/FRAME/MOV

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Definitions

Comments

1. Let a moving reference frame ${\Lambda }_t\left(A,u,v,w\right)$ .
2. For each time t, the frame position and orientation are determined via its original position $x_A$ and a rotation (orientation) matrix $R$ .
3. Let $w$ be the instantaneous rotational velocity of $\text{Λ}$ .
4. For each time t, the local coordinates $x_l$ of a point M with respect to the frame are related to its coordinates $x_G$ into the global system, as:(1)
   $$x_G=x_A+R{x}_l$$
5. The relative displacement $u_l=x_l-x_l^0$ of M between time 0 and t, with respect to the frame is related to its displacement with regard to the global system, as:(2)
   $$u_G=u_A+\left(R-R^0\right)x_l+R{u}_l$$
6. The relative velocity of M with respect to the frame is related to its velocity with regard to the global system, as:(3)
   $$R{v}_l=v_G-v_e$$

   Where, $v_e=v_A+\omega \times AM$ is the driving velocity; that is the velocity of the point coincident with M at time t and fixed with respect to the reference frame.

7. The relative acceleration of M with respect to the frame M is related to its acceleration with regard to the global system, as:(4)
   $$R{\gamma }_l={\gamma }_G-{\gamma }_e-{\gamma }_c$$

   Where, ${\gamma }_e={\gamma }_A+\raisebox{1ex}{$d\omega $}\!\left/ \!\raisebox{-1ex}{$dt$}\right.\times AM+\omega \times \left(\omega \times AM\right)$ is the driving acceleration and ${\gamma }_c=2\omega \times v_{relative}$ is the acceleration, due to Coriolis forces.

8. For a moving reference frame, the reference frame position and orientation vary with time and are defined by N₁, N₂ and N₃.
   The axis definition depends on the input for Dir.

   • When Dir=X

     node_ID₁ amd node_ID₂ define $X\text{'}$

     node_ID₁ and node_ID₃ define $Y"$

     • $Z\text{'}=X\text{'}\Lambda Y"$
     • $Y\text{'}=Z^{\prime }\Lambda X^{\prime }$

     
     
     Figure 1. Moving System with Dir=X

   • When Dir=Y

     node_ID₁ and node_ID₂ define $Y\text{'}$

     node_ID₁ and node_ID₃ define $Z"$

     • $X\text{'}=Z^{″}\Lambda Y^{\prime }$
     • $Z\text{'}=X^{\prime }\Lambda Y^{\prime }$

     
     
     Figure 2. Moving System with DIR=Y

   • Case Dir=Z

     node_ID₁ and node_ID₂ define $Z\text{'}$

     node_ID₁ and node_ID₃ define $X"$

     • $Y^{\prime }=X^{″}\Lambda Z^{\prime }$
     • $X^{\prime }=Y^{\prime }\Lambda Z^{\prime }$
     The skew is defined by $X\text{'}$ , $Y\text{'}$ and $Z\text{'}$

     
     
     Figure 3. Moving System with DIR=Z

9. In a 2D analysis N₁ and N₂ define $Y\text{'}$ .
10. Reference frame identifier must be different from all skew identifiers.