PSI - Issue 38

386 Amaury Chabod et al. / Procedia Structural Integrity 38 (2022) 382–392 Amaury CHABOD / Structural Integrity Procedia 00 (2021) 000 – 000 [ ] ∑ ̈ ( ){ } + [ ] ∑ ̇ ( ){ } + [ ] ∑ ( ){ } = { } For any mode n, we have that: { } [ ]{ } ̈ ( ) + { } [ ]{ } ̇ ( ) + { } [ ]{ } ( ) = { } { } From equation (7), we obtain for any mode, thanks to diagonality, a scalar expression: ̈ ( ) + ̇ ( ) + ( ) = (t) (14) Using the definition of the pulsation as given by equation 15, we obtain a series of independent differential equations: ² = (15) ̈ ( ) + ̇ ( ) + ² ( ) = ( ) (16) The quantity ( ) is defined as modal force vector and ( ) as modal coordinates or generalized displacements. ( ) = { } { } (17) Consequently, the result of a transient analysis is obtained as a series of modal coordinates, using the projection in the modal basis. Stresses { }( ) are reconstructed, combining linearly modal coordinates ( ) and modal stresses { } . This process is referred to as modal superposition. { } = [ ][ ]{ } , where [D] is Hooke ’s law matrix and [B] is a matrix containing the derivation of element shape functions. Replacing {u} with equation (9) yields: { } = [ ][ ] ∑ ( ){ } (18) Using: { } = [ ][ ]{ } (19) It follows that: { }( ) = ∑ ( ){ } (20) Using modal coordinates to retrieve stress time history will greatly reduce the amount of result data needed, as described in Fig. 3. Indeed, stresses are only output for n modes on the whole structure, whereas they should be read for all time steps in case of reading results from a direct transient analysis. (12) (13) 5

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