Path Analysis
Theory, text, illustrations, and editing by Ken Sasaki
4-bar path analysis by Peter Ejvinsson
Spanish Version translated by Antonio Osuna
Additional translation and edition for the web by José Rubio
“Linkage” suspension simulation by Gergely Kovacs
© Kenneth M. Sasaki 2001, all rights reserved
{The authors welcome the reposting or reprinting of this page or any part of it, so long as full credit is given to the authors}
| Index |
Chapter II - Some Useful Suspension Related Mechanics
Chapter IV - Wheel Path Analyses of Some Existing Models
Chapter V - Flawed Theories and Bogus Marketing
back to Mountain Bike Rear Suspension
| Introduction |
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Bicycle physics. Path Analysis All the suspension systems in the open. their features, their virtues and defects. The theory and practice of the Mountain Bike suspension. The physics concepts under detail. The ultimate tool to discover the best purchase option based on your riding style in the bike. |
What is Path Analysis?
It's a research work about the performance of the majority of the full suspension systems in the market. The work has a scientific and practiccal focus that by means of numerous examples and illustrations, makes clear how the physics theory is applied to the real designs. The report is oriented to show which are the main characteristics of each suspension system, making the election of the bike easier, based on your own riding style. The conclusions of the report are surprising and demonstrate how there is less difference between the designs in the market and how the election must to be done based on the riding style, the general performance of each system and specially the test of the bike.
Linkage 2, suspension simulator
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Linkage2 is a powerful suspension simulation software which shows the movement of the suspension in a Windows environment. You can watch the movement of each pivot and things as important as the forces applied to the rear shock in order to estimate the progressive response of the suspension or the variations in the chain length, which gives an idea of the interference between pedaling and suspension. |
Linkage has all you need you evaluate a suspension system. Among its features we highlight:
Linkage comes with a library of more than 50 commercial models. You can create your awn models from photos of the bike for 7 different suspension systems:
Get inside the amazing world of this suspension tech work divided into five chapters and some annexes.
| Chapter I - Primary Concerns |
Path Analysis (PA) is a qualitative method for analyzing the pedaling, braking, and shock absorption characteristics of full suspension frames. The objective is to allow anyone to determine the true merits of any suspension design claims with regard to these characteristics. While the principles here apply in general, we have focused on the non-URT (this usually means bottom bracket on the front triangle and we will use the term to mean so in this work unless otherwise stated), since these designs constitute the lion’s share of bikes produced these days.
Most theories on bicycle suspensions one sees attempt to find “the proper pivot point or points” which will make the frame shock non-reactive to pedaling at equilibrium or “sag” (in fact, it is impossible for any frame geometry to do this either exactly or universally, and getting close in any particular case will introduce other problems). A precise quantitative treatment of suspension geometries is a very involved process that requires significant assumptions, even in the most simple of cases. A number of simple theories purport to find correct geometries that eliminate rear shock activation at sag, but to this author’s knowledge, none are sound (this excludes the Giant NRS which is meant to be run with no sag). We look at a few of these at the end of the work to demonstrate how PA can be used.
The consumer should be most concerned about getting past industry hype. So rather then spending a lot of time trying to identify proper pivot locations and so forth (beyond an intuitive understanding), we will focus on the issue of what performance characteristics are achievable with viable suspension designs and how one might achieve them in principle. This analysis method is intended as a consumer tool that will allow one to accurately judge marketing claims, as well as the relative merits of suspension designs and theories.
We want this work to be useful to those with absolutely no technical background, so we present the “Main Conclusions” up front in this first chapter. Those with no technical background should also find the “Bogus Marketing” section completely accessible.
In chapters II, III, IV, and V we have included recommendations for reading and difficulty for each numbered section, stating:
Read this section if:
You wish to accomplish “this or that” objectives. This section is of “such and such” importance.
Skip this section if:
You are not interested in “this or that” objectives.
This section is “stated difficulty rating”.
Here, “stated difficulty rating” varies among: Not difficult, less difficult, moderately difficult, more difficult, and most difficult. The level of difficulty is referenced to a typical person with about a year of good solid college physics.
We hope that this will help those with a less technical background navigate around the more difficult and less necessary sections. Those with a less technical background should still find the not difficult and less difficult sections, as well as the conclusions from all sections, understandable.
About the second chapter, “Some Useful Suspension-Related Mechanics”:
We have chosen methods of analysis with an eye toward keeping math to a minimum, but some basic physics knowledge is unavoidable. Much of this chapter is not necessary if one merely wishes to use Path Analysis to evaluate different suspension designs (which is the main intent of this paper). For this application, we strongly recommend only the “Reference Frames” section.
We have included this chapter mainly as a physics primer for those who wish to rigorously verify Path Analysis and delve more deeply into bicycle suspension physics. If one wishes to independently verify the validity of PA, the “Some Important Concepts” section is most critical to understand. The concluding statements in “An Intuitive Look at Forces and Torques” are also of value. We reemphasize; much of the rest of this chapter is related to Path Analysis only in establishing finer points and is not truly necessary (although, the knowledge will be useful to anyone contemplating bicycle design and some of it will make PA more accessible). One will also need certain concepts from this chapter if one wishes to completely understand certain flaws in some of the theories detailed in the fifth chapter, “Flawed Theories and Bogus Marketing”. Most important among these is the “Center of Mass” (CM) concept, as applied to forces acting through the various wheels and cogs within a bicycle. To this author’s knowledge, this concept has previously been unknown in the bicycle industry.
About the third chapter, “Path Analysis”:
This is where the main theory is presented. We don’t consider any of the sections to be more then moderately difficult. We suggest that all readers read all sections, even if some things are not clear.
About the fourth chapter, “Wheel Path Analyses of Some Existing Models”:
This chapter contains the cad drawings by Peter Ejvinsson. These drawings are most informative in conveying information about what is out there at the present time.
Most of the major design types that are more then trivial to evaluate are covered. For the most part, the material in this chapter is extremely easy to understand, the one exception being parts of the “The Virtual Pivot Point (VPP)” section.
In addition, a “Linkage data” link to Gergely Kovacs’ Linkage suspension simulation program (see below) is provided in each frame’s section (for which .ltx data files have been pre-made). This program displays the most important characteristics of each frame. Clicking on the link in each frame’s section will automatically bring up data on that frame (note that the Linkage program must first be installed, again, see below).
About the fifth chapter, “Flawed Theories and Bogus Marketing”:
The original motivation for the production of this work was the ubiquity of false theories emanating from bicycle manufacturers and industry magazines, and circulating in bicycle-related web sites. We have thus devoted considerable space to demonstrating the flaws in some well-known and widely accepted theories.
Some of the false theories and marketing are associated with well-known names. This has made the work somewhat controversial. However, we note here, as well as in the chapter, that in all cases involving false theories, vigorous efforts were made to contact and discuss matters with the associated parties, before the release of this work.
One of the oldest and most respected of full suspension frame manufacturers has warned this author that the bike industry is very small and generally not kind to “realists”. He also warned that some “retaliation” should be expected and indeed, there has been some.
We are committed to exposing industry hype and nonsense, and to giving the public the best possible chance to make informed decisions, so we will not be deterred by retaliation. While we feel it unfortunate that some of this information has caused a good deal of consternation to some who have already made some very expensive purchases, we will continue with the circulation of this information for the greater public good.
We also note that, generally, the feedback from the industry has been positive; including, we are told, positive comments from one of Renault’s senior suspension engineers.
About the “Glossary”:
At this time, the “Glossary” has been done to explain terms in this first “Primary Concerns” chapter that may not be familiar to those new to mountain biking. We have not provided a detailed account of scientific terms in the later sections because of time constraints. We hope that those venturing into these sections will have adequate prior knowledge or know how to obtain such knowledge from more fundamental sources, or that the bold-written essential information will suffice to give a reasonable understanding. In the future, we hope to provide a more detailed account of scientific terminology.
About the “Linkage” suspension simulation program:
Linkage has been created by Gergely Kovacs to produce the most important information about any 4-bar rear suspension that one might want to consider.
A version of linkage has been included as part of the PA package. The Linkage2 software as well as the source code are also downloadable from the Linkage web site at http://www.bikecheck.com/. The Linkage web site may contain a more updated version of linkage, since Gergely maintains that site personally.
The authors are all avid bikers, who also have technical and/or language skills. We have provided this work freely to the public with the hope that it will benefit consumers and others interested in the workings of bicycles. A short biography and picture may be found on the “About the Authors” page, for those authors who have provided the information.
The authors wish to thank Prof. Curtis Collins, Ola Helenius (Ola H.), and Ray Scruggs (Derby) for their kind suggestions and help in finding errors. Thanks also to Drakon El Elfo for working on the link structure to the Spanish version.
The authors would like to especially thank J.I. Baeza (aka Sikander) and Jose R. Rubio, the editor of http://www.mtbcomprador.com/, for hosting both the English and Spanish versions of this work.
This work will be updated from time to time to reflect current technology. Updated versions will be posted on the above web sites at the links below:
The English version may be found at: http://perso.wanadoo.es/jibsna/mtb_susp_en and http://www.mtbcomprador.com/pa/english.
The Spanish version may be found at: http://www.mtbcomprador.com/pa/spanish.
We welcome and appreciate all accounts of errors and suggested additions regarding this work that anyone cares to send to bicycle_physics@yahoo.com. We apologize if we are unable to answer all correspondence, due to time constraints. Thank you.
1) All measures of suspension performance depend almost entirely on the paths of the following components relative to any reference frame defined by one of the bicycle frame members: Handlebars, seat, bottom bracket (BB), front and rear wheel axles, shock mounts, and rear brake. This is the central idea of this work and is explored in the third chapter, “Path Analysis”. The following are the main conclusions that may be drawn from this statement.
The first thing that most people are concerned about with dual suspension bikes is efficiency under pedaling; generally assuming a seated rider position and a bike on smooth ground. But there are compromises that must be made in trying to attain this goal and most other goals associated with dual suspension performance. In reality, the right geometry for any one person will depend on that person's body type (mass distribution), riding style, sensitivity to various phenomena associated with dual suspension motion (such as bump feedback), desired ride quality (comfort, efficiency, etc.), and even the type of terrain in one's backyard.
No geometry is right for everyone and no frame can achieve for any one person every goal generally desired in dual suspension performance. At suspension equilibrium (natural sag) or any other position in travel, any of the common suspension “types” (mono-pivots, various 4-bar configurations, etc.) can be as non-reactive to pedaling as any other during seated pedaling. However, no geometry can be completely “neutral” throughout a pedaling cycle, without friction. The deviation from neutral can be made small and a good suspension geometry with the right amount of friction can effectively limit oscillations, while remaining supple enough to absorb significant bumps. However, we note that no geometry is perfect in this respect as a warning against all theories purporting to provide a neutral geometry, in principle, without any qualifications.
A word on marketing: No manufacturer of a bike or frame designed to run at sag (some bikes such as the Giant NRS are meant to run at no sag) is going to market its products with a valid, quantitative theory for constructing rear suspension geometry – telling you why their pivots or whatever are in the right place. The ideas and formulae would simply be too complicated to make a good marketing tool. This author has never seen a valid, quantitative, run-at-sag suspension theory put out by any company, though quite a number (some of which we examine later in chapter five, “Flawed Theories and Bogus Marketing”) market bikes under dubious claims and false theories – some asserting that you can have it all. If any manufacturer or sales person tells you that you can have it all, run away!
Our advice is to ignore all suspension theories and other claims put forth by frame manufacturers and industry magazines, and base your buying decisions exclusively on experimentation. That is, make your decisions by test riding the bikes, even if it is just a parking lot test (you can get a lot from a parking lot test). Ignore all marketing!
Almost all dual suspension bicycles these days are non-URTs. The following comments will apply to these designs, as well as the mono-pivot-equivalent i-Drive, made by the GT bicycle company.
The most important thing to look at in assessing any non-URT frame's potential performance is the path that the rear wheel axle travels relative to the main triangle, as the suspension compresses (the main triangle is defined by the seat, handlebars, and bottom bracket). The mechanism that produces the path is not important beyond helping one determine what the path is. In particular, mechanisms that produce similar paths will perform similarly (“suspension rate” aside).
At any moment in time, the rear axle path tangent will primarily determine suspension performance. In particular, within any small segment of any non-URT suspension's travel, that suspension will behave like some mono-pivot with main pivot that gives the same path tangent. Suspension rate (spring stiffening) is also significant to suspension performance and is determined for the most part by the paths that the shock mounts travel relative to the main triangle. Shorter travel suspensions tend to have higher rates that increase more drastically as the suspensions move through travel (in part due to the fact that many use air shocks these days). However, most frames mate well with their stock shocks, and all common suspension types can achieve the really useful rates (linear or rising). So rate is only a real issue for those wishing to swap different shocks in and out of a given frame.
All this is not to say that all non-URTs will behave the same. Different geometries will certainly have different characteristics. But this depends almost entirely on the specified component paths. It does not depend on the suspension type.
For example, 4-bar “A” may have a rear axle path curvature substantially different from 4-bar “B”, yet A's path may be very close to mono-pivot “C” (circular, with a particular radius and center). Under pedaling and shock absorption, A and C will perform similarly to each other, but differently from B (suspension rate aside).
Almost all non-URTs on the market today have circular rear axle paths out to two or three decimal places, in inches. As a result, the radius and center of curvature primarily determine suspension performance in most non-URTs.
The majority of rear axle paths are of similar radius to conventional mono-pivots. The 4-bar paths plotted in “Typical Horst Link Designs”, which encompass most of the major chain stay pivot design configurations, are all very circular and of conventional radius.
The Giant NRS, The Rocky Mountain ETS-X70, and Cannondale Scalpel have tight radii of curvature, centered inside the rear wheel radius. The Giant NRS and The Rocky Mountain ETS-X70 achieve this through their link configurations. The Scalpel achieves this through the localized bending of its chainstays, about half way between the BB and rear axle. The Scalpel mimics a design we proposed some time ago, called the split-pivot mono. “Soft-tail” designs also have tight radii of curvature, but we consider the length of travel too short for this consideration to be of significance in these designs.
4-bar designs with closely spaced pivots near the frame center can achieve significantly variable path curvature. At the moment, the “The Virtual Pivot Point (VPP)” concept, conceived by Outland and soon to be re-introduced by Santa Cruz and Intense, is unique among viable concepts in its capability to produce significantly variable curvature. However, current examples do not take any real advantage of the possibilities.
Closely spaced 4-bar pivots can also achieve wider curvature then is possible in a mono-pivot. The positive travel sections of the current “The Virtual Pivot Point (VPP)” bikes contain such a curvature and the Schwinn Rocket 88 also claims such.
However, closely spaced pivot locations near the highly stressed bottom bracket area may come with a larger tradeoff between weight and durability, as the links and pivots in this area must be more heavily built.
One unique frame design, which we have yet to evaluate, is the Maverick. At the moment, it is very expensive and still hard to find. We hope to include it in these pages soon.
i) Paths and Shock Absorption (“coasting” situations).
We handle only coasting situations here, since shock ramifications for pedaling and braking will be handled specifically in those sections. A bicycle suspension may be suddenly compressed by the ground either through wheel contact with an obstacle such as a rock or from the impact of a drop-off. In general, we believe that a widely curved rear axle path running slightly up and back is the best solution. Tight curves, either circular or varying are generally inferior for shock absorption. However, this deficiency may be mitigated to some degree by having the path tangent tilting backward through all or most of the travel (for example, having a high main pivot, either real or virtual), as is the case in the The Rocky Mountain ETS-X70 and, substantially, the Giant NRS. One might also find that short travel designs such as the Cannondale Scalpel do not have enough travel for this deficiency to be significant.
In the case of a drop-off, the situation is obvious; a linear path will offer the smoothest, most consistent compliance.
In the case of an obstacle, the bump force will be up and back relative to the frame, so the initial tangent should be up and back. The direction of the bump force will turn more vertical as the bike clears objects of “ride-able” size, so a widely curving path turning slightly upward at the top should be ideal.
Experimentation will determine the most desirable path incline and radius of curvature.
Rising suspension rates benefit short travel designs, since this will allow better initial compliance, while reducing the probability of hard bottom-outs.
The rear axle path tangent will determine how the suspension will react to pedaling at any given moment. This means that, neglecting friction in the mechanism, each particular geometry will have its maximum effectiveness only in certain “ideal” gears (from a practical standpoint, this could mean one gear or several). The further the gearing from ideal, the more reactive any suspension geometry will be.
For a given deviation away from ideal gearing, “suspension rate” (spring stiffening) will determine the amount of reaction from a pedal stroke. Shorter travel suspensions tend to be less reactive to pedaling then longer travel versions, since short travel designs should have higher, more rising rates. However, the difference between linear and rising rates will be small in the shallow regions of travel where pedaling is affected. In practice, the actual rates in these shallow regions will largely be a function of the total travel, or rear axle path length.
It is common these days for designers to take into account the slight tendency of a bike to fold or “squat” under acceleration. To do this, one adjusts the rear axle path so as to increase, by some significant amount, the distance of the rear axle from the bottom bracket (BB) as the suspension compresses. This allows chain tension (mainly) to counter the squat. But this also creates significant bump feedback. We want to be clear on one thing: There is no free lunch here. Have an increasing effective chain length between the cogs – get some degree of bump feedback.
The one area where some multi-links (this usually means 4-bars) may have a slight benefit over conventional mono-pivots is in bump feedback to the pedals.
4-bars offer the possibility of both variable curvature and tight, circular curvature as the rear axle moves relative to the main triangle. Both of these possibilities allow for a center of curvature inside the rear wheel radius. Tight curvature above equilibrium allows the suspension to counter squat at equilibrium, while more effectively limiting feedback. Bikes with tight circular curvature should be run with little or no sag to prevent problems from feedback under suspension extension.
However, as we have noted, 4-bars on the market today do not provide significantly varying curvature and only the Giant NRS, The Rocky Mountain ETS-X70, and Cannondale Scalpel, have significantly tighter curvature (though one might find that the ETS-X70 does not have a small enough radius, nor the Scalpel enough travel for this to be significant for him or her, with regard to pedaling).
This means that almost all non-URT designs on the market today (with the three exceptions) must make essentially the same compromises between feedback and anti-squat. Some prefer the rearward tilting axle path and the generally increased efficiency provided by the anti-squat. Others prefer the smoother pedaling over bumps provided by a more vertical path tangent at sag.
We have seen that rearward axle path tangents at equilibrium should offer some advantage while pedaling over smooth terrain and during shock absorption while coasting. However, this will also produce bump feedback while pedaling over bumps. Many riders say that they are very sensitive to this trade off, even to the point where differences of less then an inch in main pivot locations are noticeable. Some prefer the generally efficient rearward tangents, while others want the smoother pedaling vertical tangents. So we have a compromise with which to deal.
We have also noted that tight curves above equilibrium, whether circular or varying, may help reduce the bump feedback of a rearward tangent. However, curves tight enough to make a significant difference in the shallow regions of travel where riders are likely to be pedaling may produce inferior bump performance deeper into the travel, since wide curvature should be best for shock absorption. Though again, designs with rearward paths through travel, such as the The Rocky Mountain ETS-X70 and, for the most part, the Giant NRS, may mitigate this compromise to one degree or another.
Furthermore, while variable curvature has its allure, the potential for a bigger tradeoff between weight and durability in comparison to conventional 4-bars, due to closely spaced pivots near the bottom bracket, shows that variable curvature designs may not be without their compromises (this in addition to the compromises noted above involving tight curvature).
This furthers a theme that we revisit throughout this work – there are no “right paths” or “right pivot points”. We have seen this in the mass distribution considerations of having riders with different body types. We have seen this in the fact that no geometry can be completely non-reactive through a pedal stroke, without the help of friction. And now we see it again in the fact that there are choices that must be made, depending on what type of suspension performance one wants.
Human beings can be surprisingly sensitive to physical situations. The author finds that a difference of just two millimeters in the height of a road bike seat gives the feel of a completely different bike. So we are not surprised to find that some people hold small geometric differences as important and we must assume these positions to be legitimate.
However, we must note that some people experience “have-it-all” performance in some designs from manufacturers that claim such performance (though this is certainly not the case with most experienced riders that this author has encountered). Since we have seen that have-it-all performance is impossible, we must conclude that either the powerful psychosomatic phenomenon is at work or that some of the considerations that we have been exploring are not all that discernable to some people, or perhaps it is a little of both.
All of this makes the question of suspension performance largely philosophical. So to continue another theme, we again suggest that test riding be done to determine what performance characteristics are right or even discernable for each person, even if it is just a parking lot test (you can get a lot from a parking lot test).
In the final analysis, none of the major suspension types has a clear advantage over the others. There are lots of happy mono-pivot owners out there (including those with mono-pivot-like Ventanas and Rocky Elements) and there are lots of happy 4-bar owners out there. This pretty much says it all.
The biggest question regarding braking in dual suspension bikes is whether or not 4-bars rear-brake better then mono-pivots.
There is a very well established myth (well-propagated by the magazines) that mono-pivot shocks will lock under rear braking. This is known as “Brake Induced Shock Lockout” or BISL. We have demonstrated this to be false.
We have also demonstrated through experiment that mono-pivots do not significantly extend or compress under braking.
We have even demonstrated that certain 4-bar designs, such as the Jamis Dakars and the Psycle Werks Wild Hare, should brake almost exactly the same as a mono-pivot with identical main pivot location. We have seen many Dakar and Wild Hare reviews from Mountain Bike Action, Bicycling, and several other industry magazines. That none of the reviews mentions BISL in these 4-bar designs is a good indication that it does not actually exist.
Most 4-bars extend from natural sag under smooth-surface braking a bit more then equivalently main pivoted mono-pivots, establishing a new equilibrium position and rate. Some, such as the Yeti AS-R, compress under smooth-surface braking relative to equivalently main pivoted mono-pivots. In addition, changing frame geometry through travel, due for example to bump compression, may cause the braking effect to change, further altering the effective suspension rate of a 4-bar.
But again, none of this leads to the conclusion that 4-bars brake better then mono-pivots in general, since a mono-pivot could well have (and some probably do) the same rate under both braking and pedaling as most 4-bars under braking.
The biggest consideration is the relation of the rider's body mass to the wheels and what it will do under braking. This author believes that between most of the designs, the differences are just not enough to merit a general statement.
Some people find 4-bars to brake better, but others do not, though we have seen no double-blind tests. In the end, the small differences between some designs may be significant enough for some people to feel a difference. But in general, we suspect that this is again just a case of the very well established psychosomatic phenomenon. This would not be the first time that people have been told that something is so and many have experienced what they expect (this is why placebos cure illness). Or perhaps it is again a little of both.
We also have no doubt that the BISL myth has been propagated by some in the interest of selling more expensive 4-bar designs. We see no $ 2,000 mono-pivots. In the near future, we hope to do a double-blind experiment to see once and for all if there is a difference between 4-bars and mono-pivots, under braking. We will publish any results in subsequent editions of this work.
In any case, our advice here, as always, is to make your decisions through testing the bikes, if possible.
What we have stated above regarding 4-bar linkages also applies to floating rear brake systems, since a floating brake will give a bike the same rear braking character as a 4-bar with the linkage geometry of the floating brake. Imparting the character of its linkage geometry is the only thing a floating rear brake does, for good or ill. This will, for example, give a typical mono-pivot suspension the tendency to extend under braking, rather then its inherently neutral character.
I would like to close this section with a segment from an open letter I published some time ago:
You will find people who worship most major designs out there and others who despise these same. There is a good reason for this. Most of it only exists in people's minds. Many people hate mono-pivots, but revere Ventanas, not knowing that a Ventana is essentially a mono-pivot with linkages that act as shock tuning (with respect to pedaling at least). There are some differences in the major designs, and some small advantages here and there, but in the end it is mostly academic. Stick with the major concepts and one will not change your life over the others. Of course I am speaking of comparing bikes within particular categories, not comparing free ride to XC or downhill.
This is the conclusion I have drawn from my model. Most people build a model and use it to sell one idea or another. My contention is that the four or five major, basic forms CAN all work about as well as the others.
In the end, execution is far more important. A quality company is far more important then a slight difference in the position of a pivot. And make sure the bike fits right. This applies to intended use (be realistic) as well as body weight and body dimensions.
Most of us with propellers on our heads just like to talk about this stuff because we enjoy applying the skills we have acquired to our hobby (though some obviously have religion). Let me close with a piece of a conversation I had with a professor of mechanical engineering, with whom I have discussed my theory several times:
I said that in the end, the simplest designs are the best. He responded, that this is almost always the case.
For most people this means a basic mono-pivot, or a basic 4-bar with the pivot on the chain stay or the seat stay. Some are obviously swearing that you need a link on the chain stay, but don't tell that to people who own Ventanas or Rockies.
Keep it simple and go with your gut feeling; you have to like the bike when you look at it (with whatever standards you really find important).
Good luck
Ken Sasaki.
| Chapter II - Some Useful Suspension Related Mechanics |
Read this section if:
You want to verify for yourself the validity of the “Path Analysis Main Assertions.” and understand some of the related analysis in the fifth chapter, “Flawed Theories and Bogus Marketing.”.
We strongly recommend at least reading the “Reference Frames” subsection. It will be very useful to understand this basic physical concept in later sections.
Skip this section (except the “Reference Frames” subsection) if:
You will accept the Path Analysis main assertions and are just interested in using PA to make conclusions about what suspensions can do and comparisons between various bikes.
This section is moderately difficult.
Fully understanding PA for bicycles requires some important concepts. We strongly suggest that those wishing to fully understand PA spend some time becoming familiar with these concepts as most erroneous suspension theories involve the neglect or misunderstanding of one or more of these, “Center of Mass” in particular.
In order to analyze any physical situation, we must create a reference frame. This is usually represented by of a set of coordinates in space, consisting of a mutually perpendicular set of lines, or “axes”, with common intersection. The place where the axes cross defines the origin, or zero point. We usually give names to each axis, such as “x-axis” or “y-axis”. Depending on the sort of information in which we are interested, coordinates could consist of one, two, three, or even more axes (though more then three axes depict more then the normal spatial dimensions and obviously cannot be pictured).
Often, we assign units of length along each axis. Each point in space lies along a line perpendicular to any given axis. Points may thus be defined by the set of numbers corresponding to the points along the axes through which these perpendicular lines pass. This system is called a rectangular coordinate system. Figure 2.1) shows a 2-dimensional rectangular coordinate system. The axes are colored gold, with black unit markers indicating length (exactly what the length scale is in this case is not important, but usually it will be specified). A particular point (3,-2) is noted in the lower right quadrant. The x-coordinate is usually specified first, as it is here. A bicycle frame (and some other things that we don’t need to worry about at this point) is pictured in the coordinate system, with the main pivot located at the origin.
Figure 2.1)
Reference frames may be defined by objects such as the earth or pieces of a bicycle frame. That is, we treat our coordinate system as if it were attached to the defining object. If the defining object is undergoing a linear acceleration, or has angular velocity (meaning it is rotating), then so will the reference frame. In this case, we call the reference frame “non-inertial” or “accelerated”.
If the coordinate system in Figure 2.1) were attached to the earth, then over time, the pictured frame (as part of a bicycle) would move with respect to the coordinate system. Since the earth is only undergoing very small accelerations we consider it (essentially) an inertial reference frame, for most practical purposes. If the coordinate system were attached to the main triangle, then the positions of other objects, such as the rear suspension members, would be defined by how they move about the main triangle. Since the main triangle is often undergoing significant accelerations, we consider it a non-inertial reference.
One thing to note is that in non-inertial reference frames, fictitious forces and torques can appear due to frame acceleration, the most well known of which is the “centrifugal” force of a rotating frame. If one is riding on one of those carnival rides that spin round and round, one feels as if there is a force (like gravity) pulling one out from the center of rotation and pinning one up against the constraining wall of the ride. This is the centrifugal force, which is only apparent.
The centrifugal force should not be confused with the “centripetal” force, which is the force of the wall causing one to deviate from a linear path and thus to rotate in a circle. The centripetal force is a real force. The centripetal force acts on the rider and points in, towards the center of rotation. The centrifugal force seems to act on the rider in a direction pointing out, away from the center of rotation.
Sometimes it is only important to define a reference by some object, but not important to define where the origin is located or any length scale. In this case, we may define the reference frame by naming some object, without specify anything else. For example, we may specify the reference frame of the bicycle main triangle. We do this when we want to consider how other objects move with respect to the object defining the reference, but don’t care about particular distance scales and so forth.
Each degree of freedom denotes an independent way in which a body can move. A completely free body has six degrees of freedom. Given standard rectangular coordinates, a free body can translate in any of the three coordinate directions and it can rotate around the three coordinate directions.
PA makes use of the degree of freedom limitations on bicycle components. For example, in the reference frame of the ground, a dual suspension bicycle main triangle has three relevant degrees of freedom while the bike is traveling in a straight line. It can translate horizontally and vertically, and it can rotate, all in the plane defined by the rear wheel. The balance of the rider limits the other degrees of freedom. If we fix the main triangle in space, relevant bicycle components only have at most one degree of freedom.
C) “Nature Varies Smoothly” (NVS).
The equations describing the laws of nature are continuous relations (usually stated as functions). The value on one side will not jump discontinuously as the parameters on the other side vary continuously.
(This excluding the quantum realm - very small, very big, very cold, etc.)
As a result, if we imagine a pivot position varying smoothly along some arm in a mechanism, the equations of motion will vary smoothly also. That is, the physical situation (laws) will not jump at some point. The way the mechanism behaves will change continuously.
One of the most difficult things physics students have to grasp is when and how to make approximations. The simplest form of approximation is that involving quantities much larger or smaller then other relevant quantities in a given physical situation. We will give two examples of proper and improper approximations by this method, using mass.
Consider the mass “M1” which is much, much bigger then the non-zero masses “M2” and “M3” in Figure 2.2 A). The masses move without friction in one dimension. In considering the motions of all bodies, approximating M2 and M3 as having zero mass would not be useful, since information about the interactions of the small bodies would be eliminated. On the other hand, if we approximate M1 as infinite, useful calculations may still be done (this is often done when considering human-sized objects interacting on the earth).
In Figure 2.2 B), we have the opposite situation from Figure 2.2 A). Here, we might very well neglect the mass of M2 in quantifying the results of M1 colliding with both M2 and M3.
These two examples show that, in certain situations, the odd entity may be approximated.
Figure 2.2)
Next, consider Figure 2.3 A), which depicts a block of mass M connected by a rope to a suspended wheel that is of very small mass relative to the block and spins with negligible friction. If we want to quantify what happens when the block is suddenly dropped, we could not ignore the mass of the wheel, since our approximations would then describe a non-physical situation – namely the unconstrained angular acceleration of a zero-mass body. On the other hand, if we were to ask the angular acceleration of the wheel in the “Atwood” device in Figure 2.3 B), we could get a pretty good answer while ignoring the mass of the wheel, if both M and m are large compared to the wheel.
For a further discussion of this topic, see the “Mass Approximation.” section in “P.A. Basics.”
Figure 2.3)
The CM of a solid body, or system of bodies, is the weighted average, spatial distribution of all mass in the system.
For example, the CM of a symmetric object, such as a wheel, is at the center or axle.
For us, the most important fact regarding the center of mass is that a force applied to any part of the body will cause a parallel acceleration at the center of mass. For example, a force applied to a wheel somewhere along its radius, in the plane of the wheel, will cause acceleration at the axle parallel to the force. For a wheel in free space, this means that the wheel will start translating in the direction of the external force, as well as rotating. Figure 2.4) shows this situation.
Figure 2.4)
To get an idea of how this might be applied, consider the following question, which I call “The Pole and Wheel”:
A pole is attached to the ground via a hinge with negligible friction. A wheel of mass m is attached to the top of the pole via an axle that also has no significant friction. A rope of negligible mass is wound around the wheel's circumference with the end hanging toward the ground on the right hand side of the wheel. All of this is balanced at equilibrium, with the pole pointing vertically from the ground (what we have here is really just the rear of a mono-pivot attached to the ground).
How should you pull on the rope, in order that the pole will not fall? Should you pull vertically; left or right of vertical? Should you pull through the pivot?
Answer: You should pull vertically, to the extent that one may assume the mass of the earth to be effectively infinite. To be precise, one should pull almost vertically, but juuuuust slightly to the left of vertical for the pole not to fall, since the mass of the earth is not truly infinite (for those interested in why the line is not exactly vertical, consider conservation of angular momentum).
Pull left of vertical and the pole will fall left, and analogously for the right (assuming the mass of the earth to be infinite). Pull through the pivot in particular and the pole will fall left. Figure 2.5) diagrams the situation.
Figure 2.5)
The key is to realize that the tension in the rope induces a force at the edge of the wheel, which in turn will induce a parallel force at the axle, just as theory predicts.
The result follows.
One may look at this system as a mono-pivot bicycle with the earth as a giant front triangle.
Note: The Pole and Wheel question has been extremely difficult for most people. Even most physics professors do not get it right the first time, and none of the well-known bicycle suspension designers has realized this in the past. However, if one wishes to understand the forces present within a pedaled bicycle, this concept is essential.
If the reader is having difficulty with this issue, our suggestion is to conduct the experiment. A good way to do this is to take the front wheel holder from a Yakima (or other) car-mounted bike rack and remove all of the hardware. This is your pole. Attach a bike wheel and conduct the experiment. You will see that if the line of force at the wheel's edge goes though the bottom of the pole, then the pole and wheel will topple over. If the line of force at the wheel's edge is essentially vertical, then the pole and wheel will not topple over.
Those wishing a full presentation of the methods used here can find one in “Classical Dynamics of Particles and Systems”, by Marion, 1970. The center of mass equation of motion due to an external force is on page 68. The derivation starts on page 67.
In the previous example, we have considered a single force on the wheel, with the rotational inertia of the wheel opposing the external force. However, we may also consider multiple forces acting on the wheel. If there are at least two forces creating opposing torques on the wheel, about the axle, and the mass of the wheel is small compared to other quantities, then we may ignore the wheel mass. The simplest example is that of the Atwood machine in Figure 2.3 B). While the machine is in motion, we may take the tension T in the rope on each side to be equal, if we neglect the mass of the wheel. The force at the wheel axle is 2T; both force vectors external to the wheel pointing in the same direction. If the external forces on the wheel are not pointing in the same direction, the total force at the axle will be the vector sum.
Figure 2.6) shows the forces acting at the axle of a negligible mass wheel, which is experiencing multiple external forces at two different radii.
Figure 2.6)
Later, in Figure 2.12) of “An Intuitive Look at Forces and Torques.”, we will consider an example of this, with the crank being an example of the wheel, and the pedal stroke from the rider and chain tension being the external forces. Those wishing to understand the calculations associated with Figure 2.12) should keep Figure 2.6 in mind.
If a wheel or a crank is mounted coaxially to a pivot in some mechanism, it does not matter how the object is mounted physically. In a bicycle, the rear wheel could be physically mounted to the seat stay or chain stay, and the crank could be mounted to the main triangle or the seat stay – none of this matters. The physical situation will be the same in all cases as long as the specified objects and pivots are coaxial.
Imagine a mechanism that has two rigid components (possibly among other things). Two rigid arms, attached to the components by pivots, connect these two component sides. An example would be a 4-bar suspension linkage. In this case, one component could be the main triangle and the other the rear link.
Next, fix a reference frame to the first component, which in our example is the main triangle. At any given time when the arms and the other component (rear link) move about the main triangle, we can calculate the path tangents for all points in motion on these objects by constructing the IC. We do this by drawing lines through the two pivots on either side of each arm. If the arms are linear structures, then the axes will determine our lines. The point where the two lines cross is the IC. The path tangent of any point in motion is perpendicular to the line between the IC and the point (obviously all of this is in a single plane).
Figure 2.7) shows four bars, with the red dots representing pivots. The light blue lines are the line segments defined by the upper and lower pairs of pivots. The black dot at the intersection of the blue lines is the IC of side B and the adjacent arms moving in the reference frame of side A. The green mark represents an object (such as a wheel axle) on side B, with the dark blue line representing the object's path tangent as it moves about side A. The dark blue tangent is perpendicular to the line through the IC and green object.
Figure 2.7)
The idea is that for a small angle dq, the movements of our two arms produce the same paths as if component B and the arms were rotating about the IC.
WARNING! The IC is not a “virtual pivot”. In general, it is constantly in motion, unlike a pivot. In Figure 2.7), as side B moves about side A, the IC will constantly change, as will the dark blue line representing the green object's path tangent. Many errors in suspension theories result from ascribing to an IC the attributes of a pivot. The IC gives useful information, but only for an instant of time, thus the name, instant center. By contrast, a pivot might be referred to as a constant center.
Figure 2.8) depicts a 4-bar suspension frame. The blue lines reveal various instant center positions for the rear suspension in the reference frame of the main triangle. The red curve plots the IC movement as the suspension moves through its travel. If the distance from the rear wheel axle to a fixed point remains almost constant through suspension travel, that is if the rear axle path is almost circular, then we can consider the fixed point a “virtual pivot” for the rear axle. The light green lines in Figure 2.8) reveal that a “virtual pivot” exists for the rear wheel axle, in the reference frame of the front triangle. Note that any virtual pivots will be unique to each point on the rear link. That is, the virtual pivot for the rear axle in this case will not be one for any other points on the rear link of significant distance from the rear axle. Other positions on the suspension may have virtual pivots, or they may not, since the distance deviation from any fixed points may be too large for useful approximations.
Figure 2.8)
Read this section if:
You want a semi-qualitative analysis of forces and torques going on within a suspension bicycle. Understanding everything in this section is not important to understanding Path Analysis. This is just for people who want to go a little deeper.
Read just the conclusions in this section (written bold) if:
You want just the conclusions of the analysis for application to other sections. The conclusions should not be too difficult to understand, so we suggest that one at least give them a quick read. Whatever one does not understand probably will not matter too much, but one might pick up some useful information for the trouble.
Skip this section if:
You are just interested in using Path Analysis to make conclusions and comparisons regarding various bikes.
This section is among the two most difficult in the work.
We here do an intuitive study of forces and torques in a mono-pivot non-URT to understand the things of which a suspension theory must account. This will help us further understand what goes on in a suspension and the limitations on what any viable design can really accomplish. We try to keep the math to a minimum, however we will be making some minimal calculations to demonstrate certain solutions in principle. Those with a less technical background can ignore the calculations and look directly at the conclusions, which are written bold.
The most important lesson of this section is that mass distribution is an important consideration in the physics of full suspension bicycles. No quantitative theory can be correct without this consideration.
It is common practice to take no reaction of the rear shock to pedaling as the goal, so we will follow.
Figure 2.9) shows the front and rear triangles of a “coasting” mono-pivot, with the various forces acting on them, minus friction in the hubs and air, which we neglect (the forces are not drawn to scale). “CM” indicates the rough position for the rider/main triangle center of mass.
Figure 2.9)
All forces sum to zero when there is no pedaling. For this reason, we need examine only those forces and torques that appear as the result of a pedal stroke. Figure 2.10) shows the picture without the coasting forces drawn.
Figure 2.10)
There are a number of ways one might go about analyzing this situation. We will use the torque equation:
1) I*a = å t.
Here, I is the moment of inertia of the body in question, a is the angular acceleration, and å t is the sum of the torques on the body. This is the angular analogue to ma = å F. Using equation 1), we will examine what issues are involved in keeping the torque balance between the main triangle and swingarm, about the main pivot, as close to zero as possible.
For precise calculation, this method is not very useful, since some of the torques are not easy to state explicitly and all of the torques are time-dependant (all except chain torque depending on the positions and movements of the two frame members, which will change with time through the pedal stroke). We can thus glimpse the complexity of any completely rigorous analysis. But for us and our mainly intuitive study, this method will be very useful, since we can use it to explore a number of interesting points with minimal math.
We start with a comment on chain force.
One must be very careful when thinking about lines of force in that magnitude, direction, and location are all important. Even equivalent gear ratios generally produce lines of force that differ in magnitude and direction as well as location.
Figure 2.11) shows a drive train with two possibilities for a 1-1 gearing. L is the crank lever, R1 and R2 are the respective cog radii, and T1 and T2 are the chain tensions for each case.
Figure 2.11)
If a force F is induced at L with resulting tension in the chain (examining one case at a time), the resulting torque equation for the crank is (assuming a non-URT just for ease of calculation):
2) I*a = F*L - T1*r = F*L - T2*R
Thus, T1/R = T2/r. That is, the chain tension decreases as the front cog radius increases – a rather interesting result. So, even the two 1-1 situations will generally not produce equivalent results for suspension activation under pedaling. This actually should not surprise us, since the energy transmitted through the system should be the same in both cases. Energy can be expressed as T*d, where T is the chain tension and d is the length of the chain that passes by some fixed point like the seat tube. Since a greater chain length is pulled in a bigger chain ring for a given rotation of the crank, we need a lesser force to keep the energy constant.
Continuing: Figure 2.12) shows the diagram for the calculations to follow. The partially pictured triangle represents the main triangle, to which the crank is attached. The lower horizontal line represents the swing arm of length SL. R is the front cog radius and r is the rear cog radius. L is the length of the crank arm. F is the force of a pedal stroke. T is the resulting chain tension. h is the perpendicular distance from the chain to the pivot, D is the perpendicular distance from the pivot to the force line at the BB induced by the chain tension, and d is the perpendicular distance from the pivot to the line through the rear axle that is parallel to the chain tension. q is the angle of the chain from the swingarm axis.
Recall the Center of Mass/force phenomenon described in Figure 2.6) of the “Center of Mass” section; it applies both to the proactive force F acting on the crank and to the reactive force T from the chain, both with results at the center.
We have not pictured all of the forces present on the suspension members, but only those induced as a result of a pedal stroke that are relevant for our calculations. We assume that the crank axle and main pivot are close together relative to frame size. (A few frames such as the The Rocky Mountain ETS-X70 and the i-Drive differ from this significantly, but this will not impact on the relevant points, and all conclusions will be valid for all suspension frames). This allows us to approximate the force at the pivot from the crank axle as if the two were coaxial. We also assume that the crank mass is negligible. This will allow us to equate forces on the chain ring and crank arm to resulting forces at the crank axle.
Figure 2.12)
In the following calculations, the reference frame for the main triangle torque equation is centered at the suspension main pivot and does not rotate (with respect to the earth). The reference frame for the swing arm torque equation is centered at the rear axle and also does not rotate. Since both reference frames do not rotate, the bodies will stay at a static angle to each other if their angular accelerations are the same (the initial angular velocities are both zero). IF and IS are, respectively, the main triangle moment of inertia about the main pivot and the swing arm moment of inertia about the rear axle. tFi and tSi denote the various torques on the respective bodies about their coordinate origins, which include the torques due to the chain, fork/front wheel (friction and inertia), bicycle acceleration (the most commonly recognized of which is squat), the crank rotation and lower rotating parts of the rider's body, and also the torques due to the interactions of the two frame members [we do some further work with these interaction torques in Appendix A) “PCL Problems; Some Further Calculations.”, should anyone have questions as to exactly what these are].
{An aside: One should not be too concerned about the following detail, but the astute reader will note that we are using two different non-inertial reference frames for each bicycle frame member. The bicycle acceleration and interaction torques are the fictitious torques in these reference frames.}
The torque equations for the rider/main triangle and swing arm are then, respectively:
and
Achieving the stated goal of minimizing suspension reaction to pedaling generally involves finding the best place for the main pivot relative to the chain force line for an assumed condition (mass distribution, etc.). To do this, one must express the chain torque in terms of h and solve for this quantity in the proper equation. Fortunately the chain and pedal torques are easy to state in equation and will allow us to get a formula in principle for the desired relation of pivot and chain.
Let tFC denote the torque on the main triangle due to pedaling and the resulting chain tension.
Noting that with a negligible crank mass, F = T*R/L, the torque on the main triangle due to the pedal stroke and resulting chain tension is:
5) tFC = F*L – T*D = T*R - T*D = T*(R-D) = T*h.
So we see that, neglecting the mass of the crank, the torque on the main triangle from pedaling is just as if we had been pulling on the chain from a point on the main triangle that is a perpendicular distance h above the pivot – a very interesting result. However, one must be very careful not to take this result too far; as we have seen, for a given pedal force, the larger the radius of the front cog, the lower will be the chain tension.
Let tSC denote the torque on the swingarm from the chain (again, ultimately from pedal force). Again, since in practice the pivot is relatively close to the BB compared to the frame size, we approximate the force on the swing arm at the pivot as that of the chain force induced at the BB (these values will be very close for typical frames). With this approximation, we have:
To have the torque balance between the main triangle and swingarm about the main pivot equal to zero (to get no reaction of the rear shock), we want the front and rear triangles to rotate en unison – that is, we want the aF = aS. Solving for the a's in equations 3) and 4) and setting the two expressions equal to each other we get:
This is the zero torque balance formula for the main pivot position, relative to the chain line for a non-URT mono-pivot (with pivot not too far from the BB compared to the size of the frame – again, almost always the case).
One might conclude that h depends on T, as T appears in the denominator of the last two terms. We state without proof that T will appear as a factor in all of the torques, just as it did for the chain torque, with the exception of that resulting from the fork. So with the noted fork exception, h does not depend on T. [In “PCL Problems; Some Further Calculations.”, we give an example of how the torques, for the most part, eliminate T from equation 8).]
We may draw the following conclusions from equation 8):
First, notice that the moments of inertia for both bodies are in all terms. This tells us that it will be impossible to construct any sort of a quantitative suspension theory without taking into account mass and its distribution. Mass distribution will be of equal consideration for all other suspension types. This rules out certain “Special Point” Theories, such as the most naive “Pivot at the Chain Line” (PCL) theories.
The second thing we notice is that since the torque values are time-dependant, h will also be time-dependant through the pedal stroke. We thus see that there is no single “proper pivot point” (or points), exactly, through an entire pedal stroke. In addition, we note as a matter of intuition, that as the rider makes a pedal stroke, the system of frame members, on average, will rotate back relative to the rear axle (a ¹ 0). Between pedal strokes, the frame members will fall back down, and not in such a way as to keep the rear shock inactive without help from friction in the pivots. This further tells us that it is impossible for any rear suspension geometry to be completely non-reactive to pedaling, without static friction.
The time-dependant nature of our mono-pivot situation is also common to all other suspension types, since frame member orientation changes through the pedal stroke in all of these bikes as well. In particular, mono-pivots can approximate a zero torque balance about the main pivot as well as any 4-bar, through the pedal stroke.
The effects of changing frame member orientation are relatively small, but we note them as a warning against all theories that purport to completely eliminate shock activation to pedaling, in principle, based on geometry (even if there is an assumption for mass distribution), such as “Special Point” Theories.
Since the frame orientation effects are relatively small, a single geometry can behave relatively uniformly through the pedal stroke. Suspension geometry can thus keep pedal effects on the shock to a minimum, on average, and let friction do the rest. Pedaling effects on the rear shock can be made small compared to any significant bump, so a good suspension with the right amount of friction can effectively control oscillations, while remaining supple enough to absorb any significant bump.
Lastly, as a matter of intuition, we note that in any suspension, the less the rear shock extends during a pedal stroke, the more the front shock extends. There will be loss to friction either way. The ideal proportion of front and rear shock activation will be that which minimizes sympathetic oscillation.
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This section is more difficult.
We don't want to spend too much time here, since this is probably the last issue about which a consumer should worry. All non-URT types can achieve all of the really useful suspension rates and most frames out there pair up fairly well with the stock shocks. Rate is only really a consideration for those who are real suspension wonks (yes, the author is a suspension wonk and if you are reading this, you are in danger of wonk-hood also) wanting to swap between coil and air shocks, which generally have different internal rates. Pairing a falling rate frame with a linear coil shock or an extremely rising rate frame with an air shock might not have acceptable results.
However, we do refer to suspension rate in other parts of this work, so we will look at the most important considerations.
All springs have “rates” and a suspension is just a type of spring.
Define a coordinate x as the direction in which a spring compresses. The “spring rate” is a function of x, and describes the amount of force with which the spring will tend toward equilibrium at any point of compression or extension away from equilibrium. The steeper the rate function, the more a spring will resist additional movement the further it is moved from equilibrium. For a typical coil spring near equilibrium, the rate function is almost linear. If the rate function is concave up, then the spring has a rising rate; that is, the additional force needed to further compress the spring at each point will increase as the spring goes through its travel. If it is concave down, then the spring has a falling rate, with analogous results. Figure 2.13) shows a graph with each type of rate.
Figure 2.13)
The rate of a bicycle suspension is composed of the internal rate of the shock and the rate inherent in the suspension geometry.
Internal shock rates range from near linear to rising. Coil springs tend to have more linear rates, while air springs tend to have rising rates. All frames may be fitted with a range of shocks, which these days generally have one of two lengths and standard mounts. We will not consider shocks further, since they are not an inherent feature of frame geometry.
The contribution to rate from suspension geometry is determined by the way in which the shock mounts, front and rear wheel axles, and main triangle move relative to one another. The front wheel axle establishes frame orientation to the ground but generally may be neglected, since bottom brackets are almost universally 13" ± .5" from the ground without rider (given a typical assumption for the fork “Crown to Axle Length” or “CAL”). Thus, the rear wheel and BB largely determine the frame orientation to the ground.
Again, we don't want to spend too much time on this, so we will give an example of the contribution from the relative movement of the shock mounts. Similar consideration must be given to the rear wheel axle relative to the rear shock mount and BB.
Establishing the main triangle as our reference frame, the rear shock mount will travel a circular path around some pivot – the main pivot in the case of a mono-pivot and the upper frame pivot in the case of a 4-bar (our following statements will apply in both cases).
If the tangent of the rear shock mount points near to the front shock mount as the suspension goes through its travel, then the relative movements of the shock mounts will have a neutral influence on suspension rate (by neutral we mean that, given a linear shock, the suspension rate will remain linear).
Figure 2.14) shows a suspension member moving in the frame of the main triangle. As the suspension compresses, if the rear mount tangent is moving into alignment with the front mount, then the path will increase the rise (decrease the fall) in rate. If it is moving out of alignment with the front mount, then the path will decrease the rise (increase the fall) in rate. This is because for a given angle of rotation, the two shock mounts move towards each other the most when the rear mount tangent is through the front mount.
Figure 2.14)
Click
to enlarge
If we are dealing with a mono-pivot, then the suspension member is the rear triangle and the rear connection will be to the rear axle. If we have a 4-bar, then the suspension member is the upper link and the rear connection will be to the rear link. In both cases, the larger the radius of the rear shock mount path, the larger will be the rate curvature due to geometry. Also, the longer the suspension member; the larger will be the magnification of the internal shock rate curvature, since the wheel will travel a greater distance for a given distance of rear shock travel.
This is most of the ballgame for a mono-pivot (minus only wheel path). For a 4-bar, one must do a similar analysis for the tangent of the upper rear pivot relative to the rear wheel axle. At any position in travel, if the tangent is pointing at the wheel axle, then the shock will compress least for a given amount of wheel travel. In most 4-bars this pivot has a path that will diminish the rate, and again, the larger the path radius of this pivot the larger will be the rate function curvature. The paths of the rear shock mount and upper rear pivot thus define the over all effect in a given 4-bar, minus wheel path.
| Chapter III - Path Analysis |
Read this section.
This is the central point of the entire work.
This section is moderately difficult.
1) All measures of suspension performance depend almost entirely on the paths of the following specified components relative to any reference frame defined by one of the bicycle frame members: Handlebars, seat, bottom bracket (BB), front and rear wheel axles, shock mounts, and rear brake.
As is explained in the “Reference Frames” section, establishing a frame member as our non-inertial reference does not mean that it will not move. It will translate and rotate, and our reference frame will move with it.
The above “specified” components will always move along paths or one-dimensional spaces in the reference frame of one of the supporting bicycle frame members, as a practical matter. The path tangents determine how any bike will behave at any point in time. The path curvatures determine how the bike will behave over time.
If we wish to compare two designs, we should identify a frame member common to both designs. The more alike the paths are in any two suspensions, in the common reference, the closer will be the performance of the frames that produce them. In practice, the bars and seat will always define the best reference and this will be the reference for all analysis in this work (though sometimes it can be interesting to see how paths compare from reference to reference). Mass and its distribution play an important role in any mechanism. However, the main triangle and rider are usually about 60 times as massive as the suspension members (not including the shock). The movement of rider/main triangle mass will depend on the movements of the main triangle components (seat, bottom bracket, and handlebars), even with a non-integrated main triangle (bottom bracket moving with respect to the bars and seat). In addition, the differences in mass movement of the suspension members between different designs with similar component paths are relatively small. This makes the ground and the rider/main triangle the only two significant masses in bicycle physics.
These mass considerations are what allow for PA. We have covered mass approximations in the “Approximation” section of “Some Important Concepts.”. However, when and how to apply approximation can be a very difficult issue, so in the “Mass Approximation” section below, we will explain in detail how mass approximation allows for PA.
Naturally, each individual rider will produce a unique mass distribution. When we say that we can determine suspension performance by the paths, we mean that we can know the performance of the frame for any set of assumptions for relevant physical quantities, such as rider mass distribution or contributions from the suspension fork.
Friction in the suspension mechanism will always act to oppose the movement of components along their paths and will ultimately be directed tangent to the path. Friction magnitude can for the most part be controlled in one type of geometry as well as another. Thus, while we might find one particular suspension bike to have a favorable amount of friction relative to another, friction does not lend any advantage to one type of suspension over another.
Note that the forces between components are critical in determining suspension performance. However, all lines of force, whether they are through the rider, the chain, or external are equally producible in all designs. They thus do not distinguish one design from another. However, it is very helpful to understand how the forces and torques act on and within a bicycle.
Frame stiffness is an important factor in bicycle performance. However, it is much more an issue for handling (a topic not covered in this work), particularly high speed cornering, then anything else. With regard to pedaling, braking, and shock absorption, one only need be wary of the very lightest frames. It has been several years since the author has been aware of any new frames on the market that are so severely under-built as to cause real problems for pedaling, braking, and shock absorption, beyond bad choices and defects in materials and manufacturing, that lead to frame failure (also not covered in this work).
This leaves geometry as the overriding issue in suspension performance regarding pedaling, braking, and shock absorption.
In most cases, the full machinery of PA is not necessary since the paths of components may determine the orientations of their supporting structures (frame members, fork, etc.). For example, the BB and seat may fully determine the main triangle, so one could simply look at that body rather then the attached components. However, in cases such as the i-Drive, the full machinery of PA is the only practical method of analysis. Analysis of the i-Drive by any other method would be extremely complicated. The power of PA will be revealed in the extreme simplicity of i-Drive analysis using this method.
We will give an analysis of the i-Drive theory, Ellsworth’s “Instant Center Tracking” (ICT) theory, and other erroneous theories at the end of this paper.
Read this section if:
You want to verify for yourself the validity of the Path Analysis main assertions and understand the details of how and why Path Analysis works.
Skip this section if:
You will accept the main assertions and are just interested in using Path Analysis to make conclusions about what suspensions can do and comparisons between various bikes.
This section is moderately difficult.
As stated above: Path Analysis works because the mass of the rider/main triangle dominates all mass in a bicycle. In addition, the differences in mass movement of the suspension members between different designs with similar component paths are relatively small. This makes the ground and the rider/main triangle the only two significant masses in bicycle physics.
Furthermore, all forces on the rear suspension members, other then those directly between suspension members (through pivots and so forth), are directed through the PA specified components. Since suspension member mass is not significant and the suspension members control motion between the two significant masses, it is enough to consider the forces between the PA specified components.
We now look at this a little more closely.
Consider Figure 3.1 A). Here, we have a main triangle and swing arm attached to a base by a pivot. This is actually the proper model with which to analyze bicycle suspensions, minus some contributions from the front fork. If the force F were calibrated to the force of gravity, you would have most of the situation for the analogous dual suspension bike, in a particular gear (no human rider could really produce such high values, but the values can be reduced by tilting the mechanism backward, into the page). Note that the main triangle can move only in a certain restricted way relative to the base and that the lower pivot, analogous to the rear axle on a bike can move only in a certain path relative to the main triangle.
{We have drawn the pictures vertically symmetrical and the linkages to form parallelograms. But the model should be taken more generally to include any typically shaped front triangle and lengths of linkage members. The model should also be considered in all reasonable positions.}
Figure 3.1)
Next, consider Figure 3.1 B). Here, we have a main triangle and a 4-bar linkage attached to a base by a pivot. This produces precisely the same paths as the mechanism in Figure 3.1 A). In fact, if we neglect the masses of the swing arm and linkage, we would have identical situations in both A) and B). Figure 3.1 C) shows both suspensions on the mechanism at once, from which we see that both suspensions will work harmoniously with one another. This foretells an analytical device that I have conceived, called a natural mirror bike, which we will discuss below in “The Natural Mirror Bike.” section. Now the question is, “Can we neglect the masses of the suspension members?”
If the mass of the swing arm were very large compared to the mass of the main triangle and the mass of the 4-bar linkage were very small compared to the main triangle, then it is easy to see that PA would not apply. In A), the main triangle would rotate around the upper pivot with relatively little motion from the swing arm when F is applied. In B), the main triangle and linkage would move very differently from A), the linkage moving more drastically then the swing arm, producing a very different physical situation. But this is not the case in a bicycle.
A typical dual suspension frame weighs about five to six pounds, without the rear shock. Of this, the rear might take up 2.5 pounds. Now in bikes for which there is any utility in comparison, the mass difference between any two types of rear suspensions is going to be less then half a pound. The rider/main triangle, on the other hand, averages about 150 pounds, at least. This leads to a mass difference of less then 0.3 % between the vast majority of mechanisms.
In addition, the movement of mass through suspension compression depends largely on the relative motion of the specified components, within the range of motion for all viable suspension bikes. Considering the movements of the links in a 4-bar linkage, one sees that the overall movement of mass is very similar to the movement of mass in a mono-pivot swing arm (though not exactly the same). The movement of mass in the GT i-Drive is almost identical to a mono-pivot, the only (insignificant) differences being the movement of the eccentric on the swing arm and of the “dogbone”.
As a result, we may neglect the suspension members and focus exclusively on the paths that they produce, as we have drawn in Figure 3.2 A) for the mechanisms in Figure 3.1). Here, we have drawn a circular path for the lower pivot about the main triangle. This contains all of the significant information concerning how the mechanisms will work. Figure 3.2 B) shows the type of motion allowed by all equivalent mechanisms.
Figure 3.2)
We have demonstrated these principles for a 4-bar vs. a mono-pivot, but they apply in general, since the masses of suspension members will always weigh about the same as the examples here.
B) Forces Between Linearly Constrained Particles.
Suspension performance is determined essentially by the relative movements of the PA-specified components. This is because the interfaces to the external world (the wheels) are identical in all bicycles for which there is a utility in comparison. We can thus do our analysis based entirely on these internal workings and neglect any external interactions (with the ground for example). This simplifies matters in that our analysis may involve less degrees of freedom.
As stated above, the specified bicycle components move along paths or one-dimensional spaces, assuming a reference frame attached to one of the supporting frame members. How each moves will depend on the sum of forces exerted between it and the other components in the system. So lets get some idea about how to treat objects moving along such paths by looking at some examples.
Suppose you have an x,y-axis. A particle, such as a marble, is restricted via some mechanism to move freely along the diagonal in the first quadrant. Any number of mechanisms could achieve the restricted movement. Now suppose there was a force F pushing on the particle in the x-direction. The equations of motion for the particle would involve F*cos (45°), that is, the direction of motion, but not the mechanism that restricts the degrees of freedom. Figure 3.3) shows the path, force, and force component along the path.
Figure 3.3)
Two particles move along paths relative to one another. If a force is induced between them, it will cause both particles to accelerate along their respective paths in the direction corresponding to the tangent component of the force along the path. Figure 3.4 A) shows this scenario
Figure 3.4)
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to enlarge
Next consider two wheel axles that are restricted to travel along the same paths relative to one another as the particles in A), with axles at the same points. The same magnitude of force as in A) is exerted at the wheel edges. Figure 3.4 B) shows this situation.
Notice that the forces at the axles are in the direction of the force line between the cogs, which is different from the force direction on the particles in A). The forces are also of a different magnitude due to the inertia of the wheels. Particularly important is that these two forces at the axles are not co-linear. The components along the paths in B are in the same directions as those in A, but will generally be of different magnitudes due to both the differences in overall force direction and magnitude at the axles.
Now consider a particle such as a wheel axle that is restricted to travel along a particular path relative to other components in a mechanism (a main triangle for example). If forces are exerted by the other components on this axle, by whatever means, the axle will tend to move in the direction along its path that corresponds to the tangent component of the sum of the forces. The magnitude of the tangent component determines the motion of the axle. Similar considerations exist for the other components.
If one component's path in a mechanism is a function of another's path (in part or whole), then it does not matter how that relation is achieved; any mechanism will produce the same physical results. An example of such a functional relationship is that between a seat, handle bars, and bottom bracket, which define the main triangle. An essentially equivalent functional relationship is produced in GT's i-Drive.
As a result, if we want to consider performance in non-URTS, we need only examine the paths of the wheel axles, shock mounts, and brake, relative to the main triangle.
Figure 3.5) shows a rear axle path in relation to a main triangle for a non-URT. The gray lines denote several possible axle paths. The red line shows a line of force (through the chain). The green arrow shows the force induced at the wheel axle. The blue arrow shows the component of force along the wheel path at the axle.
Figure 3.5)
*** Here is a central point. In the small neighborhood around the axle denoted by the orange lines, the paths are identical. However, above the neighborhood they diverge wildly, one being circular and the others being of more radical curvature. We pictured this to emphasize that the path tangent is what counts at any moment in time. Other aspects of the paths have no bearing on what happens in our small neighborhood around the axle. In little neighborhoods around all points in a path, all suspensions with similar paths in that neighborhood behave similarly; in particular, they behave like some mono-pivot. When we shrink the neighborhoods to zero, we see that the tangent to the path determines suspension behavior at any point in the path. ***
This might seem strange if we consider mechanisms that produce paths with very different radii of curvature. But remember, it is what happens as the suspensions move away from the particular points on the paths with common tangents that make the situations differ. Greatly differing curvature will produce swiftly diverging physical situations.
The following example should allay concerns about whether or not the tangent can really contain all of the information necessary to evaluate a given situation regardless of path curvature.
Figures 3.6 A) and B) depict main triangles attached at different points to swing arms of different length. The paths that these swingarms produce are of different radii, but have the same path tangents at the initial locations of the base/swingarm pivots. We depict the mechanisms as being horizontal and viewed from above, so that we may start at equilibrium before a force F is applied to the main triangle. Without loss of generality, we choose the swing arms to be aligned along the y-axis.
Figure 3.6)
The force F may be applied at any point, in any direction. We chose to place it in such a way that the direction extends between the two frame/swingarm pivot locations in the different mechanisms because this is the situation that is most likely to cause concern.
Neglecting swingarm mass, we see that the x-component of F, Fx, will cause essentially identical initial movements of the main triangles in the two mechanisms, since this component is perpendicular to the swing arm. We also see that the y-component of F, Fy, will have the same lever arm about both of the pivots in both cases and thus also will cause identical initial movements of the main triangles. This means that the initial movements of both mechanisms due to the total force will be identical. The two situations will diverge as the swing arm/base pivot paths diverge outside of the initial little neighborhood around the initial positions. But in the initial positions, the physical situations are identical.
In reality, a swing arm as large as the one in Figure 3.6 A) might weigh a few pounds more then the one in B) (though the difference would still be less then 2% of the rider/main triangle mass). But such swing arms do not exist in real bikes. 4-bar frame members that can produce a path curvature similar to the one in A) do not weigh substantially more then ordinary mono-pivot swingarms and the mass tends to move in a generally similar fashion over all. So as we observed earlier, neglecting the masses of the suspension components is a good approximation in our analysis.
Read this section.
It is not technically difficult and the “Natural Mirror” conceptual device is the most easily understood confirmation for the validity of Path Analysis.
The best intuitive confirmation that one can have for the validity of Path Analysis is to imagine putting two different suspension mechanisms on one bike simultaneously. There would be no conflict between them as long as the component paths were the same for both mechanisms. Shortly after I first published the “simultaneous suspension”, a particular version of my idea was proposed that would have one side of a bike constructed from a mono-pivot and the other from a 4-bar with a circular rear axle path. More generally, we may construct a bike with two different suspension mechanisms on either side, each having the same component paths. I will refer to such a bike as a “natural mirror” or simply “mirror” suspension bike, since the true nature of each suspension is mirrored on the other side.
We can include the paths of all components as part of a natural mirror analysis, or only those for which we may have a particular interest. For example, if we wish to compare only wheel paths, we may imagine pairing up frames with identical wheel paths and it will not matter whether other components, such as the shock mounts, also have identical paths.
In evaluating the validity of a theory, physicists often examine certain “obvious” cases to see if the theory makes sense. Here we examine several designs with circular rear axle paths, to demonstrate that they will all perform identically under pedaling (suspension rate adjustments accounted for in the last example).
Suppose we start on one side of a mirror bike with a 4-bar suspension in which the “bars” determine a parallelogram – that is, the upper arm is equal in length to the lower arm and the forward arm (the main triangle between the two forward pivots) is equal to the rear. We call this a “parallel” 4-bar. The wheel path (both for pivot on the chain stay or seat stay) is circular.
On the other side of the bike, we can use a mono-pivot, with main pivot at the same height above the 4-bar main pivot as the wheel axle is above the 4-bar rear pivot.
We refer to this bike as a “parallel/mono” mirror and both sides produce the same path.
We could even make the shock mounts have equivalent paths by mounting a shock to the mono-pivot in the same way that we mount a shock to one of the horizontal 4-bar arms. Each side of the bike will perform exactly the same as the other. Figure 3.7) shows both of the above suspensions from the side.
Figure 3.7)
Let us now consider another 4-bar. This time though, we will make the two forward pivots coaxial to produce what we call a “pp-coaxial” 4-bar. The pivots will still be physically attached to the main triangle separately and thus the suspension will constitute a true 4-bar. The wheel in this case also has a circular path and thus this suspension can be put in with either of the other two. We will call this 4-bar combined with a mono-pivot a “pp-coaxial/mono” mirror. See Figure 3.8) for this example.
Figure 3.8)
Lastly, let us consider a 4-bar with rear wheel mounted coaxially with the rear lower pivot. It does not matter whether the rear wheel is mounted physically to the chain stay or the seat stay, both will behave the same, as the wheel will have the same path. We will call a mirror bike with this suspension and a mono-pivot a “wp-coaxial/mono” mirror. The configurations of the 4-bar upper links contribute only to the suspension rate in this case. Adjusting the relative paths of the shock mounts as well as the “internal” rates of the shocks may be done to more or less match the over all suspension rates of the two sides. Figure 3.9) shows this mechanism.
Figure 3.9)
All of these are examples of very different suspension configurations that will behave exactly the same while not under braking (shock tuning accounted in the last case), because the rear axle paths are the same – namely circular. The shock mount paths in the first two examples are not identical in space, but are identical in relative motion and so cause no conflict.
Read this section.
This section explains the important considerations involved in most of the full suspension frames built today.
This section is less difficult, except in one or two places perhaps, and is of great use to consumers.
{A technical note about the pictures in this segment: The main triangles are not drawn to scale and the paths are not meant to represent solutions for any particular real-world situations or as endorsements for any particular designs – they are constructed merely to illustrate the points.}
Before analyzing paths, we make a few general comments on some other issues.
Both major suspension types (mono-pivots and 4-bars) may be as light or as strong as any dual suspension bike can viably be, examples of both having found success in XC and DH. Both types can also achieve comparable lateral stiffness for a given frame mass.
Mono-pivots are a bit simpler of design, but most of today's 4-bars are about as reliable.
Some 4-bars offer adjustable travel and geometry. This is equally possible with a mono-pivot, but as of this writing, mono-pivot manufacturers have yet to answer in a substantial way.
A) Mono-Pivot and 4-Bar Rear Axle Paths.
4-bar rear axle paths can deviate from those of conventional mono-pivots in three different ways:
First, mono-pivot paths will always be circular about the main pivot. 4-bars can have varying path curvature. The easiest way to see this is to consider Figure 3.10 A). Draw the complete circular path of the upper rear pivot as the upper link rotates about the upper front pivot. Next, consider the path of the lower rear pivot, as the upper rear pivot goes through its revolutions. This lower pivot will move back and forth along a circular arc. By “‘Nature Varies Smoothly' (NVS).”, we see that the paths of points along the rear link will be much like the upper link circle for points close to the upper link, and much like the lower link arc for points close to the lower link. The paths will gradually change from one to the other as the points vary along the rear link. An axle mounted somewhere in between will have varying path curvature.
Figure 3.10)
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to enlarge
At the moment, the “The Virtual Pivot Point (VPP).” concept, conceived by Outland and soon to be re-introduced by Santa Cruz and Intense, is unique among viable concepts in its capability to produce significantly variable curvature. However, as we will see, current examples do not take any real advantage of the possibilities.
Secondly, conventional mono-pivots do not have main pivots located within the wheel radius. This limits the minimum radius of axle path curvature to at least the size of the rear wheel.
A mono-pivot could achieve a tighter curvature only if the pivot were split into right and left. We have proposed such a bike, which we call the “split-pivot mono”. This design is in fact viable and will have the added benefit of a more stable pivot. Figure 3.10 C) shows the tight curvature. We thus do not consider tight curvature to be an inherent advantage of 4-bars over mono-pivots, since the split-pivot mono can achieve the path. Although no such bikes are in current production, the split-pivot mono was the motivation behind Cannondale's new Scalpel.
4-bars can achieve a tightly curved path centered inside the rear wheel radius. Figures 3.10 A) and B) show a 4-bar with the same tangent as our split-pivot mono both at equilibrium and compression. We have achieved our example by having the IC move backward as it moves down. This is essentially the Giant NRS design. The Rocky Mountain ETS-X70 also uses a linkage system to create a center of curvature inside the wheel radius.
And thirdly, mono-pivots will always have pivots located within the body of the bicycle frame. 4-bars can achieve a more widely curved path, as is the case in the current “Virtual Pivot Point (VPP).” designs.
B) Shock Absorption (“coasting” situations).
We handle only coasting situations here, since suspension issues related to pedaling and braking will be handled specifically in those sections.
A bicycle suspension may be suddenly compressed by the ground either through wheel contact with an obstacle such as a rock or from the impact of a drop-off. In general, we believe that a widely curved rear axle path running slightly up and back is the best solution. Tight curves, either circular or varying are generally inferior for shock absorption. However, this deficiency may be mitigated to some degree by having the path tangent tilting backward through all or most of travel (for example, having a high main pivot, either real or virtual), as is the case in the The Rocky Mountain ETS-X70 and, substantially, the Giant NRS. One might also find that short travel designs such as the Cannondale Scalpel do not have enough travel for this deficiency to be significant.
In the case of a drop-off, the situation is obvious. A linear path will offer the smoothest, most consistent compliance.
In the case of an obstacle, the bump force will be up and back relative to the frame, so the initial tangent should be up and back. The direction of the force will turn more vertical as the bike clears objects of “ride-able” size, so a widely curving path turning slightly upward should be ideal. Experiment should determine the path incline and radius of curvature that produces the best result on average.
Rising rates benefit short travel designs, since this will allow better initial compliance, while reducing the probability of hard bottom-outs.
Non-URT generally means BB on the front triangle. These bikes dominate the industry these days and most are either mono-pivots or 4-bars. Here we examine pedaling of non-URTs, asking specifically, “Are there any relative merits between the mono-pivot and 4-bar design concepts under pedaling, and if so, what are the considerations involved?” In asking this fundamental and rather popular question, we will get a good general idea of what attributes really effect non-URT pedaling performance.
We observed in Figure 3.5) of the “Forces Between Linearly Constrained Particles.” section that the component path tangents determine how any suspension will perform at any point in time.
This means that, neglecting friction in the mechanism, each particular geometry will have its maximum effectiveness only in certain “ideal” gears (from a practical standpoint, this could mean one gear or several). Any others sets of gears will produce different forces on the mechanism, leading to different components of force along the tangents. The further the gearing from ideal, the more reactive any suspension geometry will be.
For a given deviation away from ideal gearing, “suspension rate” (spring stiffening) will determine the amount of reaction from a pedal stroke. Shorter travel suspensions tend to be less reactive to pedaling then longer travel versions, since short travel designs should have higher more rising rates (in part due to the fact that many use air shocks these days). In practice, the actual rates in the shallow regions of travel where pedaling will be affected will largely be a function of the total travel length of the rear wheel path.
Most frames mate well with their stock shocks, and all common suspension types can achieve the really useful rates (linear or rising). So rate is only a real issue for those wishing to swap different coil and air shocks in and out of a given frame.
Since rate in the shallow regions of travel will largely be a function of total wheel path length and is of secondary importance to most people, we will not further consider rate here. We refer those still interested, to the “Suspension Rate.” section in chapter II.
Any comment on frame performance must be made with respect to a range of forks, just as is the case with rider mass. So an assumption must be made for fork characteristics. In addition, dual suspension bottom brackets (BBs) are almost universally 13" ± .5" from the ground without rider, given a typical assumption for the fork “Crown to Axle Length” or “CAL”. Thus, the rear wheel and BB largely determine the frame orientation to the ground. So, after noting the required fork assumption, we can neglect the front wheel without much problem. [If one is uncomfortable with this, then one may certainly consider the front wheel axle path. This and the rear axle path will determine orientation of the main triangle to the ground (again, a CAL assumption must be made)].
We see then that the pedaling performance of any non-URT will be determined largely by the rear axle path (including the length, which will give us a good idea of the rate influence).
So PA can become very simplified for certain types of frames and certain types of analyses. This simplified version of PA has been known for some time and been used by numerous designers in the past.
Given that any sort of design can produce all possible rear axle tangents, potential differences between the two non-URT types, and between the various individual non-URT designs in general, will have to come from differing possibilities as the rear axles move through their paths. So we now examine whether or not viable varying curvature, tight curvature, and/or wide curvature offer significant performance advantages or drawbacks.
First, let us consider what might be an ideal path to minimize suspension reaction to pedaling.
For ease of discussion, we will assume 1-1 gearing. With this gearing, there will be no feedback to the pedals as the suspension goes through its travel, if we have a circular path centered at the BB. That is, the distance from the BB to the rear axle must be constant, as in a mono-pivot with main pivot coaxial to the BB. (If the gearing is larger, then the distance must increase to eliminate feedback, while the opposite is true for smaller gearing.) Figure 3.11 A) shows this wheel path:
Figure 3.11)
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to enlarge
Continuing with 1-1: To counter squat and some compressive chain effect at equilibrium, the path tangent must have a negative slope (be tilted back counterclockwise from vertical). This will counter squat with chain force and by altering the effect of bike acceleration on the swing arm. Figure 3.11 B) shows this situation, retaining the overall circular wheel path. But now we have a situation with some bump feedback to the pedals.
The significance of feedback is very debated. It certainly achieves its greatest significance away from ideal gearing, usually in the smaller gears where chain-length growth (between the two cogs) is increased and the effect of smooth wheel spin disruption on the crank will be magnified by a large “reverse gearing”. However, we want to be very clear on one point. There is no free lunch – to any degree that you have anti-squat, you will also have some degree of bump feedback, regardless of what certain manufacturers claim.
The vast majority of experienced riders give great importance to a smooth pedal stroke, so feedback is generally seen as something to be minimized if possible. Figure 3.11 C) shows a type of path that would allow the chain to counter squat at equilibrium, while limiting bump feedback to the pedals. This path would have a small segment around equilibrium of just the right negative slope, with BB centered circular segments above and below. Most would consider this the ideal situation. In principle, a 4-bar can also achieve a path similar to the stated ideal by having a progressively “tightening curvature” as the suspension compresses through its travel. The subtle features of the Figure 3.11 C) path would be lost, but the broader shape would not be grossly different. This would allow the suspension to control squat at equilibrium, while providing less feedback then a conventional radius circular path, both above and below (below being a lesser issue, since most of the travel is above).
So we see that certain variable curvature paths can offer an advantage with regard to pedaling, in principle.
“The Virtual Pivot Point (VPP).” design concepts are capable of producing an “S-shaped” path somewhat similar to the region around equilibrium for the path in Figure 3.11 C). These designs would obviously also be capable of producing “tightening curvature” paths. To date, the Outland designs are the only bikes we know of claiming significantly varying curvature. Again, unfortunately, the current examples do not take any real advantage of these possibilities.
A tightly curved circular path above equilibrium can provide an anti-squat path tangent, while curving up more sharply to reduce feedback during compression deep into travel. Such a design should be run with little or no sag, since kick-forward during suspension extension may become an issue. The Giant NRS, the The Rocky Mountain ETS-X70, the Cannondale Scalpel, and the “split-pivot mono” described above (not in production) are examples of tight curvature designs (though one might find that the ETS-X70 does not have a small enough radius, nor the Scalpel enough travel for this to be significant for him or her).
Wide paths would offer no advantages with regard to pedaling, since they offer no special path tangents and do not address the issue of anti-squat verses feedback. There may be those who desire anti-squat throughout the travel and consider feedback an acceptable price to pay. For these people, wide curves may offer a perceived advantage under pedaling. However, we feel that this is not a wise position. During large compressions from an obstacle, such as a rock, the rider will be kicked forward when the rear tire encounters the obstacle; so squat is not the issue during this type of suspension compression. The impact of a drop-off will compress the suspension regardless and the rider is likely to be standing (thus creating a completely different pedaling situation from that for which any bike will be designed), so again, squat is not an issue.
We have seen that rearward axle path tangents at equilibrium should offer some advantage while pedaling over smooth terrain and during shock absorption while coasting. However, this will also produce bump feedback while pedaling over bumps. So we have a tradeoff. Many riders say that they are very sensitive to this trade off, even to the point where differences of less then an inch in main pivot locations are noticeable. Some prefer the generally efficient rearward tangents, while others want the smoother pedaling, more vertical tangents. So we have a compromise with which to deal.
We have also noted that tight curves above equilibrium, whether circular or varying, may help with reducing the bump feedback of a rearward tangent. However, curves tight enough to make a significant difference in the shallow regions of travel where riders are likely to be pedaling may produce inferior bump performance deeper into the travel, since wide curvature should be best for shock absorption. Though again, designs with rearward paths through travel, such as the The Rocky Mountain ETS-X70 and, for the most part, the Giant NRS, may mitigate this compromise to one degree or another.
Furthermore, while variable curvature has its allure, in practice it requires closely spaced pivots near the frame center. As a result, links and pivots in the highly stressed bottom bracket area must be more heavily built. This leads to bigger tradeoffs between weight and durability then in conventional 4-bars. So variable curvature designs are not without their compromises.
This furthers a theme that we have revisited throughout this work – there are no “right paths” or “right pivot points”. We have seen this in mass distribution considerations of having riders with different body types. We have seen this in the fact that no geometry can be completely non-reactive through a pedal stroke, without the help of friction. And now we see it again in the fact that there are choices that must be made, depending on what type of suspension performance one wants.
Human beings can be surprisingly sensitive to physical situations. This author finds that a difference of just two millimeters in the height of a road bike seat makes for the feel of a completely different bike. So we are not surprised to find that some people hold small geometric differences as important and we must assume these positions to be legitimate.
However, we must note that some people experience “have-it-all” performance in some designs from manufacturers that claim such performance (though this is certainly not the case with most experienced riders that this author has encountered). Since we have seen that have-it-all performance is impossible, we must conclude that either the powerful psychosomatic phenomenon is at work or that some of the considerations that we have been exploring are not all that discernable to some people, or perhaps it is a little of both.
Al
