PSI - Issue 38

A. Débarbouillé et al. / Procedia Structural Integrity 38 (2022) 342–351 A. De´barbouille´ / Fatigue Design (2022) 1– ??

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The virtual power of acceleration is calculated thanks to the kinetic energy U k , the mass m k of the body S k and the inertial matrix I B k S k expressed in the marker B k of the body S k : U k = 1 2 m k ˙ x t k ˙ x k + t Ω S k / 0 (6) The virtual power of acceleration is obtained by the derivation according to the time of kinetic energy in the equation (6) : A k = ( ˙ q ∗ k ) t   d dt ∂ U k ∂ ˙ q − ∂ U k ∂ q   = ( ˙ q ∗ k ) t    M k    ¨ x k ¨ p k    + J k    (7) With : 1 2 Ω t S k / 0 I B k S k According to the principle of virtual power, P k , = A k whatever the value of virtual speed i.e. P k , = j P k , j . According to the equations (5) and (7) the unconstrained accelerations are then given by : M k ¨ q k = j D k , j F k , j − J k (8) 2.3. The constraints equations In the multi-body model, there are constraints between bodies. In addition other constraints are intro duced by the need to have a normalized quaternion to calculate rotation matrix, rotation rate, J k and M k . We choose the method of Lagrange multiplier in order to take the constraints into account. In this case the mass matrix M k is non-invertible. The quaternion adds a new degree of freedom: it’s norm. After adding the unit norm constraint for the quaternion to the matrix M k is becomes invertible. The residuals of the constraint quaternion norm of the body S k is defined by the function g p : g p ( q k ) = p t k p k − 1 (9) The Lagrangian method use the second derivation order of g p in the time domain : The equation (10) is added to the equation (8) and the term W p , k is added to represent stabilization term of constraint of quaternion norm : M k   ¨ q k λ p , k   = F k With M k =    M k G t p , k G p , k 0    and F k =    j D k , j F k , j − J k L p , k + W p , k    (11) Where M k the mass matrix of body S k , F k the generalized load applied to the body S k and λ p , k the Lagrange multiplier. The stabilization term W p , k added in the equation (11) tends to compensate the devi ation thanks to Baumgarte’s stabilization method, with ω and ξ chosen according to P. Flores (2011). The stabilization term is given by : W p , k = − 2 ξω ˙ g p ( q k ) − ω 2 g p ( q k ) (12) The constraints between bodies defined by g B ( q ) = G B ¨ q − L B = 0 n c × 1 where n c is the number of constraints. They are added to the equation system defined by the equations (11) of each body S k . ¨ g p ( q k ) = G p , k ¨ p k − L p , k with G p , k = 2 ˙ p t k and L p , k = 2 p t k ¨ p k (10) M k =    m k I 0 2 × 2 2 0 2 × 2 t I B k S k 4 Φ p k Φ p k    and J k =    8 Φ 0 2 × 1 ˙ p k t I B k S k Φ p k ˙ p k   