Need help modelling bicycle tire track paths... | Mathematics | Page 1

wicked wyvernBOT Feb 22, 2024, 7:36 PM

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warm prism Feb 22, 2024, 7:37 PM

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It is for a maths IA incase you were wondering.

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lethal lagoon Feb 22, 2024, 7:54 PM

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You mean the paths of two wheels on a bike set X distance a part, as they turn etc?

warm prism Feb 22, 2024, 7:55 PM

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Here is some context -

lethal lagoon Feb 22, 2024, 7:55 PM

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Can you assume not smearing etc and single point of contact with the ground? (no tire footprint or pressure in the tube etc)

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What's the frame of reference for X and Y?

warm prism Feb 22, 2024, 7:56 PM

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I assume both points of contact for the ground at all times

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The tire tracks will be modelled as functions

lethal lagoon Feb 22, 2024, 7:56 PM

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I'm I looking down on the bike, and x,y is on a coordinate plane from an over head view?

warm prism Feb 22, 2024, 7:57 PM

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Basically yeah

lethal lagoon Feb 22, 2024, 7:57 PM

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oh lol.. exactly

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ok so riding in positive x direction

warm prism Feb 22, 2024, 7:57 PM

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Yeah

lethal lagoon Feb 22, 2024, 7:58 PM

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So they calling "a" the distance between front and back wheel?

warm prism Feb 22, 2024, 7:58 PM

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I get the relationship is that the tangent of the back wheel will always point to the front wheel at any given time

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yeah a is the wheel base

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Just a constant

lethal lagoon Feb 22, 2024, 7:59 PM

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Ty big help

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The center of rotation is important... as it turns about this point... assumed in middle of frame

warm prism Feb 22, 2024, 8:00 PM

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I used a tractrix function to demonstrate this relationship but then am lost on how they arrived at this function

lethal lagoon Feb 22, 2024, 8:00 PM

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Rotates around this point I mean

warm prism Feb 22, 2024, 8:00 PM

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They glossed over it in the paper i found

lethal lagoon Feb 22, 2024, 8:01 PM

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This is for a class?

warm prism Feb 22, 2024, 8:01 PM

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I think the principle is that they can generate a unique function at any time for one of the wheels and determine thefunction of the other one

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No its a research project

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Just stuck on this particular bit

lethal lagoon Feb 22, 2024, 8:04 PM

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Reading wiki on tractrix functions atm...

warm prism Feb 22, 2024, 8:04 PM

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I see

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I have some time dw

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Im gonna link the paper i found

lethal lagoon Feb 22, 2024, 8:05 PM

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https://en.wikipedia.org/wiki/Tractrix

Tractrix

In geometry, a tractrix (from Latin trahere 'to pull, drag'; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the tractor) that moves at a right angle to the initial line between the object and the puller at an infinite...

warm prism Feb 22, 2024, 8:05 PM

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https://tlakoba.w3.uvm.edu/AppliedUGMath/final_talks_previous_years/BicycleTracks.pdf

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Cool diagrams I made

lethal lagoon Feb 22, 2024, 8:09 PM

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Yeah makes for cool plots

warm prism Feb 22, 2024, 8:10 PM

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So any ideas?

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So far I've differentiated the tractirx -

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with respect to y

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lethal lagoon Feb 22, 2024, 8:12 PM

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Oh I see... that "aln" was confusing me, but it's "a * Log" lol

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Reading this now.... https://www.researchgate.net/publication/314620122_Kinematic_Analysis_of_the_Planar_Motion_of_Vehicles_When_Travelling_Along_Tractrix_Curves

warm prism Feb 22, 2024, 8:14 PM

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I see

lethal lagoon Feb 22, 2024, 8:14 PM

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Think that section 2 explains it better?

warm prism Feb 22, 2024, 8:14 PM

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Trying to get a copy 1 sec

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Nvm what i just said

lethal lagoon Feb 22, 2024, 8:15 PM

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Oh good, they talk about centrode and rotation about it... glad my gut instincts were good for something lol

warm prism Feb 22, 2024, 8:16 PM

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Yeah I see, what do you think about their working in this one, which was my original source - https://tlakoba.w3.uvm.edu/AppliedUGMath/final_talks_previous_years/BicycleTracks.pdf

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Just the first modelling equation section

lethal lagoon Feb 22, 2024, 8:18 PM

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Oh interesting, the equations have something to do with the tangent from this rotation point to the evolve blah blah...

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Nutz! Glad you don't have to prove this...

warm prism Feb 22, 2024, 8:20 PM

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Yeah reading through it too

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Yeah lol

lethal lagoon Feb 22, 2024, 8:22 PM

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Well they certainly walk through it all... somewhere in there lies your answer lol... that bottom section is circular tractrix which I don't think you need to worry about if the bike is displacing in a straight path

warm prism Feb 22, 2024, 8:23 PM

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I see

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Would you have any ideas though on how they arrive at this in this paper https://tlakoba.w3.uvm.edu/AppliedUGMath/final_talks_previous_years/BicycleTracks.pdf. From a tractrix?

lethal lagoon Feb 22, 2024, 8:25 PM

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Which page was that in your paper?

warm prism Feb 22, 2024, 8:26 PM

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Page 3

lethal lagoon Feb 22, 2024, 8:27 PM

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ty found it

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He cited the source from [9] Dan Velleman, Joseph D. E. Konhauser, and Stan Wagon, Which way did the bicycle go?, MAA, 1996.

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So in that reference explains how they got to that equation?

warm prism Feb 22, 2024, 8:30 PM

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Ill see

lethal lagoon Feb 22, 2024, 8:31 PM

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Happy hunting lol... but sorry I can't walk you through it myself, too complicated

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Very interesting subject though! I think those two equations you can use to plot x,y for the back tire being pulled by the front

warm prism Feb 22, 2024, 8:32 PM

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Abit too complicated for me too

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Thanks for the helps anyways

lethal lagoon Feb 22, 2024, 8:32 PM

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np, cool subject!

warm prism Feb 22, 2024, 8:32 PM

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Cheers

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#Need help modelling bicycle tire track paths...