Tailstock set-over for straight-through taper turning A cylindrical workpiece is to be turned to a uniform taper along its entire center-to-center length. If the large and small diameters are D and d respectively, what tailstock set-over is required (assume taper over full length between centers)?

Difficulty: Medium

Correct Answer: (D - d) / 2

Explanation:


Introduction / Context:
Taper turning between centers can be achieved by offsetting the tailstock. This method tilts the work axis relative to the lathe axis, producing a linear change in radius along the workpiece length. Knowing the correct set-over is crucial to obtain the target diameters.



Given Data / Assumptions:

  • Large diameter at one end: D.
  • Small diameter at the other end: d.
  • Taper is uniform over the full center distance (effective turned length equals the distance between centers).
  • Small offset angles (typical for taper turning) are assumed.


Concept / Approach:
For a full-length uniform taper produced by tailstock set-over, the radial change from one end to the other equals the change in radius, which is (D - d)/2. The tailstock set-over equals this radius difference because the work axis must be shifted laterally by exactly half the total change in diameter to align the conical surface with the tool path.



Step-by-Step Solution:

Define diameter change across the length: ΔD = D - d.Convert to radius change: ΔR = ΔD / 2.For taper over full length between centers, required lateral set-over equals the change in radius: set-over = (D - d) / 2.


Verification / Alternative check:
Graphic construction of the axis shift shows that when the work axis is offset by ΔR, the tool tracing parallel to the lathe axis generates the required cone from radius R1 to R2 along the whole length.



Why Other Options Are Wrong:
(D - d): doubles the required set-over; would overshoot the taper.(D + d)/2: unrelated to taper geometry.(D^2 - d^2)/(2L): relates to slope if L is included, but not the simple full-length set-over value.tan^-1[(D - d)/L]: this is an angle; the question asks for linear set-over.



Common Pitfalls:
Confusing set-over distance with taper angle; neglecting the fact that set-over equals radius change for full-length tapers.



Final Answer:

(D - d) / 2

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