AntiPogo
Rear Suspension 
An article by Charlie Ollinger
26May2007 
This study was done to determine what arrangement of rear
suspension works to eliminate the tendency for a suspended HPV to “pogo” while
the rider pedals. Specifically, it
focuses on a rear suspension, consisting of a simple swingarm mounting a chain
driven wheel. The study is a series of
free body diagrams, which are used to account for, and determine values of, the
forces acting on the mechanical system.
These free body diagrams were created as parametric sketches
in the Unigraphics NX CAD system. For
those unfamiliar with the technique, a parametric sketch looks like a
dimensioned part drawing. The lines and
curves are defined to have relationships  “constraints”  such as parallel,
perpendicular, tangent, etc, and the dimensions may be changed, either
numerically using a slider, which will then change the drawing. In these free body diagrams, the force
vectors are represented as lines with a length of one inch for each 10 pounds
of force. In this way, they may be
visually understood, and also added headtotail to determine vector sums. While the diagrams shown here are taken from
the sketches, they have had most of the cryptic details removed for clarity. 
The first diagram resolves the external forces acting on the
bike. Only the wheels, ground, center of
gravity, and swingarm are shown. First,
the weight (Wt) is distributed between the wheels according to the location of
the center of gravity (c.g.) between the wheels to give the weight on the rear wheel (Wr), and the
weight on the front wheel (Wf). With the
addition of acceleration (Acc), a weight shift is calculated and added to the
rear wheel and subtracted from the front wheel.
The acceleration vector is also reacted at the rear tire contact, and
added to the existing vector to get the rear wheel total force, Rt. The total force on the front wheel is shown
as Ft. 

In the next diagram, the forces acting on the swingarm are
determined, for the zero acceleration condition. The rear wheel force, Rt, is applied at the
axle, and a spring force (Sh) is applied at a shock mounting location on the
arm. The magnitude of the spring force
is calculated by taking the component normal to the swingarm of both the axle
force and the spring force, and balancing the moment about the swingarm
pivot. These forces are then added at
the pivot to get the force vector reaction at the pivot (P). The object of this diagram is to get a spring
force to match in studies done under acceleration – if we know the spring force
is equal in both the static and the accelerated conditions, we know we won’t
have pogoing. 

The short dash lines connected to the Wr and Sh vector are
the respective parallel and normal components, and the long dash lines attached
to the P vector are copies of the Wr and Sh vectors, being added to create
P. These lines are constrained according
to their definitions, so that they always respond appropriately. In the parametric sketch, the slope of the
swingarm, or the location of the shock, may be varied, and the force vectors
will automatically move around in response to the change. 
Now we can add the acceleration. In this next diagram, we have the total
forces acting on the tire contacts. A
chain force is added at the sprocket, with the correct force corresponding to
the acceleration given the sprocket and wheel size. The chain force, Ch, and the rear tire force,
Rt, are added at the axle, as shown by the force labeled Axle.

This Axle force and the shock force are added
at the pivot to get force P. If the
chain force direction had been defined, a new shock force would be
developed. As we are looking for the
solution where the shock force is the same as the no acceleration case, the
sketch has the shock force constrained to be equal to the earlier calculation,
and the chain angle is left to float, being the last unresolved parameter. The sketch resolves everything, giving the
chain angle as shown in the diagram.
Notice the crosshair shown where the chain vector intersects the swingarm centerline. 


For the next sketch, the swingarm angle has been set to a
steeper slope. All the same forces are
used as in the previous diagram, but they resolve differently in this
case. Notice the chain line is now
crossing the swingarm farther back, and a new crosshair is shown at that
intersection. We now have two points of
intersection between the chain and the swingarm, and we draw a line through
them. Extended, this line passes through
the rear tire contact, and a point above the front tire contact at the height
of the c.g. 
One more diagram is shown, to demonstrate that the points
define a line, and not some curve. In
this case, the acceleration is different, the shock position is changed, the
swingarm is shorter, and we have a third angle for the swingarm. Still the chain line and the swingarm
centerline intersect on the diagonal.
When using the parametric sketch, the value the swingarm angle can be changed
using a slider control, and the intersection point will run up and down the
diagonal, even moving forward of the front wheel. If any other parameter – such
as acceleration, shock position, or swingarm length – is changed, the
intersection point stays fixed on the diagonal. 

So what is this diagonal?
It is the locus of all points where the chain line and the swingarm
centerline intersect, such that the shock force equals its magnitude in the no
acceleration case. No variation in the
shock force means no pogo. A locus like
this is often called a metacentric curve.
In most mechanical systems, a metacentric curve will be curved, and that
curve will be different for other values of the other parameters of the
system. In this case, it happens to be
the diagonal line of a box defined by the wheelbase and the center of
gravity. Indeed, if we look at the
metacentric curve for the chainline/swingarm intersection when the bike
bounces, it will be curved, and will not follow the diagonal.
And how is it used?
When designing a suspended bike, the location of an idler or jackshaft
should be coordinated with the slope of the swingarm such that the intersection
point lies somewhere along the diagonal.
It is easy to see that for a different sprocket, or a different load on
the bike, the swingarm and/or the chain will be misaligned – meaning the
intersection will not hit the diagonal and some pogo force will develop. In practice, these forces will be small, and
if the suspension is tuned to not resonate at the pedal cadence frequency,
and/or the shock is well damped, no pogo effect will be noticed. The design condition for the suspension
should use the most typical load, and a lower gear that will be associated with
higher acceleration forces. In this way,
the worst geometry will only occur when acceleration forces are lowest.
Also, this study also shows
that as the acceleration varies, so does the contact force
on the front wheel – meaning that it may pogo. There
are no forces available to employ in countering this effect.
However, once again, proper tuning and damping should make
this unnoticeable.
One final observation: pogo, even on a poorly designed
suspension, is an effect of nonuniform acceleration. If a rider could apply his power smoothly
(and he can learn to do so, to a large degree, with practice), even a bad
suspension will only squat, and not pogo. 
