Email Questions and Answers

Topics:

Bike Balancing
Balancing at low speeds
Countersteering and bike speed
Center of mass
Treating Stop Signs as Yield Signs
Riding no-hands
Unicycles
Robot Bicycles
Physics References


Bike balancing
Question:

How do you balance a bicycle?

Answer:

 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 small turns.

Riding a unicycle is very similar.

Balancing at low speeds
Question:

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.

Answer:
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 riding slow.

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 to trackstand!
 

Countersteering and bike speed
Question:

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!

Answer:

    My article 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) - (Trail-steering)-(Castering)

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 countersteer.

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."

      Thus
      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

 Center of mass
Question:

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 this:

 Weight Redistribution

     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 the wheels.

     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 your article in that it helps initiate the countersteer.

 Any comments on the physics of what's going on would be appreciated.

Answer:

    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 seat!

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.

 
Treating Stop Signs as Yield Signs
Question:

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.
 
Riding no-hands
Question:

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?

Answer:

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.
 
Riding a Unicycle

Question:

How do you ride a unicycle?

Answer:

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.

Robot Bicycles
Question:

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 to turn.

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.

Answer:

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.

Physics References
Question:

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.

Answer:

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).