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Suspension, path analysis

Introduction

At this site there is a copy of a site concerning suspension, where one uses "path-analysis" to explain behaviour of suspension. This "PA-site" contains many constructions, reasonings and computations, which all concern mountainbikes. Though the "PA-site" is specific for mountainbikes, it is also interesting for recumbent design.
At the copy "PA-site" also a program can be downloaded, that shows behaviour of suspension for a lot of different mountainbikes.

At mountainbikes many different suspension-systems are applied, which are difficult to compare. At the path-analysis-method it is stated, that suspension-properties of a bicycle majorly depend on the paths of the frontwheel axle, rearwheel axle and pivots relative to the reference-frame. Bicycles with comparable paths should have about the same suspension-behaviour.
For many mountainbikes (and also recumbents) it is enough to compare only the path of the rear wheel axle, because they all behave like the single-pivot-construction.
Path-analysis makes it possible, to compare very complicate mountainbike-constructions.
For a lot of "bogus-marketing" is "proven", that it is not correct..

We now consider two such proofs of the "PA-site" which contradict the theory of this site.

Optimal points

At this site at the link Compute pogo(squat) at recumbent there is an explanation, how squat, caused by varying pedal-power of the cyclist, can be eliminated at a recumbent by an optimal position of the middle chainwheel. Within the given constraints there are some solutions, where the chainline is running through the pivot (PCL) and still squat is eliminated.

At two paragraphs of "path-analysis" it is proven, that such an optimal point for no-squat is impossible.

At appendix paragraph PCL Problems – Some Further Calculations. (PCL stands for Pivot at the Chain Line) one positions bicycle and cyclist horizontal on a friction wheel equiped home-trainer. For this situation it is proven, that the swing arm cannot stay at the same position at varying pedalforce, if the chainline is running through the pivot.
Unfortanately this laboratorium-situation eliminates some essential forces, which will exist at a real-life-bike. E.g. because the rearwheel axle is clamped to the home-trainer, there is no forward driving force through the swing arm and frame, caused by pedalforce. Therefore also the downward-force at the pivot (squat), caused by acceleration of the center of gravity of the bicycle is totally missing at this model.

At chapter II paragraph 2 An Intuitive Look at Forces and Torques it is proven more generally and with a lot of formulas: there is no special point possible.
Some remarks:

We come to the conclusion, the used models differ a lot from reality. The varying pedalforce gives alternating acceleration of the bike (at large pedalforce) or slowing down (by air resistance, wheel resistance or gravity when climbing hills, at small pedalforce). All resulting varying forces are essentially and should be present at a model. The proof that there is no special point, is not correct.
Therefore also part of the arguments against some mountainbike-constructions are not correct e.g. at Ellsworth’s “Instant Center Tracking” (ICT).

Squat and pedalkickback

The "PA-site" contains a lot of theories and constructions, but often it is not clear, whether one tries to eliminate squat/bob/pogo or pedalkickback or both.
Also the role of the chainline, which can vary a lot depending on the chosen gear and the accompanying size of front and rear chain wheels, is almost totally neglected.
Nevertheless an impressive quantity of interesting material is collected.

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