1. J.Cugnoni, LMAF/EPFL, 2011

2.  Goal: ◦ extract natural resonnance frequencies and eigen modes of a structure  Problem statement ◦ Dynamics equations (free vibration = no force) M u’’ + K u = 0 ◦ Class of solution: u = U e i  t ◦ Find the eigen modes U and eigen values =  2 of the problem (K -  M  U = 0  Boundary conditions: ◦ Only zero-displacement boundary conditions are allowed but not necessary ◦ Imposed degrees of freedom => the K and M matrices already contain the boundary conditions (elimination of imposed degrees of freedom)

3. Elimination of imposed DoFs: example for statics After elimination of imposed DoFs In modal analysis, imposed DoFs are eliminated from the stiffness and mass matrix So only K ff and M ff are used

4. Without boundary conditions K is singular and to solve the modal problem, K -1 is needed. But M is always positive definite, so we can « cheat » a bit and introduce a virtual frequency  shift in the problem to get a positive definite K’ matrix: So to solve a modal analysis problem with at least one free rigid body motion; you will need to define a frequency shift  To be sure to have all modes in the solution  should be taken between 0 and the  1 2 where  1 2 denotes the lowest « flexible » eigenfrequency.

5. Be careful with symmetries in modal analysis: If you use symmetries in modal analysis, you will obtain only the eigenmodes that statisfy the symmetry condition but you will miss all the anti-symmetric modes and their eigen frequencies !!! So in most cases geometrical and material symmetries should not be considered to build a modal analysis model of a structure Symmetry planeSymmetric modesAnti - Symmetric modes OK NOT OK

6.  Goal: ◦ Extract the maximum compressive force before elastic instability occure using a linearized theory (small perturbation)  Problem statement ◦ In the initial state (can include a preload P) the stiffness matrix of the system is K 0. But the apparent stiffness matrix changes if the part is deformed. The change of stiffness with geometrical configuration change is represented by the geometrical stiffness matrix K g associated to a loading Q of arbitratry magnitude ◦ The system becomes unstable when the applied force F = P + Q ◦ Where is obtained from the buckling eigenvalue equation: (K 0 + K g ) U = 0 U represents the buckling modes and is called the buckling force multiplier