DYNAMICS  THEORY

Example of rotating rigid bodies with
moving point on one body 

Previously, multiple rigid bodies that were connected with pin joints were analyzed. This section considers the same rigid body motion, but now a point on one of the rigid bodies is allowed to moved and two coordinate systems are used to model the motion. A good example of this type of motion would be two links connected with a sliding connection, as shown in the diagram.
To analyze a rotating rigid body with a moving point we need to start with the basic relative position vector equation and take a time derivative. To help with modeling complex systems, two coordinate systems are used: XY is fixed and xy rotates in the XY system. Note, xy system is not fixed to the body.




Position vectors 

Start with the basic position vector,
r_{B} = r_{A} + r_{B/A}
The first time derivative gives,
d(r_{B})/dt = d(r_{A} + r_{B/A})/dt
v_{B} = v_{A} + d(r_{B/A})/dt
A simple derivative for r_{B/A} is not correct since the coordinate system, xy, also is changing which must be accounted for when taking a derivative. 



Time Derivative of i and j
Velocity vectors


Using x_{B} and y_{B} as the scalar components of r_{B/A} gives,
v_{B} = v_{A} +
d(x_{A}i + y_{B}j)/dt
It is important to note that di/dt ≠ 0 and dj/dt ≠ 0 since i and j can rotate. Using the chain rule, this can be expanded as
The unit vectors, i and j, can rotate so their time derivatives are di/dt = Ω × i and dj/dt = Ω × j. Also, note that dx_{B}/dt i and dy_{B}/dt j is simply the relative velocity of B with respect to A. This gives
v_{B} = v_{A} + (v_{B/A})_{rel} + x_{B}Ωj +
y_{B}Ωi
The last two terms can be converted to a cross product to give

v_{B} = v_{A} + Ω × r_{B/A} + (v_{B/A})_{rel} 

This is the classic form of the relative motion equation using rotating axes. The key term is Ω × r_{B/A} which relates the xy coordinate system with the XY coordinate system. 





Coordinate Transformation



Since two coordinates are involved with most rotating coordinate system problems, they need to be related. In other words, one coordinate system needs to be mapped into the other, and vce 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 descripbe one system in the other, just like is done for a point or vector. For example, the i unitvector i can be described in the coordinate system XY as
i = cosθ I + sinθ J
likewise, the j unitvector is
j = sinθ I + cosθ 





These relationships can be inverted to describe I and J using i and j unitvectors
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 xysystem is rotated an angle θ relative to the XYsystem. 


