# 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 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° 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%

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
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
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.