NC080612DP01 - Matrix expression of the torsor transposition (Matrix cross product).doc

Rév. juin 2008

Note de Calcul Matrix expression of the torsor transposition (Matrix cross product) Résumé : Matrix expression to change the orientation of a torsor to be used in codes for forces and moments calculation.

1. Introduction Vectorial product (or vector cross product) is used in mechanics to transfer a torsor from a base to another which has distinct origin and angle between axis X, Y and Z. This operator is particulary used in codes to transpose thes torsors on a same space to apply the fundamental principle of dynamic which requires that torsors must be in a same space.

fig 1. Cross product representation

2. Matrix expressions 2.1. Global transfer matrix [M] General expression of global transfer matrix [M] (6x6) from coordinate system R1 to R2 with respective origins O1 and O2 (see on Appendix A) coud be written as :

  PR1 ⇒ R2  [0]     M R1 ⇒ R2  =   PR ⇒ R  ∗  ∆ O O   PR ⇒ R    1 2   1 2   1 2   2.2. Single rotation matrix [P] [P] is the single rotation matrix (3x3); its expression depends to angular shift between coordinate systems R1 to R2. For a rotation around x axis of an angle θ, rotation matrix [P] (3x3) will be as described below :

0 1  [ PNN ' ] = 0 cos θ 0 sin θ

 − sin θ  cos θ  0

2.3. Single translation matrix [∆ ∆] [∆] is a skew-symmetric (or antisymmetric) single transfer matrix (3x3); it depends to coordinates of vector

O1O2 in coordinate system R1 :

[∆ ] O1O 2

David PERRIN

z − y 0  = − z 0 x   y − x 0 

1/2

NC080612DP01 - Matrix expression of the torsor transposition (Matrix cross product).doc

Rév. juin 2008

2.4. Matrix expression of the vectorial product The torsor

{T

}

S1 → S 2 O , R 1 1

representing the resultant of loads R and moments M of solid S1 to solid S2 on origin

point O1 of coordinate system R1 is :

{T

}

S1 →S 2 O , R 1 1

 R Si →S j  =  M O1 ,Si →S j  R 1

The definition of

{T

{T

}

[ [

] ]

S1 →S 2 O , R 2 2

{T

from the expression of

}

S1 →S 2 O , R 1 1

 PR ⇒ R ∗ R S1→S2 1 2 =  PR1⇒ R2 ∗ M O1 ,S1→S2 + PR1⇒ R2 ∗ R S1 →S2 ∧ O1O2

}

S1 →S 2 O , R 2 2

](

[

is :

   R 2

)

Where O1O2 = x ⋅ x1 + y ⋅ y1 + z ⋅ z1 is the vector of origins of R1 and R2 in coordinates systems R1 and [P] the corresponding rotation matrix (3x3) depending to angular shift between coordinate systems R1 to R2. Vector product for moment could be altered :

[P

]∗ (R

Where

[∆ ] is the single translation matrix (3x3) of vector O O

R1 ⇒ R2

S1 →S 2

) [

](

]) [

[

] ([

]

∧ O1O2 = PR1⇒ R2 ∗ R S1→S2 * − ∆ O1O 2 = PR1⇒ R2 ∗ ∆ O1O 2 * R S1 →S2

O1O 2

1

[∆ ] O1O 2

2

)

coordinates in R1 :

z − y 0  = − z 0 x   y − x 0 

We have finally :

] ]

([

Defining the single vector

F O1 ,R1 = R Si →S j

{T

}

S1 →S 2 O , R 2 2

[ [

 PR ⇒ R ∗ R S1→S2   1 2  =   PR1⇒ R2 ∗ M O1 ,S1→S2 + PR1⇒ R2 * ∆ O1O 2 ∗ R S1→S2  R 2

(

][

])

M O1 ,Si →S j

)

R1

with size (6x1) which contains both loads and

moments, the expression could be resumed as following single matrix product :

[

]

F O2 ,R2 = M R1⇒ R2 ∗ F O1 ,R1 With :

[M

] = [P [P ]∗ [∆] ] [P[0] ]   R1⇒ R2

R1 ⇒ R2

R1 ⇒ R2

O1O2

R1⇒ R2

Bibliography : [1]

Wikipedia.- Article « Cross product ».- http://en.wikipedia.org/wiki/Cross_product

David PERRIN

2/2