chaparral wrote:
codygo wrote:
It's a bit silly to say that an instability has nothing to do with natural frequencies and is due to a bifurcation.
A bifurcation is math-speak for a 'forking' of the solutions of a system as a function of certain parameters, and within this function lies some natural frequncies inherent to the system. The complex conjugate pair comes from underdamped solutions of a system, since the discriminant is less than 0.
In airplanes you get coupling between roll and yaw resulting in certain instabilities of various magnitudes, and you can also have what's called pilot induced oscillations, where some delay in reaction and overshooting response creates the instability. In cycling, there may be both contributions in any given case of 'speed wobbles'. I would suspect rider delays in balance in response to some lateral road disturbance as the main culprit though.
Bifurcation does appear to be a better description of the phenomon than resonance. If speed wobble is a resonance, then it would happen at a variety of speeds if the bumps in the road were at a certain frequency. And it would also go away when you went even faster. But there are people that experience consistently at a certain speed no matter the road surface and also it just gets worse and worse as speed increases.
Now just like aerodymanic flutter, the stiffness and mass affect when it occurs even though it is not resonance. Also the riders inputs affect it, since a a large input near the critical speed will cause it to appear at a lower speed, once again showing the difference from resonance.
I want to clarify that all bifurcations that result in oscillations exhibit resonance, but not all resonance is the result of a bifurcation. A non-resonating bifurcation would be a system that has several stable solutions; a stalled airfoil may have a stable, lower, lift coefficient that requires a lower angle of attack than initiated the stall to reattach. An example of a resonance that does not arise from bifurcation is a simple pendulum.
The example you give of a rider and road frequencies is not valid because it assumes that only forcing frequencies that match a natural frequency can initiate oscillations. A disturbance need only exist to destabilize a system and that system can oscillate about a stable mean with constant, growing, or decaying amplitude.
As a model that assumes the rider's hand's aren't directly creating the destabilizing steering response, this would mean that the rider's torso behaves as a damped upside-down pendulum that is cantilevered over the front tube and pivoted at the seat. This is more accurate the less input there is from the rider to the cockpit. In a limit, assuming a very stiff rider with no hands on the cockpit, there will be no phase difference from top tube to rider torso and the rider will follow a yawed path into the ground. Once the rider's torso is free to respond to a disturbance, the cyclist-bicycle system may oscillate.
Slowman's article states that we don't really know the cause of "speed wobble," but vehicle dynamics theory is advanced enough to tackle the problem for the cyclist-bicycle system with rigor. I'm sure this stuff is out in journals or books for motorcycle vehicle dynamics.
