1 / 5

Solving eigenvalue problems : modal analysis & buckling

Solving eigenvalue problems : modal analysis & buckling. J.Cugnoni , LMAF/EPFL, 2012. 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

emma
Download Presentation

Solving eigenvalue problems : modal analysis & buckling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Solvingeigenvalueproblems: modal analysis & buckling J.Cugnoni, LMAF/EPFL, 2012

  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 • Cannot impose force or non-zerodisplacement as wesolve a free vibration problem

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

  4. 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

  5. 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

More Related