path analysis.pdf

8/14/2019 Path Analysis.pdf

1/138

Path Analysis.

A Bicycle Rear Suspension Analysis Method.

Theory, text, illustrations, and editing by Ken Sasaki.

4bar path analysis by Peter !"#insson.

Spanish $ersiontranslated by Antonio %suna.

&'inkage( suspension si)ulation by *ergely Ko#acs.

Kenneth M. Sasaki 2001, all rights reserved.

+The authors elco)e the reposting or reprinting o- this page or any part

o- it, so long as -ull credit is gi#en to the authors.

/ontents 0links pro#ided1.

2. Pri)ary /oncerns.

1) 1) Objectives.

2) 2) Main Conclusions.

) ) !eneral Co""ents.

#) #)$on%&'(s.i) i) aths and Shock bsor*tion.ii) ii) aths and edaling.iii) iii)Co"*ro"ises.iv) iv) aths and #raking.

C) C) n O*en +etter.

22. So)e 3se-ul SuspensionRelated Mechanics.

1) 1) So"e "*ortant Conce*ts.


2/138

) ) -'eerence /ra"es.

#) #) -egrees o /reedo".

C) C) -$ature aries S"oothl3 4$S).

) ) -**ro5i"ation.

6) 6) -Center o Mass 4CM).

/) /) -Coa5ial Condition.

!) !) -nstant Center 4C).

2) 2) n ntuitive +ook at /orces and (or7ues.

8) 8) Sus*ension 'ate.

222. Path Analysis.

1) 1) ath nal3sis Main ssertions.

2) 2) .. #asics.

) ) Mass **ro5i"ation.

#) #) /orces #et9een +inearl3 Constrained articles.

8) 8) (he $atural Mirror #ike.

:) :) aths and eror"ance.

) ) Mono%ivot and :%bar 'ear 5le aths.

#) #) Shock bsor*tion 4-coasting situations).

C) C) edaling 4$on%&'(s onl3).

) ) Co"*ro"ises.

6) 6) #raking.

$2. heel Path Analyses o- So)e !xisting Models.


3/138

1) 1) (3*ical ;orst +ink esigns.

2) 2) (he !iant $'S.

8) 8)

(he 'ock3 Mountain 6(S) >) (he irtual ivot oint 4).

?) ?) dditional +inkage ata.

$. 5laed Theories and Bogus Marketing.

1) 1) -#rake nduced Shock +ockout 4#S+).

2) 2) /alse Clai"s or /loating #rakes.

8) 8) -S*ecial oint (heories.

) ) -ivot at the Chain +ine 4C+).

#) #) 6lls9orth@s -nstant Center (racking 4C().

:) :) -nternal /orce (heories.

) ) -i%rive A er*etual Motion MachineBB

) #) -Chain +ine oes $ot Matter in a &'(.

>) >) #ogus Marketing.

Appendices.

) ) C+ roble"s A So"e /urther Calculations.

#) #) !lossar3.

About the Authors.


4/138

Path Analysis.

/hapter 2 Pri)ary /oncerns.








%b"ecti#es.ath nal3sis 4) is a 7ualitative "ethod or anal3Ding the *edaling, braking,and shock absor*tion characteristics o ull sus*ension ra"es. (he objective isto allo9 an3one to deter"ine the true "erits o an3 sus*ension design clai"s9ith regard to these characteristics. Ehile the *rinci*les here a**l3 in general,9e have ocused on the non%&'( 4this usuall3 "eans botto" bracket on theront triangle and 9e 9ill use the ter" to "ean so in this 9ork unless other9isestated), since these designs constitute the lion@s share o bikes *roduced theseda3s.Most theories on bic3cle sus*ensions one sees atte"*t to ind -the *ro*er *ivot

*oint or *oints 9hich 9ill "ake the ra"e shock non%reactive to *edaling ate7uilibriu" or -sag 4in act, it is i"*ossible or an3 ra"e geo"etr3 to do thiseither e5actl3 or universall3, and getting close in an3 *articular case 9illintroduce other *roble"s). *recise 7uantitative treat"ent o sus*ensiongeo"etries is a ver3 involved *rocess that re7uires signiicant assu"*tions,even in the "ost si"*le o cases. nu"ber o si"*le theories *ur*ort to indcorrect geo"etries that eli"inate rear shock activation at sag, but to this

author@s kno9ledge, none are sound 4this e5cludes the !iant $'S 9hich is"eant to be run 9ith no sag). Ee look at a e9 o these at the end o the 9orkto de"onstrate ho9 can be used.


5/138

(he consu"er should be "ost concerned about getting *ast industr3 h3*e. Sorather then s*ending a lot o ti"e tr3ing to identi3 *ro*er *ivot locations and soorth 4be3ond an intuitive understanding), 9e 9ill ocus on the issue o 9hat

*eror"ance characteristics are achievable 9ith viable sus*ension designs andho9 one "ight achieve the" in *rinci*le. This analysis )ethod is intended asa consu)er tool that ill allo one to accurately "udge )arketing clai)s,

as ell as the relati#e )erits o- suspension designs and theories.Ee 9ant this 9ork to be useul to those 9ith absolutel3 no technical

background, so 9e *resent the -Main Conclusions u* ront in this irst cha*ter.(hose 9ith no technical background should also ind the -#ogus Marketingsection co"*letel3 accessible.

n cha*ters 22, 222, 2$, and $9e have included reco""endations or readingand diicult3 or each nu"bered section, statingFRead this section i-6Gou 9ish to acco"*lish -this or that objectives. (his section is o -such andsuch i"*ortance.Skip this section i-6

Gou are not interested in -this or that objectives.(his section is -stated diicult3 rating.;ere, -stated diicult3 rating varies a"ongF $ot diicult, less diicult,"oderatel3 diicult, "ore diicult, and "ost diicult. (he level o diicult3 isreerenced to a t3*ical *erson 9ith about a 3ear o good solid college *h3sics.

Ee ho*e that this 9ill hel* those 9ith a less technical background navigatearound the "ore diicult and less necessar3 sections. (hose 9ith a lesstechnical background should still ind the not diicult and less diicultsections, as 9ell as the conclusions ro" all sections, understandable.bout the second cha*ter, -So)e 3se-ul SuspensionRelated MechanicsFEe have chosen "ethods o anal3sis 9ith an e3e to9ard kee*ing "ath to a"ini"u", but so"e basic *h3sics kno9ledge is unavoidable.

Much o this cha*ter is not necessar3 i one "erel3 9ishes to use ath nal3sisto evaluate dierent sus*ension designs 49hich is the "ain intent o this *a*er).


6/138

/or this a**lication, 9e strongl3 reco""end onl3 the -'eerence /ra"essection.Ee have included this cha*ter "ainl3 as a *h3sics *ri"er or those 9ho 9ish to

rigorousl3 veri3 ath nal3sis and delve "ore dee*l3 into bic3cle sus*ension*h3sics. one 9ishes to inde*endentl3 veri3 the validit3 o , the -So"e"*ortant Conce*ts section is "ost critical to understand. (he concludingstate"ents in -n ntuitive +ook at /orces and (or7ues are also o value. Eeree"*hasiDeH "uch o the rest o this cha*ter is related to ath nal3sis onl3 inestablishing iner *oints and is not trul3 necessar3 4although, the kno9ledge9ill be useul to an3one conte"*lating bic3cle design and so"e o it 9ill "ake "ore accessible).One 9ill also need certain conce*ts ro" this cha*ter i one 9ishes to

co"*letel3 understand certain la9s in so"e o the theories detailed in the ithcha*ter, -5laed Theories and Bogus Marketing. Most i"*ortant a"ongthese is the -Center o Mass 4CM)conce*t, as a**lied to orces acting throughthe various 9heels and cogs 9ithin a bic3cle. (o this author@s kno9ledge, thisconce*t has *reviousl3 been unkno9n in the bic3cle industr3.bout the third cha*ter, -Path AnalysisF(his is 9here the "ain theor3 is *resented. Ee don@t consider an3 o the

sections to be "ore then "oderatel3 diicult. Ee suggest that all readers readall sections, even i so"e things are not clear.bout the ourth cha*ter, -heel Path Analyses o- So)e !xisting ModelsF(his cha*ter contains the cad dra9ings b3 eter 6jvinsson. (hese dra9ings are"ost inor"ative in conve3ing inor"ation about 9hat is out there at the

*resent ti"e.

Most o the "ajor design t3*es that are "ore then trivial to evaluate arecovered. /or the "ost *art, the "aterial in this cha*ter is e5tre"el3 eas3 tounderstand, the one e5ce*tion being *arts o the -(he irtual ivot oint4) section.n addition, a -+inkage data link to !ergel3 Kovacs@ -+inkage sus*ensionsi"ulation *rogra" 4see belo9) is *rovided in each ra"e@s section 4or 9hichlt5 data iles have been *re%"ade). (his *rogra" dis*la3s the "ost i"*ortantcharacteristics o each ra"e. Clicking on the link in each ra"e@s section 9illauto"aticall3 bring u* data on that ra"e 4note that the +inkage *rogra" "ust

irst be installed, again, see belo9).


7/138

bout the ith cha*ter, -5laed Theories and Bogus MarketingF(he original "otivation or the *roduction o this 9ork 9as the ubi7uit3 o alsetheories e"anating ro" bic3cle "anuacturers and industr3 "agaDines, and

circulating in bic3cle%related 9eb sites. Ee have thus devoted considerables*ace to de"onstrating the la9s in so"e 9ell%kno9n and 9idel3 acce*tedtheories.So"e o the alse theories and "arketing are associated 9ith 9ell%kno9nna"es. (his has "ade the 9ork so"e9hat controversial. ;o9ever, 9e notehere, as 9ell as in the cha*ter, that in all cases involving alse theories, vigorouseorts 9ere "ade to contact and discuss "atters 9ith the associated *arties,

beore the release o this 9ork.

One o the oldest and "ost res*ected o ull sus*ension ra"e "anuacturershas 9arned this author that the bike industr3 is ver3 s"all and generall3 notkind to -realists. ;e also 9arned that so"e -retaliation should be e5*ectedand indeed, there has been so"e.Ee are co""itted to e5*osing industr3 h3*e and nonsense, and to giving the

*ublic the best *ossible chance to "ake inor"ed decisions, so 9e 9ill not bedeterred b3 retaliation. Ehile 9e eel it unortunate that so"e o thisinor"ation has caused a good deal o consternation to so"e 9ho have alread3

"ade so"e ver3 e5*ensive *urchases, 9e 9ill continue 9ith the circulation othis inor"ation or the greater *ublic good.Ee also note that, generall3, the eedback ro" the industr3 has been *ositiveHincluding, 9e are told, *ositive co""ents ro" one o 'enault@s seniorsus*ension engineers.bout the -!lossar3Ft this ti"e, the -!lossar3 has been done to e5*lain ter"s in this irst-Pri)ary /oncerns cha*ter that "a3 not be a"iliar to those ne9 to "ountain

biking. Ee have not *rovided a detailed account o scientiic ter"s in the latersections because o ti"e constraints. Ee ho*e that those venturing into thesesections 9ill have ade7uate *rior kno9ledge or kno9 ho9 to obtain suchkno9ledge ro" "ore unda"ental sources, or that the bold%9ritten essentialinor"ation 9ill suice to give a reasonable understanding. n the uture, 9eho*e to *rovide a "ore detailed account o scientiic ter"inolog3.bout the -+inkage sus*ension si"ulation *rogra"F

+inkage has been created b3 !ergel3 Kovacs to *roduce the "ost i"*ortantinor"ation about an3 :%bar rear sus*ension that one "ight 9ant to concider.


8/138

version o linkage has been included as *art o the *ackage. (he+inkage2 sot9are as 9ell as the source code are also do9nloadable ro" the+inkage 9eb site at htt*FII999.angelire.co"IjaDDIlinkageI. (he +inkage 9eb

site "a3 contain a "ore u*dated version o linkage, since !ergel3 "aintainsthat site *ersonall3.(o use +inkage, one "ust irst do9nload the sel e5tracting Di* ile and instalthe *rogra". (his can be done b3 clicking on an3 o the -+inkage links in this

*age. One "a3 then vie9 sus*ension designs that are currentl3 on the "arketor in*ut the di"ensions or an3 other linkage coniguration that one desires.(he co"bination o theor3 *resented in the te5t here and this *rogra" shouldallo9 an3 user to develo*e a keen intuition or co"*aring the *edaling and

shock absorbtion characteristics o al"ost all ull sus*ension designs.(he instructionsor using +inkage"a3 be ound in the *rogra" b3 clicking onthe -so"e hel* button.(o install +inkage, click on one o these -+inkage links. Gou 9ill be guidedthrough a series o dialogue bo5es. (he installation deaults 9ill install the+inkage older, containing the *rogra" and lt5 data iles to CFJrogra"/ilesJ+inkage2. 3ou like, 3ou can s*eci3 another location. Shortcuts to

various +inkageentities, as 9ell as the 9eb *age 9ill also be installed to the-Start "enu, under -rogra"s.ter the +inkage*rogra" has been installed, +inkagedata "a3 be called u* inthree dierent 9a3sF

1) 1) s noted above, the +inkage *rogra" has been integrated into thete5t in each ra"e@s section in /hapter 2$, or 9hich an -lt5 data ile4see the instructions) has been "ade. #3 clicking on the -+inkage datalink, the "ost i"*ortant characteristics o the ra"e in 7uestion 9ill besho9n.

2) 2) One "a3 go into the +inkage2 director3, 9hich is created on one@s

co"*uter u*on installation, and call the *rogra" b3 clicking on the+inkage.e5e icon. One "a3 then o*en an e5isting lt5 ile b3 clicking the-O*en button or create a ne9 ile b3 clicking the -$e9 button.

8) 8) One "a3 go into the +inkage2 director3 and click directl3 on an lt5

data ile icon or a ra"e o interest.

bout the uthorsF


9/138

(he authors are all avid bikers, 9ho also have technical andIor language skills.Ee have *rovided this 9ork reel3 to the *ublic 9ith the ho*e that it 9ill

beneit consu"ers and others interested in the 9orkings o bic3cles. shortbiogra*h3 and *icture "a3 be ound on the -bout the uthors *age, or those

authors 9ho have *rovided the inor"ation.(he authors 9ish to thank ro. Curtis Collins, Ola ;elenius4Ola ;.), and 'a3Scruggs 4erb3) or their kind suggestions and hel* in inding errors. (hanksalso to rakon 6l 6loor 9orking on the link structure to the S*anish version.(he authors 9ould like to es*eciall3 thank .. #aeDa4aka Sikander) and ose '.'ubio, the editor o htt*FII999."tbco"*rador.co"I, or hosting both the6nglish and S*anish versions o this 9ork.

(his 9ork 9ill be u*dated ro" ti"e to ti"e to relect current technolog3.&*dated versions 9ill be *osted on the above 9eb sites at the links belo9F(he 6nglish version "a3 be ound atFhtt*FII*erso.9anadoo.esIjibsnaI"tbLsus*Lenand

htt*FII999."tbco"*rador.co"I*aIenglish .

(he S*anish version "a3 be ound atFhtt*FII999."tbco"*rador.co"I*aIs*anishandhtt*FII*erso.9anadoo.esIjibsnaI"tbLsus*Ls* .

link to the above *ages, as 9ell as other 9orks b3 Ken Sasaki, "a3 be oundat ;tt*FII999.*h3sics.ucsb.eduIsasakiEe 9elco"e and a**reciate all accounts o errors and suggested additionsregarding this 9ork that an3one cares to send 4bic3cleL*h3sicsN3ahoo.co").Ee a*ologiDe i 9e are unable to ans9er all corres*ondence, due to ti"econstraints. (hank 3ou.

Main /onclusions.


10/138

1) 71 All )easures o- suspension per-or)ance depend al)ost entirely

on the paths o- the -olloing co)ponents relati#e to any re-erence

-ra)e de-ined by one o- the bicycle -ra)e )e)bers6 8andlebars,

seat, botto) bracket 0BB1, -ront and rear heel axles, shock )ounts,

and rear brake.(his is the central idea o this 9ork and is e5*lored in the third cha*ter, -PathAnalysis. (he ollo9ing are the "ain conclusions that "a3 be dra9n ro" thisstate"ent.

) ) !eneral Co""ents.(he irst thing that "ost *eo*le are concerned about 9ith dual sus*ension bikesis eicienc3 under *edalingH generall3 assu"ing a seated rider *osition and a

bike on s"ooth ground. #ut there are co"*ro"ises that "ust be "ade in tr3ingto attain this goal and "ost other goals associated 9ith dual sus*ension

*eror"ance. n realit3, the right geo"etr3 or an3 one *erson 9ill de*end onthat *erson@s bod3 t3*e 4"ass distribution), riding st3le, sensitivit3 to various

*heno"ena associated 9ith dual sus*ension "otion 4such as bu"* eedback),desired ride 7ualit3 4co"ort, eicienc3, etc.), and even the t3*e o terrain inone@s back3ard.

9o geo)etry is right -or e#eryone and no -ra)e can achie#e -or any one

person e#ery goal generally desired in dual suspension per-or)ance.At suspension e:uilibriu) 0natural sag1 or any other position in tra#el, any

o- the co))on suspension &types( 0)onopi#ots, #arious 4bar

con-igurations, etc.1 can be as nonreacti#e to pedaling as any other during

seated pedaling. ;o9ever, no geo"etr3 can be co"*letel3 -neutralthroughout a *edaling c3cle, 9ithout riction. (he deviation ro" neutral can

be "ade s"all and a good sus*ension geo"etr3 9ith the right a"ount oriction can eectivel3 li"it oscillations, 9hile re"aining su**le enough to

absorb signiicant bu"*s. ;o9ever, 9e note that no geo"etr3 is *erect in thisres*ect as a 9arning against all theories *ur*orting to *rovide a neutralgeo"etr3, in *rinci*le, 9ithout an3 7ualiications. 9ord on "arketingF

$o "anuacturer o a bike or ra"e designed to run at sag 4so"e bikes such asthe !iant $'Sare "eant to run at no sag) is going to "arket its *roducts 9ith avalid, 7uantitative theor3 or constructing rear sus*ension geo"etr3 A telling3ou 9h3 their *ivots or 9hatever are in the right *lace. (he ideas and or"ulae

9ould si"*l3 be too co"*licated to "ake a good "arketing tool. (his authorhas never seen a valid, 7uantitative, run%at%sag sus*ension theor3 *ut out b3 an3


11/138

co"*an3, though 7uite a nu"ber 4so"e o 9hich 9e e5a"ine later in cha*terive, -5laed Theories and Bogus Marketing) "arket bikes under dubiousclai"s and alse theories A so"e asserting that 3ou can have it all. 2- any)anu-acturer or sales person tells you that you can ha#e it all, run aay;

%ur ad#ice is to ignore all suspension theories and other clai)s put -orthby -ra)e )anu-acturers and industry )aga


12/138

/or e5a"*le, :%bar - "a3 have a rear a5le *ath curvature substantiall3dierent ro" :%bar -#, 3et @s *ath "a3 be ver3 close to "ono%*ivot -C4circular, 9ith a *articular radius and center). &nder *edaling and shockabsor*tion, and C 9ill *eror" si"ilarl3 to each other, but dierentl3 ro" #

4sus*ension rate aside).l"ost all non%&'(s on the "arket toda3 have circular rear a5le *aths out tot9o or three deci"al *laces, in inches. s a result, the radius and center ocurvature *ri"aril3 deter"ine sus*ension *eror"ance in "ost non%&'(s.(he "ajorit3 o rear a5le *aths are o si"ilar radius to conventional "ono%

*ivots. (he :%bar *aths *lotted in -(3*ical ;orst +ink esigns, 9hichenco"*ass "ost o the "ajor chain sta3 *ivot design conigurations, are all ver3circular and o conventional radius.

(he !iant $'S, (he 'ock3 Mountain 6(S%


13/138

Ee handle onl3 coasting situations here, since shock ra"iications or *edalingand braking 9ill be handled s*eciicall3 in those sections.

bic3cle sus*ension "a3 be suddenl3 co"*ressed b3 the ground either through9heel contact 9ith an obstacle such as a rock or ro" the i"*act o a dro*%o.n general, 9e believe that a 9idel3 curved rear a5le *ath running slightl3 u*and back is the best solution. (ight curves, either circular or var3ing aregenerall3 inerior or shock absor*tion. ;o9ever, this deicienc3 "a3 be"itigated to so"e degree b3 having the *ath tangent tilting back9ard throughall or "ost o the travel 4or e5a"*le, having a high "ain *ivot, either real orvirtual), as is the case in the (he 'ock3 Mountain 6(S%


14/138

rates in these shallo9 regions 9ill largel3 be a unction o the total travel, orrear a5le *ath length.t is co""on these da3s or designers to take into account the slight tendenc3 o

a bike to old or -s7uat under acceleration. (o do this, one adjusts the reara5le *ath so as to increase, b3 so"e signiicant a"ount, the distance o the reara5le ro" the botto" bracket 4##) as the sus*ension co"*resses. (his allo9schain tension 4"ainl3) to counter the s7uat. #ut this also creates signiicant

bu"* eedback. Ee 9ant to be clear on one thingF (here is no ree lunch here.;ave an increasing eective chain length bet9een the cogs A get so"e degreeo bu"* eedback.(he one area 9here so"e "ulti%links 4this usuall3 "eans :%bars) "a3 have aslightbeneit over conventional "ono%*ivots is in bu"* eedback to the *edals.

:%bars oer the *ossibilit3 o both variable curvature and tight, circularcurvature as the rear a5le "oves relative to the "ain triangle. #oth o these

*ossibilities allo9 or a center o curvature inside the rear 9heel radius. (ightcurvature above e7uilibriu" allo9s the sus*ension to counter s7uat ate7uilibriu", 9hile "ore eectivel3 li"iting eedback. #ikes 9ith tight circularcurvature should be run 9ith little or no sag to *revent *roble"s ro" eedbackunder sus*ension e5tension.

;o9ever, as 9e have noted, :%bars on the "arket toda3 do not *rovidesigniicantl3 var3ing curvature and onl3 the !iant $'S, (he 'ock3 Mountain6(S%


15/138

9ant the s"oother *edaling vertical tangents. So 9e have a co"*ro"ise 9ith9hich to deal.Ee have also noted that tight curves above e7uilibriu", 9hether circular or

var3ing, "a3 hel* reduce the bu"* eedback o a rear9ard tangent. ;o9ever,curves tight enough to "ake a signiicant dierence in the shallo9 regions otravel 9here riders are likel3 to be *edaling "a3 *roduce inerior bu"*

*eror"ance dee*er into the travel, since 9ide curvature should be best orshock absor*tion. (hough again, designs 9ith rear9ard *aths through travel,such as the (he 'ock3 Mountain 6(S%


16/138

2n the -inal analysis, none o- the )a"or suspension types has a clear

ad#antage o#er the others. There are lots o- happy )onopi#ot oners out

there 0including those ith )onopi#otlike $entanas and Rocky !le)ents1

and there are lots o- happy 4bar oners out there. This pretty )uch saysit all.

iv) iv) aths and #raking.(he biggest 7uestion regarding braking in dual sus*ension bikes is 9hether ornot :%bars rear%brake better then "ono%*ivots.(here is a ver3 9ell established "3th 49ell%*ro*agated b3 the "agaDines) that"ono%*ivot shocks 9ill lock under rear braking. (his is kno9n as -#rake

nduced Shock +ockout or #S+. Ee have de"onstrated this to be alse.Ee have also de"onstrated through e5*eri"ent that "ono%*ivots do notsigniicantl3 e5tend or co"*ress under braking.Ee have even de"onstrated that certain :%bar designs, such as the a"is akarsand the s3cle Eerks Eild ;are, should brake al"ost e5actl3 the sa"e as a"ono%*ivot 9ith identical "ain *ivot location. Ee have seen "an3 akar andEild ;are revie9sro" Mountain #ike ction, #ic3cling, and several other

industr3 "agaDines. (hat none o the revie9s "entions #S+ in these :%bardesigns is a good indication that it does not actuall3 e5ist.Most :%bars e5tend ro" natural sag under s"ooth%surace braking a bit "orethen e7uivalentl3 "ain *ivoted "ono%*ivots, establishing a ne9 e7uilibriu"

*osition and rate. So"e, such as the Geti S%', co"*ress under s"ooth%surace braking relative to e7uivalentl3 "ain *ivoted "ono%*ivots. n addition,changing ra"e geo"etr3 through travel, due or e5a"*le to bu"*co"*ression, "a3 cause the braking eect to change, urther altering theeective sus*ension rate o a :%bar.#ut again, none o this leads to the conclusion that :%bars brake better then"ono%*ivots in general, since a "ono%*ivot could 9ell have 4and so"e

*robabl3 do) the sa"e rate under both braking and *edaling as "ost :%barsunder braking.(he biggest consideration is the relation o the rider@s bod3 "ass to the 9heelsand 9hat it 9ill do under braking. (his author believes that bet9een "ost othe designs, the dierences are just not enough to "erit a general state"ent.

So"e *eo*le ind :%bars to brake better, but others do not, though 9e have seenno double%blind tests. n the end, the s"all dierences bet9een so"e designs


17/138

"a3 be signiicant enough or so"e *eo*le to eel a dierence. #ut in general,9e sus*ect that this is again just a case o the ver3 9ell established

*s3choso"atic *heno"enon. (his 9ould not be the irst ti"e that *eo*le havebeen told that so"ething is so and "an3 have e5*erienced 9hat the3 e5*ect

4this is 9h3 *lacebos cure illness). Or *erha*s it is again a little o both.Ee also have no doubt that the #S+ "3th has been *ro*agated b3 so"e in theinterest o selling "ore e5*ensive :%bar designs. Ee see no P 2,000 "ono%

*ivots.n the near uture, 9e ho*e to do a double%blind e5*eri"ent to see once and orall i there is a dierence bet9een :%bars and "ono%*ivots, under braking. Ee9ill *ublish an3 results in subse7uent editions o this 9ork.

n an3 case, our advice here, as al9a3s, is to "ake 3our decisions throughtesting the bikes, i *ossible.Ehat 9e have stated above regarding :%bar linkages also a**lies to loating rear

brake s3ste"s, since a loating brake 9ill give a bike the sa"e rear brakingcharacter as a :%bar 9ith the linkage geo"etr3 o the loating brake. "*artingthe character o its linkage geo"etr3 is the onl3 thing a loating rear brake does,or good or ill. (his 9ill, or e5a"*le, give a t3*ical "ono%*ivot sus*ension thetendenc3 to e5tend under braking, rather then its inherentl3 neutral character.

C) C) n O*en +etter.

9ould like to close this section 9ith a seg"ent ro" an o*en letter *ublishedso"e ti"e agoFGou 9ill ind *eo*le 9ho 9orshi* "ost "ajor designs out there and others 9hodes*ise these sa"e. (here is a good reason or this. Most o it onl3 e5ists in

*eo*leQs "inds. Man3 *eo*le hate "ono%*ivots, but revere entanas, notkno9ing that a entana is essentiall3 a "ono%*ivot 9ith linkages that act asshock tuning 49ith res*ect to *edaling at least). (here are so"e dierences inthe "ajor designs, and so"e s"all advantages here and there, but in the end it is"ostl3 acade"ic. Stick 9ith the "ajor conce*ts and one 9ill not change 3ourlie over the others. O course a" s*eaking o co"*aring bikes 9ithin

*articular categories, not co"*aring ree ride to


18/138

bike its right. (his a**lies to intended use 4be realistic) as 9ell as bod3 9eightand bod3 di"ensions.Most o us 9ith *ro*ellers on our heads just like to talk about this stu because

9e enjo3 a**l3ing the skills 9e have ac7uired to our hobb3 4though so"eobviousl3 have religion). +et "e close 9ith a *iece o a conversation had 9itha *roessor o "echanical engineering, 9ith 9ho" have discussed "3 theor3several ti"esF said that in the end, the si"*lest designs are the best. ;e res*onded, that this isal"ost al9a3s the case./or "ost *eo*le this "eans a basic "ono%*ivot, or a basic :%bar 9ith the *ivoton the chain sta3 or the seat sta3. So"e are obviousl3 s9earing that 3ou need a

link on the chain sta3, but donQt tell that to *eo*le 9ho o9n entanas or'ockies.Kee* it si"*le and go 9ith 3our gut eelingH 3ou have to like the bike 9hen 3oulook at it 49ith 9hatever standards 3ou reall3 ind i"*ortant).!ood luck,Ken Sasaki.


19/138


20/138

Path Analysis.

/hapter 22 So)e 3se-ul Suspensionrelated Mechanics.








So)e 2)portant /oncepts.Read this section i-6Gou 9ant to veri3 or 3oursel the validit3 o the -ath nal3sis Mainssertions. and understand so"e o the related anal3sis in the ith cha*ter,-5laed Theories and Bogus Marketing..Ee strongl3 reco""end at least reading the -'eerence /ra"essubsection. t9ill be ver3 useul to understand this basic *h3sical conce*t in later sections.Skip this section 0except the -'eerence /ra"essubsection1 i-6Gou 9ill acce*t the ath nal3sis "ain assertions and are just interested inusing to "ake conclusions about 9hat sus*ensions can do and co"*arisons

bet9een various bikes.(his section is "oderatel3 diicult.


21/138

/ull3 understanding or bic3cles re7uires so"e i"*ortant conce*ts. Eestrongl3 suggest that those 9ishing to ull3 understand s*end so"e ti"e

beco"ing a"iliar 9ith these conce*ts as "ost erroneous sus*ension theoriesinvolve the neglect or "isunderstanding o one or "ore o these, -Center o

Mass in *articular.) ) -'eerence /ra"es.

n order to anal3De an3 *h3sical situation, 9e "ust create a reerence ra"e.(his is usuall3 re*resented b3 o a set o coordinates in s*ace, consisting o a"utuall3 *er*endicular set o lines, or -a5es, 9ith co""on intersection. (he

*lace 9here the a5es cross deines the origin, or Dero *oint. Ee usuall3 givena"es to each a5is, such as -5%a5is or -3%a5is. e*ending on the sort oinor"ation in 9hich 9e are interested, coordinates could consist o one, t9o,

three, or even "ore a5es 4though "ore then three a5es de*ict "ore then thenor"al s*atial di"ensions and obviousl3 cannot be *ictured).Oten, 9e assign units o length along each a5is. 6ach *oint in s*ace lies alonga line *er*endicular to an3 given a5is. oints "a3 thus be deined b3 the set onu"bers corres*onding to the *oints along the a5es through 9hich these

*er*endicular lines *ass. (his s3ste" is called a rectangular coordinate s3ste"./igure 2.1) sho9s a 2%di"ensional rectangular coordinate s3ste". (he a5es are

colored gold, 9ith black unit "arkers indicating length 4e5actl3 9hat the lengthscale is in this case is not i"*ortant, but usuall3 it 9ill be s*eciied).

*articular *oint 48,%2) is noted in the lo9er right 7uadrant. (he 5%coordinate isusuall3 s*eciied irst, as it is here. bic3cle ra"e 4and so"e other things that9e don@t need to 9orr3 about at this *oint) is *ictured in the coordinate s3ste",9ith the "ain *ivot located at the origin.

/igure 2.1)


22/138

'eerence ra"es "a3 be deined b3 objects such as the earth or *ieces o a

bic3cle ra"e. (hat is, 9e treat our coordinate s3ste" as i it 9ere attached tothe deining object. the deining object is undergoing a linear acceleration, orhas angular velocit3 4"eaning it is rotating), then so 9ill the reerence ra"e.n this case, 9e call the reerence ra"e -non%inertial or -accelerated. the coordinate s3ste" in /igure 2.1) 9ere attached to the earth, then overti"e, the *ictured ra"e 4as *art o a bic3cle) 9ould "ove 9ith res*ect to thecoordinate s3ste". Since the earth is onl3 undergoing ver3 s"all accelerations9e consider it 4essentiall3) an inertial reerence ra"e, or "ost *ractical

*ur*oses. the coordinate s3ste" 9ere attached to the "ain triangle, then the*ositions o other objects, such as the rear sus*ension "e"bers, 9ould bedeined b3 ho9 the3 "ove about the "ain triangle. Since the "ain triangle isoten undergoing signiicant accelerations, 9e consider it a non%inertialreerence.

One thing to note is that in non%inertial reerence ra"es, ictitious orces andtor7ues can a**ear due to ra"e acceleration, the "ost 9ell kno9n o 9hich isthe -centriugal orce o a rotating ra"e. one is riding on one o thosecarnival rides that s*in round and round, one eels as i there is a orce 4likegravit3) *ulling one out ro" the center o rotation and *inning one u* againstthe constraining 9all o the ride. (his is the centriugal orce, 9hich is onl3a**arent.(he centriugal orce should not be conused 9ith the -centri*etal orce, 9hich

is the orce o the 9all causing one to deviate ro" a linear *ath and thus torotate in a circle. (he centri*etal orce is a real orce. (he centri*etal orce actson the rider and *oints in, to9ards the center o rotation. (he centriugal orce


23/138

see"s to act on the rider in a direction *ointing out, a9a3 ro" the center orotation.So"eti"es it is onl3 i"*ortant to deine a reerence b3 so"e object, but not

i"*ortant to deine 9here the origin is located or an3 length scale. n this case,9e "a3 deine the reerence ra"e b3 na"ing so"e object, 9ithout s*eci3an3thing else. /or e5a"*le, 9e "a3 s*eci3 the reerence ra"e o the bic3cle"ain triangle. Ee do this 9hen 9e 9ant to consider ho9 other objects "ove9ith res*ect to the object deining the reerence, but don@t care about *articulardistance scales and so orth.

#) #) -egrees o /reedo".6ach degree o reedo" denotes an inde*endent 9a3 in 9hich a bod3 can "ove.

co"*letel3 ree bod3 has si5 degrees o reedo". !iven standard rectangularcoordinates, a ree bod3 can translate in an3 o the three coordinate directionsand it can rotate around the three coordinate directions. "akes use o the degree o reedo" li"itations on bic3cle co"*onents. /ore5a"*le, in the reerence ra"e o the ground, a dual sus*ension bic3cle "aintriangle has three relevant degrees o reedo" 9hile the bike is traveling in astraight line. t can translate horiDontall3 and verticall3, and it can rotate, all inthe *lane deined b3 the rear 9heel. (he balance o the rider li"its the other

degrees o reedo". 9e i5 the "ain triangle in s*ace, relevant bic3cleco"*onents onl3 have at "ost one degree o reedo".

C) C) -$ature aries S"oothl3 4$S).(he e7uations describing the la9s o nature are continuous relations 4usuall3stated as unctions). (he value on one side 9ill not ju"* discontinuousl3 as the

*ara"eters on the other side var3 continuousl3.4(his e5cluding the 7uantu" real" % ver3 s"all, ver3 big, ver3 cold, etc.)s a result, i 9e i"agine a *ivot *osition var3ing s"oothl3 along so"e ar" ina "echanis", the e7uations o "otion 9ill var3 s"oothl3 also. (hat is, the

*h3sical situation 4la9s) 9ill not ju"* at so"e *oint. (he 9a3 the "echanis"behaves 9ill change continuousl3.

) ) -**ro5i"ation.One o the "ost diicult things *h3sics students have to gras* is 9hen and ho9

to "ake a**ro5i"ations. (he si"*lest or" o a**ro5i"ation is that involving7uantities "uch larger or s"aller then other relevant 7uantities in a given


24/138

*h3sical situation. Ee 9ill give t9o e5a"*les o *ro*er and i"*ro*era**ro5i"ations b3 this "ethod, using "ass.Consider the "ass -M1 9hich is "uch, "uch bigger then the non%Dero "asses

-M2 and -M8 in /igure 2.2 ). (he "asses "ove 9ithout riction in onedi"ension. n considering the "otions o all bodies, a**ro5i"ating M2 and M8as having Dero "ass 9ould not be useul, since inor"ation about theinteractions o the s"all bodies 9ould be eli"inated. On the other hand, i 9ea**ro5i"ate M1 as ininite, useul calculations "a3 still be done 4this is otendone 9hen considering hu"an%siDed objects interacting on the earth).n /igure 2.2 #), 9e have the o**osite situation ro" /igure 2.2 ). ;ere, 9e"ight ver3 9ell neglect the "ass o M2 in 7uanti3ing the results o M1colliding 9ith both M2 and M8.

(hese t9o e5a"*les sho9 that, in certain situations, the odd entit3 "a3 bea**ro5i"ated.

/igure 2.2)

$e5t, consider /igure 2.8 ), 9hich de*icts a block o "ass M connected b3 aro*e to a sus*ended 9heel that is o ver3 s"all "ass relative to the block ands*ins 9ith negligible riction. 9e 9ant to 7uanti3 9hat ha**ens 9hen the

block is suddenl3 dro**ed, 9e could not ignore the "ass o the 9heel, since oura**ro5i"ations 9ould then describe a non%*h3sical situation A na"el3 theunconstrained angular acceleration o a Dero%"ass bod3. On the other hand, i9e 9ere to ask the angular acceleration o the 9heel in the -t9ood device in/igure 2.8 #), 9e could get a *rett3 good ans9er 9hile ignoring the "ass o the9heel, i both M and " are large co"*ared to the 9heel./or a urther discussion o this to*ic, see the -Mass **ro5i"ation. section in-.. #asics.

/igure 2.8)


25/138

6) 6) -Center o Mass 4CM).

(he CM o a solid bod3, or s3ste" o bodies, is the 9eighted average, s*atialdistribution o all "ass in the s3ste"./or e5a"*le, the CM o a s3""etric object, such as a 9heel, is at the center ora5le./or us, the "ost i"*ortant act regarding the center o "ass is that a orcea**lied to an3 *art o the bod3 9ill cause a *arallel acceleration at the center o"ass. /or e5a"*le, a orce a**lied to a 9heel so"e9here along its radius, in

the *lane o the 9heel, 9ill cause acceleration at the a5le *arallel to the orce./or a 9heel in ree s*ace, this "eans that the 9heel 9ill start translating in thedirection o the e5ternal orce, as 9ell as rotating. /igure 2.:) sho9s thissituation.

/igure 2.:)


26/138

(o get an idea o ho9 this "ight be a**lied, consider the ollo9ing 7uestion,9hich call -(he ole and EheelF

*ole is attached to the ground via a hinge 9ith negligible riction. 9heel o"ass " is attached to the to* o the *ole via an a5le that also has no signiicantriction. ro*e o negligible "ass is 9ound around the 9heel@s circu"erence9ith the end hanging to9ard the ground on the right hand side o the 9heel. llo this is balanced at e7uilibriu", 9ith the *ole *ointing verticall3 ro" theground 49hat 9e have here is reall3 just the rear o a "ono%*ivot attached to theground).

;o9 should 3ou *ull on the ro*e, in order that the *ole 9ill not allB Should3ou *ull verticall3H let or right o verticalB Should 3ou *ull through the *ivotBns9erFGou should *ull verticall3, to the e5tent that one "a3 assu"e the "ass o the earth to beeectivel3 ininite. (o be *recise, one should *ull al"ost verticall3, but juuuuust slightl3 tothe let o vertical or the *ole not to all, since the "ass o the earth is not trul3 ininite 4orthose interested in 9h3 the line is not e5actl3 vertical, consider conservation o angular"o"entu").

ull let o vertical and the *ole 9ill all let, and analogousl3 or the right4assu"ing the "ass o the earth to be ininite). ull through the *ivot in

*articular and the *ole 9ill all let. /igure 2.>) diagra"s the situation.

/igure 2.>)


27/138

(he ke3 is to realiDe that the tension in the ro*e induces a orce at the edge othe 9heel, 9hich in turn 9ill induce a *arallel orce at the a5le, just as theor3

*redicts. (he result ollo9s.One "a3 look at this s3ste" as a "ono%*ivot bic3cle 9ith the earth as a giant ront triangle.

9ote6 The Pole and heel :uestion has been extre)ely di--icult -or )ost people. !#en

)ost physics pro-essors do not get it right the -irst ti)e, and none o- the ellknon

bicycle suspension designers has realiyna)ics o- Particles and Syste)s(, by Marion, 7@. The center o- )ass e:uation o-

)otion due to an external -orce is on page CD. The deri#ation starts on page C.

n the *revious e5a"*le, 9e have considered a single orce on the 9heel, 9ith the rotationalinertia o the 9heel o**osing the e5ternal orce. ;o9ever, 9e "a3 also consider "ulti*leorces acting on the 9heel. there are at least t9o orces creating o**osing tor7ues on the9heel, about the a5le, and the "ass o the 9heel is s"all co"*ared to other 7uantities, then9e "a3 ignore the 9heel "ass. (he si"*lest e5a"*le is that o the t9ood "achine in/igure 2.8 #). Ehile the "achine is in "otion, 9e "a3 take the tension ( in the ro*e on eachside to be e7ual, i 9e neglect the "ass o the 9heel. (he orce at the 9heel a5le is 2(H both

orce vectors e5ternal to the 9heel *ointing in the sa"e direction. the e5ternal orces on the9heel are not *ointing in the sa"e direction, the total orce at the a5le 9ill be the vector su".


28/138

/igure 2.?) sho9s the orces acting at the a5le o a negligible "ass 9heel,9hich is e5*eriencing "ulti*le e5ternal orces at t9o dierent radii.

/igure 2.?)

'ater, in 5igure E.7E1o- &An 2ntuiti#e 'ook at 5orces and Tor:ues.(, e

ill consider an exa)ple o- this, ith the crank being an exa)ple o- the

heel, and the pedal stroke -ro) the rider and chain tension being the

external -orces. Those ishing to understand the calculations associated

ith 5igure E.7E1should keep 5igure E.C in )ind.1) /) -Coa5ial Condition

a 9heel or a crank is "ounted coa5iall3 to a *ivot in so"e "echanis", it does not "atterho9 the object is "ounted *h3sicall3. n a bic3cle, the rear 9heel could be *h3sicall3"ounted to the seat sta3 or chain sta3, and the crank could be "ounted to the "ain triangle orthe seat sta3 A none o this "atters. (he *h3sical situation 9ill be the sa"e in all cases as longas the s*eciied objects and *ivots are coa5ial.

/) !) -nstant Center 4C).

"agine a "echanis" that has t9o rigid co"*onents 4*ossibl3 a"ong otherthings). (9o rigid ar"s, attached to the co"*onents b3 *ivots, connect theset9o co"*onent sides. n e5a"*le 9ould be a :%bar sus*ension linkage. n thiscase, one co"*onent could be the "ain triangle and the other the rear link.

$e5t, i5 a reerence ra"e to the irst co"*onent, 9hich in our e5a"*le is the"ain triangle. t an3 given ti"e 9hen the ar"s and the other co"*onent 4rearlink) "ove about the "ain triangle, 9e can calculate the *ath tangents or all

*oints in "otion on these objects b3 constructing the C. Ee do this b3dra9ing lines through the t9o *ivots on either side o each ar". the ar"s arelinear structures, then the a5es 9ill deter"ine our lines. (he *oint 9here the


29/138

t9o lines cross is the C. (he *ath tangent o an3 *oint in "otion is*er*endicular to the line bet9een the C and the *oint 4obviousl3 all o this is ina single *lane).

/igure 2.=) sho9s our bars, 9ith the red dots re*resenting *ivots. (he lightblue lines are the line seg"ents deined b3 the u**er and lo9er *airs o *ivots.(he black dot at the intersection o the blue lines is the C o side # and theadjacent ar"s "oving in the reerence ra"e o side . (he green "arkre*resents an object 4such as a 9heel a5le) on side #, 9ith the dark blue linere*resenting the object@s *ath tangent as it "oves about side . (he dark bluetangent is *er*endicular to the line through the C and green object.

/igure 2.=)

(he idea is that or a s"all angle dq, the "ove"ents o our t9o ar"s *roduce

the sa"e *aths as i co"*onent # and the ar"s 9ere rotating about the C.E'$$! (he C is not a -virtual *ivot. n general, it is constantl3 in"otion, unlike a *ivot. n /igure 2.=), as side # "oves about side , the C 9illconstantl3 change, as 9ill the dark blue line re*resenting the green object@s *athtangent. Man3 errors in sus*ension theories result ro" ascribing to an C theattributes o a *ivot. (he C gives useul inor"ation, but onl3 or an instant oti"e, thus the na"e, instant center. #3 contrast, a *ivot "ight be reerred to asa constant center.

/igure 2.) de*icts a :%bar sus*ension ra"e. (he blue lines reveal variousinstant center *ositions or the rear sus*ension in the reerence ra"e o the"ain triangle. (he red curve *lots the C "ove"ent as the sus*ension "ovesthrough its travel. the distance ro" the rear 9heel a5le to a i5ed *ointre"ains al"ost constant through sus*ension travel, that is i the rear a5le *ath isal"ost circular, then 9e can consider the i5ed *oint a -virtual *ivot or therear a5le. (he light green lines in /igure 2.) reveal that a -virtual *ivot e5istsor the rear 9heel a5le, in the reerence ra"e o the ront triangle. $ote thatan3 virtual *ivots 9ill be uni7ue to each *oint on the rear link. (hat is, the

virtual *ivot or the rear a5le in this case 9ill not be one or an3 other *oints onthe rear link o signiicant distance ro" the rear a5le. Other *ositions on the


30/138

sus*ension "a3 have virtual *ivots, or the3 "a3 not, since the distancedeviation ro" an3 i5ed *oints "a3 be too large or useul a**ro5i"ations.

/igure 2.)

An 2ntuiti#e 'ook at 5orces and Tor:ues.Read this section i-6Gou 9ant a se"i%7ualitative anal3sis o orces and tor7ues going on 9ithin asus*ension bic3cle. &nderstanding ever3thing in this section is not i"*ortant tounderstanding ath nal3sis. (his is just or *eo*le 9ho 9ant to go a littledee*er.Read "ust the conclusions in this section 0ritten bold1 i-6Gou 9ant just the conclusions o the anal3sis or a**lication to other sections.(he conclusions should not be too diicult to understand, so 9e suggest thatone at least give the" a 7uick read. Ehatever one does not understand

*robabl3 9ill not "atter too "uch, but one "ight *ick u* so"e useulinor"ation or the trouble.Skip this section i-6

Gou are just interested in using ath nal3sis to "ake conclusions andco"*arisons regarding various bikes.


31/138

(his section is a"ong the t9o "ost diicult in the 9ork.

Ee here do an intuitive stud3 o orces and tor7ues in a "ono%*ivot non%&'(to understand the things o 9hich a sus*ension theor3 "ust account. (his 9illhel* us urther understand 9hat goes on in a sus*ension and the li"itations on9hat an3 viable design can reall3 acco"*lish. Ee tr3 to kee* the "ath to a"ini"u", ho9ever 9e 9ill be "aking so"e "ini"al calculations tode"onstrate certain solutions in *rinci*le. (hose 9ith a less technical

background can ignore the calculations and look directl3 at the conclusions,9hich are ritten bold.The )ost i)portant lesson o- this section is that )ass distribution is an

i)portant consideration in the physics o- -ull suspension bicycles. 9o:uantitati#e theory can be correct ithout this consideration.t is co""on *ractice to take no reaction o the rear shock to *edaling as thegoal, so 9e 9ill ollo9./igure 2.R) sho9s the ront and rear triangles o a -coasting "ono%*ivot, 9iththe various orces acting on the", "inus riction in the hubs and air, 9hich 9eneglect 4the orces are not dra9n to scale). -CM indicates the rough *osition

or the riderI"ain triangle center o "ass.

/igure 2.R)

ll orces su" to Dero 9hen there is no *edaling. /or this reason, 9e neede5a"ine onl3 those orces and tor7ues that a**ear as the result o a *edal stroke./igure 2.10) sho9s the *icture 9ithout the coasting orces dra9n.

/igure 2.10)


32/138

(here are a nu"ber o 9a3s one "ight go about anal3Ding this situation. Ee

9ill use the tor7ue e7uationF

1) 1) 2*a t.

;ere, 2 is the "o"ent o inertia o the bod3 in 7uestion, a is the angular

acceleration, and tis the su" o the tor7ues on the bod3. (his is the angular

analogue to "a 5. &sing e7uation 1), 9e 9ill e5a"ine 9hat issues areinvolved in kee*ing the tor7ue balance bet9een the "ain triangle ands9ingar", about the "ain *ivot, as close to Dero as *ossible.

/or *recise calculation, this "ethod is not ver3 useul, since so"e o the tor7uesare not eas3 to state e5*licitl3 and all o the tor7ues are ti"e%de*endant 4alle5ce*t chain tor7ue de*ending on the *ositions and "ove"ents o the t9ora"e "e"bers, 9hich 9ill change 9ith ti"e through the *edal stroke). Ee canthus gli"*se the co"*le5it3 o an3 co"*letel3 rigorous anal3sis. #ut or usand our "ainl3 intuitive stud3, this "ethod 9ill be ver3 useul, since 9e can useit to e5*lore a nu"ber o interesting *oints 9ith "ini"al "ath.

Ee start 9ith a co""ent on chain orce.One "ust be ver3 careul 9hen thinking about lines o orce in that "agnitude,direction, and location are all i"*ortant. 6ven e7uivalent gear ratios generall3

*roduce lines o orce that dier in "agnitude and direction as 9ell as location./igure 2.11) sho9s a drive train 9ith t9o *ossibilities or a 1%1 gearing. + isthe crank lever, '1and '2are the res*ective cog radii, and (1 and (2 are thechain tensions or each case.

/igure 2.11)


33/138

a orce / is induced at + 9ith resulting tension in the chain 4e5a"ining onecase at a ti"e), the resulting tor7ue e7uation or the crank is 4assu"ing a non%&'( just or ease o calculation)F

2) 2) *a /*+ % (1*r /*+ % (2*'Thus, T7FR G TEFr. That is, the chain tension decreases as the -ront cog

radius increases H a rather interesting result. So, even the t9o 1%1 situations9ill generall3 not *roduce e7uivalent results or sus*ension activation under

*edaling. (his actuall3 should not sur*rise us, since the energ3 trans"ittedthrough the s3ste" should be the sa"e in both cases. 6nerg3 can be e5*ressed

as (*d, 9here ( is the chain tension and d is the length o the chain that *assesb3 so"e i5ed *oint like the seat tube. Since a greater chain length is *ulled ina bigger chain ring or a given rotation o the crank, 9e need a lesser orce tokee* the energ3 constant.ContinuingF /igure 2.12) sho9s the diagra" or the calculations to ollo9. (he

*artiall3 *ictured triangle re*resents the "ain triangle, to 9hich the crank isattached. (he lo9er horiDontal line re*resents the s9ing ar" o length S+. ' is

the ront cog radius and r is the rear cog radius. + is the length o the crank ar"./ is the orce o a *edal stroke. ( is the resulting chain tension. h is the

*er*endicular distance ro" the chain to the *ivot, is the *er*endiculardistance ro" the *ivot to the orce line at the ## induced b3 the chain tension,and d is the *er*endicular distance ro" the *ivot to the line through the rear

a5le that is *arallel to the chain tension. qis the angle o the chain ro" thes9ingar" a5is.Recall the /enter o- MassF-orce pheno)enon described in 5igure E.C1o- the

&/enter o- Mass( sectionI it applies both to the proacti#e -orce 5 acting onthe crank and to the reacti#e -orce T -ro) the chain, both ith results at

the center.


34/138

Ee have not *ictured all o the orces *resent on the sus*ension "e"bers, butonl3 those induced as a result o a *edal stroke that are relevant or ourcalculations.

Ee assu"e that the crank a5le and "ain *ivot are close together relative tora"e siDe. 4 e9 ra"es such as the (he 'ock3 Mountain 6(S%


35/138

include the tor7ues due to the chain, orkIront 9heel 4riction and inertia),bic3cle acceleration 4the "ost co""onl3 recogniDed o 9hich is s7uat), thecrank rotation and lo9er rotating *arts o the rider@s bod3, and also the tor7uesdue to the interactions o the t9o ra"e "e"bers T9e do so"e urther 9ork

9ith these interaction tor7ues in **endi5 ) -C+ roble"sH So"e /urtherCalculations., should an3one have 7uestions as to e5actl3 9hat these areU.Vn asideF One should not be too concerned about the ollo9ing detail, but theastute reader 9ill note that 9e are using t9o dierent non%inertial reerencera"es or each bic3cle ra"e "e"ber. (he bic3cle acceleration and interactiontor7ues are the ictitious tor7ues in these reerence ra"es.W(he tor7ue e7uations or the riderI"ain triangle and s9ing ar" are then,res*ectivel3F

*i

/i//XY,8)

and

*i

SiSSXY,:)

.

chieving the stated goal o "ini"iDing sus*ension reaction to *edaling generall3 involvesinding the best *lace or the "ain *ivot relative to the chain orce line or an assu"edcondition 4"ass distribution, etc.). (o do this, one "ust e5*ress the chain tor7ue in ter"s o hand solve or this 7uantit3 in the *ro*er e7uation. /ortunatel3 the chain and *edal tor7ues areeas3 to state in e7uation and 9ill allo9 us to get a or"ula in *rinci*le or the desired relationo *ivot and chain.

+et t/Cdenote the tor7ue on the "ain triangle due to *edaling and the resultingchain tension.

$oting that 9ith a negligible crank "ass, / (*'I+, the tor7ue on the "aintriangle due to the *edal stroke and resulting chain tension isF

J1 J1 t/C /*+ A (* (*' % (* (*4'%) (*h.So e see that, neglecting the )ass o- the crank, the tor:ue on the )ain

triangle -ro) pedaling is "ust as i- e had been pulling on the chain -ro) a

point on the )ain triangle that is a perpendicular distance h abo#e the

pi#ot H a #ery interesting result 0see %la 8elenius -or an interestingintuiti#e argu)ent -or this result, though e are not exactly sure here it is

on his site1. 8oe#er, one )ust be #ery care-ul not to take this result too


36/138

-arI as e ha#e seen, -or a gi#en pedal -orce, the larger the radius o- the

-ront cog, the loer ill be the chain tension.

+et tSCdenote the tor7ue on the s9ingar" ro" the chain 4again, ulti"atel3

ro" *edal orce). gain, since in *ractice the *ivot is relativel3 close to the## co"*ared to the ra"e siDe, 9e a**ro5i"ate the orce on the s9ing ar" atthe *ivot as that o the chain orce induced at the ## 4these values 9ill be ver3close or t3*ical ra"es). Eith this a**ro5i"ation, 9e haveF

d(S

d(S)4sin(S X?)

+++SC ***q**

(o have the tor7ue balance bet9een the "ain triangle and s9ingar" about the

"ain *ivot e7ual to Dero 4to get no reaction o the rear shock), 9e 9ant the rontand rear triangles to rotate en unison A that is, 9e 9ant the a/ aS. Solving or

the a@s in e7uations 8) and :) and setting the t9o e5*ressions e7ual to eachother 9e getF

S

CiSi

S

CiSi

/

Ci/i

,

Xh)4r(

,

Xd(

,

Xh(=)

*

*

*

.

$o9 solving or h, 9e getF

),

,41(

X

),

,41(

X

),

,41

rhO)

S

/

Ci/i

/

S

CiSi

/

S *

*

.(his is the Dero tor7ue balance or"ula or the "ain *ivot *osition, relative tothe chain line or a non%&'( "ono%*ivot 49ith *ivot not too ar ro" the ##co"*ared to the siDe o the ra"e A again, al"ost al9a3s the case).One "ight conclude that h de*ends on (, as ( a**ears in the deno"inator o thelast t9o ter"s. Ee state 9ithout *roo that ( 9ill a**ear as a actor in all o thetor7ues, just as it did or the chain tor7ue, 9ith the e5ce*tion o that resultingro" the ork. So 9ith the noted ork e5ce*tion, h does not de*end on (. Tn-C+ roble"sH So"e /urther Calculations., 9e give an e5a"*le o ho9 thetor7ues, or the "ost *art, eli"inate ( ro" e7uation ).UEe "a3 dra9 the ollo9ing conclusions ro" e7uation )F

5irst, notice that the )o)ents o- inertia -or both bodies are in all ter)s.

This tells us that it ill be i)possible to construct any sort o- a :uantitati#e


37/138

suspension theory ithout taking into account )ass and its distribution.

Mass distribution ill be o- e:ual consideration -or all other suspension

types. This rules out certain -S*ecial oint (heories, such as the )ostnai#e -ivot at the Chain +ine 4C+)theories.

(he second thing 9e notice is that since the tor7ue values are ti"e%de*endant, h9ill also be ti"e%de*endant through the *edal stroke. Ee thus see that there isno single -*ro*er *ivot *oint 4or *oints), e5actl3, through an entire *edalstroke. n addition, 9e note as a "atter o intuition, that as the rider "akes a

*edal stroke, the s3ste" o ra"e "e"bers, on average, 9ill rotate back relative

to the rear a5le 4a0). #et9een *edal strokes, the ra"e "e"bers 9ill allback do9n, and not in such a 9a3 as to kee* the rear shock inactive 9ithouthel* ro" riction in the *ivots. (his urther tells us that it is i"*ossible or an3rear sus*ension geo"etr3 to be co"*letel3 non%reactive to *edaling, 9ithoutstatic riction.

The ti)edependant nature o- our )onopi#ot situation is also co))on to

all other suspension types, since -ra)e )e)ber orientation changes

through the pedal stroke in all o- these bikes as ell. 2n particular, )ono

pi#ots can approxi)ate a


38/138

Read this section i-6Gou 9ant a ath nal3sis *ers*ective on sus*ension tuning.

Skip this section i-6Gou are not interested.(his section is "ore diicult.

Ee don@t 9ant to s*end too "uch ti"e here, since this is *robabl3 the last issueabout 9hich a consu"er should 9orr3. ll non%&'( t3*es can achieve all o

the reall3 useul sus*ension rates and "ost ra"es out there *air u* airl3 9ell9ith the stock shocks. 'ate is onl3 reall3 a consideration or those 9ho are realsus*ension 9onks 43es, the author is a sus*ension 9onk and i 3ou are readingthis, 3ou are in danger o 9onk%hood also) 9anting to s9a* bet9een coil and airshocks, 9hich generall3 have dierent internal rates. airing a alling ratera"e 9ith a linear coil shock or an e5tre"el3 rising rate ra"e 9ith an airshock "ight not have acce*table results.;o9ever, 9e do reer to sus*ension rate in other *arts o this 9ork, so 9e 9ill

look at the "ost i"*ortant considerations.ll s*rings have -rates and a sus*ension is just a t3*e o s*ring.eine a coordinate 5 as the direction in 9hich a s*ring co"*resses. (he-s*ring rate is a unction o 5, and describes the a"ount o orce 9ith 9hichthe s*ring 9ill tend to9ard e7uilibriu" at an3 *oint o co"*ression ore5tension a9a3 ro" e7uilibriu". (he stee*er the rate unction, the "ore as*ring 9ill resist additional "ove"ent the urther it is "oved ro" e7uilibriu"./or a t3*ical coil s*ring near e7uilibriu", the rate unction is al"ost linear. the rate unction is concave u*, then the s*ring has a rising rateH that is, theadditional orce needed to urther co"*ress the s*ring at each *oint 9illincrease as the s*ring goes through its travel. it is concave do9n, then thes*ring has a alling rate, 9ith analogous results. /igure 2.18) sho9s a gra*h9ith each t3*e o rate.

/igure 2.18)


39/138

0.00 10.005

0.00

10.00

/orce

'ising rate

+inear rate

/alling rate

(he rate o a bic3cle sus*ension is co"*osed o the internal rate o the shock

and the rate inherent in the sus*ension geo"etr3.nternal shock rates range ro" near linear to rising. Coil s*rings tend to have"ore linear rates, 9hile air s*rings tend to have rising rates. ll ra"es "a3 beitted 9ith a range o shocks, 9hich these da3s generall3 have one o t9olengths and standard "ounts. Ee 9ill not consider shocks urther, since the3are not an inherent eature o ra"e geo"etr3.

(he contribution to rate ro" sus*ension geo"etr3 is deter"ined b3 the 9a3 in9hich the shock "ounts, ront and rear 9heel a5les, and "ain triangle "overelative to one another. (he ront 9heel a5le establishes ra"e orientation tothe ground but generall3 "a3 be neglected, since botto" brackets are al"ostuniversall3 18Z [ .>Z ro" the ground 9ithout rider 4given a t3*ical assu"*tionor the ork -Cro9n to 5le +ength or -C+). (hus, the rear 9heel and ##largel3 deter"ine the ra"e orientation to the ground.gain, 9e don@t 9ant to s*end too "uch ti"e on this, so 9e 9ill give ane5a"*le o the contribution ro" the relative "ove"ent o the shock "ounts.Si"ilar consideration "ust be given to the rear 9heel a5le relative to the rear

shock "ount and ##.6stablishing the "ain triangle as our reerence ra"e, the rear shock "ount 9illtravel a circular *ath around so"e *ivot A the "ain *ivot in the case o a "ono%

*ivot and the u**er ra"e *ivot in the case o a :%bar 4our ollo9ing state"ents9ill a**l3 in both cases). the tangent o the rear shock "ount *oints near to the ront shock "ount asthe sus*ension goes through its travel, then the relative "ove"ents o the shock

"ounts 9ill have a neutral inluence on sus*ension rate 4b3 neutral 9e "eanthat, given a linear shock, the sus*ension rate 9ill re"ain linear).


40/138

/igure 2.1:) sho9s a sus*ension "e"ber "oving in the ra"e o the "aintriangle. s the sus*ension co"*resses, i the rear "ount tangent is "ovinginto align"ent 9ith the ront "ount, then the *ath 9ill increase the rise4decrease the all) in rate. it is "oving out o align"ent 9ith the ront "ount,

then the *ath 9ill decrease the rise 4increase the all) in rate. (his is because ora given angle o rotation, the t9o shock "ounts "ove to9ards each other the"ost 9hen the rear "ount tangent is through the ront "ount.

/igure 2.1:)

9e are dealing 9ith a "ono%*ivot, then the sus*ension "e"ber is the reartriangle and the rear connection 9ill be to the rear a5le. 9e have a :%bar, then

the sus*ension "e"ber is the u**er link and the rear connection 9ill be to therear link. n both cases, the larger the radius o the rear shock "ount *ath, thelarger 9ill be the rate curvature due to geo"etr3. lso, the longer thesus*ension "e"berH the larger 9ill be the "agniication o the internal shockrate curvature, since the 9heel 9ill travel a greater distance or a given distanceo rear shock travel.(his is "ost o the ballga"e or a "ono%*ivot 4"inus onl3 9heel *ath). /or a:%bar, one "ust do a si"ilar anal3sis or the tangent o the u**er rear *ivotrelative to the rear 9heel a5le. t an3 *osition in travel, i the tangent is

*ointing at the 9heel a5le, then the shock 9ill co"*ress least or a givena"ount o 9heel travel. n "ost :%bars this *ivot has a *ath that 9ill di"inishthe rate, and again, the larger the *ath radius o this *ivot the larger 9ill be therate unction curvature. (he *aths o the rear shock "ount and u**er rear *ivotthus deine the over all eect in a given :%bar, "inus 9heel *ath.


41/138

Path Analysis.

/hapter 222 Path Analysis.








Path Analysis Main Assertions.Read this section.(his is the central *oint o the entire 9ork.(his section is "oderatel3 diicult.

1) 71 All )easures o- suspension per-or)ance depend al)ost entirely

on the paths o- the -olloing speci-ied co)ponents relati#e to any

re-erence -ra)e de-ined by one o- the bicycle -ra)e )e)bers6

8andlebars, seat, botto) bracket 0BB1, -ront and rear heel axles,

shock )ounts, and rear brake.s is e5*lained in the -'eerence /ra"essection, establishing a ra"e "e"beras our non%inertial reerence does not "ean that it 9ill not "ove. t 9illtranslate and rotate, and our reerence ra"e 9ill "ove 9ith it.


42/138

(he above -s*eciied co"*onents 9ill al9a3s "ove along *aths or one%di"ensional s*aces in the reerence ra"e o one o the su**orting bic3clera"e "e"bers, as a *ractical "atter. The path tangents deter)ine ho anybike ill beha#e at any point in ti)e. The path cur#atures deter)ine ho

the bike ill beha#e o#er ti)e. 9e 9ish to co"*are t9o designs, 9e should identi3 a ra"e "e"berco""on to both designs. (he "ore alike the *aths are in an3 t9o sus*ensions,in the co""on reerence, the closer 9ill be the *eror"ance o the ra"es that

*roduce the". 2n practice, the bars and seat ill alays de-ine the bestre-erence and this ill be the re-erence -or all analysis in this ork4thoughso"eti"es it can be interesting to see ho9 *aths co"*are ro" reerence toreerence).

Mass and its distribution *la3 an i"*ortant role in an3 "echanis". ;o9ever,the "ain triangle and rider are usuall3 about ?0 ti"es as "assive as thesus*ension "e"bers 4not including the shock). (he "ove"ent o riderI"aintriangle "ass 9ill de*end on the "ove"ents o the "ain triangle co"*onents4seat, botto" bracket, and handlebars), even 9ith a non%integrated "ain triangle4botto" bracket "oving 9ith res*ect to the bars and seat). n addition, thedierences in "ass "ove"ent o the sus*ension "e"bers bet9een dierentdesigns 9ith si"ilar co"*onent *aths are relativel3 s"all. (his "akes theground and the riderI"ain triangle the onl3 t9o signiicant "asses in bic3cle

*h3sics.(hese "ass considerations are 9hat allo9 or . Ee have covered "assa**ro5i"ations in the -**ro5i"ationsection o -So"e "*ortant Conce*ts..;o9ever, 9hen and ho9 to a**l3 a**ro5i"ation can be a ver3 diicult issue,so in the -Mass **ro5i"ation section belo9, 9e 9ill e5*lain in detail ho9"ass a**ro5i"ation allo9s or .

$aturall3, each individual rider 9ill *roduce a uni7ue "ass distribution. Ehen

9e sa3 that 9e can deter"ine sus*ension *eror"ance b3 the *aths, 9e "eanthat 9e can kno9 the *eror"ance o the ra"e or an3 set o assu"*tions orrelevant *h3sical 7uantities, such as rider "ass distribution or contributionsro" the sus*ension ork./riction in the sus*ension "echanis" 9ill al9a3s act to o**ose the "ove"ento co"*onents along their *aths and 9ill ulti"atel3 be directed tangent to the

*ath. /riction "agnitude can or the "ost *art be controlled in one t3*e ogeo"etr3 as 9ell as another. (hus, 9hile 9e "ight ind one *articularsus*ension bike to have a avorable a"ount o riction relative to another,

riction does not lend an3 advantage to one t3*e o sus*ension over another.


43/138

$ote that the orces bet9een co"*onents are critical in deter"ining sus*ension*eror"ance. ;o9ever, all lines o orce, 9hether the3 are through the rider,the chain, or e5ternal are e7uall3 *roducible in all designs. (he3 thus do notdistinguish one design ro" another. ;o9ever, it is ver3 hel*ul to understand

ho9 the orces and tor7ues act on and 9ithin a bic3cle./ra"e stiness is an i"*ortant actor in bic3cle *eror"ance. ;o9ever, it is"uch "ore an issue or handling 4a to*ic not covered in this 9ork), *articularl3high s*eed cornering, then an3thing else. Eith regard to *edaling, braking, andshock absor*tion, one onl3 need be 9ar3 o the ver3 lightest ra"es. t has beenseveral 3ears since the author has been a9are o an3 ne9 ra"es on the "arketthat are so severel3 under%built as to cause real *roble"s or *edaling, braking,and shock absor*tion, be3ond bad choices and deects in "aterials and"anuacturing, that lead to ra"e ailure 4also not covered in this 9ork).

(his leaves geo"etr3 as the overriding issue in sus*ension *eror"anceregarding *edaling, braking, and shock absor*tion.n "ost cases, the ull "achiner3 o is not necessar3 since the *aths oco"*onents "a3 deter"ine the orientations o their su**orting structures4ra"e "e"bers, ork, etc.). /or e5a"*le, the ## and seat "a3 ull3 deter"inethe "ain triangle, so one could si"*l3 look at that bod3 rather then the attachedco"*onents.

;o9ever, in cases such as the i%rive, the ull "achiner3 o is the onl3

*ractical "ethod o anal3sis. nal3sis o the i%riveb3 an3 other "ethod9ould be e5tre"el3 co"*licated. (he *o9er o 9ill be revealed in thee5tre"e si"*licit3 o i%riveanal3sis using this "ethod.Ee 9ill give an anal3sis o the i%rive theor3, 6lls9orth@s -nstant Center(racking 4C()theor3, and other erroneous theories at the end o this *a*er.

P.A. Basics.

Read this section i-6Gou 9ant to veri3 or 3oursel the validit3 o the ath nal3sis "ain assertionsand understand the details o ho9 and 9h3 ath nal3sis 9orks.Skip this section i-6

Gou 9ill acce*t the "ain assertions and are just interested in using athnal3sis to "ake conclusions about 9hat sus*ensions can do and co"*arisons

bet9een various bikes.


44/138

(his section is "oderatel3 diicult.

) ) Mass **ro5i"ation.s stated aboveFath nal3sis 9orks because the "ass o the riderI"ain triangle do"inates all"ass in a bic3cle. n addition, the dierences in "ass "ove"ent o thesus*ension "e"bers bet9een dierent designs 9ith si"ilar co"*onent *athsare relativel3 s"all. (his "akes the ground and the riderI"ain triangle the onl3t9o signiicant "asses in bic3cle *h3sics.

/urther"ore, all orces on the rear sus*ension "e"bers, other then thosedirectl3 bet9een sus*ension "e"bers 4through *ivots and so orth), are directedthrough the s*eciied co"*onents. Since sus*ension "e"ber "ass is notsigniicant and the sus*ension "e"bers control "otion bet9een the t9osigniicant "asses, it is enough to consider the orces bet9een the s*eciiedco"*onents.Ee no9 look at this a little "ore closel3.

Consider /igure 8.1 ). ;ere, 9e have a "ain triangle and s9ing ar" attachedto a base b3 a *ivot. (his is actuall3 the *ro*er "odel 9ith 9hich to anal3De

bic3cle sus*ensions, "inus so"e contributions ro" the ront ork. the orce/ 9ere calibrated to the orce o gravit3, 3ou 9ould have "ost o the situationor the analogous dual sus*ension bike, in a *articular gear 4no hu"an ridercould reall3 *roduce such high values, but the values can be reduced b3 tiltingthe "echanis" back9ard, into the *age). $ote that the "ain triangle can "oveonl3 in a certain restricted 9a3 relative to the base and that the lo9er *ivot,analogous to the rear a5le on a bike can "ove onl3 in a certain pathrelative tothe "ain triangle.VEe have dra9n the *ictures verticall3 s3""etrical and the linkages to or"

*arallelogra"s. #ut the "odel should be taken "ore generall3 to include an3t3*icall3 sha*ed ront triangle and lengths o linkage "e"bers. (he "odelshould also be considered in all reasonable *ositions.W

/igure 8.1)


45/138

$e5t, consider /igure 8.1 #). ;ere, 9e have a "ain triangle and a :%bar linkageattached to a base b3 a *ivot. (his *roduces *recisel3 the sa"e *aths as the

"echanis" in /igure 8.1 ). n act, i 9e neglect the "asses o the s9ing ar"and linkage, 9e 9ould have identical situations in both ) and #). /igure 8.1C) sho9s both sus*ensions on the "echanis" at once, ro" 9hich 9e see that

both sus*ensions 9ill 9ork har"oniousl3 9ith one another. (his oretells ananal3tical device that have conceived, called a natural "irror bike, 9hich 9e9ill discuss belo9 in -(he $atural Mirror #ike. section. $o9 the 7uestion is,-Can 9e neglect the "asses o the sus*ension "e"bersB the "ass o the s9ing ar" 9ere ver3 large co"*ared to the "ass o the "aintriangle and the "ass o the :%bar linkage 9ere ver3 s"all co"*ared to the "ain

triangle, then it is eas3 to see that 9ould nota**l3. n ), the "ain triangle9ould rotate around the u**er *ivot 9ith relativel3 little "otion ro" the s9ingar" 9hen / is a**lied. n #), the "ain triangle and linkage 9ould "ove ver3dierentl3 ro" ), the linkage "oving "ore drasticall3 then the s9ing ar",

*roducing a ver3 dierent *h3sical situation. #ut this is not the case in abic3cle. t3*ical dual sus*ension ra"e 9eighs about ive to si5 *ounds, 9ithout therear shock. O this, the rear "ight take u* 2.> *ounds. $o9 in bikes or 9hich

there is an3 utilit3 in co"*arison, the "ass dierence bet9een an3 t9o t3*es orear sus*ensions is going to be less then hal a *ound. (he riderI"ain triangle,


46/138

on the other hand, averages about 1>0 *ounds, at least. (his leads to a "assdierence o less then 0.8 \ bet9een the vast "ajorit3 o "echanis"s.n addition, the "ove"ent o "ass through sus*ension co"*ression de*ends

largel3 on the relative "otion o the s*eciied co"*onents, 9ithin the range o"otion or all viable sus*ension bikes. Considering the "ove"ents o the linksin a :%bar linkage, one sees that the overall "ove"ent o "ass is ver3 si"ilar tothe "ove"ent o "ass in a "ono%*ivot s9ing ar" 4though not e5actl3 thesa"e). (he "ove"ent o "ass in the !( i%riveis al"ost identical to a "ono%

*ivot, the onl3 4insigniicant) dierences being the "ove"ent o the eccentricon the s9ing ar" and o the -dogbone.s a result, 9e "a3 neglect the sus*ension "e"bers and ocus e5clusivel3 onthe *aths that the3 *roduce, as 9e have dra9n in /igure 8.2 ) or the

"echanis"s in /igure 8.1). ;ere, 9e have dra9n a circular *ath or the lo9er*ivot about the "ain triangle. (his contains all o the signiicant inor"ationconcerning ho9 the "echanis"s 9ill 9ork. /igure 8.2 #) sho9s the t3*e o"otion allo9ed b3 all e7uivalent "echanis"s.

/igure 8.2)

Ee have de"onstrated these *rinci*les or a :%bar vs. a "ono%*ivot, but the3a**l3 in general, since the "asses o sus*ension "e"bers 9ill al9a3s 9eighabout the sa"e as the e5a"*les here.


47/138

#) #) /orces #et9een +inearl3 Constrained articles.

Sus*ension *eror"ance is deter"ined essentiall3 b3 the relati#e"ove"ents othe %s*eciied co"*onents. (his is because the interaces to the e5ternal

9orld 4the 9heels) are identical in all bic3cles or 9hich there is a utilit3 inco"*arison. Ee can thus do our anal3sis based entirel3 on these internal9orkings and neglect an3 e5ternal interactions 49ith the ground or e5a"*le).(his si"*liies "atters in that our anal3sis "a3 involve less degrees o reedo".s stated above, the s*eciied bic3cle co"*onents "ove along *aths or one%di"ensional s*aces, assu"ing a reerence ra"e attached to one o thesu**orting ra"e "e"bers. ;o9 each "oves 9ill de*end on the su" o orcese5erted bet9een it and the other co"*onents in the s3ste". So lets get so"eidea about ho9 to treat objects "oving along such *aths b3 looking at so"e

e5a"*les.Su**ose 3ou have an 5,3%a5is. *article, such as a "arble, is restricted viaso"e "echanis" to "ove reel3 along the diagonal in the irst 7uadrant. n3nu"ber o "echanis"s could achieve the restricted "ove"ent. $o9 su**osethere 9as a orce / *ushing on the *article in the 5%direction. (he e7uations o

"otion or the *article 9ould involve /*cos 4:>), that is, the direction o"otion, but not the "echanis" that restricts the degrees o reedo". /igure 8.8)sho9s the *ath, orce, and orce co"*onent along the *ath.

/igure 8.8)

(9o *articles "ove along *aths relative to one another. a orce is induced

bet9een the", it 9ill cause both *articles to accelerate along their res*ective*aths in the direction corres*onding to the tangent co"*onent o the orce alongthe *ath. /igure 8.: ) sho9s this scenario.

/igure 8.:)


48/138

$e5t consider t9o 9heel a5les that are restricted to travel along the sa"e *athsrelative to one another as the *articles in ), 9ith a5les at the sa"e *oints. (hesa"e "agnitude o orce as in ) is e5erted at the 9heel edges. /igure 8.: #)sho9s this situation.

$otice that the orces at the a5les are in the direction o the orce line bet9eenthe cogs, 9hich is dierent ro" the orce direction on the *articles in ). (heorces are also o a dierent "agnitude due to the inertia o the 9heels.articularl3 i"*ortant is that these t9o orces at the a5les are not co%linear. (heco"*onents along the *aths in # are in the sa"e directions as those in , but9ill generall3 be o dierent "agnitudes due to both the dierences in overallorce direction and "agnitude at the a5les.

$o9 consider a *article such as a 9heel a5le that is restricted to travel along a*articular *ath relative to other co"*onents in a "echanis" 4a "ain triangle ore5a"*le). orces are e5erted b3 the other co"*onents on this a5le, b39hatever "eans, the a5le 9ill tend to "ove in the direction along its *ath thatcorres*onds to the tangent co"*onent o the su" o the orces. (he "agnitudeo the tangent co"*onent deter"ines the "otion o the a5le. Si"ilar

considerations e5ist or the other co"*onents. one co"*onent@s *ath in a "echanis" is a unction o another@s *ath 4in *artor 9hole), then it does not "atter ho9 that relation is achievedH an3 "echanis"9ill *roduce the sa"e *h3sical results. n e5a"*le o such a unctionalrelationshi* is that bet9een a seat, handle bars, and botto" bracket, 9hichdeine the "ain triangle. n essentiall3 e7uivalent unctional relationshi* is

*roduced in !(@s i%rive.

As a result, i- e ant to consider per-or)ance in non3RTS, e need onlyexa)ine the paths o- the heel axles, shock )ounts, and brake, relati#e to

the )ain triangle.


49/138

/igure 8.>) sho9s a rear a5le *ath in relation to a "ain triangle or a non%&'(.(he gra3 lines denote several *ossible a5le *aths. (he red line sho9s a line oorce 4through the chain). (he green arro9 sho9s the orce induced at the

9heel a5le. (he blue arro9 sho9s the co"*onent o orce along the 9heel *athat the a5le.

/igure 8.>)

;ere is a central *oint. n the s"all neighborhood around the a5le denoted

b3 the orange lines, the *aths are identical. ;o9ever, above the neighborhoodthe3 diverge 9ildl3, one being circular and the others being o "ore radicalcurvature. Ee *ictured this to e"*hasiDe that the *ath tangent is 9hat counts atan3 "o"ent in ti"e. Other as*ects o the *aths have no bearing on 9hatha**ens in our s"all neighborhood around the a5le. 2n little neighborhoodsaround all points in a path, all suspensions ith si)ilar paths in that

neighborhood beha#e si)ilarlyI in particular, they beha#e like so)e )ono

pi#ot. Ehen 9e shrink the neighborhoods to Dero, 9e see that the tangent tothe *ath deter"ines sus*ension behavior at an3 *oint in the *ath. (his "ight see" strange i 9e consider "echanis"s that *roduce *aths 9ithver3 dierent radii o curvature. #ut re"e"ber, it is 9hat ha**ens as thesus*ensions "ove a9a3 ro" the *articular *oints on the *aths 9ith co""ontangents that "ake the situations dier. !reatl3 diering curvature 9ill

*roduce s9itl3 diverging *h3sical situations.

(he ollo9ing e5a"*le should alla3 concerns about 9hether or not the tangentcan reall3 contain all o the inor"ation necessar3 to evaluate a given situationregardless o *ath curvature.


50/138

/igures 8.? ) and #) de*ict "ain triangles attached at dierent *oints to s9ingar"s o dierent length. (he *aths that these s9ingar"s *roduce are odierent radii, but have the sa"e *ath tangents at the initial locations o the

baseIs9ingar" *ivots. Ee de*ict the "echanis"s as being horiDontal andvie9ed ro" above, so that 9e "a3 start at e7uilibriu" beore a orce / isa**lied to the "ain triangle. Eithout loss o generalit3, 9e choose the s9ingar"s to be aligned along the 3%a5is.

/igure 8.?)

(he orce / "a3 be a**lied at an3 *oint, in an3 direction. Ee chose to *lace it

in such a 9a3 that the direction e5tends bet9een the t9o ra"eIs9ingar" *ivotlocations in the dierent "echanis"s because this is the situation that is "ostlikel3 to cause concern.

$eglecting s9ingar" "ass, 9e see that the 5%co"*onent o /, /5, 9ill causeessentiall3 identical initial "ove"ents o the "ain triangles in the t9o"echanis"s, since this co"*onent is *er*endicular to the s9ing ar". Ee alsosee that the 3%co"*onent o /, /3, 9ill have the sa"e lever ar" about both othe *ivots in both cases and thus also 9ill cause identical initial "ove"ents o

the "ain triangles. (his "eans that the initial "ove"ents o both "echanis"sdue to the total orce 9ill be identical. (he t9o situations 9ill diverge as thes9ing ar"Ibase *ivot *aths diverge outside o the initial little neighborhood


51/138

around the initial *ositions. #ut in the initial *ositions, the *h3sical situationsare identical.n realit3, a s9ing ar" as large as the one in /igure 8.? ) "ight 9eigh a e9

*ounds "ore then the one in #) 4though the dierence 9ould still be less then2\ o the riderI"ain triangle "ass). #ut such s9ing ar"s do not e5ist in realbikes. :%bar ra"e "e"bers that can *roduce a *ath curvature si"ilar to the onein ) do not 9eigh substantiall3 "ore then ordinar3 "ono%*ivot s9ingar"s andthe "ass tends to "ove in a generall3 si"ilar ashion over all. So as 9eobserved earlier, neglecting the "asses o the sus*ension co"*onents is a gooda**ro5i"ation in our anal3sis.

The 9atural Mirror Bike.

Read this section.t is not technicall3 diicult and the -$atural Mirror conce*tual device is the"ost easil3 understood conir"ation or the validit3 o ath nal3sis.

(he best intuitive conir"ation that one can have or the validit3 o ath

nal3sis is to i"agine *utting t9o dierent sus*ension "echanis"s on one bikesi"ultaneousl3. (here 9ould be no conlict bet9een the" as long as theco"*onent *aths 9ere the sa"e or both "echanis"s. Shortl3 ater irst

*ublished the -si"ultaneous sus*ension, a *articular version o "3 idea 9as*ro*osedthat 9ould have one side o a bike constructed ro" a "ono%*ivot andthe other ro" a :%bar 9ith a circular rear a5le *ath. More generall3, 9e "a3construct a bike 9ith t9o dierent sus*ension "echanis"s on either side, eachhaving the sa"e co"*onent *aths. 9ill reer to such a bike as a -natural"irror or si"*l3 -"irror sus*ension bike, since the true nature o eachsus*ension is "irrored on the other side.Ee can include the *aths o all co"*onents as *art o a natural "irror anal3sis,or onl3 those or 9hich 9e "a3 have a *articular interest. /or e5a"*le, i 9e9ish to co"*are onl3 9heel *aths, 9e "a3 i"agine *airing u* ra"es 9ithidentical 9heel *aths and it 9ill not "atter 9hether other co"*onents, such asthe shock "ounts, also have identical *aths.n evaluating the validit3 o a theor3, *h3sicists oten e5a"ine certain-obvious cases to see i the theor3 "akes sense. ;ere 9e e5a"ine several

designs 9ith circular rear a5le *aths, to de"onstrate that the3 9ill all *eror"identicall3 under *edaling 4sus*ension rate adjust"ents accounted or in the laste5a"*le).


52/138

Su**ose 9e start on one side o a "irror bike 9ith a :%bar sus*ension in 9hichthe -bars deter"ine a *arallelogra" A that is, the u**er ar" is e7ual in lengthto the lo9er ar" and the or9ard ar" 4the "ain triangle bet9een the t9o

or9ard *ivots) is e7ual to the rear. Ee call this a -*arallel :%bar. (he 9heel*ath 4both or *ivot on the chain sta3 or seat sta3) is circular.On the other side o the bike, 9e can use a "ono%*ivot, 9ith "ain *ivot at thesa"e height above the :%bar "ain *ivot as the 9heel a5le is above the :%bar rear

*ivot.Ee reer to this bike as a -*arallelI"ono "irror and both sides *roduce thesa"e *ath.

Ee could even "ake the shock "ounts have e7uivalent *aths b3 "ounting ashock to the "ono%*ivot in the sa"e 9a3 that 9e "ount a shock to one o thehoriDontal :%bar ar"s. 6ach side o the bike 9ill *eror" e5actl3 the sa"e asthe other. /igure 8.=) sho9s both o the above sus*ensions ro" the side.

/igure 8.=)

+et us no9 consider another :%bar. (his ti"e though, 9e 9ill "ake the t9oor9ard *ivots coa5ial to *roduce 9hat 9e call a -**%coa5ial :%bar. (he

*ivots 9ill still be *h3sicall3 attached to the "ain triangle se*aratel3 and thusthe sus*ension 9ill constitute a true :%bar. (he 9heel in this case also has acircular *ath and thus this sus*ension can be *ut in 9ith either o the other t9o.Ee 9ill call this :%bar co"bined 9ith a "ono%*ivot a -**%coa5ialI"ono"irror. See /igure 8.) or this e5a"*le.

/igure 8.)


53/138

+astl3, let us consider a :%bar 9ith rear 9heel "ounted coa5iall3 9ith the rear

lo9er *ivot. t does not "atter 9hether the rear 9heel is "ounted *h3sicall3 tothe chain sta3 or the seat sta3, both 9ill behave the sa"e, as the 9heel 9ill havethe sa"e *ath. Ee 9ill call a "irror bike 9ith this sus*ension and a "ono%

*ivot a -9*%coa5ialI"ono "irror. (he conigurations o the :%bar u**er linkscontribute onl3 to the sus*ension rate in this case. djusting the relative *athso the shock "ounts as 9ell as the -internal rates o the shocks "a3 be done to"ore or less "atch the over all sus*ension rates o the t9o sides. /igure 8.R)sho9s this "echanis".

/igure 8.R)

ll o these are e5a"*les o ver3 dierent sus*ension conigurations that 9ill

behave e5actl3 the sa"e 9hile not under braking 4shock tuning accounted in thelast case), because the rear a5le *aths are the sa"e A na"el3 circular. (heshock "ount *aths in the irst t9o e5a"*les are not identical in s*ace, but areidentical in relative "otion and so cause no conlict.


54/138

Paths and Per-or)ance.

Read this section.

(his section e5*lains the i"*ortant considerations involved in "ost o the ullsus*ension ra"es built toda3.(his section is less diicult, e5ce*t in one or t9o *laces *erha*s, and is o greatuse to consu"ers.

V technical note about the *ictures in this seg"entF (he "ain triangles are notdra9n to scale and the *aths are not "eant to re*resent solutions or an3

*articular real%9orld situations or as endorse"ents or an3 *articular designs Athe3 are constructed "erel3 to illustrate the *oints.W#eore anal3Ding *aths, 9e "ake a e9 general co""ents on so"e other issues.#oth "ajor sus*ension t3*es 4"ono%*ivots and :%bars) "a3 be as light or asstrong as an3 dual sus*ension bike can viabl3 be, e5a"*les o both havingound success in


55/138

arc or *oints close to the lo9er link. (he *aths 9ill graduall3 change ro" oneto the other as the *oints var3 along the rear link. n a5le "ounted so"e9herein bet9een 9ill have var3ing *ath curvature.

/igure 8.10)

t the "o"ent, the -(he irtual ivot oint 4). conce*t, conceived b3Outland and soon to be re%introduced b3 Santa CruD and ntense, is uni7uea"ong viable conce*ts in its ca*abilit3 to *roduce signiicantl3 variablecurvature. ;o9ever, as 9e 9ill see, current e5a"*les do not take an3 realadvantage o the *ossibilities.Secondl3, conventional "ono%*ivots do not have "ain *ivots located 9ithin the9heel radius. (his li"its the "ini"u" radius o a5le *ath curvature to at leastthe siDe o the rear 9heel. "ono%*ivot could achieve a tighter curvature onl3 i the *ivot 9ere s*lit intoright and let. Ee have *ro*osed such a bike, 9hich 9e call the -s*lit%*ivot"ono. (his design is in act viable and 9ill have the added beneit o a "orestable *ivot. /igure 8.10 C) sho9s the tight curvature. Ee thus do notconsider tight curvature to be an inherent advantage o :%bars over "ono%

*ivots, since the s*lit%*ivot "ono can achieve the *ath. lthough no such bikesare in current *roduction, the s*lit%*ivot "ono 9as the "otivation behindCannondale@s ne9 Scal*el.

:%bars can achieve a tightl3 curved *ath centered inside the rear 9heel radius./igures 8.10 ) and #) sho9 a :%bar 9ith the sa"e tangent as our s*lit%*ivot


56/138

"ono both at e7uilibriu" and co"*ression. Ee have achieved our e5a"*le b3having the C "ove back9ard as it "oves do9n. (his is essentiall3 the !iant

$'S design. (he 'ock3 Mountain 6(S%


57/138

so, 9hat are the considerations involvedB n asking this unda"ental andrather *o*ular 7uestion, 9e 9ill get a good general idea o 9hat attributes reall3eect non%&'( *edaling *eror"ance.

Ee observed in /igure 8.>) o the -/orces #et9een +inearl3 Constrainedarticles. section that the co"*onent *ath tangents deter"ine ho9 an3sus*ension 9ill *eror" at an3 *oint in ti"e.

(his "eans that, neglecting riction in the "echanis", each *articular geo"etr39ill have its "a5i"u" eectiveness onl3 in certain -ideal gears 4ro" a

*ractical stand*oint, this could "ean one gear or several). n3 others sets ogears 9ill *roduce dierent orces on the "echanis", leading to dierentco"*onents o orce along the tangents. (he urther the gearing ro" ideal, the"ore reactive an3 sus*ension geo"etr3 9ill be.

/or a given deviation a9a3 ro" ideal gearing, -sus*ension rate 4s*ringstiening) 9ill deter"ine the a"ount o reaction ro" a *edal stroke. Shortertravel sus*ensions tend to be less reactive to *edaling then longer travelversions, since short travel designs should have higher "ore rising rates 4in *artdue to the act that "an3 use air shocks these da3s). n *ractice, the actual ratesin the shallo9 regions o travel 9here *edaling 9ill be aected 9ill largel3 be aunction o the total travel length o the rear 9heel *ath.

Most ra"es "ate 9ell 9ith their stock shocks, and all co))on suspensiontypes can achie#e the really use-ul rates 0linear or rising1. So rate is onl3 areal issue or those 9ishing to s9a* dierent coil and air shocks in and out o agiven ra"e.Since rate in the shallo9 regions o travel 9ill largel3 be a unction o total9heel *ath length and is o secondar3 i"*ortance to "ost *eo*le, 9e 9ill noturther consider rate here. Ee reer those still interested, to the -Sus*ension'ate. section in cha*ter .

n3 co""ent on ra"e *eror"ance "ust be "ade 9ith res*ect to a range oorks, just as is the case 9ith rider "ass. So an assu"*tion "ust be "ade orork characteristics. n addition, dual sus*ension botto" brackets 4##s) areal"ost universall3 18Z [ .>Z ro" the ground 9ithout rider, given a t3*icalassu"*tion or the ork -Cro9n to 5le +ength or -C+. (hus, the rear9heel and ## largel3 deter"ine the ra"e orientation to the ground. So, aternoting the re7uired o

path analysis.pdf

Documents