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How to calculate bicycle acceleration

  1. Jun 7, 2011 #1
    I am designing a bicycle (theoretically), and i have my max torque on the crank 245Nm

    (Ignor this unless you think iv made a mathematical mistake)

    (1400N*.175m), which provides a torque on the drive cog, (245Nm/0.085m)= 2882N which is the tension in the chain, multiplied by the raduis of the back cog 0.035= 100Nm of torque at the back cog, dividing this by the radius of my back wheel i get 305N

    Is this 305 N my linear force forward? and divided by the total mass 120kg my acceleration 2.54 m/s^2
    I know I have no friction forces ie rolling resistance or air resistance friction at bearing etc.
    I thought I understood moments of inertia but i didnt know how to include the mass of the person plus bicycle in them, any help would be much appreciated
  2. jcsd
  3. Jun 7, 2011 #2
    The relationship between translational acceleration and rotational acceleration comes from the relation between translational speed and the corresponding rotary velocity (V (m/s) = r (m) * rotational speed (rev/s) ). This relationship gives a link between simultaneous equations for the translational and rotary accelerations (F=ma & T=I* alpha).

    In vehicle engineering the common practice is to relate the rotational acceleration back to the translational acceleration as an added mass to be accelerated, rather than keep using the simultaneous equations.

    Do remember that components that go through gear units (e.g. the chain, drive cog and pedals) are spun at a higher speed than the ground wheels, their initial mass is increased by the square of the gear ratio.
  4. Jun 7, 2011 #3
    so what equation do I use if I want to relate the mass of the rider into the rotational acceleration of the back wheel?

    basically the moment of inertia for my back wheel is 0.1965KgM^2 and my torque is 100Nm
    but can I just add the moment of inertia for the crank and the front wheel, assume the 2 cogs are negligable, but I still have to accelerate the total mass and how does that fit in?
    Last edited: Jun 7, 2011
  5. Jun 7, 2011 #4
    I would not relate the mass of the rider, bike frame, etc. to the rotation mass of the back wheel. I would relate the rotary mass (total) to the translational mass as most analyses are likely to be dealing with translational speeds and accelerations.

    To relate one type of mass to the other it is done using the set of simultaneous equations noted above. Typically the rotational mass is less likely to change and once a solution is obtained, the ratio of the effect on acceleration of rotational inertia versus the translational mass of the bicycle (without rider) can then be used as an increase in the bike's translational mass (e.g. rotating inertia increases the bike's mass by 2%). Rider mass would then be added separately to the total mass for analysis.

    If you try to relate everything to the back wheel, you will also need to relate the front wheel's translational and rotary mass to the back wheel.
  6. Jun 7, 2011 #5


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    Johnk690: Your answer in post 1 is close, but slightly incorrect, because you omitted the rotational inertia of your two wheels. Neglecting bearing friction, rolling resistance, and wind drag, the bicycle forward acceleration is

    a = T/(r*m1 + 2*I/r) = 100/(0.328*120 + 2*0.1965/0.328) = 2.47 m/s^2,

    where T = drive torque applied to rear wheel, r = wheel radius, m1 = total mass of bicycle plus human, and I = mass moment of inertia of each wheel (kg*m^2).
  7. Jun 8, 2011 #6
    Great info... Thanks :)
  8. Jun 8, 2011 #7


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    You might also want to keep in mind that the torque output from the human will vary as a function of crank angle. This is because the rider will only be able to apply maximum force straight down on to the pedals.

    When the cranks are perpendicular to the ground, the torque will be at a minimum (probably close to zero) and when they are parallel to the ground your torque will be the calculated maximum.
    Last edited: Jun 8, 2011
  9. Jun 8, 2011 #8
    Thanks nvn, that actually makes sense to me!!!

    I also appreciate the DickLs comments but they weren't what I was looking for.

    As for mech eng, I know I got the 1400N experimentally mainly for calculating the stress in the crank but had been using it to get the acceleration.

    I think figures for someone of 105 kg capable of cycling 30km/h would have an average pedal force of 61N (and thats including all frictions) but this would actually be each foot pedalling 156N through 70degrees, at a cadence of 100rev/min

    this website seems helpful for experimental data

  10. Jun 8, 2011 #9


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    You might consider doing your analysis from average power output rather than torque output. Given that professional cyclists put out something like 300-400W, you can calculate lot of useful stuff.

    For example, if you know their cadence (for example 100 rpm, 10.5 rad/s), and you know the average sustained power output (let's assume 350 watts), you can calculate the average sustained torque output (I calculate 33.4 N*m).

    Additionally if you know power output as a function of time you can do some other things. Say for example a person can "burst" at 500W for 10 seconds, and then slow to 300W output after 10 seconds), you can calculate things like acceleration as a function of time using conservation of energy.

    Energy analysis is important also because given a constant power output acceleration will decrease as a function of speed. Kinetic energy is based on the velocity squared, so for example it takes three times the energy to accelerate from 5 to 10 m/s as it does to accelerate from 0 to 5 m/s.
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