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 2015t0W28 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