Shaft Taper Angle and Tailstock Set-Over Calculation

This problem focuses on the calculation of the taper angle and the required tailstock set-over for a shaft with a tapered portion. The tailstock set-over method is a fundamental technique used in lathe operations to achieve precise tapered surfaces.

Tailstock Set-Over Method Explained

The tailstock set-over method is commonly used to machine long and gradual tapers on a workpiece mounted between centers on a lathe. It involves intentionally displacing the tailstock center horizontally from the headstock center. This offset causes the cutting tool, which moves parallel to the lathe axis, to cut along a line that is inclined relative to the workpiece axis, thereby forming a taper. The amount of set-over is crucial for achieving the desired taper dimensions.

Tapered Shaft Parameters

Let's identify the given dimensions of the shaft and its tapered section:

• Total length of the shaft, \(L = 90 \text{ mm}\)
• Length of the tapered portion, \(l = 55 \text{ mm}\)
• Larger diameter of the taper, \(D = 80 \text{ mm}\)
• Smaller diameter of the taper, \(d = 65 \text{ mm}\)

Taper Angle Calculation Details

The taper angle (\(\alpha\)) is determined by the difference in diameters over the length of the taper. We use the formula for the tangent of half the taper angle:

\[ \tan\left(\frac{\alpha}{2}\right) = \frac{\text{Difference in Radii}}{\text{Length of Taper}} = \frac{\frac{D}{2} - \frac{d}{2}}{l} = \frac{D - d}{2l} \]

Substitute the given values into the formula:

\[ \tan\left(\frac{\alpha}{2}\right) = \frac{80 \text{ mm} - 65 \text{ mm}}{2 \times 55 \text{ mm}} \]

\[ \tan\left(\frac{\alpha}{2}\right) = \frac{15 \text{ mm}}{110 \text{ mm}} \]

\[ \tan\left(\frac{\alpha}{2}\right) = \frac{3}{22} \]

\[ \tan\left(\frac{\alpha}{2}\right) \approx 0.1363636 \]

To find the half taper angle \(\frac{\alpha}{2}\), we apply the inverse tangent function:

\[ \frac{\alpha}{2} = \arctan(0.1363636) \]

\[ \frac{\alpha}{2} \approx 7.766^\circ \]

The full taper angle \(\alpha\) is simply twice the half taper angle:

\[ \alpha = 2 \times 7.766^\circ \]

\[ \alpha \approx 15.532^\circ \]

To express this in degrees and minutes, we convert the decimal part of the degree:

\[ 0.532^\circ \times 60 \text{ minutes/degree} \approx 31.92 \text{ minutes} \]

Rounding to the nearest minute, we get \(32'\). Therefore, the taper angle is approximately \(15^\circ 32'\).

Tailstock Set-Over Calculation Steps

The tailstock set-over (\(S\)) is calculated based on the total length of the workpiece and the dimensions of the tapered section. The formula for the set-over in the tailstock set-over method is:

\[ S = \frac{L}{l} \times \frac{D - d}{2} \]

Let's substitute the known values into this formula:

\[ S = \frac{90 \text{ mm}}{55 \text{ mm}} \times \frac{80 \text{ mm} - 65 \text{ mm}}{2} \]

\[ S = \frac{90}{55} \times \frac{15}{2} \]

Simplify the fractions to perform the calculation:

\[ S = \frac{18}{11} \times \frac{15}{2} \]

\[ S = \frac{9 \times 15}{11} \]

\[ S = \frac{135}{11} \]

\[ S \approx 12.2727 \text{ mm} \]

Rounding to two decimal places, the tailstock set-over is approximately \(12.27 \text{ mm}\). When comparing this value to the options provided, \(12.16 \text{ mm}\) is the closest numerical match for the set-over, while the taper angle matches exactly.

Shaft Solution Summary

Based on the calculations for the given tapered shaft:

• The taper angle is approximately \(15^\circ 32'\).
• The tailstock set-over is approximately \(12.16 \text{ mm}\) (considering the closest available option).

These values align with option 1 provided.