## Front Wheel Wobble

### Mechanical Resonance (the theoretical discussion part)

A mechanical resonant system has mass (e.g. a pendulum weight) and a restoring force pushing the mass back to equilibrium (e.g. gravity). The resonant frequency is determined by the mass and restoring force. There's usually also some form of damping, think of a pendulum under water.

When a periodic input (e.g. regular push) is a applied to a resonant system (i.e. the pendulum), the output amplitude (i.e. back and forth motion) is maximum at the resonant frequency. There's a pretty good discussion about resonance effects at Splung or more interactively at the University of Salford. This is a 'driven resonant system' (i.e. periodic input, some amplitude of output at the same frequency as input).

The steady-state output amplitude depends on two things, the input, and the damping. For an input with a particular frequency and amplitude, the output amplitude is such that the energy removed per cycle by the damping (e.g the water) equals the energy input per cycle (e.g the regular push). If the damping is proportional to velocity squared (typical for fluids), it inherently compensates for any increased input amplitude.

There's also a system transient response. Think of giving the underwater pendulum a single push. There will be a few cycles of oscillation at the resonant frequency which gradually die out. The more damping, the quicker it peters out.

The steady state and transient responses are related, really two sides of the same coin. If you have large output amplitude for a given periodic input near resonance, you will also have quite a few cycles of oscillation at resonance after a transient.

### Concours Wobble (the real world problem)

For a motorcycle, the mass is the fork assembly (forks, wheel, tire, bars, etc) and the restoring torque is the side-slip force of the tire when the wheel is offset from the direction of travel times the trail. The main damping mechanisms which limit the amplitude are the tire hysteresis (energy absorption) and the hands and arms of the rider. Which is why you see more wobble with hands off.

There's a site which covers motorcycle wobble resonant frequency very well, see Vibration Modes (heavy bikes 4 Hz ranging to lighter bikes 9Hz). The resonant frequency goes up with trail and tire stiffness and down with the mass of the front fork assembly. The neat thing is that all motorcycles have a wobble resonant frequency, not just a particular Concours with a particular front tire.

In our case, the system input is the dynamic unbalance of the front wheel, see "Dynamic unbalance explained". At 100 kph, the wheel turns 7 times per second, i.e., the input frequency is 7 Hz, at 80 kph 5.6 Hz, 60 kph 4.2 Hz, etc. What we observe and are concerned about is wobble amplitude, particularly if it keeps increasing.

Firstly, if your front wheel is perfectly dynamically balanced, you should not see any wobble at any speed. If there is some front wheel dynamic unbalance, you should see some hands-off wobble near the resonant frequency. I balance my wheels statically, not dynamically, so there's probably at least some dynamic unbalance input. When the Avon was new, there was no perceptible hands-off wobble, now with the tire worn, it's back. The maximum hands-off amplitude on my machine is at around 80 kph, or 5.6 Hz.

If the damping is sufficient, the hands-off wobble amplitude at resonance should reach a stable, hopefully small amplitude. If the damping is insufficient, the wobble amplitude keeps increasing with each cycle at or near resonance. It is eventually limited either by the laying on of hands, or by the forks banging into the travel stops.

Transient response occurs if you're riding along (at any speed, not necessarily near resonance) and something imparts a sudden twist to the fork assembly (e.g. a bump in a turn). The resulting oscillation lasts quite a few cycles if the damping is light. You notice this as an unsettling 'twitch' at the handlebars. If you have perceptible hands-off wobble, you will definitely have underdamped transient response. However, if you have no wobble, you don't know for sure. It may be well damped, or you may have no wheel unbalance input - you have to try it by riding no hands and giving the bar a bit of a shove.

### Old wives' tales

A fork brace cures wobble. Twaddle. A fork brace has no impact on damping, tire side-slip restoring force or trail. There's still a resonant frequency and the amplitude still depends on input and damping.

A brand-X tire cures wobble. Balderdash but not completely. A more energy-absorbent tire (e.g. bias ply) which damps more will reduce amplitude of the wobble. Or Tire-X may simply be better dynamically balanced than Tire-Y. So there's still a resonant frequency but the input may be less or the damping may be more.

Cupping causes wobble. Horsepucky. There are numerous cups around the tire circumference. So the input frequency is much higher than the wheel rotational frequency and way above resonance at any practical riding speed.

### Now for some that I think are true

There's more wobble when the tire is worn. Agree because it has less hysteresis and therefore damps less. And cupping is evidence of wear.

Wobble causes increased front tire wear. Agree, because twice per revolution, the tire is 'in a turn'. You'll see off-center wear on both sides of the tire, 180° out of phase. This is particularly costly for me, as a lot of my miles on the BRP are right around the 80kph resonant point.

Tightening the steering head bearing cures wobble. Agree (sort of), but I don't like depending on it. You have to set the preload high enough so that there's deformation of the rollers and races, as the friction comes from the energy absorption on deformation/rebound.

Also, it's a critical adjustment, too tight and steering is impaired, too loose and there's wobble. But if it's carefully adjusted, it may reduce wobble amplitude to imperceptible. However, if anything changes (e.g. tire wears and damps less), the head bearing friction will need to be adjusted again. There's no automatic compensation, as there is with fluid damping.

Steering head friction is approximately constant independent of both position and velocity. Unfortunately, even though this increased friction is added to impede fast inputs (i.e. wobble), it's also felt for slow inputs (i.e. normal steering).

### Conclusion

I think a steering damper is the way to go.