Centreless grinding kinematics: actual axial feed expression In centerless grinding, with regulating wheel diameter d (in meters), speed n (rev/min), and inclination angle α of the regulating wheel axis, what is the actual feed rate imparted to the work?

Introduction / Context:In centerless grinding, the regulating wheel controls both the rotational speed and the axial feed of the workpiece. The axial feed results from inclining the regulating wheel axis by a small angle α relative to the work axis.

Given Data / Assumptions:

• d = diameter of the regulating wheel.
• n = rotational speed of the regulating wheel in rev/min.
• α = inclination angle of the regulating wheel axis.
• No slip assumption at the work–regulating wheel interface.

Concept / Approach:The peripheral (tangential) speed of the regulating wheel is v = π d n. The axial feed component is the projection of this velocity along the work axis, determined by the inclination angle α.

Step-by-Step Solution:Compute peripheral speed: v = π * d * n.Resolve along work axis: feed = v * sin α.Therefore, actual axial feed = π d n sin α.

Verification / Alternative check:Dimensional check: d (m) * n (rev/min) gives m/min when multiplied by π; sin α is dimensionless.

Why Other Options Are Wrong:

• π d and π d n omit the angular component; they are circumferential, not axial.
• cos α projects onto the transverse direction, not along the axis.
• 2 π d n tan α is not the standard kinematic decomposition for this geometry.

Common Pitfalls:Using cos α instead of sin α; mixing wheel diameter units; forgetting to express d in meters for consistent units.

Final Answer:π d n sin α