PSI - Issue 80

Luke Wyatt et al. / Procedia Structural Integrity 80 (2026) 31–42

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4 L. Wyatt et. al. / Structural Integrity Procedia 00 (2023) 000–000 where M , K and F are the global mass, sti ff ness and forcing matrices, and V and ¨ V are the generalised displace ments and their accelerations. Assuming the motion to be harmonic, the natural frequencies, ω i and vibration modes X i can be found by solving the eigenproblem  K − ω 2 i M  X i = 0 , (12) where i = 1 , 2 , ..., n dof , where n dof is the number of DOFs of the global system. The kinetic energy of an element, E k , is given by where ρ is the density of the plate [8]. The element mass matrix, M E , can then be approximated by the VEM functions as M E =  E ρ N T    t 3 12 0 0 0 t 3 12 0 0 0 t    N dE (14) where N is the collection of the VEM approximation functions for the generalised displacements. As N is not explicitly known over the element area, similarly to the calculation of the local element sti ff ness, K E , the functions are projected onto a polynomial subspace, and stabilised by an approximate polynomial form [9]. M E can therefore be approximated as where ˜ Π 0 and Π 0 are L 2 orthogonal projections onto the spaces P k ( E ) and V h | E respectively, and N P is the matrix of polynomial trial functions up to order k . It can be shown that for k = 1 elements Π 0 coincides with Π ∇ , the H 1 orthogonal projection, which has been calculated for K E [10]. Therefore only the integral of polynomial functions over the element needs to be performed, and so M E can be found with minimal extra computational e ff ort after the calculation of K E . The M can be assembled from the element matrices and Equation 12 can be solved to find the frequencies and mode shapes of the system. For an initially strained Mindlin plate element, the energy due to the in plane loads, U G , over the volume of the element, V E , neglecting powers higher than two in the displacement gradients, assuming small deflections, can be written as U G =  V E σ 0 T ε L dV E (17) = 1 2  E t ( ∇· w ) T ˆ σ 0 ( ∇· w ) dE + 1 2  E t 3 12 ( ∇· θ x ) T ˆ σ 0 ( ∇· θ x ) dE + 1 2  E t 3 12 ( ∇· θ y ) T ˆ σ 0 ( ∇· θ y ) dE , (18) 4.2. Buckling Analysis E k = 1 2  E ρ  t ˙ w 2 + t 3 12  ˙ θ 2 x + ˙ θ 2 y   dE , (13) M E = M C + M S (15) = ˜ Π 0 T  E N T P    t 3 12 0 0 0 t 3 12 0 0 0 t    N P dE ˜ Π 0 + 1 2 tr ( M C )( I − Π 0 ) T ( I − Π 0 ) , (16)