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DYNAMICS - THEORY


Position Vectors
 

The acceleration of a moving point on a rotating rigid body is more complex than its velocity since there are tangent and normal terms.

To determine the acceleration of a rotating rigid body with a moving point start with the relative velocity vector equation that was derived in the previous Rotating Coordinates:Velocities section. To help with modeling complex systems, two coordinate systems are used: X-Y is fixed and x-y rotates in the X-Y system. Note, x-y system is not fixed to the body.

     


Velocity Vectors


Time Derivative of i and j


Acceleration Vectors

 

Start with the relative velocity equation for a moving frame of reference:

     vB = vA + Ω × rB/A + (vB/A)rel

Differentiate with respect to time:

   aB = aA + dΩ/dt × rB/A + Ω × drB/A/dt + dvB/A/dt

Similar to velocities, the x-y coordinate system is rotating inside the X-Y coordinate system. This means the time derivatives of the unit vectors, di/dt and dj/dt are not zero but

     di/dt = Ω × i
     dj/dt = Ω × j

Thus, the time derivative of the position vector, rB/A is

     

Putting all the terms together gives,

 
aB =aA + dΩ/dt × rB/A + Ω × (Ω × rB/A
+ 2 Ω × (vB/A)rel + (aB/A)rel
 



   
    Coordinate Transformation

   

Since two coordinates are involved with most rotating coordinate system problems, the two systems need to be related. In other words, one coordinate system needs to be mapped into the other, and vice versa. Relating two coordinate system is a common task in most all engineering fields, not just dynamics.

     

Velocity vectors
 

The easiest way to relate coordinate systems is to describe one system in the other, just like is done for a point or vector. For example, the i unit-vector i can be described in the coordinate system XY as

   i = cosθ I + sinθ J

likewise, the j unit-vector is

   j = -sinθ I + cosθ

     
   

These relationships can be inverted to describe I and J using i and j unit-vectors

   I =  cosθ I - sinθ J
   J = sinθ I  + cosθ J

     
   

These transformation equations are commonly written in matrix form, and called the 2D coordinate transformation matrix

 
 

They can be used for any two coordinate systems where the xy-system is rotated an angle θ relative to the XY-system.

     
   
 
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