Counter-steering -- how it works: Part 5 -- lean-angle force dynamics

August 23, 5:13 PMDC Motorcycle Travel ExaminerMark Poesch

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A static model of a motorcycle leaning (speed = 0 mph) shows that the force vector for the mass of the bike and rider always points directly down through the tire contact patch.  As the bike tilts over, the force vector at the contact patch continues to point down and remains substantially equal to mass of the bike and rider * gravity (reduced slightly by the bike’s acceleration toward the ground due to gravity) – at least, until additional forces are produced from the reaction of the controls and side of the bike colliding with the ground!

• Counter-steering revisited: Redux -- how it works
• Counter-steering -- how it works: Part 2 -- force causing acceleration
• Counter-steering -- how it works: Part 3 -- balancing torques
• Counter-steering -- how it works: Part 4 -- lessons learned
• Counter-steering -- how it works: Part 5 -- lean-angle force dynamics

A pack of riders take a curve during their MotoGP race, with right to left: Fiat Yamaha Team's Spanish MotoGP rider Jorge Lorenzo, Repsol Honda Team's Spanish MotoGP rider Dani Pedrosa, San Carlo Honda Gresini's Spanish MotoGP rider Toni Elias and Fiat Yamaha Team's Italian MotoGP rider Valentino Rossi enter Goddards during the early laps of the MotoGP Grand Prix at the British Motorcycle Grand Prix at the Donington Circuit, in Donington, England, Sunday July 26, 2009. (AP Photo/Tom Hevezi)

So, at speeds above 10-15 mph, why doesn’t this vector continue to point down when the bike is in motion, regardless of its lean angle?  Perhaps the change in the orientation of this vector is a reaction to the turn?  Or, perhaps, more generally, the vector grows laterally relative to the shift in the momentum of the bike as it leans and then arcs into the curve.

The complete set of dynamic equations describing this process are complex, and are beyond the scope of this article.  According to http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics:

Lateral dynamics [are] complicated, requiring three-dimensional, multibody dynamic analysis with at least two generalized coordinates to analyze. At the minimum, two coupled, second-order differential equations are required to capture the principle motions. Exact solutions are not possible, and numerical methods must be used instead. Competing theories of how bikes balance can still be found in print and online. On the other hand, as shown in later sections, much longitudinal dynamic analysis can be accomplished simply with planar kinetics and just one coordinate.

The analyses of motorcycle dynamics in these articles are trivial first-order solutions focused only on a simplified model of the dominate forces and torques.  Competing theories notwithstanding, in this simplified model, the force vector due to mass of the bike closely follows the lean angle of the bike as a result of the shift in momentum of the bike and the rider.  Initially, this shift is small, and produces relatively small reactive centripetal force (lateral force into the turn).

As the motorcycle approaches the idealized radius of the turn, the continuous change in momentum due to acceleration of the bike into the turn (centripetal acceleration) results in the lateral component of the force vector due to gravity, described previously by the function:

Reactive Force = Mass * Gravity * tan( CG Lean Angle )

So, here we at last discover our chicken and the egg!  The change in momentum of the motorcycle (“speed and the radius of the turn you are riding”) causes the lateral reactive force at the tires.   The reactive force at the tire drives the bike into the turn.  Which came first?

As is true in many dynamic systems, the answer is “both”!   An iterative linear solution of the forces would show a stable feedback loop with one force generating the other, until the idealized reactive force shown above is generated.  But, the input that starts the feedback loop comes from counter-steering, and the lean that it invokes.

Confused yet?  Many are!  The fact is, the bike leans as a result of counter-steering to the outside of the turn.  As a result of that lean, forces accumulate.  Those forces interact to turn the mototcycle into a radius defined as:

Radius = Speed^2 / ( Mass * Gravity * tan( CG Lean Angle ) )

Perhaps it is simpler just to remember: At counter-steering speeds on a motorcycle (above 10-15 mph), push the left grip to go left, and push the right grip to go right!

Coming soon: Counter-steering – how it works: Part 6 – lessons learned from riding 2-up

For more information:
http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics#Lateral_motion_theory