Geometry

Trail Calculator

It turns out that when designing a recumbent one should start with rider ergonomics, rather than the head tube angle. Trail becomes a function of where the handle bars and front wheels are placed, not the other way round. The following calculator computes trail lengths for different wheel sizes based on the range of pivot angles and ratios derived from the two surveys below.

Wheel Size:	700 X 56 700 X 50 700 X 44 700 X 38 700 X 35 700 X 32 700 X 28 700 X 25 700 X 23 700 X 20 27 X 1 3/8 27 X 1 1/4 27 X 1 1/8 27 X 1 26 X 2.125 26 X 1.9 26 X 1.5 26 X 1.25 26 X 1.0 26 x 1/650C 26 X 1 3/8 24 24 x 1 20 X 1.75 20 X 1 1/4 18 x 1.5 17 x 1 1/4 16 x 1 3/8 16 x 1.5
Pivot Angle:	89° 88° 87° 86° 85° 84° 83° 82° 81° 80° 79° 78° 77° 76° 75° 74° 73° 72° 71° 70° 69° 68° 67° 66° 65° 64° 63° 62° 61° 60° 59° 58° 57° 56° 55° 54° 53° 52° 51° 50° 49° 48° 47° 46° 45°
Ratio:	100% 99% 98% 97% 96% 95% 94% 93% 92% 91% 90% 89% 88% 87% 86% 85% 84% 83% 82% 81% 80% 79% 78% 77% 76% 75% 74% 73% 72% 71% 70% 69% 68% 67% 66% 65% 64% 63% 62% 61% 60% 59% 58% 57% 56% 55% 54% 53% 52% 51% 50% 49% 48% 47% 46% 45% 44% 43% 42% 41% 40% 39% 38% 37% 36% 35% 34% 33% 32% 31% 30% 29% 28% 27% 26% 25% 24% 23% 22% 21% 20% 19% 18% 17% 16% 15% 14% 13% 12% 11% 10% 9% 8% 7% 6% 5% 4% 3% 2% 1%
Metric:
Fork Offset (rake):
Mechanical Trail:
Trail:
Maximum Trail:

 

The calculator output was compared with my original CAD plots, so I can confirm its accuracy. It should take some of the guess work out of assessing trail lengths for different sized wheels and pivot angles. It gives you a range of appropriate values within which any design you are contemplating should reside. They may also be helpful if you are interested in replicating a design for say a 26" wheel, but would prefer to scale it down to a 20" wheel. All you need do is find the ratio by looking up the trail that is closest for the pivot angle for the 26" wheel, then select the 20" wheel and you will get the new trail value for the same ratio.

What is Trail?

Trail is the measurement between the front wheel contact patch and the intersection of the ground plane with the steering axis (head tube pivot). The above diagram illustrates how trail is measured on a typical recumbent. Too much trail causes the front wheel to flop from side to side. Too little trail and the steering becomes twitchy. However, the wheelbase and distance from the rider CoG to the rear wheel also influences handling. So for instance, a twitchy setup may be desirable on a bike with a longer wheelbase.

There are actually quite a few measurements that relate to trail.

• Pivot Angle: is the angle between the steering axis and the ground plane.
• Fork Offset: is the perpendicular distance by which the front axle is ahead of the steering axis (a common but more confusing name for this is rake).
• Mechanical Trail: is the perpendicular distance from the front-wheel contact patch to the steering axis (described as positive when the axis is ahead of the contact patch).
• Trail: is the distance with respect to the ground plane from the front-wheel contact patch to the steering axis (this is the more commonly used measurement).

What do I actually know about trail? Well, before I started this project, very little, and judging by the responses I got from my requests for help from online forums, with respect to FWD moving BB recumbent bike/delta trike design, there was a great deal of uncertainty as to what geometry worked best and why. I myself made the serious mistake of using someone else's recommended pivot angle and trail settings without understanding that differences in wheel size matter. Then I discovered that the self centering effect influences the front wheel trail because it lifts the rear of the frame by over 60mm as it tilts. So I have set about trying to learn what I can from what is known and then extrapolating my best guess from there.

Tail Survey

Type	Pivot Angle	Fork Offset	Mechanical Trail
Touring	72°	50.5	55.5
	72°	50.7	55.2
	72°	47.5	58.5
	73°	57.9	42.3
Road Racing	73°	57.4	42.8
	74°	50.0	44.5
	74°	66.9	27.6
	74.5°	55.1	36.5
Track Racing	75°	52.1	36.7
	75°	65.4	23.4

On page 274 of Bicycle Science (ISBN 0-262-73154-1) <mitpress.mit.edu> is the above table, which represents a survey of the Mechanical Trail for typical diamond frame bicycles. The measurements are in mm and all bicycles included in this table have a wheel radius of 343mm.

The numbers obviously must work, but by themselves they seem somewhat meaningless, that is until you plot them...

 

Here are the min and max mechanical trail values plotted for the 72° pivot angle of the touring bikes.

Here are the min and max mechanical trail values plotted for the 73° pivot angle of the touring and road bikes.

Here are the min and max mechanical trail values plotted for the 74° pivot angle of the road bikes.

Here are the mechanical trail value plotted for the 74.5° pivot angle of the road bike.

Here are the min and max mechanical trail values plotted for the 75° pivot angle of the track bikes.

I believe that a definite pattern has emerged here -- all the trail plots appear to fall within a narrow band between the zero Trail to zero Fork Offset boundaries. It would be interesting to convert these values to ratios relative to this maximum trail value for each pivot angle.

Type	Pivot Angle	Mechanical Trail	Max Trail	Ratio
Touring	72°	55.5	106	0.52
	72°	55.2	106	0.52
	72°	58.5	106	0.55
	73°	42.3	100.3	0.42
Road Racing	73°	42.8	100.3	0.43
	74°	44.5	94.5	0.47
	74°	27.6	94.5	0.28
	74.5°	36.5	91.7	0.40
Track Racing	75°	36.7	88.8	0.41
	75°	23.4	88.8	0.26
Minimum				0.26
Average				0.43
Maximum				0.55

 

Here we have taken the Maximum Trail to be the perpendicular distance from zero Trail to zero Fork Offset, then we have calculated the ratio of the mechanical trail compared to this maximum trail value. The average ratio turns out to be 0.43 which corresponds to road racing. The maximum ratio being 0.55 which corresponds to touring and the minimum ration being 0.26 which corresponds to track racing. It would seem to me that the optimum trail for a FWD moving BB recumbent bike/delta trike design would likely lie somewhere within this range of ratios.

But before we start calculating the optimum trail, we need to decide on an optimum pivot angle.

Pivot Angle Survey

FWD Moving BB	Pivot Angle	FW Radius
TT Clone	62° - 65°	26"
Speculoos	55° - 60°	24"

The above table is a sample of Pivot Angles for FWD moving BB recumbent bikes.

FWD Twist Chain	Pivot Angle	FW Radius
Nitro Clone	70°	16"
Hachi	70°	650C
Trilenium	76° - 80°	20"

The above table is a sample of Pivot Angles for FWD twist chain recumbent bikes.

RWD	Pivot Angle	FW Radius
Baron Clone	73°	20"
Cycloratio	78°	20"

The above table is a sample of Pivot angles for RWD recumbent bikes.

While this survey is certainly not definitive, it illustrates the range of angles currently being used in various drive train designs by home builders. I think that one could safely assume that the ideal pivot angle for any recumbent may lie somewhere between 55° and 80°, depending on the drivetrain configuration -- that's a big range!

So which is best? At this stage, I am not so much concerned with which angle is best, but making sure that the trail setting is appropriate for the selected Pivot Angle and wheel size. Nevertheless, if the surveys above are anything to go by, angles closer to 67° may be better for larger wheels or higher speeds when combined with shorter trail lengths, whereas angles closer to 60° may be better for smaller wheels or touring (carrying loads) when combined with longer trail lengths, but that is just an educated guess. Trail selections can also influenced by wheelbase.

As I said at the beginning, when designing a recumbent one should start with rider ergonomics, rather than the head tube angle. Trail becomes a function of where the handle bars and front wheels are placed, not the other way round.

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