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DYNAMICS - THEORY

    Translational Motion

   

The translational motion of a rigid body is defined in terms of the acceleration of the body's center of gravity relative to an inertial reference frame XYZ is

     ΣF = macg

The scalar form of this equation is

 
ΣFx = macg-x
ΣFy = macg-y
ΣFz = macg-z
 

These equations are the same regardless if the problem is 1D, 2D or 3D.

     
    Rotational Motion

   

In addition to the translational motion equation, the equations for the rotational motion of a body that is moving in 3-D with an angular velocity ω relative to XYZ need to be derived.

For this purpose, it is convenient to introduce a second coordinate system xyz that translates with the rigid body and has an angular velocity Ω relative to XYZ. Note that Ω is not necessarily equal to ω.

By definition, the sum of the moments about a fixed point O, or the center of gravity cg, is equal to the time rate of change of total angular momentum about that point:

     

Because xyz may now be rotating, the previously derived angular acceleration equation needs to be used again,

     dA/dt = (dA/dt)xyz + (Ω × A)

This equation can be used to relate the time derivative of the angular momentum with respect to XYZ. This gives,

     

Depending on the motion of the body, there are three cases for the angular velocity Ω of the xyz axes that need to be examined.

     
    Case 1: Ω = 0


Case 1 - Reference Axis Motion
 

In the case Ω = 0, the xyz axes only translate with the body, thus the derivative of the angular momentum in XYZ is the same as in xyz,

     

The body may have an angular velocity ω about xyz. This form is useful if ω is known at a given instant: specifically, when the body is oriented such that xyz represent the principal axes of the body. As the body rotates, however, the moments and products of inertia about xyz will change as a function of time. Thus, this method can be cumbersome.

     
    Case 2: Ω = ω


Case 2 - Reference Axis Motion
 

The xyz axes may be chosen so that they are fixed and rotate with the body, i.e. Ω = ω.

The derivative of the angular momentum in XYZ is

     

where ω = ωxi + ωxj + ωxk
         H = Hxi + Hyj + Hzk
        

Substitute and expand the cross product:

     

                (1)

     
   

The xyz axes can be oriented in the rigid body so that they are always the principal axes of inertia:

     Ixx = Ix       Iyy = Iy        Izz = Iz   

     Ixy = Iyz = Ixz = 0

Equations 1 reduce to

 
               (2a
               (2b
                (2c
 

These are commonly referred to as the Euler equations of motion. These equations are only valid when summed about a fixed point O or the center of gravity.

     
    Case 3: Ω ≠ ω


Case 3 - Reference Axis Motion

 

In some cases, it is convenient for the xyz axes to have an angular velocity Ω that is different from the angular velocity ω of the body. This is useful when analyzing the motion of spinning tops or gyroscopes. The derivative of the angular momentum in XYZ is

     

where

     Ω = Ωxi + Ωyj + Ωzk

Substitute and expand the cross products gives,

 
            (3a
            (3b
             (3c
 
     
   
 
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