Geometry
How Does Jetrike Work?
The Jetrike self centering effect works because the force of the rider pushing down on the inside wheel forces it up. This pivots the rocker arm which forces the outside wheel to move down. The geometry ensures that the outside wheel moves down further than the inside wheel moves up.
On a normal two wheeled bike, the rider CoG rotates in an arc about the contact patch. On the Jetrike, the rider CoG rotates in an arc about the front contact patch, while at the same time moving in collinear path away from it. This causes the height of the CoG to remain level with the ground plane while the seat height is raised slightly.
Here we have a frontal view of the Jetrike when the seat level.
Now we have a frontal view of the Jetrike when it is tilting. Notice that the seat height has risen slightly, and the COG while tilting around the center contact patch has moved outwards so its motion remains level to the ground plane. The idea is that when the trike is stationary and the rider is sitting still, there is more force required to lean the trike than there is to keep it centered, so the trike will stay level. If the rider however shifts their weight by leaning to one side or the other, the trike will tilt. The geometry has a natural limit that causes the tilt to lock at about 27°. When the trike is at tilt lock (fully tilted) the rider can shift their weight back towards the center by leaning inwards and the trike will center -- thats the theory at least.
The Simulation
A simulation was used in an attempt to isolate and understand the impact of different angles and lengths in the design. The graphic above shows part of the simulation. The seat height is 250mm and the CoG is 250mm above that. The simulation shows that at an angle of 27° the seat height has raised 27mm while the CoG remains almost level with the ground plane. Another interesting side effect of the geometry is that the track increased from 700 to 785mm, improving the cornering stability. Notice also at that this tilt angle, the line between the CoG and the outside contact patch falls well inside the tipping point.
The simulation is driven by a parametric sketch that isolates the swing and rocker arms on two independent planes and provides relations and dimensions to interactively drive the simulation. Click here to see a screen capture of the fully annotated simulation geometry, and here to see it animated (600K).
- Angle A: this is the angle of the swing arm when measured against a vector perpendicular to the ground plane. When this angle is less than 90° it increases the self-center effect, greater than 90° lessens it. This angle also influences the relationship between LO and LI (described below). When this angle is 70°, LO is remains less than LI for the entire tilt range.
- Angle B: when this angle is greater than 90° it can increase the self-center effect, less than 90° lessens it.
- Angle C: when this angle is at 110° the self-center effect is about optimum, at 180° it has no influence and less than 180° lessens it.
- Length D: as this length increases, the maximum tilt increases, but the self-centering effect is diminished, an optimal length seems to be between 1x and 1.5x the wheel radius, depending on the track.
- Length E: this length should be should be about 1/3 of Length D.
- Length F: this length obviously must be less than the track. However the longer it is, the heavier the design will ultimately be. I recommend you keep this length between 150mm and 300mm.
- Length G: this length should be at least equal to Length E, but a longer length is actually required to reduce the lateral movement of the rod ends (described below).
From the simulation, the variable that was found to be most effective in generating the self-centering effect was Angle C. However all the other variables were found to interact with Angle C to form a critical relationship.
As long as this angle between the swing arm elbow and the rocker arm plane (Angle B) is maintained, you can rotate the rocker arm plane around Pivot H and have your rocker arm plane on whatever axis is convenient for your design. In my case I chose a horizontal plane, you could just as easily have a vertical or diagonal rocker arm plane as illustrated by the gray outline above.
On its own, Angle C set at about 110° will cause the outside swing arm to tilt down about 55° while the inside swing arm tilts up 26° and raise the seat height by 27mm. However, one would expect that this self centering effect should not work at all, because when the trike is at speed and tilting into a corner, the mechanical advantage of the road pushing up against the outside swing arm should cause the inside swing arm to push down at over 2 times the force. If this were to happen, it would easily overcome the riders weight, cause the tilting trike to center, and most likely flip over, ejecting the rider.
It turns out however that this anticipated machanical advantage is neutralized by the angle of the swing arms to the ground plane. At full tilt, the length from the pivot to the load on the inside swing arm (LI) is actually longer than the length from the pivot to the load on the outside swing arm (LO). So at speed when the trike is tilting into a corner, the force of the road pushing against the swing arms is actually helping to tilt the trike, despite the fact that the seat height is being raised. Angle A however has a big influence on this dynamic.
When Angle A is less than 90°, LO becomes greater than LI. By how much and for how long depends on just how much less than 90° Angle A is, but only a few degrees has a big impact. The optimum value has been found to be about 85°. The following snapshots of the simulation providing the details:
- 0°: LO == LI (279 == 279)
- 9°: LO > LI (278 > 270)
- 19°: LO == LI (253 > 253)
- 28°: LO < LI (161 > 240)
With Angle A set to 85°, when the trike is centered and level, LO == LI. As the trike starts to lean, while the CoG is outside the tipping point, LO > LI which helps the trike self center. This occurs up until the CoG crosses the tipping point, which occurs at about 19°, at which point LO == LI again. As the trike tilts further, and the CoG moves inside the tipping point, LO < LI which helps the trike to lean. So the net effect of Angle A set to 85°, is to help the trike self-center when centering, and lean, when leaning.
Another more technical issue that must also be addressed by any design is the physical limits of rod ends themselves. M8 rod ends can only pivot 14° laterally, so the tilt mechanism must keep them within these limits to avoid damaging them. Offsets R & S have been incorporated into the simulation to ensure that from tilt lock to tilt lock, the lateral movement of the rod ends remains within these limits.
It is important to note, that while investigating and simulating this design, I came across many permutations that had exactly the opposite effect. They caused the rider CoG to move toward the contact patch as they leaned. A trike with this kind of geometry would be almost impossible to ride, because it would keep wanting to flop uncontrollably to either side. The rider weight on the outside wheel in this case is pulling the inside wheel up, so it would be as though you were trying to balance the trike on the outside wheel. According to my simulation, Bram Smit's design suffers mildly from this malady, and I would suggest that designs with these undesirably properties be avoided.
It was actually this discovery that lead me to conclude, if a geometry existed that moved the rider toward the contact patch, then there must be one that moved them the other away (maybe not empirical, but intuitive). That being said, I found a lot more permutations that moved the rider towards the contact patch, and only a few that moved them away. It is actually quite easy to adjust many of the variables in the list to do just that. So without a simulation of your own to play with, I would strongly advise against straying too far from the recommended values.
Recommended Geometry
Angle A 83° Angle B 95° Angle C 110° Length D 280mm Length E 85mm Length F 180mm Length G 200mm Offset R 15mm Offset S 10mm Track 700mm Seat Height 250mm max (or lower) Backrest Angle 25° - 35° Rear Wheel 20" Data Must Be Considered Experimental
NOTE: This table once had several other variations that relied on Angle C being 90°, after experimenting with this angle on Jetrike I can no longer recommend them.
The table above provides a set of recommended values that should work as expected in a variety of designs. While these reccomendations are informed by the limited experience I have with my own trike, the data has only been derived from simulation, and must be considered experimental at this stage. To make these easier to adapt to your own projects I have created a set of bare bones diagrams that you can download here
. The drawings have dimensions in both metric and [imperial] (in square brackets).
If you come up with a design that you want to build that has significant differences, email me with your variables, I will be happy to plug them into my simulation and validate them for you.
Copyright © 2007 Henry Thomas
