同济大学：《建筑结构抗震》课程教学资源（课件讲稿）Chapter 3 Seismic Responses of SDOF and MDOF（2/2）

Outline Dynamics 3.1 Free Vibration of SDOF Systems Chapter 3 3.2 Forced Vibration of SDOF Systems 3-3 NumericalAnalysis of Seismic Response of SDOF Seismic Responses of Systems ( SDOF and MDOF Free Vibration of MDOF Systems 3-7 Response Specturm Method of MDOF 3-8 Earthquake Action and Responses of MDOF 3.9 Time History Method of MDO 2015/1020 国 20151020 Review Review Base excitati From the moment.(k-1) 4 x(t))) R+25ax+02x=-3 Then Method: nearaNewmak-andWilomethod to solve the differentialissue. 国 大01so0 Displacement spectra T-0.5s Computation of the response spectrum 1

1 Chapter 3 Seismic Responses of SDOF and MDOF 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 1 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 2 Outline 3.1 Free Vibration of SDOF Systems 3.2 Forced Vibration of SDOF Systems 3.3 Numerical Analysis of Seismic Response of SDOF 3.4 Response Spectrum of SDOF 3.5 Response of Nonlinear SDOF Systems （*） 3.6 Free Vibration of MDOF Systems 3.7 Response Specturm Method of MDOF 3.8 Earthquake Action and Responses of MDOF 3.9 Time History Method of MDOF Dynamics 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  2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 3 Base excitation (Earthquakes) x ground mass k c g x  From the moment, (k-1) Derivate the moment, k Method: linear acceleration, Newmark-β, and Wilson-θ method to solve the differential issue. -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 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 4 Assumption: linear acceleration increase No. 5 No. 6

Velocity Spectra Acceleration spectra Relationship between spectra ST,5=S(C=S①，5) S(T)=TxS,(T)=TxS.(T.5) 2π 4π reqyin raians/secon (r) ==frequency en cycles/second (Hertz) T2=1/f=period in seconds ( cofficientof critica damping 194 3.4 Response Spectrum of SDOF ent Thceratcanetodoatheyr8r8o0ed 2 dT 大 2015t0/2

2 No. 7 No. 8 No. 9 No. 10 No. 11 3.4 Response Spectrum of SDOF A response spectrum is simply a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series of oscillators of varying natural frequency, that are forced into motion by the same base vibration or shock. The resulting plot can be used to pick off the response of any linear system, given its natural frequency of oscillation. One such use is in assessing the peak response of a SDOF to earthquakes. 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 12

3.4 Response Spectrum of SDOF 3.4 Response Spectrum of SDOF spectrum in Chinese code GB50011-2010 a=.o--s可a combined to producea desn spectrum 01T 20151020 1.Seismic Effect Coefficient,a (地晨加速度影响系数) 2t8 Coomc1et人 只 ais defined as the ratio of the horizontal seismic action to the gravity of a SDOF elasticsystem. Intensity 6 7(7.5)8(85)9 斗t k0.050.10.15020.3004 4c005g01g0,15g02g0.3g04g8 大用 2015/10w20 20151020 3.Dynamic Coefficient 1.Seismic Effect Coefficient,a (动力系数) S, (地震加速度影响系数) 004 (0.12) 016(024)0.2 Seismic Effect Coefficient a-s- =kB 20110y20 3

3 3.4 Response Spectrum of SDOF Time [sec] 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Acceleration [cm/sec2]40 20 0 -20 -40 Time [sec] 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Acceleration [g] 40 20 0 -20 -40 Time [sec] 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Acceleration [cm/sec2]40 20 0 -20 -40 combined to produce a design spectrum 3.4 Response Spectrum of SDOF Design spectrum in Chinese code 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 14 GB 50011-2010） α- T Curve α is defined as the ratio of the horizontal seismic action to the gravity of a SDOF elastic system. G α G g S ＝ g S F mS mg a a a           k  x S g x g S α g a g a max max   2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 15 1. Seismic Effect Coefficient, α (地震加速度影响系数) 2. Earthquake Coefficient （地震系数） There exists some corresponding relation between earthquake intensity and earthquake coefficient. Intensity 6 7(7.5) 8(8.5) 9 k 0.05 0.1(0.15) 0.2(0.30) 0.4 Acc. 0.05g 0.1g(0.15g) 0.2g(0.3g) 0.4g 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 16 g x k g max   3. Dynamic Coefficient （动力系数） It is the ratio of maximum absolute acceleration to maximum ground motion acceleration for SDOF system, reflecting how much the maximum absolute acceleration is amplified due to dynamic effect. max g a x S  ＝ 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 17 Seismic Effect Coefficient 18 αmax Intensity 6 7 8 9 Frequently Occurred Eq. 0.04 0.08（0.12） 0.16（0.24） 0.32 Rare Occurred Eq. —— 0.50（0.72） 0.90（1.20） 1.40 1. Seismic Effect Coefficient, α (地震加速度影响系数)

4.Site Predominant Period场地卓越周期 Earthquake wave is influenced with soil profile Active faults nearby, Regional geology influenc sidence(下陷性地震) 2015y10w20 几大生大 30151020 4.Site Predominant Period,Ta To emress site characteristics and the distance to epicanter Based on Chinese Seismic Code (0012010). Site Predominant oup 025.0.350.450.65 @ 3.4 Response Spectrum of SDOF 3.5 Response of Nonlinear SDOF Systems (* Response spectra are very useful tools 1 Nonlinearity of Materia If we know a singie story building's natural period,and the 2.Motion Equations of SDOF Nonlinear ding const n ste,we can find the s System(K e the the Nonlinear Motion Equations 4.Hystorosis(回)Model If we can resolve a MDOF to a family of equivalent SDOF skeletor system,then we canc e multi-story buldng's Eq.loads using 大

4 4. Site Predominant Period 场地卓越周期 Tg 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 19 Earthquake wave is influenced with soil profile: • Active faults nearby, • Soil dynamic response • Regional geology influence • Liquefaction • Landslide • Subsidence (下陷性地震) 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 20 Site Category Site Group I II III IV Group 1 0.25 0.35 0.45 0.65 Group 2 0.30 0.40 0.55 0.75 Group 3 0.35 0.45 0.65 0.90 Site Predominant Period Tg （s） 4. Site Predominant Period, Tg To express site characteristics and the distance to epicanter, It is divided to three groups, Group 1 to 3 for near-earthquake (近震) to far-earthquake （远震）. Based on Chinese Seismic Code (GB 50011-2010), 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 21    0.3 6 0.05 0.9       4 32 0.05 1 0.02     ς    0.08 1.6 0.05 2 1     Attenuation index number declined slope coefficient Damping adjusting coefficient • Response spectra are very useful tools If we know a single story building’s natural period, and the building construction site, we can find the seismic effective coefficient to determine the equivalent static force (earthquake load), the we can combine the Eq. load with other loads. Then we can check the system if it meets the requirements. If we can resolve a MDOF to a family of equivalent SDOF system, then we can calculate multi-story building’s Eq. loads using Response Spectra of SDOF. 3.4 Response Spectrum of SDOF 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 23 1. Nonlinearity of Materials 2. Motion Equations of SDOF Nonlinear System (K(t)) 3. Solutions of Nonlinear Motion Equations 4. Hysteresis (滞回) Model ( ; skeleton curve （骨架曲线） and hysteretic curve （滞回曲线）)   ,M ,F  3.5 Response of Nonlinear SDOF Systems （*）

3.5 Response of Nonlinear SDOF Systems (* 3.5 Response of Nonlinear SDOF Systems (* tic curve skeleton curve 国 20110/20 20151020 3.5 Response of Nonlinear SDOF Systems (* 3.5 Response of Nonlinear SDOF Systems (* Then, ‘的h四o ∫(x)=k(x)x mt+ct+kt}xt上-m,) mt,)+ct)+t小上一成) m成+c+k()x=一n成，() 4,=式，)-，b4t={-4- 2015/1020 国 20151020 Dynamic Incremental Equllibrium: 3.5 Response of Nonlinear SDOF Systems ( m4+c4+k4=m4优 Basodon 核=3M+二A·d (+e是小小=-m4+++u+ We can0--+ Simplified as. k△x=△P· Then, 兴4小-号血-玩 国 20151020 5

5 Hysteretic curve 3.5 Response of Nonlinear SDOF Systems （*） 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 25 Skeleton curve Hysteretic curve Loading and un-loading is in different way 3.5 Response of Nonlinear SDOF Systems （*） 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 26 kt 27 Then, f x k x x s  ＝   mx+cx+k x x mx t   ＝－ g   3.5 Response of Nonlinear SDOF Systems （*） 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn  But, at moment, t, if is very small, during tk-1 to t k , is a constant value.            k 1 k 1 k 1 k 1 g k 1 mx t cx t k x t x t ＝－mx t                       k k k k g k m  x  t  cx  t  k x t  x t ＝－m  x  t • if,             k k-1 k k-1 k k-1 Δx＝x t  x t ，Δx  ＝x  t  x  t ，Δ x  ＝  x  t   x  t     g g k g k-1 k k 1 Δx ＝x t x t ，Δt＝t t       3.5 Response of Nonlinear SDOF Systems （*） t kt 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 28 g mΔ x   cΔx   kΔΔ mΔ x  Based on Δx Δt 2 1 Δx＝x Δt k 1        2 2 k 1 k 1 Δx Δt 6 1 x Δt 2 1 Δx＝x  Δt          We can get, 2x x Δt Δt 3 Δx Δt 6 Δx＝ 2  k1  k1    Then,              x  Δt 2 1 3 x Δt 3ΔΔ Δx＝x Δt k 1 k 1 k 1     x Δt 2 1 Δx 3x Δt 3 ＝ k 1 k 1        2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 29 Dynamic Incremental Equilibrium:   2 k 1 k 1 3xk-1 x c t 3c 2x x t t 3m x t 6 m     （            k 1 k x m xg x t 2 1        ） ＝－ 2x x t t 3m k x m x t 3 c t 6 m 2 g k-1  k-1               ＝－            x  t 2 1 c 3x k-1 k-1     k x＝P Simplified as, Equivalent Stiffness Equivalent Restoring Force 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 30 3.5 Response of Nonlinear SDOF Systems （*）

3.5 Response of Nonlinear SDOF Systems (* 3.6 Free Vibration of MDOF Systems Idealized three-lines semi-degradation restoring model (健想的“半退化三线型恢复力模型) MDOF Systems Response to Base E 201510w2 30151020 Review Review Single-degree-of-freedom osallators and 0-2 且 Veraton PenodT 3.6 Free Vibration of MDOF Systems 3.6 Free Vibration of MDOF Systems .Multi-degree-of-freedom systems Systems havingndegrees of freedom will have: Fundamental Vbration Period for Buldings The basic penod is tunc ental perod 风，大大

6 0 px py  K1 K2 K12 p K3 Idealized three-lines semi-degradation restoring model (理想的“半退化三线型”恢复力模型) 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 31 3.5 Response of Nonlinear SDOF Systems （*） 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 32 time, sec 0.0 0.2 0.4 0.6 0.8 Vibration Period T W g K  2  Single-degree-of-freedom oscillators W K T 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 33 First￾Mode Shape Third￾Mode Shape Second￾Mode Shape • Linear response can be viewed in terms of individual modal responses. Idealized Model Actual Building • Multi-story buildings can be idealized and analyzed as multi-degree-of-freedom systems. 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 34  Multi-degree-of-freedom systems  Systems having n degrees of freedom will have:  n modes of vibration;  n natural frequencies.  Response will, in general, be some linear combination of response in these modes  Common approach is to treat the n-DOF system as n SDOF systems instead  The basic period is fundamental period. 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 35 Fundamental Vibration Period for Buildings 2015/10/20 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 36 3.6 Free Vibration of MDOF Systems

3.6 Free Vibration of MDOF Systems 3.6 Free Vibration of MDOF Systems 。 么：04+Ke-s0m 网0+K+K0-K)=-0 a+么2=-属i0-m0-KI0小-明=0 一国 2011020 20151020 3.6 Free Vibration of MDOF Systems 3.6 Free Vibration of MDOF Systems 6-6 MDOF: [M]ob-ic]o-[1=-[M1050 5o-[jso--. Raleigh domping or6 脚9 [c]=a[M]+a[K] ow=28 四-骨 2015/10W20 国、 30151020 3.6 Free Vibration of MDOF Systems 思略 Modal Analysis 然后想力法童加，再转执到几何坐标系中。 誉兴 肉级号瑰条笑持国广又标（国有银题 新肉整受窝路华方从健 报型更加后在转换到几何坐标。·。 转来转去 7

7 Formulation of the Equation of Motion and Selection of the Dynamic Degrees of Freedom Assumption: • Beams and floors are concentrated to a mass particle • Axial deformations of the beams and columns are neglected • Axial load on columns are neglected 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 37 .. .. 1 1 1 1 11 12 1 1 2 2 1 [ ( ) ( )] ( ) [ ( ) ( )] g I S S S f m x t x t f f f K x t K x t x t           .. .. 1 1 1 1 1 1 1 2 2 2 1 ( ) ( ) ( ) ( ) ( ) 0 g I S f f m x t m x t K x t K x t K x t         .. .. 1 1 1 2 1 2 2 1 m x t K K x t K x t m x t ( ) ( ) ( ) ( ) ( )      g .. .. 2 2 2 2 2 2 2 1 ( ) ( ) [ ( ) ( )] 0 g I S f f m x t m x t K x t x t        .. .. 2 2 2 2 2 1 2 m x t K x t K x t m x t ( ) ( ) ( ) ( )     g 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 38 .. .. 1 1 1 2 2 1 1 .. .. 2 2 2 2 2 2 0 0 ( ) ( ) ( ) 0 0 ( ) ( ) ( ) g g m K K K m x t x t x t m K K m x t x t x t                                                   Define:       1 2 1 2 2 2 2 1 2 0 = 0 ( ) ( ) ( ) m M m K K K K K K x t x t x t                            .. .. 1 .. 2 ( ) ( ) ( ) 1 1 x t x t x t I                            .. .. M x t K x t M I x t ( ) ( ) ( )    g 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 39 MDOF: Raleigh damping             .. . .. M x t C x t K x t M I x t ( ) ( ) ( ) ( )     g C a M a K    0 1     1 2 1 2 2 1 0 2 2 2 1 2 ( )             2 2 1 1 1 2 2 2 1 2( )           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 40 Modal Analysis  Overview of the method:  The equations of motions, when transformed to modal coordinates, become uncoupled.  The response in each mode can be computed independently of the other modes by solving an SDOF system with the vibration properties of that mode.  Modal responses are combined to obtain 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 41 思路 42 • 由几何坐标（节点位移）转换到广义坐标（固有振型 的振幅）的转换来完成。 • 由于固有振型的正交性，运动方程可解耦，从而使多 自由度方程变为多个单自由度方程进行求解。 • 振型叠加后在转换到几何坐标。 将多自由度体系以一种方法，转换到单自由度体系，然 后用单自由度体系的方法求解单自由度的振型和反应， 然后想办法叠加，再转换到几何坐标系中。 转来转去

3.6 Free Vibration of MDOF Systems 6Free Vibration of MDOF Systems 坐标转换(Raylcigh-Ritz法) [M伐+[C]+[K]{x(}=-[Mx,) -M]05.0 {x}=[x]a} ()-(x[M])-ix1-i {()》=[x]{)明 {(}=[x]{} =r,, Orthogonality of Modes Geometry(几何坐标)mdardites(广义坐标】 :Mx1=0和{xKxl-0 20151020 301510y20 3.6 Free Vibration of MDOF Systems XKX9-X9()-X19.() XHMI x写as+aMW-a+axIw],, Divided by) modal equations Express thejmode contribute to general response {a,+(a+ae,)4,+e,{9,=y, 几大林情 3.6 Free Vibration of MDOF Systems 3.6 Free Vibration of MDOF Systems Modal Analysis The modal equations wllbe uncoupled and redues We define: M,i。+C4.+K,g.=P0 .Generalized modal mass: Or in matrix form: .Generalized modal stiffness: Mi+Cir+Ku=P(t) .Generalized modal damping:C. .Generalized modal forces: P.()=p 大木 20110w2 2015t0W2

8 M x t C x t K x t M I x t             ＝－   g   坐标转换 （Rayleigh-Ritz法） x t = q t   X    令: Geometry (几何坐标) modal coordinates (广义坐标) x t = t   X q    x t = t   X q    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 43                          1 2 1 ( ) ( ) M t M ( ) ( ) M t T T j j i n i n T j j j q t q t X X q X X X X q t q t X X q                                                     g   M t C t K q t M I x t T T T j j j T j X X q X X q X X X   ＝－   M 0 K =0        T T j j i i X X X X  和 Orthogonality of Modes 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 44                        2 K q t K q t K q t T T T j j j j j j j j X X X X X X                     2 1 0 0 1 ( K ) q t ( ) q t T T j j j j j X M X X M X         Divided by   M  T j j X X            2 2 0 1 g t ( + ) t q t - x t j j j j j j q q        ＝ 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 45 modal equations Express the j mode contribute to general response. 46 Mode Participant Coefficient 振型参与系数             n i 1 n i 1 2 i ji i ji j T j T j j m X m X X M X X M I ＝ ＝  ＝ ＝ 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 46 Modal Analysis  We define:  Generalized modal mass:  Generalized modal stiffness:  Generalized modal damping:  Generalized modal forces: n T Mn  n m n T Kn  n k n T Cn  n c P (t) (t) T n  n p 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 47  The modal equations will be uncoupled and reduces to:  Or in matrix form: M is the diagonal matrix of the generalized modal mass K is the diagonal matrix of the generalized modal stiffness C is diagonal matrix of the generalized modal damping P(t) is diagonal matrix of the generalized modal forces M q C q K q P (t) n n  n n  n n  n   Mu  Cu   Ku  P(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 48

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 9

9  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