This preview shows page 1. Sign up to view the full content.
Unformatted text preview: From the archives (April 1970, pages 34–40) The stability of
the bicycle
David E. H. Jones
Tired of quantum electrodynamics, Brillouin zones, Regge poles? Try this
old, unsolved problem in dynamics: How does a bike work?
David E. H. Jones took bachelor’s and doctor’s degrees in chemistry at Imperial College, London. When this article first appeared,
he was a spectroscopist at ICI in England. Almost everyone can ride a bicycle, yet apparently
no one knows how they do it. I believe that the apparent simplicity and ease of the trick conceals much unrecognized subtlety, and I have spent some time and effort trying to discover
the reasons for the bicycle’s stability. Published theory on the
topic is sketchy and presented mainly without experimental
verification. In my investigations I hoped to identify the stabilizing features of normal bicycles by constructing abnormal
ones lacking selected features (see figure 1). The failure of
early unridable bicycles led me to a careful consideration of
steering geometry, from which—with the aid of computer
calculations—I designed and constructed an inherently unstable bicycle. stably for just this reason. A bicycle is thus assumed to be
merely a hoop with a trailer.
The lightness of the front wheel distresses some theorists, who feel that the precession forces are inadequate to stabilize a heavily laden bicycle.2,3 K. I. T. Richardson4 allows
both theories and suggests that the rider himself twists the
front wheel to generate precession, hence staying upright. A
theory of the hoop and bicycle on gyroscopic principles is
given by R. H. Pearsall5 who includes many rotational moments and derives a complex fourthorder differential equation of motion. This is not rigorously solved but demonstrates
on general grounds the possibility of selfrighting in a gyroscopically stable bicycle. The nature of the problem A nongyroscopic bicycle Most mechanics textbooks or treatises on bicycles either ignore the matter of their stability or treat it as fairly trivial. The
bicycle is assumed to be balanced by the action of its rider
who, if he feels the vehicle falling, steers into the direction of
fall and so traverses a curved trajectory of such a radius as to
generate enough centrifugal force to correct the fall. This theory is well formalized mathematically by S. Timoshenko and
D. H. Young,1 who derive the equation of motion of an idealized bicycle, neglecting rotational moments, and demonstrate that a falling bicycle can be saved by proper steering
of the front wheel. The theory explains, for example, that the
ridability of a bicycle depends crucially on the freedom of
the front forks to swivel (if they are locked, even dead ahead,
the bicycle can not be ridden), that the faster a bicycle moves
the easier it is to ride (because a smaller steering adjustment
is needed to create the centrifugal correction), and that it can
not be balanced when stationary.
Nevertheless this theory can not be true, or at least it can
not be the whole truth. You experience a powerful sense,
when riding a bicycle fast, that it is inherently stable and
could not fall over even if you wanted it to. Also a bicycle
pushed and released riderless will stay up on its own, traveling in a long curve and finally collapsing after about 20 seconds, compared to the 2 seconds it would take if static.
Clearly the machine has a large measure of selfstability.
The next level of sophistication in current bicyclestability theory invokes the gyroscopic action of the front
wheel. If the bike tilts, the front wheel precesses about the
steering axis and steers it in a curve that, as before, counteracts the tilt. The appeal of this theory is that its action is perfectly exemplified by a rolling hoop, which indeed can run It was with vague knowledge of these simple bicycle theories
that I began my series of experiments on bicycle stability. It
occurred to me that it would be fun to make an unridable bicycle, which by canceling the forces of stability would baffle
the most experienced rider. I therefore modified a standard
bicycle by mounting on the front fork a second wheel, clear
of the ground, arranged so that I could spin it against the real
front wheel and so oppose the gyroscopic effect. This creation, “Unridable Bicycle MK I” (URB I), unaccountably
failed; it could be easily ridden, both with the extra wheel
spinning at high speed in either direction and with it stationary. Its “feel” was a bit strange, a fact I attributed to the
increased moment of inertia about the front forks, but it did
not tax my (average) riding skill even at low speeds. The result resolves the ambiguity admitted by Richardson: The gyroscopic action plays very little part in the riding of a bicycle
at normally low speeds.
This unexpected result puzzled me. If the bicycle, as
seemed likely, is a hoop with a trailer but is not gyroscopic,
perhaps the hoop is not gyroscopic either? I repeated the experiment of URB I on a hoop by constructing one with an inner
counterrotating member, and this collapsed gratifyingly
when I tried to roll it. The hoop is a bona fide gyroscope.
Then I tried to run URB I without a rider, and its behavior was quite unambiguous. With the extra wheel spinning
against the road wheels, it collapsed as ineptly as my nongyroscopic hoop; with it spinning the same way it showed a
dramatic slowspeed stability, running uncannily in a slow,
sedate circle before bowing to the inevitable collapse.
These results almost satisfied me. The light, riderless bicycle is stabilized by gyroscopic action, whereas the heavier © 2006 American Institute of Physics, S0031922806090405 September 2006 Physics Today 51 a
b c
Figure 1. Unridable bicycles. David Jones is seen here with
three of his experimental machines, two of which turned out
to be ridable after all. (a) At the top of this page is URB I,
with its extra counterrotating front wheel that tests the gyroscopic theories of bicycle stability. (b) At left is URB III, whose
reversed front forks give it great stability when pushed and
released riderless. (c) URB IV (immediately above) has its
front wheel mounted ahead of the usual position and comes
nearest to being “unridable.” I was thus led to suspect the existence of another force at
work in the moving bicycle. More theories ridden model is not—it requires constant rider effort to maintain its stability. A combination of the simple theories accounts neatly for all the facts. But the problem of why a ridden bicycle feels so stable, if in fact it is not, remains. There
was one more crucial test: Could URB I be ridden in its disrotatory mode “hands off”? For about the only sensible theory for riding with “no hands” supposes that the rider tilts
the frame by angular body movements and thus steers by the
resulting frontwheel precession.3
Gingerly, and with great trepidation, I tried the experiment—downhill, to avoid complicating the effort with pedaling. URB I is not an easy bicycle to ride “hands off” even
with the front wheel static; it somehow lacks balance and responsiveness. In the disrotatory mode it was almost impossible and invited continual disaster, but it could, just, be done.
52 September 2006 Physics Today In the preliminary stages of this investigation, I had pestered
all my acquaintances to suggest a theory of the bicycle. Apart
from the two popular theories that I have mentioned already,
I obtained four others, which I shall call theories 3, 4, 5, and 6:
3. The bicycle is kept upright by the thickness of its tires
(that is, it is a thin steamroller).
4. When the bicycle leans, the point of contact of the
front tire moves to one side of the plane of the wheel, creating a frictional torque twisting the wheel into the lean and
stabilizing the bicycle, as before, by centrifugal action.
5. The contact point of the bicycle’s front tire is ahead of
the steering axis. Turning the front wheel therefore moves the
contact point with the turn, and the rider uses this effect,
when he finds himself leaning, to move his baseline back underneath his center of gravity.
6. The contact point of the bicycle’s tire is behind the
steering axis. As a result, when the bicycle leans a torque is
developed that turns the front wheel.
I suspect that theory 3 is not really serious. Theories 5
and 6 raise the question of steering geometry, which I was
later to look at in this work—note that the gyro theory is silent
on why all front forks are angled and all front forks project
forward from them. To test this matter I made URB II.
www.physicstoday.org Figure 2. Frontfork geometry. On the left is a normal bicycle. The center shows URB III with reversed forks giving a negative
front projection, and on the right is URB IV with extended front projection. URB II had a thin front wheel, only one inch in diameter (an adapted furniture castor) mounted dead in line with
the steering axis, to test any steeringgeometry theory. It
looked a ludicrous contraption. URB II was indeed hard to
ride, and collapsed readily when released, but this was at
least in part because it could negotiate no bump more than
half an inch high. The little front wheel also got nearly red
hot when traveling fast.
I abandoned URB II as inconclusive, but preferred theory 6 to theory 5, because in all actual bicycles the front
wheel’s contact point is behind the intersection of the steering axis with the ground. Theory 6 is also advocated by the
only author who supports his hypothesis
with actual measurements.6 But I could not
see why this force should vanish, as it has
to, once the bicycle is traveling in its equilibrium curve. I had grave suspicions of
theory 4, for surely this torque acting across
less than half the width of the tire would
have a very small moment, and would depend crucially on the degree of inflation of
the tire. Besides, I did not want nasty variable frictional forces intruding into the pure,
austere Newtonian bicycle theory towards
which I was groping. and riderless bikes. It appeared that the weights made the bicycle a little less stable, and the counterrotating wheel still
threw it over almost immediately. But the brutal effects on
the hapless machine as it repeatedly crashed to earth with its
burden had me straightening bent members and removing
broken spokes after almost every run.
It occurred to me to remove the handlebars to reduce the
moment of inertia about the steering axis; this meant removing the concrete slabs and the brake assembly, which incidentally enabled the front wheel to be turned through 180°
on the steering axis, reversing the frontfork geometry (see
figure 2). I had tried this experiment once before, calling the Steering geometry
The real importance of steering geometry
was brought home to me very dramatically.
I had just completed a distressing series of
experiments involving loading URB I, with
or without its extra gyro wheel, with some
30 pounds of concrete slabs and sending it
hurtling about an empty parking lot (there
are some tests one can not responsibly carry
out on public roads). The idea was to see if
the extra weights—projecting from the front
of the frame to have the maximal effect on
the front wheel—would prevent the gyro effect from stabilizing the bicycle, as anticipated from the difference between ridden
www.physicstoday.org Figure 3. A tricky trigonometrical problem. We need to know H, the
height of the forkpoint from the ground, for a leaning bicycle. Subroutine
BICYC calculates both the vertical height and the height in the plane of
the bike. September 2006 Physics Today 53 Figure 4. Computerized bicycles. These data, from
BICYC output, show that the minimal height of the forkpoint occurs nearer to the straightahead position for
greater angles of lean. Note also that dH/dα varies
linearly with lean angle L for small L. Curves, computed for typical steering geometry (20° radii front
projection), are vertically staggered for clarity. Figure 5. Stable and unstable bicycles. On this plot of
fork angle versus front projection the d 2H/dαdL lines
are lines of constant stability. Shaded area shows the
unstable region. Point 1 is a normal modern bicycle;
2 is a racing bike. Points 3 and 4 are highwheelers
(or “pennyfarthings”) from the 1870s. Point 5 is an
1887 Rudge machine, and 6 is a Lawson “Safety” of
1879. Point 7 is URB III, and point 8, the only unstable
bicycle, is of course URB IV. 54 September 2006 Physics Today result URB III; that machine had been strangely awkward to
wheel or ride, and I had noted this result as showing that
steering geometry was somehow significant. Idly I reversed
the forks of the bike and pushed it away, expecting it to collapse quickly. Incredibly, it ran on for yards before falling
over! Further tests showed that this new riderless bicycle was
amazingly stable. It did not merely run in a curve in response
to an imposed lean, but actively righted itself—a thing no
hoop or gyro could do. The bumps and jolts of its progress
did not imperil it, but only as it slowly lost speed did it become unstable. Then it often weaved from side to side, leaning first one way and then the other before it finally fell over.
This experiment convinced me that the forces of stability
were “hunting”—overcorrecting the lean at each weave and
ultimately causing collapse. Once or twice the riderless disrotatory URB I had shown momentary signs of the same behavior in its brief doomed career.
Why does steering geometry matter? One obvious effect
is seen by wheeling a bicycle along, holding it only by the
saddle. It is easy to steer the machine by tilting the frame,
when the front wheel automatically steers into the lean. This
is not a gyroscopic effect, because it occurs even if the bike is
stationary. A little study shows that it occurs because the center of gravity of a tilted bicycle can fall if the wheel twists out
of line. So here was a new theory of bicycle stability—the
steering is so angled that as the bike leans, the front wheel
steers into the lean to minimize the machine’s gravitational
potential energy. To check this theory I had to examine the
implications of steering geometry very seriously indeed. Computerized bicycles
It turns out that defining the height of the forkpoint of a bicycle in terms of the steering geometry and angles of lean and
of steer (figure 3) is a remarkably tricky little problem. In fact
I gave it up after a few attempts and instead wrote a Fortran
subroutine, “BICYC,” that solved the simultaneous trigonometrical equations iteratively and generated all the required
dimensions for me. Armed with BICYC, I could now create
all sorts of mad bicycles on the computer and put them
through their steerandlean paces. The first few runs were
most encouraging; they showed that with normal bicycle
geometry, tilting the frame did indeed ensure that the center
of gravity had its minimal elevation with the wheel twisted
into the tilt. This had the makings of a really good theory. I
hoped to prove that, for the observed steering geometry, the
steering angle for minimal centerofgravity height increased
with the angle of lean by just the factor needed to provide
perfect centrifugal stability, and that was why all bicycles
have more or less the same steering geometry. As for the
strange behavior of URB III, awkward to ride but incredibly
stable if riderless, perhaps BICYC would provide a clue.
But further calculations shattered my hopes. Even with
the bicycle dead upright, the forkpoint fell as the wheel
turned out of plane (thus neatly disproving the contention of
reference 7 that a bicycle tends to run true because its center
of gravity rises with any turn out of plane), and the minimal
height occurred at an absurdly large steering angle, 60°. Even
worse, as the bike tilted, this minimum occurred at angles
nearer and nearer the straightahead position (figure 4) until
at 40° of tilt the most stable position was only 10° out of plane
(these values are all for a typical observed steering geometry). Clearly the tilting wheel never reaches its minimalenergy position, and the minimum can not be significant for
determining the stability of the bicycle.
I looked instead at the slope of the height versus
steeringangle curve at zero steering angle, because this slope
www.physicstoday.org Figure 6. Sideways force on front tire produces a torque
about the steering axis, so tending to lower the center of
gravity of the bicycle. is proportional to the twisting torque on the front wheel of a
tilted bike. Then, if H is the height of the forkpoint, the torque
varies as −dH/dα at small values of α, the steering angle.
The curves in figure 4 show clearly that dH/dα varies linearly with lean angle L for small angles of lean. The more the
bike leans, the bigger is the twisting torque, as required. The
constant of proportionality for this relationship is d2H/dαdL,
and the sign convention I adopted implies that a bicycle is
stable if this parameter is negative. That is, for stability the
forkpoint falls as the wheel turns into the lean when the bike
is tilted.
I therefore computed d2H/dαdL for a wide range of steering geometries, and drew lines of constant stability on a diagram connecting the two parameters of steering geometry—
the angle of the frontfork steering axis and the projection of
the wheel center ahead of this axis. I then plotted on my stability diagram all the bicycles I could find—ranging from
many existing models to old highwheeled “pennyfarthings”
to see if they supported the theory.
The results (figure 5) were immensely gratifying. All the
bicycles I plotted have geometries that fall into the stable region. The older bikes are rather scattered but the modern ones are all near the onset of instability defined by the
d2H/dαdL = 0 line. This is immediately understandable. A
very stable control system responds sluggishly to perturbation, whereas one nearer to instability is more responsive;
modern bicycle design has emphasized nimbleness and maneuverability. Best of all, URB III comes out much more stable than any commercial bike. This result explains both its
wonderful selfrighting properties and also why it is difficult
to ride—it is too stable to be steered. An inert rider with no
balancing reflexes and no preferred direction of travel would
be happy on URB III, but its characteristics are too intense for
easy control.
This mathematical exercise also made it plain that the
centerofgravity lowering torque is developed exactly as
shown in figure 6, and is identical with that postulated in reference 6. But it does not vanish when the bicycle’s lean is in
equilibrium with centrifugal force, as therein supposed
(BICYC calculated the height of the forkpoint in the plane of
the bicycle—the “effective vertical”—to allow for this). It can
only vanish when the contact point of the front wheel is intersected by the steering axis, which BICYC shows clearly is
the condition for minimal height. There is thus an intimate
connection between the “trail” of a bicycle, as defined in figure 6, and d2H/dαdL; in fact the d2H/dαdL line in figure 4 coincides with the locus of zero trail.
Two further courses of action remained. First, I could
make URB IV with a steering geometry well inside the unstable region, and second, I had to decide what force opposes the
twisting torque on a bike’s front wheel and prevents it reaching BICYC’s predicted minimal centerofgravity position. Selfcentering
Let us consider the second point first; I was looking for some
sort of selfcentering in a bicycle’s steering. Now this is well
known in the case of fourwheeled vehicles: selfcentering is
built into all car steering systems, and various selfrighting
torques such as “pneumatic trail” are described by automobile engineers. Once again nasty variable frictional forces
were rearing their ugly heads! But how could I check whether
a bicycle wheel has selfcentering? I examined a child’s tricycle for this property, releasing it at speed and, running alongside it, giving the handlebars a blow. It certainly seemed to
recover quickly and continue in a straight line, but unfortunately the tricycle (being free of the requirement of twowheeled stability) has a different steering geometry.
So I made an experimental fixedlean bicycle by fastening an extra “outrigger” wheel to the rear of the frame, converting it to an asymmetric tricycle. Adjustment of the outrigger anchorage could impose any angle of lean on the main
frame. This machine was very interesting. Initially I gave it
15° of lean, and at rest the front wheel tilted to the 40° angle
predicted by BICYC. When in motion, however, the wheel Figure 7. Selfcentering? A bicycle
with an “outrigger” third wheel to
keep it upright was pushed and
released riderless. At the point
shown the handlebars were
knocked out of true, resulting in a
change of direction and no selfcentering. The slight wave in the
track resulted from oscillations in
the framework. www.physicstoday.org September 2006 Physics Today 55 JANIS
CRYOSTATS USING
ORIGINAL SHIAPD
DISPLEXTM REFRIGERATION
SYSTEM TECHNOLOGY • Options from <10 K to 700 K
• Optical or tubular cryostats
• Complete systems
• Custom cryostats
• Wide selection of
options and accessories
• Proven technology Janis Research Company
2 Jewel Drive Wilmington, MA 01887 USA
TEL +1 978 6578750 FAX +1 978 6580349 sales@janis.com
Visit our website at www.janis.com . See www.pt.ims.ca/946822 NEW MultipleHit TDC/TimeMultiscaler. The new P 7 8 8 9 offers 100ps time resolution The P7889 analyzes the rising or falling edge of input pulses
with 100 ps time resolution. Max. input rates to 10 Gbit/s
G NEW !! TimeInterval analysis with 100 ps
G
G
G
G
G
G
G precision by analyzing both the rising and
falling edge !!
PCIboard (64 bit) with onboard FIFO for
continuous ultra fast data transfer
Continuous histogramming in the PC
at more than 30.000.000 stop events/s
Sweep range from 128ns to 58 years in
increments of 6.4 ns
Multihit capable; No Deadtime,
No Missed Events, No Double Counting
WINDOWS 2000/XP based operating software included
Two onboard ultrafast discriminators
(+/2V input range)
16 Taginputs, Options: oven controlled
oscillator and Drivers for LabVIEW and LINUX Other versions with 250 ps to 2 ns time resolution are also
available. For four inputs see Model P7888. For more details and applications visit our site:
www.fastcomtec.com
In the USA call 8314295227
For information contact: ComTec GmbH
FAST ComTec GmbH, Grünwalder Weg 28a
D82041 Oberhaching/Germany WebPage: http://www.fastcomtec.com
email: support@fastcomtec.com
Phone: +498966518050, FAX: +498966518040 See www.pt.ims.ca/946823 tended to straighten out, and the faster the bike was pushed
the straighter did the front wheel become. Even if the machine was released at speed with the front wheel dead ahead
it turned to the “equilibrium” angle for that speed and lean—
another blow for gyro theory, for with the lean fixed there can
be no precessional torque to turn the wheel. So clearly there
is a selfcentering force at work. It is unlikely to be pneumatic
trail, for the equilibrium steering angle for given conditions
appears unaltered by complete deflation of the front tire.
Now I had encountered a very attractive form of selfcentering action, not depending directly on variable frictional
forces, while trying the naive experiment of pushing a bicycle backwards. Of course it collapsed at once because the two
wheels travel in diverging directions. In forward travel the
converse applies and the paths of the two wheels converge.
So, if the front wheel runs naturally in the line of its own
plane, the trailing frame and rear wheel will swing into line
behind it along a tractrix, by straightforward geometry. To an
observer on the bike, however, it will appear that selfcentering is occurring (though it is the rest of the bike and
not the front wheel that is swinging).
I modified my outrigger tricycle to hold the main frame
as nearly upright as possible, so that it ran in a straight line.
Then, first soaking the front wheel in water to leave a track,
I pushed it up to speed, released it, and, running alongside,
thumped the handlebars out of true. Looking at the bike, it
seemed evident that the wheel swung back to dead ahead.
But the track (figure 7) showed what I hoped to find—a sharp
angle with no trace of directional recovery. The bicycle has
only geometrical castor stability to provide its selfcentering. Success at last!
This test completed the ingredients for a more complete theory of the bicycle. In addition to the rider’s skill and the gyroscopic forces, there are, acting on the front wheel, the
centerofgravity lowering torque (figure 6) and the castoring
forces; the heavier the bicycle’s load the more important these
become. I have not yet formalized all these contributions into
a mathematical theory of the bicycle, so perhaps there are surprises still in store; but at least all the principles have been
experimentally checked.
I made URB IV by moving the front wheel of my bicycle
just four inches ahead of its normal position, setting the system well into the unstable region. It was indeed very dodgy
to ride, though not as impossible as I had hoped—perhaps
my skill had increased in the course of this study. URB IV had
negligible selfstability and crashed gratifyingly to the
ground when released at speed.
It seems a lot of tortuous effort to produce in the end a
machine of absolutely no utility whatsoever, but that sets me
firmly in the mainstream of modern technology. At least I will
have no intention of foisting the product onto a longsuffering public in the name of progress. References
1. S. Timoshenko, D. H. Young, Advanced Dynamics, McGrawHill,
New York (1948), p. 239.
2. A. Gray, A Treatise on Gyrostatics and Rotational Motion, Dover,
New York (1959), p. 146.
3. J. P. den Hartog, Mechanics, Dover, New York (1961), p. 328.
4. K. I. T. Richardson, The Gyroscope Applied, Hutchinson, London
(1954), p. 42.
5. R. H. Pearsall, Proc. Inst. Automobile Eng. 17, 395 (1922).
6. R. A. WilsonJones, Proc. Inst. Mech. Eng. (Automobile division),
1951–52, p. 191.
7. Encyclopaedia Britannica (1957 edition), entry under “Bicycle.” 56 September 2006 Physics Today ...
View
Full
Document
This note was uploaded on 11/23/2011 for the course PHYSICS 7221 taught by Professor Gonzalez during the Fall '09 term at LSU.
 Fall '09
 gonzalez
 mechanics, Work

Click to edit the document details