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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3

with Euler or Cardan angles after derivating the rotation matrix in the space or time domain. The quaternion p J. C. K. Chou, (1992) contain the informations of u and θ and is given by p = cos θ 2 + i u sin θ 2 . It denotes the rotation between the reference marker, called B 0 = e 0 x , e 0 y , e 0 z and the marker B k = e k x , e k y , e k z of the solid S k considered. The quaternion p is a number with four dimensions, three of which are purely imaginary. Then the quaternion can be expressed as a vector p = ( p 0 , p t v ) t with p 0 the real part and p v = ( p 1 , p 2 , p 3 ) t the imaginary part. By definition, p t p = 1. In order to process similarly the 2D models and 3D models, the 2D axis of rotation u is normal to the 2D model plane. The 2D quaternion p is thus simplified as following p = cos θ 2 , sin θ 2 t = ( p 0 , p 1 ) t . To build the model with quaternion representation for the prediction step of the Kalman filter, we need to use the specific function denoted Φ ( . ) , the operator on vectors + ( . ) and he rotation matrix R 0 , k from B 0 to B k and defined by: Φ p = [ − p 1 p 0 ] , + p = p 0 − p 1 p 1 p 0 and R 0 , k = + p + p (1) For quaternions p a and p b , Φ ( . ) and + ( . ) respect some use-full relations: + p a + p b = + p b + p a , + p a p b = + p b p a and Φ p a p b = − Φ p b p a (2) The derivation of quaternion p according to the time domain and the rotation rate orthogonal to the plan ˙ θ are given by : ˙ p = ( ˙ p 0 ˙ p 1 ) = ˙ θ 2 − sin θ 2 cos θ 2 and ˙ θ = 2 Φ p ˙ p (3) 2.2. Multi-body equations Let x k be the gravity center position G k in the terrestrial reference frame and p k the attitude of the solid S k . Let B k be the marker associated with S k and k be the body index. Let q k = ( x t k p t k ) t be the state vector of the rigid body S k . To compute accelerations ¨ q n in the Kalman prediction, we use the principle of virtual power adapted to the quaternion representation. First, a description of the virtual kinematic and kinetic energy are given according to the quaternion. The virtual power of loads and virtual power of accelerations are then calculated. Finally we identify the virtual power of loads to the virtual power of accelerations. In addition, we have to take the kinematic and quaternion constraints into account to properly compute accelerations for the multi-body model. The virtual speed V ∗ P k , j ∈S k / 0 of a point P k , j , with j is the index of point in the solid, and the virtual rotation rate Ω ∗ S k / 0 in the body S k are : V ∗ P k , j ∈S k / 0 Ω ∗ S k / 0 = ˙ x ∗ k + 2 + p k + G k P k , j B k ˙ p ∗ k 2 Φ p k ˙ p ∗ k (4) Let F P k , j ∈S k / 0 denote the force applied at the point P k , j in the body S k and T P k , j ∈S k / 0 the torque at the point P k , j in the body S k . The virtual power of external forces P k , is the sum of virtual power of load apply in P k , j , denote P k , j : P k , j = V ∗ P k , j ∈S k / 0 t F P k , j ∈S k / 0 + Ω ∗ S k / 0 t T P k , j ∈S k / 0 = q ∗ k D k , j F k , j (5) With

=    2 + p

t    and F k , j = F P

I

0

k , j ∈S k / 0 T P k , j ∈S k / 0

2

2 × 1

D

B k

+ G k P k , j

k , j

2 Φ p k

k