Counter-steering -- how it works: Part 3 -- balancing torques

August 23, 5:15 PMDC Motorcycle Travel ExaminerMark Poesch

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The balance of a motorcycle in a turn comes from the balance between two torques.  The questions are: What are these two torques?   And, how do they so perfectly balance the motorcycle in turn, and yet allow the rider complete control of the lean angle and turn radius?

• 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

Torque is a force applied at a distance.  In this case, one torque is the due to the mass of the bike and rider, m, accelerated by gravity, g, where the distance is measured from the center of gravity of the bike and rider, to the contact patch with the ground; we’ll call this distance Dg, and the force due to gravity, Fg (m * g).  When the bike and rider are vertical, this torque is zero, or near zero.  As the bike leans over, this torque goes up until the bike hits the ground.

Spain's Toni Elias rides his Honda during the MotoGP race of the Czech Republic Motorcycle Grand Prix at the Brno circuit in Brno, Czech Republic, Sunday, Aug. 16, 2009. Elias finally placed third in the race. (AP Photo/Eckehard Schulz)

In a turn, the torque opposing the bike’s tendency to fall over is due to our phantom centrifugal "force," we’ll call this force Fc, applied across the vertical distance from the contact path to the center of gravity of the motorcycle and rider, Dc.

Now, more exactly, this torque is due to the centripetal force at the contact patch with the ground which is driving the bike toward the inside of the turn, applied over the vertical distance from the contact patch to the center of gravity of the bike and rider.  But, as we’ll see below, the solution is much more direct if we instead use the equation for centrifugal force applied at the CG across the vertical distance to the contact patch.  (From part 2, we know that these two forces are equal.)

With these variables, the equations for the opposing torques are:

Tc = Fc * Dc; where Dc may also be expressed as Dcg * cos( θ )

Tg = Fg * Dg; where Dg may also be expressed as Dcg * sin( θ )

Where Dcg is the distance from the contact patch to the center of gravity (CG) of the motorcycle and rider, and θ is the lean angle of this CG.  (The correspondence between the “c” for centrifugal and “g” for gravity, and “cg” for center-of-gravity is purely coincidental, but does help to emphasize that Dc and Dg are indeed the component distances of Dcg.)

Setting these two torques equal to each other (balancing the motorcycle and rider), yields:

Tc = Tg, or

Fc * Dc = Fg * Dg

Substituting terms yields:

Fc * Dcg * cos(  θ ) = Fg * Dcg * sin( θ )

Notice that the distance of the CG from the contact patch cancels out; it is not a factor in the balance of the bike.

Substituting m * g for Fg, and Speed^2 / Radius for Fc yields:

Speed^2 / Radius * cos( θ ) = m * g * sin( θ )

Rearranging terms as a function of speed and CG lean angle (θ) yields:

Radius = Speed^2 / ( m * g ) * cos( θ ) / sin ( θ )

And, everyone keeping up so far will recall that the identity for sin over cos is tan, or, in other words, that the equation can also be expressed as:

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

The same equation derived in Counter-steering – how it works: Part 2 – force causing acceleration.

In other words, the correspondence in these equations confirms that the balance of the motorcycle through the turn is due to the centripetal force driving the bike into the turn offsetting the tendency of the bike to fall over.

Or, even more exactly, the balance of the motorcycle in the turn, and the centrifugal force felt by the bike and rider, are due to the equal and opposite reactions to the acceleration toward the center of the radius of the turn, caused by the speed and lean angle of the motorcycle.

Cause: Speed, and lean angle due to counter-steering.

Effect: Turn radius and balance through the turn.

Next: Counter-steering – how it works: Part 4 – lessons learned