- #1
twowheelsbg
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Homework Statement
Recently I met a Mathcad calculation plotting bicycle path traveled.
Initial positions of tire tracks are known, their increments were calculated with
bike speed 'v', wheelbase 'p', and steering angle 'a' so via ODE system solving
front and rear tracks coordinates were derived as time functions.
All ok, but I cannot comprehend how the derivatives are found.
Here is a simple sketch explaining letters in use:
Reference frame center is R0, coinciding with rear wheel contact point initially.
Bike is oriented in x-axis, front wheel contact at F0 initially at distance 'p' from R0,
steering is set right at angle 'a' assumed constant for simplicity.
I believe both wheels follow circular path with common center C
(maybe this is my problem )
R1 and F1 are the amended positions of R0 and F0 respectively after elementary time interval dt. Below also with 'b' is indicated angle between amended bike frame R1F1 and x-axis, obviously this angle increases gradually.
Homework Equations
as per the author of the Mathcad document:
dxr/dt={2.v.cosa/(1+cosa)}.cosb
dyr/dt={2.v.cosa/(1+cosa)}.sinb
dxf/dt={2.v/(1+cosa)}.cos(a+b)
dyf/dt={2.v/(1+cosa)}.sin(a+b)
The Attempt at a Solution
Bold members are reasonable - if tire track point increment is found,
it can be amended with the bold members to find their projections over the axles x and y,
rear having angle 'b' as bike frame is displaced from x-axis so, front angle is (a+b) as front wheel is misaligned from x-axis so.
Rear track radius is CR0=p/tga,
front track radius is CF0=p/sina , as they differ with cosa that means to me -
if i find one of the increments front F0F1 or rear R0R1, amending accordingly with cosa would give me the other. So I try to find the rear one as it seems easier to me, author idea about it as seen from above to be {2.v.cosa/(1+cosa)}.
As rear contact point travels via circle with tangent speed v, angular speed comes v/CR0,
elementary sector covered comes (v/CR0).dt ,
and elementary travel of rear point comes R0R1 = 2.CR0.sin{(v/CR0).dt/2} ... which simplified to first order ( sinx to x ) comes to v.dt ...
close to v.dt.{2.cosa/(1+cosa)} but not exactly
So, I am baffled ... pls help