Bringing it all together, after all the maths.
The goal I had in my head at the start of all this was a set of operating parameters that would describe the way a given outfit would behave, like standing ¼ mile times for an indication of the performance of a car or bike.
But before I could do this, pertinent information about the outfit, such as the location in three dimensions of the Centre of Gravity, had to be determined, hence the maths.
The Physics is needed so vectors are understood, as it all comes down to vectors in the end.
Now that we know roughly were the centre of gravity is, its height above the ground and the distance to the wheel contact triangle horizontally can be measured.
The height above the ground determines the scale of the vectors, and this scale is then applied to the horizontal component to determine how much cornering force (lateral acceleration) can be applied before the sidecar wheel will lift.
As an example, for turns towards the sidecar, if the CG was at 1 meter height, and the bike pivot point was 500mm from the CG, as the acceleration of gravity is 1 gee, the lateral acceleration of ½ a gee will cause the sidecar to lift, 1000 / 9.81 = 101.9 mm per 1 meter per second squared.
This lift is much more controllable as any acceleration / deceleration vectors are along the axis of the tyre contact patches, and so have very little affect of the stability, hence the fun of “flying the chair”.
The other lift, the back wheel in the air, pivoting on the line between the sidecar wheel and the bike front wheel, requires more lateral acceleration to achieve the lift, but when lifted, is much less stable, as the acceleration / deceleration vectors are off the line of pivot.
An example, the bike is acceleration around a turn towards the sidecar, the CG moves outside the tyre triangle and the back wheel lifts, the wheel can’t drive in the air, and so the acceleration vector disappears, this moves the CG even further outside the triangle, and generates the rapid flip some have experienced.
I have taken the distance from the CG to the sidecar to bike front wheel pivot line as the shortest distance, rather than as it actually is, being much closer to the sidecar wheel, and hence longer, as I wanted the value arrived at from the two lateral accelerations to be the worst case.
I would like the towards the sidecar turn lift point to away from the sidecar turn lift point to be a performance figure of the sidecar.
If we take the away from the sidecar lift lateral acceleration vector to be 1and the away from the sidecar lift vector to be the multiple of the first vector, this figure is a direct value of the lateral stability of the sidecar.
The above example would be something like 2.5, so the back wheel lift vector is 2.5 x 0.5 so 1.25 meters.
As the centre of gravity is moved towards the sidecar, through intelligent placement of fixed components such as extra batteries, then the figure will reduce, indicating that the sidecar wheel lift point is closer to the back wheel lift point, it is debatable as to how good an idea this is, but the rider would then know an important parameter of the cornering capability of the bike.
The simple example I have shown is to show how it all works, but is slightly more complicated in practice.
The vertical height of the CG would be divided by the value of gravity, 9.81 for we metric people, being 9.81 meters per second squared.
The metric challenged can apply whatever value they are happy with, it will all work out in the end.
The result is the scale to be applied to the horizontal, mm / meter per second squared, or whatever.
This scale is then applied to the horizontal distance to the pivot point, and the result is the lateral acceleration required to reach this point in meters per second squared, or whatever.
I find that I tend to be symmetrical in my cornering speeds, if I have compromised clearance on one side, it is applied by my mind to both sides.
I ride sidecar outfits sort of the same way, it is the sidecar wheel lift point that is determining the cornering speed in both directions, unless you are much more bold than I, which is very much possible.