Which of the following expressions can determine the longitudinal feed rate in "Through Feed Centreless Grinding"?
(If the diameter of the regulating wheel = D, r.p.m. of the regulating wheel = N, angle of inclination of the regulating wheel = θ)

Explanation:

Through Feed Centreless Grinding

Centreless grinding is a machining process that uses abrasive cutting to remove material from a workpiece. It is characterized by the fact that the workpiece is not held between centers or in a chuck, but rather between two wheels: a grinding wheel and a regulating wheel. The regulating wheel controls the rotational speed and longitudinal feed of the workpiece, while the grinding wheel performs the cutting action.

Determining the Longitudinal Feed Rate:

The longitudinal feed rate in through feed centreless grinding is determined by several factors, including the diameter of the regulating wheel (D), the rotational speed of the regulating wheel (N), and the angle of inclination of the regulating wheel (θ). The correct expression for calculating the longitudinal feed rate is given by:

Longitudinal feed rate = π × D × N × Sin (θ)

Here’s a detailed explanation of why this formula is correct:

• The diameter of the regulating wheel (D) affects the linear distance the workpiece travels per revolution of the wheel. A larger diameter means a greater distance traveled per revolution.
• The rotational speed of the regulating wheel (N) determines how many revolutions the wheel makes per unit time. This directly affects the speed at which the workpiece is fed through the grinding zone.
• The angle of inclination of the regulating wheel (θ) impacts the direction and magnitude of the feed rate. The sine of the angle θ gives the proportion of the wheel's circumference that contributes to the longitudinal feed. This is because the component of the wheel's motion that contributes to the feed rate is the one parallel to the axis of the workpiece.

Combining these factors, we get the formula:

Longitudinal feed rate = π × D × N × Sin (θ)

The π term (pi) comes from the relationship between the diameter and the circumference of a circle, which is essential in translating rotational motion into linear motion.

Analysis of Other Options:

Let's analyze why the other options are incorrect:

• Option 2: Longitudinal feed rate = [π × D × N] / Sin (θ)
• This formula incorrectly places the sine function in the denominator. The sine of the angle θ should be in the numerator because it represents the component of the wheel's motion that contributes to the longitudinal feed rate. Dividing by Sin (θ) would imply that the feed rate decreases as the angle increases, which is incorrect.
• Option 3: Longitudinal feed rate = π × D × N × Cos (θ)
• This formula uses the cosine of the angle θ instead of the sine. The cosine function would give the component of the wheel's motion perpendicular to the axis of the workpiece, which does not contribute to the longitudinal feed rate. Hence, this formula is incorrect.
• Option 4: Longitudinal feed rate = [π × D × N] / Cos (θ)
• This formula incorrectly uses the cosine function and places it in the denominator. As explained for Option 3, the cosine function does not represent the longitudinal component of the wheel's motion. Dividing by Cos (θ) further complicates the formula without any justification.

Conclusion:

The correct expression for determining the longitudinal feed rate in through feed centreless grinding is:

Longitudinal feed rate = π × D × N × Sin (θ)

This formula appropriately considers the diameter and rotational speed of the regulating wheel, as well as the angle of inclination, to accurately calculate the linear feed rate of the workpiece through the grinding zone.