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Spehro Pefhany
 
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Default why greek "pi" exists in cutting speed calculation?

On Thu, 27 Nov 2003 16:05:44 GMT, the renowned Robert Nichols
wrote:

In article ,
natarajan ayyavu wrote:
:
:We have installed a Gear Shaping machine (model: similar to Lorenz
:LS400). We would like to operate at 20 mpm cutting speed while cutting
:internal splines. The intruction manual says:
:cutting speed = pi * stroke length * strokes per minute / (1000)
:
note, here stroke length = spline length + approach + over travel)
:
:I do not understand why greek pi (value = 3.1428) presents in this
:formula
:when the cutter does not follow a circular motion. Here the cutter
:follows a reciprocatory movement and shifted for relief by about 0.5mm
:away from cutting surface. To my understand, I think the formula must
:have been
:cutting speed = 2 * stroke length * strokes per minute / (1000)

For suitable units (stroke length in mm, cutting speed in meters/min),
the formula in the manual gives the _peak_ cutting speed for a cutter
whose velocity varies sinusoidally. The formula you suggest would be
correct for a cutter whose velocity was constant during the stroke and
with an instantaneous reversal at each end.


Or if you'd prefer calculus: If the cutter moves 's' strokes per
minute in a simple sinusoidal motion, then the cutter motion can be
described as:

x = x0 + 0.5 * stroke * sin(2 * pi * s * t) where t is in minutes

(sin goes +/-1 so, hence the 0.5 factor)

The speed is:
dx/dt = 0.5 * stroke * 2 * pi * s) * cos (2 * pi * 1/2 * t).

(since we know d/dx sin(a*x) = a * cos(ax) )


The maximum speed is when cos(...) = 1, or stroke * pi * s.

I think shapers are made asymmetrical so they move faster on the back
stroke.

Best regards,
Spehro Pefhany
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