Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsToday’s Objectives:Students will be able to:a) Find the homogeneous or steady state solution using analytical modal analysisAnalytical Modal AnalysisNote: Bring your laptop to class tomorrow!Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsOverview (from Pete Avitabile)Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsAnalytical Modal Analysis(MDOF free and forced vibration)EOM (1)Solve the eigenvalue problemFor eigenvalues:For eigenvectors:Mass normalize the eigenvectors:Form the modal matrix:[]{}[]{}[]{}{}FxKxCxM=++&&&Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsModal analysis (cont.)Apply a coordinate transformationInto (1) gives:Premultiply by [Φ]Tgives:Gives:Notes: – we get n uncoupled differential equations!– qi(t) is called a “principle coordinate” of, if [Φ] is mass normalized, a “normal coordinate”[][]{}[][]{}{}FqKqM=Φ+Φ&&[][][]{}[][ ][]{}[]{}FqKqMTTTΦ=ΦΦ+ΦΦ&&Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsModal analysis (cont.)Then uncoupled equations are:We know how to solve this!!How do we fine Aiand Bi? iiiiQqq =+2ω&&()()()tqtqtqPHiii+=Particular solution(depends on RHS)Homogeneous solution=Aicosωit+ BisinωitRose-Hulman Institute of TechnologyMechanical EngineeringVibrationsInitial conditions for normal coordinatesWe need Recall:But if we mass normalize the modes we have SoSo we getOne we find {q(t)} we can find {x(t)} by transforming back()()0 and 0 q q&(){}= 0q(){}= 0q&(){}[](){}tqtxΦ=Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsSummary of Analytical Modal Analysis1. Find EOM2. Solve the eigenvalue problem to find frequencies and modes• Mass normalize the modes3. Apply coordinate transformation (basically any motion can be considered as a superposition of the normal modes)• Obtain the decoupled equations of motion:• Find the generalized force: Qi(t) = [Φ]T{F(t)}• Find the initial conditions for {q(t)}:4. Solve the decoupled equations and apply IC’s to find qi(t)5. Transform back to find {x(t)}[]{}[]{}{}FxKxM=+&&[]{}[]{}XKXM =2ω[][]ΛΦ and [][][][][][][][] so Λ=ΦΦ=ΦΦ KIMTT(){}[](){}tqtxΦ=(){}(){}0 and 0 qq&iiiiQqq =+2ω&&(){}[](){}tqtxΦ=Rose-Hulman Institute of TechnologyMechanical EngineeringVibrationsExampleThe 3-DOF system shown is found to have the natural frequencies 0.40.30.1232221=ω=ω=ω and the mass normalized modal matrix: []⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=φ4083.05.02886.04083.005774.04083.05.02886.0 and mass matrix []⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=200020002M a) Using modal analysis determine the time response of each mass if the system is given the initial conditions 0)0()0()0()0(3211==== xxxx&&& and 1)0()0(32==xx b) What is the stiffness matrix for this problem? x1 x2
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