THE STABILITYOF THE BICYCLETired of quantum electrodynamics, Brillouinzones, Regge poles? Try this old, unsolved problemin dynamics-how does a bike work?

David E. H. Jones

ALMOST EVERYONE can ride a bicycle,yet apparently no one knows how theydo it. I believe that the apparentsimplicity and ease of the trick con-ceals much unrecognized subtlety, andI have spent some time and effort try-ing to discover the reasons for thebicycle's stability. Published theoryon the topic is sketchy and presentedmainly without experimental verifica-tion. In my investigations I hoped toidentify the stabilizing features ofnormal bicycles by constructing ab-normal ones lacking selected features(see figure 1). The failure of earlyunridable bicycles led me to a care-ful consideration of steering geometry,from which—with the aid of computercalculations—I designed and con-structed an inherently unstable bi-cycle.

The nature of the problem

Most mechanics textbooks or treatiseson bicycles either ignore the matter oftheir stability, or treat it as fairly triv-ial. The bicycle is assumed to bebalanced by the action of its rider who,if he feels the vehicle falling, steersinto the direction of fall and so tra-verses a curved trajectory of such aradius as to generate enough centrif-ugal force to correct the fall. This

David E. H. Jones took bachelor's anddoctor's degrees in chemistry at Im-perial College, London, and has sincealternated between the industrial andacademic life. Currently he is a spec-troscopist with ICI in England.

34 • APRIL 1970 • PHYSICS TODAY

"UNRIDABLE" BICYCLES. DavidJones is seen here with threeof his experimental machines, two ofwhich turned out to be ridable after all.At top of this page is URB I,with its extra counter-rotating front wheelthat tests the gyroscopic theoriesof bicycle stability. At left is URB III,whose reversed front forksgive it great stability when pushedand released riderless.URB IV (immediately above) has itsfront wheel mounted ahead of the usualposition and comes nearest to being"unridable." —FIG. 1

theory is well formalised mathemati-cally by S. Timoshenko and D. H.Young,1 who derive the equation ofmotion of an idealized bicycle, ne-glecting rotational moments, and dem-onstrate that a falling bicycle can besaved by proper steering of the frontwheel. The theory explains, for ex-ample, that the ridability of a bicycledepends crucially on the freedom ofthe front forks to swivel (if they arelocked, even dead ahead, the bicyclecan not be ridden), that the faster abicycle moves the easier it is to ride(because a smaller steering adjust-ment is needed to create the centrif-ugal correction) and that it can not bebalanced when stationary.

Nevertheless this theory can not betrue, or at least it can not be the wholetruth. You experience a powerfulsense, when riding a bicycle fast, thatit is inherently stable and could notfall over even if you wanted it to.Also a bicycle pushed and releasedriderless will stay up on its own,traveling in a long curve and finallycollapsing after about 20 seconds,compared to the 2 sec it would takeif static. Clearly the machine has alarge measure of self-stability.

The next level of sophistication incurrent bicycle-stability theory in-vokes the gyroscopic action of thefront wheel. If the bike tilts, thefront wheel precesses about the steer-ing axis and steers it in a curve that,

as before, counteracts the tilt. Theappeal of this theory is that its actionis perfectly exemplified by a rollinghoop, which indeed can run stably forjust this reason. A bicycle is thus as-sumed to be merely a hoop with atrailer.

The lightness of the front wheeldistresses some theorists, who feel thatthe precession forces are inadequateto stabilize a heavily laden bicycle.2"'5

K. I. T. Richardson4 allows both the-ories and suggests that the rider him-self twists the front wheel to generateprecession, hence staying upright. Atheory of the hoop and bicycle ongyroscopic principles is given by R. H.PearsalF1 who includes many rotationalmoments and derives a complexfourth-order differential equation ofmotion. This is not rigorously solvedbut demonstrates on general groundsthe possibility of self-righting in agyroscopically stable bicycle.

A non-gyroscopic liicycie

It was with vague knowledge of thesesimple bicycle theories that I beganmy series of experiments on bicyclestability. It occurred to me that itwould be fun to make an unridablebicycle, which by canceling the forcesof stability would baffle the most ex-perienced rider. I therefore modifieda standard bicycle by mounting on thefront fork a second wheel, clear of theground, arranged so that I could spin

PHYSICS TODAY APRIL 1970 • 35

Steering axis -

Forkpoint

Front projectionNegative

front projection

Extendedfront

projection

Normal bicycle URB i URB IV

FRONT-FORK GEOMETRY. On left is a normal bicycle. Center shows URB III with reversed forksgiving a negative front projection, and on right is URB IV with extended front projection. —FIG. 2

it against the real front wheel and sooppose the gyroscopic effect. Thiscreation, "Unridable Bicycle MK I"(URB I), unaccountably failed; itcould be easily ridden, both with theextra wheel spinning at high speed ineither direction and with it stationary.Its "feel" was a bit strange, a fact Iattributed to the increased moment ofinertia about the front forks, but it didnot tax my (average) riding skill evenat low speeds. The result resolves theambiguity admitted by Richardson:the gyroscopic action plays very littlepart in the riding of a bicycle at nor-mally low speeds.

This unexpected result puzzled me.If the bicycle, as seemed likely, is ahoop with a trailer but is not gyro-scopic, perhaps the hoop is not gyro-scopic either? I repeated the experi-ment of URB I on a hoop by con-structing one with an inner counter-rotating member, and this collapsedgratifyingly when I tried to roll it.The hoop is a bona fide gyroscope.

Then I tried to run URB I withouta rider, and its behavior was quite un-ambiguous. With the extra wheelspinning against the road wheels, itcollapsed as ineptly as my nongyro-scopic hoop; with it spinning the sameway it showed a dramatic slow-speedstability, running uncannily in a slow,sedate circle before bowing to the in-evitable collapse.

These results almost satisfied me.The light, riderless bicycle is stabi-lized by gyroscopic action, whereasthe heavier ridden model is not—it re-quires constant rider effort to main-

tain its stability. A combination ofthe simple theories accounts neatlyfor all the facts. But the problem ofwhy a ridden bicycle feels so stable,if in fact it is not, remains. There wasone more crucial test: Could URB Ibe ridden in its disrotatory mode"hands off"? For about the only sen-sible theory for riding with "no hands"supposes that the rider tilts the frameby angular body movements and thussteers by the resulting front-wheel pre-cession.3

Gingerly, and with great trepida-tion, I tried the experiment—downhill,to avoid complicating the effort withpedalling. URB I is not an easy bi-cycle to ride "hands off" even with thefront wheel static; it somehow lacksbalance and responsiveness. In thedisrotatory mode it was almost impos-sible and invited continual disaster,but it could, just, be done. I was thusled to suspect the existence of an-other force at work in the moving bi-cycle.

More theories

In the preliminary stages of this inves-tigation, I had pestered all my ac-quaintances to suggest a theory of thebicycle. Apart from the two populartheories that I have mentioned al-ready, I obtained four others, whichI shall call theories 3, 4, 5 and 6:

3. The bicycle is kept upright bythe thickness of its tires (that is,it is a thin steamroller).

4. When the bicycle leans, thepoint of contact of the front tiremoves to one side of the plane of

the wheel, creating a frictionaltorque twisting the wheel intothe lean and stabilizing the bi-cycle, as before, by centrifugalaction.

5. The contact point of the bicycle'sfront tire is ahead of the steeringaxis. Turning the front wheeltherefore moves the contact pointwith the turn, and the rider usesthis effect, when he finds himselfleaning, to move his baselineback underneath his center ofgravity.

6. The contact point of the bi-cycle's tire is behind the steeringaxis. As a result, when the bi-cycle leans a torque is de-veloped that turns the frontwheel.

I suspect that theory 3 is not reallyserious. Theories 5 and 6 raise thequestion of steering geometry, whichI was later to look at in this work-note that the gyro theory is silent onwhy all front forks are angled and allfront forks project forward from them,To test this matter I made URB II.

URB II had a thin front wheel, onlyone inch in diameter (an adapted fur-niture castor) mounted dead in linewith the steering axis, to test anysteering-geometry theory. It lookeda ludicrous contraption. URB II wasindeed hard to ride, and collapsedreadily when released, but this was atleast in part because it could negoti-ate no bump more than half an inchhigh. The little front wheel also gotnearly red hot when traveling fast.

I abandoned URB II as inconclu-

36 • APRIL 1970 • PHYSICS TODAY

sive, but preferred theory 6 to theory5, because in all actual bicycles thefront wheel's contact point is' behindthe intersection of the steering axiswith the ground. Theory 6 is alsoadvocated by the only author whosupports his hypothesis with actualmeasurements.11 But I could not seewhy this force should vanish, as ithas to, once the bicycle is travelingin its equilibrium curve. I had gravesuspicions of theory 4, for surely thistorque acting across less than half thewidth of the tire would have a verysmall moment, and would depend cru-cially on the degree of inflation of thetire? Besides, I did not want nastyvariable frictional forces intrudinginto the pure, austere Newtonian bi-cycle theory towards which I wasgroping.

Steering geometry

The real importance of steering geom-etry was brought home to me verydramatically. I had just completed adistressing series of experiments in-volving loading URB I, with or with-out its extra gyro wheel, with some30 pounds of concrete slabs and send-ing it hurtling about an empty park-ing lot (there are some tests one cannot responsibly carry out on publicroads). The idea was to see if the ex-tra weights—projecting from the front

of the frame to have the maximal ef-fect on the front wheel—would pre-vent the gyro effect from stabilizingthe bicycle, as anticipated from thedifference between ridden and rider-less bikes. It appeared that theweights made the bicycle a little lessstable, and the counter-rotating wheelstill threw it over almost immediately.But the brutal effects on the haplessmachine as it repeatedly crashed toearth with its burden had me straight-ening bent members and removingbroken spokes after almost every run.

It occurred to me to remove diehandlebars to reduce the moment ofinertia about the steering axis; thismeant removing the concrete slabsand the brake assembly, which inci-dentally enabled the front wheel to beturned through 180 deg on the steer-ing axis, reversing the front-fork geom-etry (see figure 2). I had tried thisexperiment once before, calling theresult URB III; that machine hadbeen strangely awkward to wheel orride, and I had noted this result asshowing that steering geometry wassomehow significant. Idly I reversedthe forks of the bike and pushed itaway, expecting it to collapse quickly.Incedibly, it ran on for yards beforefalling over! Further tests showedthat this new riderless bicycle wasamazingly stable. It did not merely

Plane ofbicycle

Vertical plane

Hub offront wheel

Heights Hcalculated by BICYC

A TRICKY TRIGONOMETRICAL PROBLEM. We need to know H, the height ofthe forkpoint from the ground, for a leaning bicycle. Subroutine BICYC calculatesboth the vertical height and the height in the plane of the bike.

run in a curve in response to an im-posed lean, but actively righted itself—a thing no hoop or gyro could do.The bumps and jolts of its progressdid not imperil it, but only as itslowly lost speed did it become un-stable. Then it often weaved fromside to side, leaning first one way andthen the other before it finally fellover. This experiment convinced methat the forces of stability were "hunt-ing"—overcorrecting the lean at eachweave and ultimately causing col-lapse. Once or twice the riderlessdisrotatory URB I had shown mo-mentary signs of the same behavior inits brief doomed career.

Why does steering geometry mat-ter? One obvious effect is seen bywheeling a bicycle along, holding itonly by the saddle. It is easy to steerthe machine by tilting the frame,when the front wheel automaticallysteers into the lean. This is not agyroscopic effect, because it occurseven if the bike is stationaiy. A littlestudy shows that it occurs because thecenter of gravity of a tilted bicycle canfall if the wheel twists out of line. Sohere was a new theory of bicycle sta-bility—the steering is so angled that asthe bike leans, the front wheel steersinto the lean to minimise the ma-chine's gravitational potential energy.To check this theory I had to examinethe implications of steering geometryveiy seriously indeed.

Computerized bicycles

It turns out that defining the height ofthe fork point of a bicycle in terms ofthe steering geometry and angles oflean and of steer (figure 3) is a re-markably tricky little problem. Infact I gave it up after a few attemptsand instead wrote a Fortran subrou-tine, "BICYC," that solved the simul-taneous trigonometrical equationsiteratively and generated all the re-quired dimensions for me. Armedwith BICYC, I could now create allsorts of mad bicycles on the computerand put them through their steer-and-lean paces. The first few runs weremost encouraging; they showed thatwith normal bicycle geometry, tiltingthe frame did indeed ensure that thecenter of gravity had its minimal ele-vation with the wheel twisted into thetilt. This had the makings of a reallygood theory. I hoped to prove that,for the observed steering geometry,the steering angle for minimal center-of-gravity height increased with theangle of lean by just the factor needed

PHYSICS TODAY • APRIL 1970 •

to provide perfect centrifugal stability,and that was why all bicycles havemore or less the same steering geom-etry. As for the strange behavior ofURB III, awkward to ride but in-credibly stable if riderless, perhapsHICYC would provide a clue.

But further calculations shatteredmy hopes. Even with the bicycledead upright, the forkpoint fell as thewheel turned out of plane (thus neatlydisproving the contention of refer-ence 7 that a bicycle tends to run truebecause its center of gravity rises withany turn out of plane), and the mini-mal height occurred at an absurdlylarge steering angle, 60 deg. Evenworse, as the bike tilted, this minimumoccurred at angles nearer and nearerthe straight-ahead position (figure 4)until at 40 deg of tilt the most stableposition was only 10 deg out of plane(these values are all for a typical ob-served steering geometry). Clearly

the tilting wheel never reaches itsminimal-energy position, and theminimum can not be significant for de-termining the stability of the bicycle.

I looked instead at the slope of theheight versus steering-angle curve atzero steering angle, because this slopeis proportional to the twisting torqueon the front wheel of a tilted bike.Then, if H is the height of the fork-point, the torque varies as —dll/da atsmall values of a, the steering angle.

The curves in figure 4 show clearlythat dU/da varies linearly with leanangle L for small angles of lean. Themore the bike leans, the bigger is thetwisting torque, as required. Theconstant of proportionality for thisrelationship is cPH/dadL, and thesign convention I adopted implies thata bicycle is stable if this parameter isnegative. That is, for stability theforkpoint falls as the wheel turns intothe lean when the bike is tilted.

30 - 2 0 -10 0 10 20STEERING ANGLE,-, (DEG)

30 40

COMPUTERIZED BICYCLES. These data, from BICYC output, show that the mini-mal height of the forkpoint occurs nearer to the straight-ahead position for greaterangles of lean. Note also that dH/da varies linearly with lean angle L for small L.Curves, computed for typical steering geometry (20-deg fork angle, 0.2 radii frontprojection), are vertically staggered for clarity. FIG. 4

I therefore computed d'2H/dadLfor a wide range of steering geom-etries, and drew lines of constant sta-bility on a diagram connecting the twoparameters of steering geometry-theangle of the front-fork steering axisand the projection of the wheel cen-ter ahead of this axis. I then plottedon my stability diagram all the bi-cycles I could find—ranging frommany existing models to old high-wheeled "penny-farthings" to see ifthey supported the theory.

The results (figure 5) were im-mensely gratifying. All the bicycles Iplotted have geometries that fall intothe stable region. The older bikes arerather scattered but the modern onesare all near the onset of instability de-fined by the d°-H/dadL = 0 line.This is immediately understandable.A very stable control system respondssluggishly to perturbation, whereasone nearer to instability is more re-sponsive; modern bicycle design hasemphasized nimbleness and maneu-verability. Best of all, URB III comesout much more stable than any com-mercial bike. This result explainsboth its wonderful self-righting prop-erties and also why it is difficult toride—it is too stable to be steered. Aninert rider with no balancing reflexesand no preferred direction of travelwould be happy on URB III, but itscharacteristics are too intense for easycontrol.

This mathematical exercise alsomade it plain that the center-of-gravitylowering torque is developed exactlyas shown in figure 6, and is identicalwith that postulated in reference 6.But it does not vanish when the bi-cycle's lean is in equilibrium withcentrifugal force, as therein supposed(BICYC calculated the height of theforkpoint in the plane of the bicycle-the "effective vertical"—to allow forthis). It can only vanish when thecontact point of the front wheel isintersected by the steering axis, whichBICYC shows clearly is the conditionfor minimal height. There is thus anintimate connection between the"trail" of a bicycle, as defined in figure6, and d-H/dadL; in fact the d-H/da-dL line in figure 4 coincides with thelocus of zero trail.

Two further courses of action re-mained. First, I could make URB IVwith a steering geometry well insidethe unstable region, and second, I hadto decide what force opposes thetwisting torque on a bike's fr°nt

wheel and prevents it reaching

38 • APRIL 1970 • PHYSICS TODAY

Lines of constant stabilitydcdL

-10

30- 0 . 4 - 0 . 2 0.0 0.2 0.4

FRONT PROJECTION (FRACTION OF WHEEL RADIUS)0.6

STABLE AND UNSTABLE BICYCLES. On this plot of fork angle versus front pro-jection the d2H/dadL lines are lines of constant stability. Grey area shows the unstableregion. Point 1 is a normal modern bicycle; 2 is a racing bike. 3 and 4 are high-wheel-ers (or "penny-farthings") from the 1870's. Point 5 is an 1887 Rudge machine, and 6is a Lawson "Safety" of 1879. Point 7 is URB III, and point 8, the only unstablebicycle, is of course URB IV. —FIG. 5

BICYC's predicted minimal center-of-gravity position.

Self-centering

Let us consider the second pointfirst; I was looking for some sort ofself-centering in a bicycle's steering.Now this is well known in the case offour-wheeled vehicles: self-centeringis built into all car steering systems,and various self-righting torques such

Fork axis

Traildistance

Sideways force on tire

as "pneumatic trail" are described byautomobile engineers. Once againnasty variable frictional forces wererearing their ugly heads! But howcould I check whether a bicycle wheelhas self-centering? I examined achild's tricycle for this property, releas-ing it at speed and, running alongsideit, giving the handlebars a blow. Itcertainly seemed to recover quicklyand continue in a straight line, butunfortunately the tricycle (being freeof the requirement of two-wheeledstability) has a different steeringgeometry.

So I made an experimental fixed-

lean bicycle by fastening an extra"outrigger" wheel to the rear of theframe, converting it to an asymmetrictricycle. Adjustment of the outriggeranchorage could impose any angle oflean on the main frame. This ma-chine was very interesting. InitiallyI gave it 15 deg of lean, and at restthe front wheel tilted to the 40-degangle predicted by BICYC. When inmotion, however, the wheel tended tostraighten out, and the faster the bikewas pushed the straighter did thefront wheel become. Even if the ma-chine was released at speed with thefront wheel dead ahead it turned tothe "equilibrium" angle for that speedand lean—another blow for gyro the-ory, for with the lean fixed there canbe no precessional torque to turn thewheel. So clearly there is a self-centering force at work. It is unlikelyto be pneumatic trail, for the equilib-rium steering angle for given condi-tions appears unaltered by completedeflation of the front tire. Now I hadencountered a very attractive form ofself-centering action, not dependingdirectly on variable frictional forces,while trying the naive experiment ofpushing a bicycle backwards. Ofcourse it collapsed at once because thetwo wheels travel in diverging direc-tions. In forward travel the converseapplies and the paths of the twowheels converge. So, .if the frontwheel runs naturally in the line of itsown plane, the trailing frame and rearwheel will swing into line behind italong a tractrix, by straightforwardgeometry. To an observer on thebike, however, it will appear that self-centering is occurring (though it isthe rest of the bike and not the frontwheel that is swinging).

I modified my outrigger tricycle tohold the main frame as nearly uprightas possible, so that it ran in a straight

. Handlebars pushed out of true

Initial straight track Subsequent track shows no self-centering

SIDEWAYS FORCE on front tire pro-duces a torque about the steering axis,so tending to lower the center of gravityof the bicycle. —FIG. 6

SELF-CENTERING? A bicycle with an "outrigger" third wheel to keep it uprightwas pushed and released riderless. At the point shown the handlebars were knockedout of true, resulting in a change of direction and no self-centering. The slight wavein the track resulted from oscillations in the framework. —FIG. 7

PHYSICS TODAY • APRIL 1970 • 39

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6. R. A. Wikon-Jones, Proc. Inst. M #Eng. (Automobile division), 1951-52page 191.

7. Encyclopaedia Britannica (1957 edi-tion), entry under "Bicycle."

line. Then, first soaking the frontwheel in water to leave a track, Ipushed it up to speed, released it, and,running alongside, thumped the han-dlebars out of true. Looking at thebike, it seemed evident that the wheelswung back to dead ahead. But thetrack (figure 7) showed what I hopedto find—a sharp angle with no traceof directional recovery. The bicyclehas only geometrical castor stability toprovide its self-centering.

Success at last!

This test completed the ingredientsfor a more complete theory of the bi-cycle. In addition to the rider's skilland the gyroscopic forces, there are,acting on the front wheel, the center-of-gravity lowering torque (figure 6)and the castoring forces; the heavierthe bicycle's load the more importantthese become. I have not yet formal-ized all these contributions into a Imathematical theory of the bicycle, so iperhaps there are surprises still in :|store; but at least all the principles |have been experimentally checked.

I made URB IV by moving thefront wheel of my bicycle just fourinches ahead of its normal position,setting the system well into the un-stable region. It was indeed verydodgy to ride, though not as impos-sible as I had hoped—perhaps my skillhad increased in the course of thisstudy. URB IV had negligible self-stability and crashed gratifyingly tothe ground when released at speed.

It seems a lot of tortuous effort toproduce in the end a machine of abso-lutely no utility whatsoever, but thatsets me firmly in the mainstream ofmodern technology. At least I willhave no intention of foisting the prod-uct onto a long-suffering public in thename of progress.

References1. S. Timoshenko, D. H. Young, Ad-

vanced Dynamics, McGraw-Hill, NewYork (1948), page 239.

2. A. Gray, A Treatise on Gyrostatics andRotational Motion, Dover, New York(1959), page 146.

3. J. P. den Hartog, Mechanics, Dover,New York (1961), page 328.

4. K. I. T. Richardson, The GyroscopeApplied, Hutchinson, London (1954),page 42.

5. R. H. Pearsall, Proc. Inst. AutomobileEng. 17, 395 (1922).

40 • APRIL 1970 • PHYSICS TODAY