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Solving eigenvalue problems : modal analysis & buckling

Solving eigenvalue problems : modal analysis & buckling. J.Cugnoni , LMAF/EPFL, 2011. Modal analysis. Goal: extract natural resonnance frequencies and eigen modes of a structure Problem statement Dynamics equations (free vibration = no force) M u ’’ + K u = 0

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Solving eigenvalue problems : modal analysis & buckling

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  1. Solvingeigenvalueproblems: modal analysis & buckling J.Cugnoni, LMAF/EPFL, 2011

  2. Modal analysis • Goal: • extractnaturalresonnancefrequencies and eigen modes of a structure • Problemstatement • Dynamics equations (free vibration = no force) M u’’ + K u = 0 • Class of solution: u = U e i w t • Find the eigen modes U and eigen values l = w2 of the problem (K - l M) U= 0 • Boundary conditions: • Onlyzero-displacementboundary conditions are allowed but not necessary • Imposeddegrees of freedom => the K and M matrices alreadycontain the boundary conditions (elimination of imposeddegrees of freedom)

  3. Modal analysis : boundary conditions Elimination of imposedDoFs: example for statics Afterelimination of imposedDoFs In modal analysis, imposedDoFs are eliminatedfrom the stiffness and mass matrix So onlyKff and Mff are used

  4. Modal analysis : free boundary conditions Withoutboundary conditions Kissingular and to solve the modal problem, K-1 isneeded. But M isalways positive definite, sowecan « cheat » a bit and introduce a virtualfrequencydl shift in the problem to get a positive definiteK’matrix: So to solve a modal analysisproblemwithat least one free rigid body motion; youwillneed to define a frequency shift dl. To be sure to have all modes in the solution dlshouldbetakenbetween 0 and the w12wherew12 denotes the lowest « flexible » eigenfrequency.

  5. Modal analysis : symmetries Be carefulwithsymmetries in modal analysis: If you use symmetries in modal analysis, youwillobtainonly the eigenmodesthatstatisfy the symmetry condition but youwill miss all the anti-symmetric modes and theireigenfrequencies !!! So in most cases geometrical and materialsymmetriesshould not beconsidered to build a modal analysis model of a structure Symmetry plane Symmetric modes Anti - Symmetric modes OK NOT OK

  6. Buckling • Goal: • Extract the maximum compressive force beforeelasticinstabilityoccureusing a linearizedtheory (small perturbation) • Problemstatement • In the initial state (caninclude a preloadP) the stiffnessmatrix of the system isK0 . But the apparent stiffnessmatrix changes if the part isdeformed. The change of stiffnesswithgeometrical configuration change isrepresented by the geometricalstiffnessmatrixKgassociated to a loadingQ of arbitratry magnitude • The system becomesunstablewhen the applied force F = P + l Q • Wherelisobtainedfrom the bucklingeigenvalueequation: (K0 + lKg ) U = 0 Urepresents the buckling modes and liscalled the buckling force multiplier

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