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see diagram

a circular disc of radius r has an initial angular velocity [tex]\omega[/tex] and rolls down a quarter circle ramp of radius R. The initial angular velocity will be very high (e.g. 10,000 RPM) causing the disc to slip. The normalised longitudinal force (i.e. longitudinal force / maximum longitudinal force) is at a maximum value of 1 at a slip percentage of 20%. The slip profile is shown in the above graph.

I am trying to derive an expression that will let me calculate the horizontal speed of the disc at a distance L (m) after the end of the quarter circle ramp, based on the friction co-efficient, mass of disc, radius of disc, radius of ramp and initial angular velocity of disc.

Relevant equations

slip ratio is defined as SR = (omega*r/V)-1 where V is the longitudinal velocity of the wheel. based on this definition SR = 0 under pure rolling.

The attempt at a solution

I am struggling massively with this problem. So far i have managed to calculate the gain in velocity from the ramp itself, but this is based on the assumption of no rolling of the disc (i.e. it effectively assumes the disc is a point mass with zero rotation). My solution for this is based on considering the disc at angle [tex]\theta[/tex] (theta) from its starting point (i.e. draw horizontal line from mass starting position to ramp curve centre and then a line between point mass and ramp curve centre, with [tex]\theta[/tex] being the angle between the two):

a = F/m

friction force Fr = [tex]\mu[/tex]N where N = mg.sin[tex]\theta[/tex]

resolving forces gives F = mg.cos[tex]\theta[/tex] - [tex]\mu[/tex]mg.sin[tex]\theta[/tex]

a = g(cos[tex]\theta[/tex]-[tex]\mu[/tex]sin[tex]\theta[/tex])

for a small change in [tex]\theta[/tex], d[tex]\theta[/tex], tangential displacement dx = Rd[tex]\theta[/tex]

a = v.dv/dx = g(cos[tex]\theta[/tex]-[tex]\mu[/tex]sin[tex]\theta[/tex])Rd[tex]\theta[/tex]

integrating this with respect to [tex]\theta[/tex] and simplifying i end up with

v = [tex]\sqrt{2Rg(1-\mu)}[/tex]

But as i say this is a massive simplification because it does not consider the rolling motion of the disc or the tractive force generated by the initial angular velocity of the disc.

The thing that is really throwing me is the slip of the wheel from the initial angular velocity. The initial slip will be very high, but will reduce over time, while at the same time the horizontal disc velocity increases due to the tractive force. I am assuming i may need to estimate the relationship between the velocity and slip and then perhaps integrate over time rather than angular position, but quite honestly im not massively hot on calculus and feel extremely lost!

The more i look at this the more complex a problem it seems to become. This is my first post and i dont expect anyone to solve this outright for me, but if someone can spot the method/approach and help guide me along the right path i would appreciate it immensely.

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# Disc on a quarter circle ramp with slip

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