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Steady Turn Plot about origin

by Dale Lukas Peterson — last modified Apr 27, 2011 11:00 PM

A plot exhibiting right handed stable steady turns with positive steer torque, near the upright zero steer configuration

Steady Turns, Stability, and Steer Torque

Black lines are level curves of steer torque.  Red and green regions indicate stability \(Re(\lambda) < 0\), with green indicating that the dominant eigenvalue is complex, red indicating that the dominant eigenvalue is real.  By 'dominant', I mean least stable -- closest to the \(j-\omega\) axis.  Note the scale on the plot is in degrees, not radians, unlike my previous plot.

Here is a right hand steady turn which is stable and requires a right hand (positive) steer torque to maintain equilibrium.  It corresponds to a lean angle of 5 degrees, although I give units below in radians:

  • lean (\(\phi\)) = 0.08726646259971647 rad
  • steer (\(\delta\)) = 0.02499354367752174 rad
  • \(T_\delta\) = 0.002383252665032476 N*m
  • \(v\) = 6.047582886660646 m/s (defined as -\dot{\theta}_f * r_f)
  • \(\dot{\theta_f}\) = -17.27880824760184 rad/s (front wheel rate)
 
Clearly this torque is only just barely positive.  For turns of lean around 10 degrees, the torque is more positive, but still only slightly so.  I suspect that for some parameter sets, the torques could be much more positive and still be stable.  This requires some searching of the parameter space.
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