A study on the curve traceed by the knife edge of the rotary lathe

Abstract

Here are studied the problems about the locus of the knife edge of rotary lathe.
The centre of the spindle of rotary lathe is taken as the origin of rectangular coordi. nate (x, y) and polar coordinate (r, θ). The knife edge is to progress on the line (y=na) parallel to x axis. The angular velocity of rotation of spindle is w, the traveling velocity of knife edge is ν and ν/w=ma.
Then the results are as follows:
1) The general equation of the locus in polar form (r, θ) is
θ=Ro√r²-n²a²+sin^-1 na/r and the equation is also written as follow.
x=(R_o-maφ)cosφ-na sinφ y=(R_o-maφ) sinφ+na cosφ
φ is parameter and shown in Fig. 3.
2) The above equation is approximately written
θ=R_o-r/ma+na/r(1+n/2m)
or
r=(R_o-maθ)+n(2m+n)/2 a₂/R_o-maθ
when na/r is small
In actual case na/r is so small that the approximate equation is sufficient.
3) The cutting velccity u is not exactly proportional to radius r and approximately shown as follow.
u=w&sdotr(1-1/2m₂₂/r₂)
4) To hold the cutting angle constant, it is sufficient when the relation m=-n
is satisfied. 5) Then the thickness of produced veneer is 2π ma.