Force diagram of pedal

Force diagram of pedal

Assume the sensors measure the torque (the bending moment) on the pedal spindle at each of the associated points,

*x*and

_{1}*x*. Forces

_{1}*F*and

_{1}*F*are applied at the points

_{2}*x*and

_{3}*x*. The goal is to determine

_{4}*F*and

_{3}*F*from the measurements

_{4}*τ*and

_{1}*τ*.

_{2}Simply enough: two equations, two unknowns (

*F*,

_{1}*F*):

_{2}*τ*=

_{1}*F*(

_{1}*x*‒

_{3}*x*) +

_{1}*F*(

_{2}*x*‒

_{4}*x*),

_{1}*τ*=

_{2}*F*(

_{1}*x*‒

_{3}*x*) +

_{2}*F*(

_{2}*x*‒

_{4}*x*).

_{2}Note the second equation is the same as the first with the

*x*'s replaced with

_{1}*x*'s, and the

_{2}*τ*replaced with

_{1}*τ*. Multiply the top equation by (

_{2}*x*‒

_{3}*x*), and the bottom by (

_{2}*x*‒

_{3}*x*):

_{1}(

*x*‒

_{3}*x*)

_{2}*τ*=

_{1}*F*(

_{1}*x*‒

_{3}*x*) (

_{1}*x*‒

_{3}*x*) +

_{2}*F*(

_{2}*x*‒

_{4}*x*) (

_{1}*x*‒

_{3}*x*),

_{2}(

*x*‒

_{3}*x*)

_{1}*τ*=

_{2}*F*(

_{1}*x*‒

_{3}*x*) (

_{2}*x*‒

_{3}*x*) +

_{1}*F*(

_{2}*x*‒

_{4}*x*) (

_{2}*x*‒

_{3}*x*),

_{1}and substract the two and combine terms:

(

*x*‒

_{3}*x*)

_{2}*τ*‒ (

_{1}*x*‒

_{3}*x*)

_{1}*τ*=

_{2}*F*[ (

_{2}*x*‒

_{2}*x*) (

_{1}*x*‒

_{4}*x*) ],

_{3}yielding the solution

*F*= [ (

_{2}*x*‒

_{3}*x*)

_{2}*τ*‒ (

_{1}*x*‒

_{3}*x*)

_{1}*τ*] / [ (

_{2}*x*‒

_{2}*x*) (

_{1}*x*‒

_{4}*x*) ].

_{3}Now for the other solution, swap

*F*and

_{1}*F*, and swap

_{2}*x*and

_{3}*x*, then move terms around to get rid of excess negatives:

_{4}*F*= [ (

_{1}*x*‒

_{4}*x*)

_{1}*τ*‒ (

_{2}*x*‒

_{4}*x*)

_{2}*τ*] / [ (

_{1}*x*‒

_{2}*x*) (

_{1}*x*‒

_{4}*x*) ].

_{3}Of primary interest is the total force, trivially adding the previous two results, and simplifying:

*F*+

_{1}*F*= (

_{2}*τ*‒

_{1}*τ*) / (

_{2}*x*‒

_{2}*x*).

_{1}Wow. That's embarrassingly simple.

Interestingly, the only position data you need to know here is the separation of the two torque sensors. This is known, of course, assuming the electronics are inserted straight into the hole, and the hole is itself straight within the spindle. Both of these are likely good approximations within the 1.5% net error budget of the Vector, since errors in orientation only contribute second order to errors in separation along the spindle axis. You don't need to know how far the electronics are from the crank arm, nor how far the pedal body is from the sensors.

Also of interest is the torque on the pedal body:

(

*F*‒

_{2}*F*) (

_{1}*x*‒

_{4}*x*) / 2 =

_{3}[ ( [

*x*+

_{4}*x*] / 2 ‒

_{3}*x*)

_{2}*τ*‒ ( [

_{1}*x*+

_{4}*x*] / 2 ‒

_{3}*x*)

_{1}*τ*] / (

_{2}*x*‒

_{2}*x*).

_{1}Wow -- I hope I got all of that correct (I rarely do). Anyway, here you need an additional bit of information, which is the separation of the center of the pedal body from the sensor. This is a parameter which would need to be determined by a calibration step. Fortunately, accuracy on this torque number probably isn't important. It's power we really want to nail down, and that depends on propulsive force, not how hard the pedal is being twisted.

Luckily you do not need to know how far anything is from the crank arm. If you did, for example inserting a spacer under the pedal to change the Q-factor would mess up the result. Perhaps more significantly, bending of the crank arm may yield an effective pivot point beyond the point where the pedal is screwed into the crank, and the position of this virtual pivot may depend on pedal force. So since the result doesn't depend on the location of the pivot point, it may be relatively insensitive to crank flex.

So what do I get out of all of this analysis? Basically that with good bending data taken at two points along the spindle, you can get the average force on the pedal body without knowing the distance to the pedal body. That's very nice. Additionally, you can derive the torque exerted on the pedal body about its center if you know the distance from the bending sensors to the center of the pedal body. This could be calibrated for maximum accuracy, or it could be more crudely estimated given the lack of relative importance of this value.

## 7 comments:

Here is something to noodle on:

What if the sensors don't measure bending moment?

Well, I suppose this shows a difference: my background in mechanics is undergraduate physics where all sensors are viewed as ideal, whereas your background is in actually making things that work :). But it seems the curvature is the only "observable" along the two axes perpendicular to the spindle (along the longitudinal axis you can measure strain directly).

http://www.roymech.co.uk/Useful_Tables/Beams/Shear_Bending.html

This is a good article, and I agree with what you are saying, but it would be more clear if you referred to the bending load on the pedal spindle as a "moment" rather than a "torque".

Thanks! The bending moment is the cross-product of a position and a force, which my freshman physics brain calls a torque, but I guess "bending moment" would be more clearly differentiated from the twisting moment applied to the pedal body.

I'm trying to wrap my head around Jim's reference, but I think what he's saying is Bernoulli's idealized cantilever fails to adequately describe a relatively short hollow tapered cylinder like a pedal spindle, at least within the 1.5% error budget, and that off-diagonal elements of the stiffness matrix are needed to really solve the problem (not just the scalar term used in the simple bending analysis). So maybe a second order calibration is needed.

The link shows shear force and bending diagrams. Note that unlike bending moment, shear force is not a function of displacement along the beam. I don't know how the measurements are oriented in the Metrigear design, but it is possible to arrange the sensors such that the electrical output is a function of shear force.

Interesting! Curiously, I initially thought the sensors would pick up total force, not a bending moment. Jim Pampodopolous, an MIT engineer who collaborated on Bicycling Science Third Edition, said it would be a bending moment and not a force measured by the sensors. That's consistent with two parallel strain gauges, oriented longitudinally, one at the center axis and one away from the center, measuring the difference in strain.

But if they were instead configured to measure shear strain, which is a change in vertical position along the axis of the pedal, then you're right that the driving force would be force, not a force times distance.

So I guess the question is then whether it's relatively easier and more reliable to measure bending or to measure shear distortion. It's a hollow tapered cyclinder, so I think one would need to run some real numbers (preferably FEA) to see if there's a strong enough shear signal without second-order contamination from bending stress, and vice-versa.

My money's would be on the bending moment. I suppose I could

ask, but it's more fun to guess.It's funny I'm so interested in this topic. It sort of gives me an excuse to exercise some basic physics on a topic (bicycles) which I really enjoy.

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