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Two-wheeler Dynamics

  1. May 17, 2007 #1
    Needed a query for the stability of two-wheelers in general.
    How can i derive the equations governing its stability. Suppose i am moving on a straight road, the normal reaction is equal to the Weight of the Vehicle and body combined- hence the vehicle doesn't topple. Suppose i steer left/right. what are the equations that govern this steering.
    What forces, moments are acting during turning and how are they balanced so that the vehicle(2-wheeler) doesn't topple.

  2. jcsd
  3. May 17, 2007 #2
    I can't answer your question myself but here is an article that considers the
    problem: “The Stability of the Bicycle,” Physics Today 23 (April), 34 (1970).
    If I remember correctly, the author shows how to construct an unridable
  4. May 18, 2007 #3
    i would want to know how do i define the movement of a Two-Wheeler on a straight road, the movement of a Two-wheeler in a Curve.
    Which mathematical equations describe it. I need an equation which at least mimics the physics of Two wheeler.
  5. May 19, 2007 #4
    "How can i derive the equations governing its stability. "

    I think you can not.

    Think about this experiment: Fasten a 50 Kg stone on top of our bicycle and just send it away. How long time will it take before it will fall down ? Is it stable ?

    I believe that the reason you can not find an equatation for the stability of a bicycle is because it is unstable. A two weeler can be compared allmost with a pendlum, just with position upside down.

    One other experiment - what about placing a ball on the tip of the nose of a dead sea lion, will it be stable or will it just fall down ?

    What about placing a ball on the tip of a nose of a living sea lion with some clue, experience and training ? Can it be balanced ? Can it be stable ?

    Rather to ask: What is the equatation for the ball on the tip of the sea lions nose that will not fall down or the stability of a bicycle, I think that the bether questions will be: What is the quatation of the way the sea lion handles the ball on tip on it's nose, or how the the rider interacts with the bicycle, (so the riding itself it stable) .. ??

    I think that there is such a disipline in mathematics and that the name of it is "dynamic systems".

    I believe that the stability of the rider and the bicycle can not be expressed in one single equatation, it is a rather complicated case that might contain allmost all the theories of "dynamic systems".

    One can find a bit more easy exsample of such a "dynamic system". Think about a automatic cruise control an a car, or even bether about youself as the cruise control. When speed is moves lower than 90 Km/h, you push the trottle so it reach 90 km/h then it might continue up to 100, you then trottle off until the speed stabilize on 90 km/h. In this case it is only a few parameters the trottle and the speed (the wind speed, the friction etc), but it ilustrates an easy example of a dynamic system. (And without a cruiscontrol or a driver, no constant speed and no stable system.)
    Last edited: May 19, 2007
  6. May 19, 2007 #5
    if a bicycle moving around is not stable then nobody would be able to ride a bicycle!

    given sufficient speed (rotational speed of the wheels), the bicycle becomes a spinning top like system, with friction involved. If you have a bicycle wheel and you give it a good spin so that it rolls forward, it would stand straight for a good while until its rotation begins to damp out. for simplicity, you can just work out the system of one bicycle wheel and you should arrive at some conditions of stability.

    edit: I think that using Euler's equation, you'll find your answer!
    the Euler's equation is:
    [tex]\dot{\vec L}+\vec\omega\times \vec L=\vec{\Gamma}[/tex]
    then you can work out the rotation tensor and get some symmetry to simplify the system.

    of course, you can also use the Lagrangian approach and crank out the kinetic energy of the wheel using the Euler's angle. (I think the second approach would be easier).
    Last edited: May 19, 2007
  7. May 19, 2007 #6


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    Two wheeled vehicle vertical stability is due to steering geometry, specifically trail. There are radio controlled motorcycles that don't require any special algorithims or gyro sensors to work, they just use a lot of trail, and a steering servo with a low resistance to small "non-commanded" movements caused by trail effect, allowing the self correction process to work.

    Trail is the distance from where the front tire pivot point axis would intercept the pavement, back to the point where the front tire actually makes contact with the pavement.

    When a bicycle is vertical, the trail creates a castor effect (like the wheels on a shopping cart), creating a tendency for the front tire to move in the direction traveled.

    When a bicycle is leaned, the trail creates a inwards steering torque force on the front tire. This is because contact patch moves laterally when steering and vice versa. Suspend a bicycle off the ground. Steer left, and the contact patch area on the front tire will move right and vice versa. This lateral motion is relativly large on a radio control motorcycle, and moderate for human controlled bicycles / motorcycles. To see this effect, a person can hold a bicycle by the rear seat and lean it over, and the front tire will steer inwards.

    Getting back to a leaned bicycle, gravity results in downwards force on the center of mass, and the pavement in turn generates an upwards force on the tires. At the front tire of a bicycle, this upwards force is applied "behind" the pivot axis and causes the front tire to steer inwards, and given suffiecient inwards steering and speed, the lean angle of the bicyle will be reduced until it is vertical again (it may overcorrect due to momentum).

    The amount of trail effect determines the minimal speed required for vertical stability. At or above this minimum speed, a bicycle will be vertically stable, auto-correcting for any reasonable amount of lean introduced. If there is excessive trail, flex, or momentum, in the bicycle, constant overcorrection can occur, resutling in speed wobble.

    Gyroscopic reaction generates lean angle stability (as opposed to vertical stability). As speed increases, a bicycle will tend to hold a lean angle and resist changes in lean angle, including the vertical stablity reaction from trail. In the case of motorcycles at high speeds, 100+mph, the lean stability dominates, and the motorcycle just holds a lean angle with no perceptible tendency to straighten up.

    To induce a lean on a bicycle, motorcycle, or unicycle, a rider steers outwards to initiate a lean. The outwards steering can be the result of a rider applying an outwards steering torque force on the handlebars (counter-steering), or the result of the rider leaning inwards, causing the motorcycle to lean outwards (since the center of mass won't move sideways without a sideways force at the contact point between tires and pavement (or a side wind)), and the self-correcting geometry causes an outwards steering reaction to the motorcycle being leaned outwards. Deliberate counter-steering is best, since leaning doesn't provide as quick as a response, and at high speeds on a motorcycle, provides no lean inducing steering response.
    Last edited: May 19, 2007
  8. May 20, 2007 #7
    From the posts abowe (tim_lou):

    "if a bicycle moving around is not stable then nobody would be able to ride a bicycle!"

    Well, try this experiment. Stand up on your feet. Lift the left leg, just to stand on the right one.

    Are you able to stand and balance with the right foot on the ground only ? I think most people can do that. If you lift your heel to stand on the tip of your right foot only, are you still able to keep the balance ? Is this because your body is stable ?

    What about sending your bicycle in a cirkular motion with a 50 kg stone (and not you) on top of it, will it fall or not fall ? Ok, try ..

    Whith the use and principles of "control of dynamic systems" and "feedback control systems" unstable systems can be made stable.

    As an exsamle can be mentioned some modern aircrafts that can not be flown at all without the use of "feedback control systems". (As an example can be mentioned the F16 that has lift on the wings and the its tail so it can not be flown a second without automatic feedback control systems.)

    When you are walking or running, playing basketball or bicycling your body (and the bicycle) will be unstable most of the time. Your body's "automatic feedback control system" will stabilize it.

    For humans and sealions this works quite automatic and by itself. For F-16's, stealth bumbers, autopilots for ordinary aircrafts, and things like that, the control system wil have to be built from technology and based on a dicipline in mathematics called something like "dynaic systems", "dynamic feedback system" or "dynamic control systems", as you like it.
  9. May 20, 2007 #8
    Of course the movement of the stearing, the angle of the bicycle and the road, the possition of the body of mass on the bicycle is a part of it's stability, but it is not all of it. The movment of the stearing, the positioning of your body, etc will be the "power organs" your built in automatic feedback control system will use to stabilize the system consisting of you and the bicycle.

    When a child is sitting up on top of a two weel bicycle for the first time will he/she just be running away in a safe way, or will the child have to learn to stibilize a bicycle first ?

    Is the bicycle stable by itself or does the safe ride ocour because the child has obtained a good enough training to aply his/hers automatic feedback control systems in a good and safe way ?
    Last edited: May 20, 2007
  10. May 20, 2007 #9
    Here is something more about it, but it doesn't need to be that complicated. Children and sealions can do it as well.

    http://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/16-31Feedback-Control-SystemsFall2001/CourseHome/index.htm [Broken]
    Last edited by a moderator: May 2, 2017
  11. May 20, 2007 #10
    It might be true that a bicycle in a circular notion might be able to balance in that cerain circular moition if a number of initial conditions comes true:

    The speed must be constant, the position of the stearing must be constant. etc.

    But the problem is that these initial conditions will not be true if you do not have one or more feedback control system to make them to come true.

    The initial conditions of a theoretical stable bicycle moving in circle will not be fullfilled as stable by itself, there will be needed feedback control systems also for this. To travel in a circular motion only, will also be rather booring.
  12. May 20, 2007 #11


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    At or faster than a minimum speed, a bicycle will have vertical stability, no rider or control needed. When a child learns how to "ride" a bicycle, the child typically learns to "steer" the bike indirectly by leaning with the body and relaxing on the handlebars. Because of the self-stability of a bicycle, the learning process is usally 5 to 10 minutes for even a very young child. The main thing is overcoming the fear of going the "minimum" speed. The smaller bicycles for very young children (3 years old and younger), have a lot of trail which reduces this minimum speed, making it very easy to drive a bicycle.

    A unicycle isn't inherently stable, and requires a longer learning process. I took about 15 to 20 minutes to get the basics, and this is faster than most. The main thing was I knew in advance to keep my weight on the seat, and light pressure on the pedals. Even more difficult would be learning to balance on a large ball (as in circus acts). You'd probably want to start on a heavier ball (more momentum) to allow for slower and cruder control inputs. Then as skill increased, a lighter ball could be used, which would require faster and more subtle control inputs.

    Yes, once above a minimum speed, for the amount of trail, a bicycle will be vertically stable by itself.

    Assuming the initial lean isn't excessive, the bicycle will straighten up. Bicycles are designed to be inherently stable (vertically) by themselves. The "feedback" control system is built into the steering geometry and is called trail. I've already explained how this works in a previous post. Adding weight up high on a bicyle will increase it's vertical stability. However, adding weight very low that results in a very low center of mass will be an issue, since there's less lever effect from the side forces of steering to change lean angle. This is an issue for the torpedo like motorcycles that run at Bonneville. Some use "training wheels" that get re-tracted once the bike is up to speed.
    Last edited: May 20, 2007
  13. May 20, 2007 #12
    "The "feedback" control system is built into the steering geometry and is called trail."

    Yes, I think you are partly right. Mechanical "things" can have such a "built in feedback prosess."

    One example is clasical aircraft design. They are all built stable around all tree axes, nose up, node down, yaw, rudder, etc.

    When it comes to the bicycle I find it hard to believe that the bicycle itself take care about all the stability of the system allone.

    The "moving system" will consist of the biker and the "bicycle". With a grown up person a lot more than 50 % of the mass of the moving system vil be related to the biker and not to the bicycle allone.

    Because of all this mass related to the biker person, it will not be of no interesst how the biker person behaves.

    If he or she forget everything about bicycling and just sit down on the top of the bike to have a lunch, I think it will not take long time before you have a crash. (I have allmost tried, and my right shoulder still hurts.)

    I think that it is right that the geometry and the feedback of the geometry of the bicycle is a part of the stability, but then the feedback and the control of the biker will also have great influence of the "total control" of the bicycle.

    If there were "crash proof" bicycles, I would like to have such a bike.
  14. May 20, 2007 #13
    By the way, the way the biker applyes his/her feedback to the bicycle is by moving his/her body and moving the stearing to the right or to the left.

    Have you tried to run and keep the balance of a bicycle with a locked front wheel ? (That can not be turned to left or to the right.) This will loch of the most important feedback from the biker to the bike so most bikers including me will make a crash.

    A cirkus artist might run his bike on a stright line, without any capability of stearing. This will be a completely unstable system with the use of body control only, to maintain the balance. He can do it, but I can not.
  15. May 20, 2007 #14


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    Obviously a rider on a bicycle genreating the wrong control inputs will cause a crash. Within a range of speed, a bicycle can be ridden hands free.

    The main issue is a rider weighs much more than a bicycle, and even a slight lean will offset the center of mass of the rider/bicycle system significantly, and beyond the point where the vertical stability of the bicycle is enough to return it to a vertical attitude.

    When riding hands free on a 10 speed type bicycle, I find I have to initiate a lean by leaning inwards, then lean back upwards, to lean the bike inwards enough to generate enough inwards steering input to hold the lean or reduce it. It's pretty instinctive if you pay attention, just lean in the direction you want the bicycle to roll to increase or decrease lean angle.

    The exception cases where moderate to high skill is required:

    If the front wheel is fixed, for example a bicycle on a wire as in the circus, the rider has to generate torque forces in order to balance the bicycle. Circus riders ride the bikes hands free and use their arms if really good, or use the typical long poll to generate corrective torque forces.

    If bicycle is not moving, steering inputs can still be used to balance the bicycle, because the contact patch moves relative to the bike due to steering inputs. Steer left and the contact patch moves to the right relative to the bike, resulting in a roll reaction to the right. It requires precision to keep the lean angle within a small range, but the response is instintive, if the bike leans left, steer left to generate a corrective right roll response. Velodrome racers, can do this indefinately.

    Trials type events often involve a spot where the bicycle / motorcyle has to come to a complete stop and be rotated (yawed) before continuing. Trials type bicycles and motorcyles have no significant dampening in the shocks, which allows the rider to hop the bicycle / motorcycle in order to rotate it while not moving forwards or backwards. The riders free up one leg and move it inwards / outwards to generate additional torque forces (beyond steering inputs) to maintain veritical stability.
    Last edited: May 20, 2007
  16. May 21, 2007 #15
    I think that in the end I think we does agree quite much :-)

    The reason that I see a "feedback control system" in any thing that in any way could be a "feedback control system" might be that I have worked withs such system for years including also with jet fighters (F16).

    To support your point of view as well I can tell one other story ..

    Some years ago I used to drive a rather heavy motorbike, a Suzuki GS1100G.

    That motorbike had that property that when speed accelerated and passed approx 110-120 km/hour the motorbike started to get completely stable.

    Futher on at the speed of 150-160 km/hour it was completely stable on the road. It was not neccessary to control it at all.

    I wondered a bit why it worked like that, and my thery was that som of the rason might be that is had rather heavy aluminium wheels that possibly got some gyro effect on that speed.

    I believe that the complete thery and mathematical control of a bike of any type is a rather complicated one. The geometry of the bike itself is a part of it. The feedback from the driver is an other part, etc, etc.

    I think that the full mathematical anlysis of an ordinary bycycle is so complex and difficult, so if anybody invented the bicycle today, without any earlier bikes to refer to, and without technical tests, experts would have said: No this invention can not work in real life, it is much to advanced and to unstable for that.

    I think that the bicycle is a rather good example of the general principle that new inventions have to be developed from a serie of real prototypes and from testing and trying.
  17. May 21, 2007 #16


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    Bicycles aren't that complicated, poorly designed ones were just more difficult to ride, but probably still easier than a unicycle, which in turn is easier than standing / moving on a large ball (rolling globe).

    You can see from the link below that on early bicycles, that the steering axis was vertical, so no inherent vertical stability, the rider had to manage the bicycle, but the main issue was going over the large front wheel.


    "Safety bicycle" showed up around 1885:

    http://www.ohtm.org/1887safety.html [Broken]
    Last edited by a moderator: May 2, 2017
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