shimmy analatics

Shimmy Analytics:

It appears as if all trikes and quads have a natural tendency to shimmy unless the proper damping provisions are included in their design.  The purpose of this section is to quantify what effects the speed that such a shimmy sets in and its amplitude and how the various design parameters affect the outcome.  With this knowledge at hand, you can determine what effect changing the weight of the wheels or their size, or the stiffness of the suspension will have on the shimmy properties before you complete your design choices. 

Primary Resonant System:

You can think about this along the following lines.  The outrigger wheels are essentially like a dumbbell in that they represent two masses (tires, rims and hubs), each independently sitting on a spring (the torsion suspension unit) suspended at the ends of a connecting frame.  As one wanders down the road, the rotating wheels bounce alone, jiggling this dumbbell at a frequency that steadily increases with speed.  At a particular speed, the unit will be jiggled at its “natural frequency”.  That is a special frequency where each cyclic jiggle will add energy to the system amplifying the resulting jiggling motion.  Above or below this frequency (speed) the motions are naturally damped and minimal.

We are all familiar with pushing a child on a swing.  The swing will have a natural cyclic motion depending on the weight of the child and the length of the swing support.  If one pushed at just the right rate (when the swing is at the end of its cycle), the child can be pushed higher and higher with each added shove.  This is adding energy at the same rate as the natural frequency of the system.  If one decides to push as some other rate, the swing will actually absorb the energy but slow down.  Wikipedia has several good entries on this natural frequency phenomenon. (http://en.wikipedia.org/wiki/Natural_frequency)

It is not too difficult to calculate the natural frequency of the outrigger assembly as we have the necessary information at our finger tips.  The formula is simple however the units of measure take a little manipulating.

The formula for the suspension’s natural frequency:  

K is the “spring constant” that measures the force necessary to move the outrigger wheel a certain distance when load is applied.  If you recall, the torsion suspension unit I initially used was rated at 250 lbs per wheel and at this load, the wheel’s axle rotated approximately 2.5 inches.  It was necessary to be certain that at least this fender clearance was provided.  The spring constant is therefore:  250lbs/2.5 inches = 100 lbs per inch.  To use this in the equation it must be expressed as pounds per foot thus K = 1200 lbs per foot.

M represents the mass (weight) of the wheel suspended at the extremity of the outrigger frame by the suspension.  This would be the tire + wheel rim + hub.    This is what will be bouncing along the road.  For my 13 inch wheel, the tire and wheel rim weigh 33 lbs and the hub weighed another 6 lbs for a total of 39 lbs.  To use this in the equation, it must be expressed as mass by dividing the weight by the gravitational constant “g“ = 32.174 feet per second².  The resulting units are called slugs (English humor I assume).  Thus M as used in the equation becomes 39 lbs/32.174 feet per second².    M = 1.21 slugs.

When one combines all of the numbers in the formula above, it turns out that the natural frequency (f₀) of the outrigger suspension is approximately 5 cycles per second.  f₀ = 5 cps.

It now remains to determine at what speed the system would be jiggled at 5 cycles per second.  I used 13 inch wheels which have an outside diameter (D) of 24 inches (2 feet).   The circumference of the wheel is therefore C= πD which calculates to be 3.14 x 2 = 6.28 feet.  The wheels will therefore advance 6.28 feet with every revolution (cycle).  At the natural frequency of 5 cycles per second, the wheels would advance 5 x 6.28 = 31.4 feet per second which equals 21.4 miles per hour.

This result almost exactly matches the actual riding experience where the notch shimmy originally set in at 22 mph.

The formula for Shimmy Velocity (in mph.) is: V_s = 0.1784 f₀ D  where the natural frequency (f₀) is expressed in cycles per second and the outside diameter of the wheel (D) is measured in inches.

Before proceeding further, it should be noted that we have just related several design choices to the speed at which a shimmy would be expected to occur.  The design choices are:

·         Wheel outside diameter (diameter D)

·         Wheel weight  (mass M)

·         Suspension stiffness (spring constant K)

In general, one would like the natural frequency and the related shimmy speed to be as high as possible since for an equal energy input, the amplitude of the disturbance will naturally diminish with increased frequency.  The smaller the amplitude of the oscillations, the more effective a damper will be.  How is this best accomplished?

From the proceeding calculations, you could conclude that the shimmy speed appears to be directly proportional to the outside diameter (D) of the wheel you choose.  As a rule of thumb, the outside diameter is approximately 1.8 times the wheel size (i.e.  a 13 inch wheel has a 24 inch outside diameter).  Unfortunately, increasing tire diameter also tends to increase the total wheel weight which reduces the natural frequency f₀ thereby offsetting some of the gain being sought.  Thus, the ideal wheel would appear to be large in diameter and very light in weight.  Motorcycle wheels, unlike trailer wheels, tend to have this property.  I leave it to you to decide how attractive such a wheel choice might be.

 

The last design choice to deal with is the stiffness of the suspension (spring constant K).  If one reduces the spring constant, making for a much softer ride, the effect would be to reduce the suspension natural frequency and the shimmy speed by the square root of the change.  The softer ride is certainly desirable however, the lowered natural frequency and correspondingly lower shimmy speed is undesirable.  When I truncated the torsion suspension by physically cutting it down from 10.5 inches to only 6 inches long, I effectively reduced the spring constant of the suspension (K) by approximately 43%.  The formula would predict that the new shimmy velocity would now be 16.2 mph.

This exactly matches the new riding experience where the notch shimmy tendency now sets in at 16 mph!

I was pleased to have achieved the softer ride that I sought; however, it came with a price.  The lower the natural frequency, the greater the amplitude of the shimmy motion that must be damped.  The adjacent graph shows that the shimmy amplitude increased by approximately 30% when the shimmy speed decreased from 21.4 mph to 16.2 mph.   The front fork damper is still effective but it must now work harder to suppress the motion.

As you can see, the shimmy speed is likely to be between 15 mph and 22 mph for the practical range of choices available to the designer.  Exactly where you come out depends on the combination of M, K and D that you chose in your own design.   There is no free lunch as desirable features work against one another … ah, such is life!

You should note that the distance between the outrigger wheels (the width of the outrigger unit) does not figure into the suspension’s natural frequency (f₀) determination assuming the outrigger frame is rigid.  As discussed later, the width does affect the total quad assembly’s own natural frequency (f_x).

It was very interesting (and predictable) to note that the original notch shimmy speed of 22 mph.  did not change when I put the outrigger wheels on my second bike (Honda Shadow).  Logically this should be expected since the fundamental features (M, K and D) that determine the suspension’s natural frequency are a property of the outrigger assembly itself, not the motorcycle.

Some builders have chosen to move the outrigger wheels forward or aft of the rear wheels axle in an effort to avoid shimmy problems.  Based on the fundamentals however, I doubt that moving the outrigger wheels in this manner would have much effect on the onset of the natural frequency induced notch shimmy.   I will include results from the builders at a later date.

Lateral Motions:

Although the proceeding calculations are useful to home in on the effect of several  design choices, the actual total dynamic system is more complicated because it involves both the up and down oscillations of the outrigger wheels at their natural frequency (f ₀) plus the induced lateral swaying motion of the entire configuration.

Before addressing the lateral motion, it is necessary to look at the front suspension properties.  As illustrated in the adjacent sketch, the front fork is ahead of the steering axis which in turn passes through the steering head and meets the ground ahead of the wheel’s contact point.  The distance from these two ground points is the “trail” which is normally from 4 to 6 inches.  The tilt back angle of the steering axis is the rake angle.  The greater the rake angle, the larger the trail.  The larger the trail, the greater the tendency to steer in a straight line as is characteristic of a chopper.

The important point is that if the steering head (along with the steering axis) is laterally pushed to the right, the front wheel will be turned to the right since the wheel twists about its contact point with the ground.  The front wheel is turned by this motion and the amount of torque exerted on the wheel depends on the size of the trail dimension which is in effect the lever arm of the lateral force.  The motion would apply a larger torque to the front wheel that had a large trail and less torque to a wheel with a small trail.  In the academic extreme, the motion would apply no torque to a wheel with zero trail.

As a consequence, a quad with large trail would have heavy steering in that it would wish to continue in a straight line, but would be shimmy prone as lateral motion of the chassis would exert more torque on the front wheel.

The next sketch illustrates the plan-view or lateral aspect of the shimmy (wobble) problem which relates to the undamped motion at the steering head.  The pivot point of the front wheel is at its contact point with the pavement.  The rear pivot point is at the contact point of the motorcycle’s rear (center) wheel.

When the bike is at the proper speed to excite the vertical oscillation of the outrigger wheels, some of that energy will find its way into the lateral swaying dimensions of the system.  The motorcycle chassis, the steering head plus steering axis and the outrigger assembly form one integral unit as they oscillate from right to left causing the steering head to be pulled right and left at the quad’s own natural frequency (f_x).

It stands to reason that the lateral force applied to the steering head during these motions, will be proportional to the width of the outrigger wheel assembly.  The wider the assembly, the more chassis torque will be generated from the disturbance emanating at the outboard wheel locations.  Trikes typically have about a four foot wide track while the outrigger I originally built has a little over five feet.  There obviously is virtue in keeping the outrigger as narrow as possible from a shimmy dynamics point of view.

The front wheel pivots about its contact point with the pavement.  Therefore, the front wheel will be turned right to left as the chassis swings right to left.  The result is a lateral shimmy at the quad’s natural frequency (f_x) which will in general, be lower than the suspension’s natural frequency f₀.   You can roughly determine the quad’s overall natural frequency by simply sitting in the saddle and oscillating the handle bars until the quad settles into a continuous rocking jiggling motion.  Use a stopwatch and see how many seconds it takes for say 50 cycles of oscillation and calculate the natural frequency in cycles per second.  After softening the suspension, my quad has a natural frequency of about 2 cycles per second per the jiggle test.

The final shimmy motion that one will experience results from a combination of these two frequencies.   It appears as if the suspension’s 4 to 5 cycle per second natural frequency (f₀) determines the velocity at which the notch shimmy will set in while the natural frequency of the entire quad (f_x) will be the  frequency at which the front end will actually shimmy or wobble (around 2  cycles per second).  The adjacent diagram illustrates the effect of combining the quad’s own low natural frequency (blue curve) with the suspension’s higher natural frequency to get the final total assembly shimmy behavior (red curve).  You can see that the peaks tend to retain the low frequency character of the quad’s own natural frequency (f_x).

As the handlebars are turned, the bike very easily twists from side to side even at rest.  There is normally very little resistance to this steering type of motion.  In other words, there is little or no damping in the lateral dimension.  High performance sport bikes have addressed this problem by building in mechanical or electronically controlled dampers of various types and many commercial trikes use similar dampers in their design.  Most trike conversions also employ a mechanical damper.

The greater the “trail”, the more exaggerated the front end lateral shimmy motion will be.  A chopper would be a poor starting point for a trike conversion from a shimmy perspective.  The greater the front end “rake angle”, the larger will be the trail distance from the contact point to the steering axis.  Cruising bikes generally have about a 30 degree rake while sport bikes have somewhat less and dirt bikes even less.  When ridden as a motorcycle, a large chopper-like rake angle results in a bike that is very stable and tracks the road well.  However, the steering feels “heavy” in that the bike wants to naturally continue in a straight line.  However, any lateral oscillations of the chassis increase the torque applied to the front wheel making it more prone to a shimmy.   

Since we are dealing with a Trike+Plus having the “on-off” feature, it is undesirable and impractical to tamper with the basic motorcycle’s front end configuration.  A far better solution is to simply install an adjustable strength damper as I mentioned earlier in the section on shimmy problems.  The effects of the damper on the bike’s handling characteristics are minimal while the effect on damping shimmy motions is substantial.  As I ride, I am still aware of the notch shimmy speed but the effect is so suppressed that it is not of concern.  When I cut the torsion suspension tube to soften the ride, I reduced the notch shimmy speed from 22 mph. to 16 mph. which slightly increased the amplitude of the motion but it remained of no practical concern.

My Conclusion:

·         Manipulating the outrigger design choices (suspension stiffness, wheel size and weight) will change the velocity at which the tendency to shimmy will set in but is unlikely to eliminate that tendency.  The notch shimmy speed is likely to be between 15 and 22 mph.  The higher, the better.

·         The notch shimmy will oscillate the front end at the entire quad’s natural frequency (about 2 cycles per second).

·         The severity of the shimmy motion will depend on the outrigger’s own natural frequency, the motorcycle’s front wheel trail distance and the outrigger’s  width  … higher natural frequency, less motion … less trail, less motion … narrower outrigger width, less motion.

·         The most practical solution to the shimmy tendency is to install an adjustable strength damper at the front fork.  This is easily done, very effective and relatively inexpensive ($50).  If the natural frequency and shimmy speed become too low, a more robust damper may be necessary to avoid sensing the inherent tendency.