PSI - Issue 38

Amaury Chabod et al. / Procedia Structural Integrity 38 (2022) 382–392 Amaury CHABOD / Structural Integrity Procedia 00 (2021) 000 – 000

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Fig. 2. Load Reconstruction Process

4.2. Linear transient analysis In the case of a dynamic analysis, a common strategy employed in Finite Element Analysis is to solve the equations of motion by means of a modal superposition technique. We will show here how we can take advantage of the modal superposition and precisely define modal coordinates. A global assumption needed is to consider the structure as being fully linear elastic. The first step is to perform a modal analysis. As an intermediate result, we ask for modal vectors { } and eigenvalues 2 , by eigenvalues decomposition (He and Fu (2001), Dhatt and Thouzot (1984), Craig and Kurdila (2006)). ([ ] − 2 [ ]){ } = 0 (5) Modal vectors { } constitute an orthogonal basis, where: { } [ ]{ } = 0 { } [ ]{ } = 0 For ≠ (6) Further, as a result of diagonalization, for each mode n we have a scalar expression for mass and stiffness: { } [ ]{ } = { } [ ]{ } = (7) The global equation of motion is given below, linking mass, stiffness and damping matrix with displacement and its derivatives: [ ]{ ̈} + [ ]{ ̇} + [ ]{ } = { } (8) We search for solutions of the following form, projected on the modal basis: For displacements: { } = ∑ ( ){ } (9) For velocity: { ̇} = ∑ ̇ ( ){ } (10) For accelerations: { ̈} = ∑ ̈ ( ){ } (11) Combining with previous equations (9), (10) and (11) yields:

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