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Geometrical Relation between Head Tube Angle, Trail and Fork Offset

Anglo-american units
metric units

An image explaining Steering Geometry data.
The input field of the variable to be calculated must be empty.
The result will be shown there.
The comma as well as the point may be used as decimal point.
Head Tube Angle  Beta degrees
Outside Diameter of the wheel in / cm
Fork Offset ( Fork Rake )  B in / cm
Trail  N in / cm
Further results from the above steering geometry data:
Trail passes zero (turns to negative values) if the steering is turned by degrees

TopOfPage | SteeringGeometryCalculator | WheelDiameterTable | Text | Formulae

Head Tube Angle Beta:
Here defined as the angle between the floor behind the front wheel and the steering axis (see image above).
 Trail N:
Distance between the contact point (center of the contact patch) of the front wheel on the floor and the point where the "virtually" elongated steering axis meets the ground.
Fork Offset ( Fork Rake ) B:
Positive values: hub preceeds the steering axis (as shown in the image). Negative values: hub follows the steering axis (fork "bent backward").
 Note:
The smaller the fork offset (the straighter the fork) the longer the trail - contrary to a wide-spread conviction.

TopOfPage | SteeringGeometryCalculator | WheelDiameterTable | Text | Formulae

Outside diameter of wheels:
A few examples measured at my own bicycles:

[Inches] ETRTO
[mm]
Outside Diameter
[mm]  [in]
16-Inch 37-349 Tire for the Brompton 420  16.53
17-Inch 32-369 Alex Moulton Bicycle Tire 430   16.9
20 x 1 1/8 28-406 Conti GrandPrix 469   18.5
20 x 1 3/8 x 1 5/8 37-406 Conti TopTouring 2000 483   19.0
20 x 1,75 47-406 Tioga CompRamp, IRC FreeStyle 498   19.6
20 x 1 1/8 28-451 IRC Roadlite 506   19.9
26 x 1 25-559 Conti GrandPrix 617   24.3
26 x 1,3 33-559 Conti Avenue 630   24.8
26 x 1,75 47-559 Conti Avenue 645   25.4
26 x 2,00 51-559 IRC PiranhaPro (Off Road Tire) 664   26.1
28 x 1 5/8 x 1 3/8 37-622 Schwalbe Marathon 694   27.3
28 x 1,75 47-622 Conti TopTouring 2000 709   27.9

TopOfPage | SteeringGeometryCalculator | WheelDiameterTable | Text | Formulae

 

The formulae this program is based on:


Beta Head Tube Angle
R Radius of the wheel
B Fork Offset
N Trail
Epsilon Steering Angle
Equation for the Wheel Radius "R"
Equation for the fork offset "B"
Equation for the trail "N"

Solving for Beta:

Substituting Substitution of "sin Beta" by an expression containing cos Beta
into the equation for fork offset B yields the following quadratic:
(R2 + N2) * cos2Beta - 2 * R * B * cos Beta + B2 - N2 = 0

=>
Equation for the head tube angle "Beta"
[1] "minus" for N < 0

The steering angle at which trail becomes zero (turns to negative values):


If B>0 (fork bent forward):
Equation for the steering angle at which trail becomes zero. B>0
If B<0 (fork bent backward):
Equation for the steering angle at which trail becomes zero. B<0
TopOfPage | SteeringGeometryCalculator | WheelDiameterTable | Text | Formulae

 

TopOfPage | SteeringGeometryCalculator | WheelDiameterTable | Text | Formulae

 

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Last modified: 31. 12. 2002  Impressum / About this page

 
 
 
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