Email Questions and Answers
at low speeds
Countersteering and bike speed
Treating Stop Signs as Yield Signs
How do you balance a bicycle?
The simplest way to explain how you balance a bicycle is to make
an analogy with the way you balance a stick on your palm. For
example, suppose that the top of the stick tilts and starts to fall to
the right. To regain balance, you move your hand to the
right, thereby moving the bottom of the stick underneath the top.
As the stick is no longer tilted, it stops falling. (Actually,
you move your hand rightward just a bit further so that the stick tilts
left a bit. This ends the rightward momentum of the stick, and allows
you to move your hand back left, closer to you, a moment later.)
If you are riding a bike, and you start to fall to the right, you do
exactly the same thing; you steer right, thereby driving the wheels
directly underneath you again. Now you are no longer tilted, so
you stop falling. (Once again, you actually overcompensate, so
that you are tilted to the left. This stops the rightward
momentum, and allows you to steer back left a moment later.)
Thus, balancing a bike is a process of making successive right and left
turns. Typically these turns are very small, but can be easily
seen in the tracks of a bike ridden on sand. The two tracks (from
the front and rear wheels) weave in and out as the rider makes these
Riding a unicycle is very similar.
Can you explain how my six-year old daughter LuLu can ride her bike in
a straight line without ever exceeding 0.5 mph? Don't you have to
have some kind of speed just to keep from falling over? Maybe she's a
freak of nature.
Suppose you've tilted over three inches. To get the wheels back
underneath you, you have to steer three inches over. If you are
riding fast, you only need to turn the front wheel slightly to get over
quickly. If you are riding slowly, however, you have to make a
very sharp turn to get over quickly. Slight turns are easier than
sharp turns to execute and control, so riding fast is easier than
People can ride very slowly, however. There's a technique
called track standing where you balance an almost stationary
bike. I learned to trackstand while waiting for lights on my
commute to work. Trackstanding is fun, and I was to lazy to
unclip from my pedals.
Velodrome bike racers are particularly good at track standing.
Typically, these are two person, short races on a steeply banked
circular track. The two racers usually go very slowly for most of
the course, and then engage in a sprint at the end. Being
slightly behind and higher on the track just before the sprint begins
can be advantageous: being behind allows the racer to accelerate
in the windshadow of the leading racer, saving energy, and being higher
allows the rider to gain speed by riding downward. So at the begining
of the race, the riders jockey for position, often coming to a complete
stop, before breaking into their sprint.
Track standing is a bit more complicated than the balancing
technique described above, but the major technique, moving the wheels
underneath you by turning, is the same. Obviously you have to
advance just a bit to move over. To stay essentially in the same
place you use the camber in the road (or the banking in the track) to
roll backwards every now and then. (Bikes build specifically for
velodrome racing, called track bikes, have an additional advantage;
they have a single fixed gear with no freewheel, so pedaling backwards
actually moves the bike backwards.)
So your daughter just has very good balance. See if she can learn
and bike speed
I'm a motorcycle enthusiast in Wisconsin and I talk to other
motorcyclists on the Compuseve motorcycle forum. We've recently
been talking about the stability/instability of motorcycles.
Looking around on the web I learned about the instabilities of capsize,
wobble and weave and learned about the effect of trail (caster). Then I
came across your article
on steering in bicycles and motorcycles. I understand maybe 60
percent of it. While I like science and math all I've had is high
school physics and a couple semesters of calculus in college - and that
was in the 60's <g>. I think I correctly understand your article
to say that the gyroscopic forces of the wheels are not very relevant
to balance and turning effort on motorcycles compared to the
stabilizing forces generated by the trail geometry. I'm wondering if
you can tell me, in laymans terms, why the countersteering force needed
to initiate a turn goes up as speed increases. Appreciate your help!
is pretty mathematical, and not easy to understand if you are not up on
your physics. So 60% is pretty good.
I confess that I've never ridden a motorcycle
(though I'm an avid bicyclist) so my conclusions about motorcycles are
based on extrapolations from my experience on bicycles, conversations
with motorcyclists, and the results of the math. But the
conclusion that the lean angle (the balance) is not
very dependent on gyroscopic action
is pretty robust. But gyroscopic action does play a role in the
torques used to steer the wheel, i.e. in the amount of torque you need
to apply to the handlebars. There's real experimental data
here...at least for bicycles. As mentioned in the article, Jones
actually build a bike with an auxiliary counterspinning "wheel" that
had no angular momentum, and it behaved about the same as a normal
bike. As for you question, its not an easy one. Many factors
interact in a complicated way to determine how much torque you need to
apply to the handlebars, and how the torques scale with velocity.
If you look at Equation 1 of my paper, you see that
(Gyroscopic Torque) + (Moment of inertia
of wheel around steering axis times acceleration of the steering angle)
= (Applied torque on handlebars) -
The sign of the gyroscopic term always tends to turn the wheel in the
required final direction. Since you are countersteering, the
torque that you need to apply works against the gyroscopic term.
Thus the bigger the gyroscopic term, the harder you have to
countersteer to compensate. And the gyroscopic term increases
linearly with velocity, so the faster you go, the harder you have to
But this is only a small part of the answer; the two last terms in the
equation are much bigger than the gyro term. Both depend on the
velocity like v^2 (this is explicit for the castering term, and comes
from the fact that the final lean angle lambda goes like v^2 for the
trail-steering term.) At the very beginning of the turn, when you need
to turn the handlebars the wrong way, the castering term resists this,
so the countersteering has to increase...like v^2. Later on, as
the wheel eventually turns in the right way, the castering term helps
out, but the trail-steering term almost cancels the castering term and
there is no simple way to predict which "wins."
1) You've got to fight the gyroscopic
term, which increases like v.
2) Initially you've got to fight the
castering term, which goes like v^2. Eventually the castering
term helps, but by then
3) You've got to fight the trail term,
which effectively goes like v^2 because of its dependence on angle.
Hope this helps
We have another discussion going on motorcycle handling and I'm
wondering if you might help us understand the physics. Basically
we're talking about the effect weighting the pegs of a motorcycle has
on its handling. On a street bike, the rider typically sits on
the seat with his feet resting on the pegs. Weighting the pegs means
starting to stand up on the pegs so that the force of the rider's
weight on the bike is at the pegs rather than at the seat.
Many riders think that weighting the pegs lowers the
center-of-gravity of the bike and rider as compared to just sitting on
the seat. I did too but now I'm not so sure. To discount
the rider shifting his body's center-of-gravity by moving around, we
imagined a bike where the seat was replaced by a board. For one
setup, we place a 200 pound metal weight on the board (seat) with bars
going down either side stopping just short othe pegs. For the
other the bars are connected to the pegs supporting the weight just
clear of the seat. We assume the center-of-gravity of the weight
and bars remains the same in both cases. Does the point on the
motorcycle where the weight is supported (low by the pegs or higher at
the seat) move the center-of-gravity of the bike and weight
combined? Is there any difference in the handling characteristics
of the motorcycle between the setups?
A noted motorcycle racing instructor, Keith Code, has written
Using the outside peg as your pivot
point--while pressure is being applied to the bars either by just
pushing or using a combination push and pull--reduces your weight on
the seat and puts the majority of your weight on that lower, outside
peg. Doesn't putting the weight on the outside peg make the bike
try to stand up? Not at all: Don't forget the gyro effect from
In fact, since your weight is now closer to
the center-of-mass for the machine, the bike is much easier to
steer. Technically, I understand the bike rotates around the
center-of-mass, so the more of the total weight that can be put at or
close to the center-of-mass, the better.
Pivot steering starts with the opposite
footpeg, moves through the torso and down to the handlebar. It
feels like power steering.
In dirt-bike motorcycles and mountain bicycles, the rider is
often up on the pegs or pedals so that his legs and arms act as
additional suspension components and he can move his body around to
effect the handling.
On road racing motorcycles (I'm not sure about bicycles), and to
some extent on the street, the rider doesn't want the bike to push
under him dirtbike style in a turn but rather wants to move his body
with the bike, often hanging off to the inside. From my own
experience, I believe Code's observations are accurate. Weighting
the outside peg seems to make steering lighter. However, I don't
think his explanation of why that is so is accurate. I wonder if
weighting the outside peg is similar to the hip thrust you discuss in
in that it helps initiate the countersteer.
Any comments on the physics of what's going on would be
First the introductory physics course answer.
So long as the connections are fixed and the rider's position does not
otherwise change, the position of the center of mass does not change
when riders shift their weight from the seat to the pegs. No if,
and or buts about it...this is an absolutely, always true
statement. So your late instincts are correct.
Now the caveat. Its in the phrase "so long as
the connections are fixed and the rider's position does not otherwise
change." Motorcyclists are not made of steel, and are not
rigid. If they slump while doing this, then their center of mass
will change. The must important "slump" would be to bend at the
waist, and I'd guess that this could be marginally significant.
If you hang off the bike to one side, you also change the center of
mass. (Imagine a ruler pivoted off the peg till it is horizontal.
Obviously its center of mass decreases.)
Actually, in the absence of a slump, the center of
mass of the bike is bound to get just a bit higher. After all, to
get your weight onto the pegs, you've got to lift your rear off the
So Keith Code's statement: “In fact, since your weight is now
closer to the center-of-mass for the machine” is wrong if you don't
slump at the same time.
What you do do when you weight the pegs unevenly is apply a
torque. You are right that this will cause the bike to steer thru
what I called hip steering and what Code calls pivot steering.
I'm fairly certain that the reason that the bike feels more responsive
when you are on the pegs is simply that you are already in position to
initiate a pivot steer. All you have to do is let up with one
leg, and all the weight will go to the other leg. Since you are
not lifting any weight, this doesn't take any energy and can be done
very quickly. If you a still on the seat, you need to lift you
rear a bit to shift your weight to a peg, and, as this takes energy, it
will take a bit longer.
There may also be a physiological factor...With you weight on the pegs,
you are tense, ready to go. On the seat, your are laid back and
relaxed. This is going to make the bike feel more
responsive on the pegs.
Stop Signs as Yield Signs
What states allow bicyclists to treat stop signs as yield signs?
Answer (courtesy Melanie Curry):
Idaho (and I thought Montana but I haven't been able to find my
original citing for this) allows bicyclists to go through stop
signs--and even red lights(!)--without stopping, if it's safe.
Suppose you're riding a bicycle, and you put your hands over your head
and lean to the right. What happens? This is not a trick
question, although it's possible that there's insufficient
information to answer the question. Does it matter how fast you're
going? I think it's safe to assume that you're going fast enough
that you don't just fall over, but other than that, does it matter?
Generally, you lead with your shoulders...the bike will go in the
direction your shoulders point. If you lean to the right, your
hips will swivel left to compensate. This will lean the bike to
the left, which, because of the bike's trail (this distance
between where the steering axis projects on to the ground and where the
front tire touches the ground,) causes the front wheel will to turn
left slightly...the wrong way. With the wheel turned left, but
your center of mass still vertical, centrifugal forces will lean you
and your bike over to the right. The act of leaning the bike over
to the right will cause gyroscopic forces in the wheel to turn the
wheel to the right. Since you are now leaned to the right as
well, everything is stable and you will go right. I don't think
that your speed matters to which way you would go if you could
maintain control, but it will be much harder to maintain control at low
speeds. For a more complete discussion, read this technical paper.
How do you ride a unicycle?
Riding a unicycle is very much like balancing a stick in the palm of
your hand. To balance the stick, you move your palm to keep it
directly underneath the stick: if the stick begins to tip forward, you
move your hand forward, etc. To balance a unicycle, you pedal the
unicycle so that the wheel is always directly underneath the unicycle
seat; if the unicycle begins to tip forward, you pedal forward,
etc. To keep the stick upright, you must continually move your
palm…there is no perfect position. Likewise, you must continually
adjust the position of the unicycle. These continual adjustments
cause the characteristic back and forth motion of the unicycle.
Its easy to pedal the unicycle backwards and forwards; to pedal to the
right or left, you twist the unicycle underneath yourself so that the
wheel points in the direction that you need to go.
Because the wheel is so light and not rotating quickly, angular
momentum is not significant.
Because bicycles have two wheels, they can only tip sideways, not
forwards or backwards. Skilled bicyclists can “trackstand”, i.e.
balance the bicycle while keeping it essentially stationary. Once
again, they have to pedal the bicycle in the direction that it starts
to tip. (Actually, bicycle are a bit more complicated because the
front wheel’s point of contact with the ground can be shifted right or
left by turning the handlebars to extreme angles. If the point of
contact is shifted to the side, there will be a torque that can help
keep the bike erect.) Its trickier to trackstand a bike than keep
a unicycle stationary, though, because most bicycles have a freewheel
that prevents them from being pedaled backwards. Typically the cyclist
points the bicycle slightly uphill so that he can pedal forwards uphill
and roll backwards downhill, thus keeps the bike approximately
stationary. Even when the road is flat, the cyclist can often
take advantage of the center hump in the roadbed.
In order to interest young people in engineering and physics I assisted
a group of teenage air cadets from Edinburgh, Scotland, to modify a
child’s bicycle to enter the BBC television programme
‘Technogames’. This is an athletics contest for robots, and one
of the events is a bicycle race of two laps round a flat oval track
length 50m. Our entry ended in disaster when it hit the sidewall of the
arena at the first bend.
The competition rules state that the robot bicycle has to be electric
powered and travel round the course under radio control. The maximum
dimensions are 1m long, 0.5m high and 0.5m wide. The max weight is 25kg.
In ignorance of your stability equations we constructed a closed loop
stabilisation system which consisted of a pendulum connected to a
potentiometer which drove a PWM servo connected to the handlebars. (I
do not know what the time constant of the pendulum to handlebar servo
loop is). Initial trials in a car park showed a divergent oscillation
of the pendulum so we added a damping pot, filled with water, to the
pendulum and in order to stiffen the whole stabilisation system fitted
a large instructional gyroscope to the bicycle frame This
gyroscope was in the horizontal plane and free to tilt about a lateral
axis. Heavy rubber bands fore and aft allowed some movement and gave a
restoring force. We managed to achieve a few test runs in a roughly
straight direction, but most tests ended by toppling over. The gyro
definitely helped to delay the crash and provided an opportunity to
teach gyro theory.
In an attempt to achieve a controlled turn we mounted the battery on a
screwed rod mounted laterally and driven by an electric drill motor.
This allowed the C of G to be moved rapidly sideways. Our theory was
that the radius of turn would be proportional to the lateral movement
of the C of G and the stability would be achieved by the closed loop
steering system that would steer in the direction of the lean until the
centrifugal force balanced the gravitational force and returned the
pendulum to a near central position of equilibrium. We never achieved a
controlled turn and lack of time prevented further experiments before
our race. The result was not unexpected, but the cadets learnt a lot
about several aspects of engineering and physics and want to have
another attempt later this year.
Notes: 1) The servo is powerful enough not to be back
driven by the natural movement of the front wheel due to the trail
angle. Would a spring compressible linkage from servo to
handlebars be better?
2) We moved the weight (battery) laterally in the direction we wanted
Having just read your paper on ‘steering in bicycles and motorcycles’ I
realise that our problem is much bigger than we first imagined and we
may have to completely redesign the control system.
We would value your advice as to how to modify our bicycle and its
control system for this years contest. Could you also indicate which
parameters should be subject to experimental adjustment as we were
overwhelmed with the number of parameters that could be altered and
unclear as to their individual effect and interaction.
As you saw in my article,
the dynamics of a bicycle are quite complicated. If we had to think
about how we ride a bike, we could never do it. (By the way, "my
equations" really aren't mine. They've been known for a long time.)
Have you ever noticed that it is easier to balance a long stick in your
palm than a short one? This is because the stick tends to fall
"faster." Likewise, a short bicycle falls over more quickly than a tall
one. So a kid's bike is actually more difficult to balance than an
adult bike. Even if the bike is being balanced by some active
mechanism, the frequency of oscillations around equilibrium will be
slower for a tall bike. And slower is easier to control. So try moving
your center of gravity up as high as possible. I know this sounds
counterintuitive, but it should help. For instance, mount the heavy
gyro above the seat, not near the ground.
Speaking of the gyro, it seems that you have its rotation axis
verticle. That means that as the bike tilts the gyro will respond by
bucking forward or back. This motion may be subtle, but it could
interact with the front wheel trail and cause difficulties. If you
mount the gyro so that it spins horizontally (with the same sense that
the wheels turn,) it will try to direct the bike left or right in the
appropriate direction to counter any tilts. This is the same mechanism
that makes a rolling hoop stable. It is more complicated for a bike
than for a hoop because the bike's got two wheels, and one can be
steered, but it should still help.
You are getting your balance signal from the pendulum, which you need
to respond quickly, and yet not oscillate. Obviously it should be
critically damped, but this slows the response down. Clearly, the
response of the pendulum should be much faster (at least a factor of
ten) than any oscillations in the bike angle. It may not be easy to
make a pendulum that responds this quickly. Yours seems to have a
length of about 1/3m. That means its period is about 1s. This is
probably too slow. You need a much shorter pendulum. I know that it is
not as pedagolically interesting, but you would probably be much better
off with a commerical tilt sensor.
Moving the battery in the direction of the desired turn is probably
backwards, at least if the handlebars have some freedom to move. When
people ride a bike, they take advantage of the natural, stabilizing
motions of the handlebars. As you point out, your servo is too strong
for the handlebars to move on their own. This certainly means that the
bike will not behave as a normal bike would. This simplifies the
system, but you lose all the stabilizing factors from the trail and the
front wheels gyro action. You may be better off with a much weaker
servo that can only apply torques.
I have become increasingly curious about the physics of cycling and
very recently started to research the topic a little as a hobby and for
additional intellectual stimulation. I was wondering if you have
come across any articles/books that explore the complete physics and
dynamics of cycling.
Read this article
by David Jones first: (David E.H. Jones, Physics
Today The Stability of
the Bicycle 23 p34-40 (1970.)
Then try this book:
Frank Whitt and David Gordon Wilson, Bicycling Science, Second edition.
(MIT Press, Cambridge, MA, 1982).