Checking Your Platform / DOB System's Center of Gravity
- a practical method of Measurement and Designing
Here is a method to show you how your scope will feel on the completed platform with roller spacing  and distance from the apex that you choose:

To start the process, go through the platform calculations using the distance of the scope's altitude bearing from the rockerbox as the CG.  Then again, go through the calculations using the scope's CG alone positioned on the rotational axis. Both ways, the calculations will give you a distance that the two rollers should be apart.  The calculations will give you the vertical height of the north runner.  You should be able to figure the distance to the apex or the origin of the virtual cone and the distance the rollers are from the base. 

Build a triangle out of 2 x 3s or 2 x 4s to simulate the platform to make your CG measurements.  With a close approximation of the final total rotational package (add the thickness of the base top boards - 3/4" or 1.5"), add shims to the triangle to get the scope at the height you plan to rotate it at.   You only need to make one triangle.  To simulate the different distances from the rotational axis, you simply slide your scope closer or further away from the apex.   Place a shim under the triangle at the point you expect the roller to be.  The shim wll act as a fulcrum point for tipping.  
Measuring the CG - click here
Tip the scope over (you might need a helper for a large scope) and balance it on one edge of the triangle (for safety reasons you might need to bolt your scope to the triangle and strap your tube to the rockerbox).  Mark the point where the vertical line crosses the centerline of the rockerbox.  Tip the scope over on the other side and do the same thing.  Where the two lines cross is the CG.  Put the scope back to vertical and measure the distance to the two crossed pencil lines.  That is the CG.  Plug that CG back into the platform design equations.  You might find that you need to test CG again if you make large changes to the design.

THE BIGGEST BENEFIT OF THIS TEST IS THAT YOU WILL GET TO FEEL HOW TIPPY YOU PLATFORM WILL FEEL BEFORE YOU BUILD IT. Move your scope all around hard and fast (like some Yahoo will do at a party).  Make sure that you try this stability test with the simulated platform tipped over to the maximum rotation you expect to have.  Put a shim under the triangle at the east or west wheel location and try again to unbalance the system. Does it feel like it wants to tip over?  If it does, move the scope away from the apex which will increase the effective distance between the rollers.  That will make your triangle bigger.  You could just spread your rollers out without changing your rotation axis position - the cost, less drive time or thicker sector.  When you find the "right stability for you, you can go back to the calculations and input the correct real distances that will give you what you want. This procedure is all just an idea right now.  I have not tried it yet. 
Balance or Not to Balance  The stresses on the system  - click here
The lower the CG, the more stable the system is.  The lower the CG relative to the rotational axis, the more stress that is place on the motor and the ground board.  Increasing the platforms distance from the apex gives more room to spread the rollers apart, therefore creating a lower CG.  One thing is sure; you never want the system CG above the rotational axis.  That would mean that the scope wants to be upside down.  Having the system CG rotate on the rotational axis or below the rotational axis is a matter of trading off other desirable properties of the platform - size, weight, stability, run time, and drive power.

In most cases, you probably want to have some margin of error that will allow the scope to fall vertical if drive power is lost. Keeping the CG slightly below rotational axis CG is probably a good idea.  It is all a matter of degree and tradeoffs.  If you have plenty of power and prefer stability to light weight, then lower the CG relative to the  rotational axis will allow for a greater stability.  Having the CG well below the rotation axis means having to build a strong ground board to hold back the twisting force of the CG acting as a pendulum swinging back and forth on the roller supports as the platform rotates. 

Many thanks to the good folks in the EQ platforms Yahoo group for their assistance in formulating the ideas presented here.  Please let me know if this procedure helps or hurts your understanding of Rotation Axis vs. Stability. vs CG.
What is the Center of Gravity?  It is the center of mass of an object. It is the point in an object where all the torques due to the force of gravity are in equilibrium.  We can put all the weight of the object at a point in its volume for calculations.

There are folks that believe in putting the platform's CG on the rotational axis and there are those that think the optical axis should be closer to the rotation axis.  Both extremes will produce working platforms (and anything in between).  There is probably NO perfect CG relative to the rotational point.  Look at this graphic to get a feel of how pulling the scope away from the apex of the virtual cone will produce a more stable platform:
Rotation Axis VS. CG VS. Stability
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