Anti-Pogo Rear Suspension
Anti-Pogo 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 head-to-tail 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 pogo-ing. 
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 non-uniform 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.

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