Why not stability?

The technical community has spent the last 100 years analyzing the stability of bicycles and motorcycles. Fabulous models have been generated and the work was exemplary. Engineers and scientists love stability because it assumes a "hands free" situation and doesn't have the complication of human interaction.

The big problem with stability is that few, if any bike riders would want such a machine. In my opinion, such a thing would be a disgusting unresponsive slug. The Wright brothers understood stability and purposefully built an unstable Wright flyer. The aviation community built most fighters as unstable machines until instrument flying became important. Maneuverability and stability are enemies.

Why do we need another theory?

The great time of the bicycle was the 1890's. The Safety bike superseded the Ordinary& and people were truly mobile for the first time.

The ordinary was the first bike with a high enough gear for true cruising. The gear came from the very large front wheel. In fact the Ordinary was also called the Penny-Farthing due to the great difference between the front wheel and the rear wheel. The rider sat on top of the large front wheel, which caused many accidents as Headers occurred during heavy braking.

The Safety bike had a chain to the rear wheel and the rider sat between the wheels. A safe, rapid, efficient transportation system was born. The rider of the safety was still high and forward. Heavy application of the front brake could still bring on the header.

Both the Safety and theOrdinary had one common characteristic. They were single speed machines. A suitable gear for cruising is too high for starting. The rider was forced to sit in a position above the cranks to attain a high-torque mode. The rider had to stand on the pedals to achieve additional torque for starting. Bike riders have been relegated to setting on a leather strap above the cranks, for more than a century.


Comfortable recumbent bicycles, with suitable seats, were inevitable when gearing became available. The rider could chose a low gear for starting and a high gear for cruising. People could move down to a safer position between the wheels. The seating position was more comfortable and allowed maximum braking with little chance of a header. Then a short circuit happened. The recumbent revolution never came about. Some say that racing bans were the cause. Perhaps that is the case. I believe that something quite different is the cause.

A custom frame builder of today&s safety bike, has well known guidelines for his frame geometry. He builds the bike, measures the geometry and happily bends the fork forward to provide the rider with proper handling qualities. A recumbent builder doesn't have that luxury. His rider is in a completely different position; the wheels are of wildly differing sizes and no guidelines exist.

The Chronicles of the Lords of the Chainring is a first attempt to provide those guidelines to the recumbent builder. I envision a time when the builder measures the geometry of his bike. He carefully notes the position of the seat relative to the rear wheel contact point. He checks the seat-back angle, and happily bends the front fork backward to provide the rider with proper handling qualities.

Time and general acceptance will tell.

Human attitude perception

Control systems are all about making corrections to errors. When we place a human in charge of a machine, the first job is to perceive an error. In the case of bikes (bicycles and motorcycles) the job is to keep the ;rubber side; down. So, the rider must perceive roll angle. Roll angle can be found by looking out and seeing the angle directly. It is also found through the handle bars. When a bike frame is rolled left or right, the handlebar tends to turn the same direction. This ;fork flop; provides a visceral, natural control that doesn;t even have to pass through the riders mind. A properly designed bike should be easily ridden with the eyes closed. Happily, fork flop can be quantified.

Flop equals the weight supported by the front wheel times Trail times the sine of beta.


Flop is measured in newtons per radian and should be in the range of 50 to 200 radians per second. Beta is the head tube angle measured from the vertical. Trail is the distance from the extension of the head tube at the ground to the contact point of the front wheel. Please see the figure.

A bike with the proper flop will be easy to ride without too much attention from the rider.

Geometry of the front fork

S is fork offset
T is trail
Beta is the head tube angle measured from the vertical

High speed maneuverability

The rider of a two wheeled vehicle controls the machine by changing roll rate about the contact point of the wheels. The change of roll rate with respect to handlebar displacement increases linearly with speed. Without some mitigating factor, riding a bike would be like playing a computer game with an automatic dial that keep increasing the sensitivity of the mouse. At some point the bike reacts so violently to the slightest displacement of the handlebar the machine is uncontrollable. The rider overreacts to the machine and induces larger and larger oscillations. The aircraft industry terms this PIO, pilot induced oscillation.

The mitigating factor to PIO is the fact that the handlebars of motorcycles and most bicycles become harder to move as the speed increases. This is called control spring, or centering force or centering spring. The control spring arises from the rotating mass of the wheels and the friction force on the front wheel contact point acting on the moment arm of trail.

Wheel mass can be included in determining control spring in computer models. It is too complicated to be discussed here. Trail, however, is somewhat simpler and can be derived by neglecting wheel mass. This is a reasonable exercise for bicycles, because their wheels are light, it is not a good model for motorcycles.

High speed bikes must control excess sensitivity by a wise choice of trail or other methods of increasing control spring. The following equations neglects the moment of inertia of the wheels and other centering forces. These could include actual springs on the handlebar or tiller/stem to generate control spring with the hands.

T = K5 (B/M)(1/(Kx*Kx) +1/(h*h) )

K5 to have a value of 2.4 kg m2 for normal steering down to 1.2 for light
Steering.
All measurements are in units of kg and meters.

B = the horizontal distance from the rear wheel contact point to the cg
h = the vertical distance from the rear wheel contact point to the cg

For a bicycle, B and h are very near the riders belly button. For motorcycles B and h will need to be measured with the rider aboard.

m = Mass of the bike and rider Kg
Kx = Longitudinal radius of gyration through the cg.
T = Trail

Longitudinal Radius of Gyration

We have made several measurements of the value of the Radius of gyration. The presumption is that the rider overwhelms the geometry of the bike, so we concentrate on Kx vs. seat back angle. The following are measurements. They are averaged over several different bikes. The position of the feet and errors in measurement make exact numbers impossible for general configurations. These values will suffice for the designer.

A vertical man/rider Kx = .5 meters (estimated)
A diamond frame rider Kx = .36 meters (one measurement)
A vertical seat recumbent Kx = .44 meters 90 degrees (one measurement)
A reclined seat recumbent Kx = .41meters 60 degrees (median of 8 measurements)
A laid back seat recumbent Kx = .35 meters 45 degrees (median of 5 measurements)
A laid back seat recumbent Kx = .31 meters 20 (median of 2 measurements)


The trail equations is much more profound than just finding a trail that will reduce sensitivity. Since it is an indication of how much trail is needed to overcome bike that is too maneuverable, it is a measurement of a machines inherent maneuverablitiy.

A bike is more maneuverable when the cg moves forward or down, or when the weight is reduced or when the heavy components are aligned horizontally. The equation ignores wheel mass, a lighter wheel will also add to maneuverability. A less maneuverable machine happens when the opposite is done.

wm.patterson@earthlink.net