) Determine the modal damping ratios Compute the element forces associated with the n- .Determine the natural frequencies and mode modal displacements t) shapes Combine the contributions of all the modes to .Compute the response in each mode by uncoupling determine the total response. the equations of motion and solving for modal coordinates9.() ，济大学土工 2015/10W20 201510/20 3.6 Free Vibration of MDOF Systems Homework Review:SDOF Rreview:MDOF m=X)[M](X) 量降贝：屑待大导土木工整华香 2015/10w20 53 99  Divide the uncoupled equations by :  Solve the above equation for the modal coordinate similar to an SDF system. Mn n n n n n n n n M P t q q q ( ) 2 2        q (t) n 3.6 Free Vibration of MDOF Systems 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 49 Modal Analysis  Displacements:  Contribution of the nth mode to the displacement is:  Combining these modal contributions gives the total displacement:       N n N n n n n t u t q t 1 1 u( ) ( )  ( ) (t) q (t) un n n u(t) 3.6 Free Vibration of MDOF Systems 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 50 Modal Analysis: Summary  Define the structural properties:  Determine the mass matrix m and the stiffness matrix k.  Determine the modal damping ratios  Determine the natural frequencies and mode shapes .  Compute the response in each mode by uncoupling the equations of motion and solving for modal coordinates . n n q (t) n 3.6 Free Vibration of MDOF Systems 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 51 Modal Analysis: Summary (cont’d)  Compute n-modal displacements un (t) .  Compute the element forces associated with the n￾modal displacements rn (t) .  Combine the contributions of all the modes to determine the total response. 3.6 Free Vibration of MDOF Systems 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 52 Homework  Review: SDOF  Rreview：MDOF      T m = X M X j 2 j j   3.6 Free Vibration of MDOF Systems 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 53