/FRAME/MOV2

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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 $\lambda$ .
4. For each time t, the local coordinates of $x_l$ 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$}\right.\times AM+\omega \times \left(\omega \times AM\right)$

   Driving acceleration

   ${\gamma }_c=2\omega \times v_{relative}$

   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 origin of the frame is defined by the position of N₁.

   node_ID₁and node_ID₂ define $Z^{\prime }$

   node_ID₁ and node_ID₃ define $X"$ (5)

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

   (6)

   $$X\text{'}=Y\text{'}\Lambda Z\text{'}$$

   
   
   Figure 1.

   Reference frame identifier must be different from all skew identifiers.