Dynamic Demo

Controls

Animation

Results

name	value	unit	description
Total accel	?	$g$	total acceleration/deceleration parallel to road surface
Slope	?	$%$
"Friction"	?	$1$	Minimum required friction coefficient to achieve given braking/accelerating force
$F_{A}$	?	$F_{G}$	accelerating force on rear wheel contact patch
$F_{B}$	?	$F_{G}$	braking force on front wheel contact patch
$F_{C}$	?	$F_{G}$	$F_{A} - F_{B}$
$F_{G, D}$	?	$F_{G}$	downhill component of gravitational force (parallel to road surface)
$F_{G, O}$	?	$F_{G}$	component of gravitational force, orthogonal to road surface
$F_{S, 1}$	?	$F_{G}$	sum of vertical forces on all front wheels' contact points from road surface
$F_{S, 2}$	?	$F_{G}$	sum of vertical forces on all rear wheels' contact points from road surface

Explanations

The drawing shows a vehicle from its left side. The vehicle is moving towards the left of the screen, i.e. its front wheel is on the left and the rear wheel on the right of the drawing.

For multi track vehicles like trikes and quads (car), the projection to 2D simplifies things considerably. For an ideal tadpole trike, the center of gravity divides the wheel distance 1/3 to 2/3 to give 1/3 of the total load under constant-speed level surface circumstances.

The center of mass of vehicle+rider+baggage together (baggage is optional) is the dark blue checkerboarded dot.

Derivations

We can decompose the gravitational force in a component orthogonal and parallel to the road surface:

{\vec{F}}_{G, D} + {\vec{F}}_{G, O} = {\vec{F}}_{G}

where

F_{G, O} := F_{G} \cos α

For convenience, we can sum up brake and accel forces:

F_{C} := F_{A} - F_{B}

As there is no movement (and thus no acceleration) of the center of mass in the direction orthogonal to the road, the sum of all vertical forces is 0:

0 = - F_{G, O} + F_{S, 1} + F_{S, 2}

As there is no rotation (and thus no angular acceleration) around the center of mass, the sum of the torques is to 0:

0 = - a_{1} F_{S, 1} + h F_{B} + a_{2} F_{S, 2} - h F_{A}

Resolving for $F_{S, 1}$ and $F_{S, 2}$ :

F_{S, 1} = \frac{a_{2} F_{G, O} - h F_{C}}{a_{1} + a_{1}} = ?

F_{S, 2} = \frac{a_{1} F_{G, O} + h F_{C}}{a_{1} + a_{1}} = ?

If any of $F_{S, n}$ calculates to a negative value, we have liftoff on that wheel and the torque balance equation does not apply any more.

Vehicle Parameters

Vehicle class: ?
Vehicle comment: ?

name	value	unit	description
$a_{1}$	?	$cm$	horizontal distance between front wheel contact point to center of mass
$a_{2}$	?	$cm$	horizontal distance between rear wheel contact point to center of mass
$h$	?	$cm$	vertical distance from road to center of mass (height)
$n_{1}$	?	$1$	number of wheels on front axle
$n_{2}$	?	$1$	number of wheels on rear axle

Notes

There should be enough explanations on this page to eventually figure out the maths, even though the explanations could be written and sketched much much better.
I have mentioned this page in an answer to this physics.stackexchange.com question on what causes the back of a bike to lift when the front brake is applied.

Lazy Developer

Colophon

TODO

Properly show liftoff condition without impossible negative F_Sn values
Verify center of mass location in the different models.
Explain which formula symbols are what, and show them in the illustration.
Add "simulation": Given a specific situation, show what the torques and accelerations would do to the vehicle during the course of the next second or so.
Determine computed extreme min/max values. Show them as numeric values, and possibly use them as constraints on the input value controls?
draw torques
display some measure of the "safety margin" on extreme downhill braking (or uphill acceleration :-)
(for tadpole trike), determine the evenness of the both front brakes required to keep the rear wheel from slipping sideways
drag and drop to change the forces, and vehicle dimensions
use better controls (mouse sliders, jQueryUI stuff?)
fix Chrome bug with control: 0.00 -> 0.05 -> 0.10 -> 0.15 -> 0.20 -> 0.15 -> 0.0: BOOM
scale drawing such that everything fits onto screen
explain the methodology and the equations used (MathML?)
Add vehicle/rider silhouette, center silhouette on drawing. Storage idea: Parametrized SVG drawing, params being: hip opening angle, hip position, upper body angle.
Add CSS etc. magic for MathML in non-MathML capable browsers (hello Chromium!)
Add legend and draw labels for: F_A acceleration force, F_B braking force, F_C combined. F_G gravitation, F_GD downhill component, F_GO orthogonal component. F_S1 surface force wheel 1, F_S2 surface force wheel2. a1 distance on ground from CG to wheel1, a2 distance on ground from CG to wheel2. h height above ground of CG. radius of wheels.
JS code cleanup
consistent naming of classes and IDs
add optional trailer
Verify liftoff detection on non-level slopes. Probably just a case of making clear that we are only showing a momentary state.
Make the whole thing controllable by keyboard
Add different perspectives: From the rear, from above.
When from-above perspective works, make differential braking possible (different forces on left and right wheel(s).
Compare two (or more) vehicles with the same input parameters
Move as many SVG element styles to <defs/> element as possible.
Magic!

DONE

Add note about F_S1 and F_S2 being about the sum of all wheels on that axis
have numeric controls generated by JS code
make ground/sky look nicer
Decompose F_G on slopes
make decelerating and accelerating forces settable
properly solve force and torque equations
add slope to ground
change vehicle dimensions on the fly (upright, SWB, LWB, low racer, tadpole trike, delta trike, car)
change wheel dimensions when changing vehicle
move drawing such that the center between the wheels is at the center of the drawing
Fix Entity 'RightArrow' not defined in Chromium
use camelCase for JS var/fun names
change dimensions in drawing to use properly useful units
Traction King™ (baggage) support
Add elongations of force vectors in drawings
Add "++" and "--" buttons to numeric controls
Add abort mechanism for intro animation
Set font-size in SVG to work in both Firefox and Chrome/Webkit