13. “Exact” Dynamic Analysis of Linear MDOF Structures• Free Vibrations of Undamped Systems–Example –Rayleigh’s Methodpygm00m =[m] 83-3-2 l76EI =[k] 3 EICIE 619 Chapter 4 – Seismic Analysis103 mlEI2,815 = mlEI0,806 = 32313. “Exact” Dynamic Analysis of Linear MDOF Structures• Free Vibrations of Undamped Systems–Example –Rayleigh’s MethodpygWe use a static load vector corresponding to the weight associated to the degrees-of-freedom. The fundamental mode of vibration corresponds to the static deformations under this static load vector. mgmg = (F) CIE 619 Chapter 4 – Seismic Analysis104511 EI 6l mg = mgmg 2 33 8 EI 6l = )( (F)[f] = (F) ][k = )( )A(33-1(1)23. “Exact” Dynamic Analysis of Linear MDOF Structures• Free Vibrations of Undamped Systems–Example –Rayleigh’s MethodpygSubstituting Rayleigh’s formula, it yields :  5 + 11 (mg) 6EImgl g )(5 + )(11 (mg) 6EImgl 2 T322321jjNj=12jjNj=11 W g W 2 T CIE 619 Chapter 4 – Seismic Analysis105 We notice that the difference between the estimate of Rayleigh’s formula and the exact fundamental period calculated in equation 4.176 is only 0.6 %. EIml 7,749 T31 EIml7,796 = T 313. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition MethodStrategy. The modal analysis (or the modal superposition method) is a very useful technique to determine the dynamic response of a linear MDOF system subjected to an arbitrary dynamic load (or a base acceleration). The governing system of differential equations is written as: where [m] = global mass matrix (usually diagonal) [c] = global damping matrix (hard to evaluate) [k] = global stiffness matrix (F(t))=dynamic load vector(F(t)) = (x)[k] + )x([c] + )x([m]  (4.239)=-(m) (r)¨xs(t).CIE 619 Chapter 4 – Seismic Analysis106(F(t)) dynamic load vector(x), (x) et (¨x) = relative displacement, velocity and acceleration vectors In general, the differential equations can be coupled by [m], [c], or [k]. In other words, these matrices are not necessarily diagonal matrices. If mij exists for i  j, the system is said to be dynamically coupled. The basic strategy of the modal analysis is to introduce a linear transformation of the variables, using the modal matrix, [A], in order to uncouple the equations of motion. (m) (r) xs(t).33. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition MethodNormal CoordinatesTo uncouple the equations of motion a linear transformation isNormal Coordinates. To uncouple the equations of motion, a linear transformation is introduced using the modal matrix, [A]. This transformation converts the differential equations from the real coordinates (x(t)) (or degrees-of-freedom) into a new system of normal coordinates (u(t)). (u(t))[A] = (x(t)) (4.240)CIE 619 Chapter 4 – Seismic Analysis1073. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition MethodUncoupling the Equations of Motion. Equation 4.240 is substituted into equation 42394.239. Equation 4.241 is premultiplied by the transpose of a specific mode shape (A(i))T : Remembering the orthogonality conditions of the mode shapes, for i  j : (F(t)) = (u)[A] [k] + )u([A] [c] + )u([A] [m]  (F(t)))A( = [k][A](u))A( + )u[c][A]()A( + )u[m][A]()A(T(i)T(i)T(i)T(i) (4.241)(4.242)CIE 619 Chapter 4 – Seismic Analysis108 and also assuming for the moment for i  j : 0 = )A([k] )A( = )A([m] )A((j)T(i)(j)T(i) 0 = )A([c] )A((j)T(i) (4.244)43. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition Methodthen only one relation inuremains in equation 4 242then, only one relation in uiremains in equation 4.242.where i mode oft coefficien damping dgeneralize = )A( [c] )A( = C i mode of mass dgeneralize(i)T(i)i = )A([m] )A( = M(i)T(i)i(t) P = u K + u C + u Miiiiiii (4.245)CIE 619 Chapter 4 – Seismic Analysis109 i mode of load dgeneralize = (F(t)) )A( = (t) Pi mode oft coefficien stiffness dgeneralize = )A( [k] )A( = KT(i)i(i)T(i)i 3. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition MethodEquation 4.245 has the same form as equation 4.64 for a SDOF system. Equation 4.245was already solved by the Duhamel’sintegral(equation 4 69)was already solved by the Duhamel s integral (equation 4.69). where d )-(t e )(P M1 = (t)ud)-(t-it0diiiiiisin i2iK=CIE 619 Chapter 4 – Seismic Analysis1102iidiiiiii-1 = M2C = Mi53. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition MethodAfter solving for each normal coordinate ui (i= 1 to N), the response of each degree-of-freedom can be determined using the same linear transformation.g The response of a specific DDOF, xi(t), is found by : (u(t))[A] = (x(t))  (t)u A = (t)xj(j)iNj=1im CIE 619 Chapter 4 – Seismic Analysis111 with Nm representing the number of modes considered in the modal analysis. In general, it is not necessary to consider all the modes of vibration and Nm < N in the modal analysis of a civil engineering structure. d )-(t e )(P MA = (t)xd)-(t-jt0jj(j)iNj=1iijjmsin3. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition MethodNecessary Conditions for Damping MatrixIn the previous section the assumptionNecessary Conditions for Damping Matrix. In the previous section, the assumption was made that the modes of vibration were orthogonal with respect to the dampingmatrix (equation 4.244). The modes of vibration which satisfy equation 4.244 are called classical normal modes. These classical normal modes ensure that the equations ofmotion are uncoupled and that the damped modes of vibration are identical to theundamped modes of vibration. In this section, we study the conditions that the damping matrix must satisfy in order to ensure the existence of classical normal modes. CIE 619 Chapter 4 – Seismic Analysis11263. “Exact” Dynamic Analysis of Linear MDOF Structures• Modal Analysis - The Modal Superposition MethodRayleigh’s