ASEN5519, Fall 2004. Homework 01 (Due: Monday, 20 September 2004)Problem: Understanding Partitioned Formulations and Interface CouplingThe present homework is to familiarize the basic partitioned formulation using two substructuresinteracting along their interfaces. First, we recall that the partitioned structure-structure interactionequations of motion may be expressed asK1+ M1D20B1000K2+ M2D20B20BT1000−Lf 10BT200−Lf 200−LTf 1−LTf 20u1u2λ1λ2uf=f1f2000(1)where D =d2dt2.Of several approaches to gaining insight into the coupling mechanisms, you are asked to performthe following eigenvalue problems and study them. For two partition problems, you may select,for example, beam-beam interactions as shown below.12345123123fFour-element beam model partitioned into two domainsPartition 1 Partition 2Coupled System(1) When each of the two partitions consists of two free-free beam elements, carry out partitioneddomain eigenvalue analysis given byFor domain 1: K1X1= M1X1Λ1For domain 2: K2X2= M2X2Λ2(2)(2) Using the relationu1= X1q1, u2= X2q2(3)transform (2) to the following coupled eigenvalue problem:AcXc= BcXcΛcAc=Λ10XT1B1000 Λ20XT2B20BT1X1000−Lf 10BT2X200−Lf 200−LTf 1−LTf 20Bc=I100000I20000 00000 00000 0000(4)Note that, once the above eigenvalue problem is carried out, the coupled global frequenciesare given by Λcand the global mode shapes are given byΦ1= X1Xc(1:n1, :), Φ2= X2Xc((1 + n1) : (n1+ n2), :)(5)where (n1, n2) are the degrees of freedom for subdomains 1 and 2, respectively. Note thatthere are a total of (n1+ n2− nf) global modes for this problem where nfis the interfaceframe (coupling) degrees of freedom.Plot the subdomain mode shapes and its local frequencies and the coupled mode shapes andthe corresponding frequencies. Can you explain how the local modes shapes are changed intocoupled mode shapes?(3) Letusobtaintheanalyticalfrequencyresponsefunctionsofthepartitioned systemsandcoupledsystem given byH1= [K1− ω2M1]−1H2= [K2− ω2M2]−1Hc= [Ac− ω2Bc]−1(6)Plot a set of representative FRFs for the partitioned systems and the coupled systems at thesame sample locations (i.e., the same nodes). Can you gain physical insight regarding theeffect of coupling by comparing the partitioned FRFs vs. those of the coupled system. Inusing Hc(ω), observe that the physically relevant portion is Hc(:, 1:(n1+ n2)). (why?)(4) Equation(1) suggests that the interface forces λ1and λ2are given byλ1= [BT1B1]−1BT1(f1− [K1+ M1D2]u1)λ2= [BT2B2]−1BT2(f2− [K2+ M2D2]u2)(7)which implies thatHλ1= K1C1HcHλ2= K2C2HcK1= [BT1B1]−1BT1(I1− [K1− ω2M1]K2= [BT2B2]−1BT2(I2− [K2− ω2M2](8)Discuss how you may gain physical insight by examining the above expressions, especiallyfor understanding the nature of interface coupling. Hint: First plot (C1Hc), (C2Hc), K1andK2individually. Then plot Hλ1and Hλ2.(5) Summarize your findings that you have learned from this homework.Help: Forthosewho mightneed helpin programming, a sample bareigenvalueproblem isprovidedalongwith beam element
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