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Hi, I'm making an electric longboard and trying to write an app for my phone to function as the remote. I've got a bunch of fancy StarTrekesque indicators on it, one of which is the pitch and roll of the deck. All the indicators have "danger zones" and turn red when they hit them, and for this the danger zone will be dynamic as your speed changes so I'm trying to find the maximum angle (roll) in degrees that the board can be tilted to turn before the wheels slip on the road based on speed. I thought I had it down, but must have done something wrong because the results don't quite make sense so here's my process (sorry for the picture being so scatterbrained haha)
So, the board turns in a circle whose radius r changes as some function of the roll, θ_{r}. More directly, the radius depends on the angle of the wheels with the midline of the deck, θ_{w} (see lower right section of picture for clarification). One can calculate this correlation based on the length between the trucks, l_{w} by using SOH (sin = opposite/adjacent) to get
sinθ_{w} = l_{w} / 2r ⇒ r = l_{w}/2sinθ_{w}
To get θ_{w} as a function of θ_{r}, I figured out it's a sinusoidal relationship to the baseplate angle θ_{b}, specifically
θ_{w} = θ_{b}sinθ_{r}
Therefore, the radius of the curve is
r = l_{w} / 2sin(θ_{b}sinθ_{r}).
Now, to get the centrifugal force, the formula is mv^{2} / r, so we can simply plug in what we got for r to get
F_{cen} = mv^{2} / (l_{w} / 2sin(θ_{b}sinθ_{r}))
which can simplify to
F_{cen} = 2mv^{2}sin(θ_{b}sinθ_{r}) / l_{w}
The force that has to counteract that is the force of friction between the wheels and the road (with axis of movement parallel to wheel's axis of rotation), the maximum of which is F_{fric} = μmg. Therefore to figure out the maximum θ_{r} (the end goal of all this), we need to
solve F_{fric} = F_{cen} for θ_{r}
Oh boy, here we go.
μmg = 2mv^{2}sin(θ_{b}sinθ_{r}) / l_{w}
First of all, the m's cancel out, meaning it's apparently mass independent (makes sense; a heavier person will produce more centrifugal force but also push harder on the ground increasing friction), so now let's just algebra the crap outta this:
μgl_{w} = 2v^{2}sin(θ_{b}sinθ_{r})
μgl_{w} / 2v^{2} = sin(θ_{b}sinθ_{r})
sin^{1}(μgl_{w} / 2v^{2}) = θ_{b}sinθ_{r}
sin^{1}(μgl_{w} / 2v^{2}) / θ_{b} = sinθ_{r}
sin^{1}(sin^{1}(μgl_{w} / 2v^{2}) / θ_{b}) = θ_{r}
Therefore,
θ_{rMax} = sin^{1}(sin^{1}(μgl_{w} / 2v^{2}) / θ_{b})
This should be the end of the problem, but when I applied it to the program and plugged in appropriate values for all the variables, it wasn't behaving properly (for example, when the coefficient of friction is set to 1 the maximum roll should be 90 regardless of speed but it still changes as the speed changes). Any ideas on what I might've done wrong? Do I need to bring calculus into this somewhere as an optimization problem? I'm at a loss...
So, the board turns in a circle whose radius r changes as some function of the roll, θ_{r}. More directly, the radius depends on the angle of the wheels with the midline of the deck, θ_{w} (see lower right section of picture for clarification). One can calculate this correlation based on the length between the trucks, l_{w} by using SOH (sin = opposite/adjacent) to get
sinθ_{w} = l_{w} / 2r ⇒ r = l_{w}/2sinθ_{w}
To get θ_{w} as a function of θ_{r}, I figured out it's a sinusoidal relationship to the baseplate angle θ_{b}, specifically
θ_{w} = θ_{b}sinθ_{r}
Therefore, the radius of the curve is
r = l_{w} / 2sin(θ_{b}sinθ_{r}).
Now, to get the centrifugal force, the formula is mv^{2} / r, so we can simply plug in what we got for r to get
F_{cen} = mv^{2} / (l_{w} / 2sin(θ_{b}sinθ_{r}))
which can simplify to
F_{cen} = 2mv^{2}sin(θ_{b}sinθ_{r}) / l_{w}
The force that has to counteract that is the force of friction between the wheels and the road (with axis of movement parallel to wheel's axis of rotation), the maximum of which is F_{fric} = μmg. Therefore to figure out the maximum θ_{r} (the end goal of all this), we need to
solve F_{fric} = F_{cen} for θ_{r}
Oh boy, here we go.
μmg = 2mv^{2}sin(θ_{b}sinθ_{r}) / l_{w}
First of all, the m's cancel out, meaning it's apparently mass independent (makes sense; a heavier person will produce more centrifugal force but also push harder on the ground increasing friction), so now let's just algebra the crap outta this:
μgl_{w} = 2v^{2}sin(θ_{b}sinθ_{r})
μgl_{w} / 2v^{2} = sin(θ_{b}sinθ_{r})
sin^{1}(μgl_{w} / 2v^{2}) = θ_{b}sinθ_{r}
sin^{1}(μgl_{w} / 2v^{2}) / θ_{b} = sinθ_{r}
sin^{1}(sin^{1}(μgl_{w} / 2v^{2}) / θ_{b}) = θ_{r}
Therefore,
θ_{rMax} = sin^{1}(sin^{1}(μgl_{w} / 2v^{2}) / θ_{b})
This should be the end of the problem, but when I applied it to the program and plugged in appropriate values for all the variables, it wasn't behaving properly (for example, when the coefficient of friction is set to 1 the maximum roll should be 90 regardless of speed but it still changes as the speed changes). Any ideas on what I might've done wrong? Do I need to bring calculus into this somewhere as an optimization problem? I'm at a loss...
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