I know that a lot of you want to link the vertical position of the front bucket load into this, but all that does is add another layer that obscures the pure physics. Generally, when deriving the equations you want to keep driving things simpler and simpler, not get more complex.
An equation I posted back in #39, was:
F * LF = R * LR
This equation has to hold in order for no additional load to be applied to the front axle. In other words, you can think of this as a way to calculate the required rear counterweight so that no added load is placed on the front axle.
Here:
F is the load on the front loader
LF is the horizontal distance from the front loader's load to the rear axle
R is the load on the three point
LR is the horizontal distance from the three point's load to the rear axle
That tells us that the rear counterbalance load must be:
R = (F * LF)/LR
So that gets it down to forces and horizontal distances, and you cannot get it simpler than that.
Now, of course LF is geometrically related to the angle of the loader, so I could rewrite it as:
LF = LA + R * cos(theta)
Where: LA is the horizontal length from the rear axle to the front loader arm pivot, R is the "radius" that the front load arcs through, and theta is the angle from horizontal. With that, the equation would become:
R = (F * (LA + R * cos(theta)))/LR
If you instead want this in terms of the height of the bucket load H, it becomes:
R = (F * (LA +H /tan(theta)))/LR
All of these equations give the same result, just using different variables. But we had to pull geometry and complexity into this to express it in terms of something other than the horizontal lever arm. Complexity is the enemy of fundamental understanding. If you require complexity to understand the physics, you blew it!
Also note in this case we can use a single variable of horizontal distance, LF. But if we want to rework the equation to get away from that most fundamental form, we now back up to something that requires three variables, LA, vertical distance (H), and angle from horizontal (theta). We'll never be able to make this depend purely on the vertical distance, because it's not in the physics. If you want this to depend purely on a single length variable, that will have to be horizontal distance LF.