Calculating Machining Time for a Drilling Operation

This problem asks us to determine the time it takes to drill a through-hole in a metal plate, considering various factors like plate thickness, drill geometry, initial position, overtravel, rotational speed, and feed rate.

To calculate the machining time for drilling, we need to find the total distance the drill must travel along its axis (the total depth of cut) and the rate at which it travels (the feed rate).

Understanding the Total Depth of Cut

The total distance the drill tip travels includes several components:

• The thickness of the plate being drilled.
• The initial distance between the drill tip and the plate surface at the start of cutting.
• The recommended overtravel distance beyond the plate's bottom surface to ensure a clean through-hole.
• The length of the conical tip of the drill bit itself, as the entire cone must pass through the material for the full diameter hole to be formed.

Calculating the Drill Tip Length

The length of the drill tip ($L_{tip}$) can be calculated using the drill diameter (D) and the drill's lip angle ($2\alpha$). The formula relates the tip length to the radius (D/2) and the cotangent of the semi-lip angle ($\alpha$).

Given:

• Drill Diameter, \(D = 20 \text{ mm}\)
• Lip Angle, \(2\alpha = 120^{\circ}\)

First, find the semi-lip angle:

\(\alpha = \frac{120^{\circ}}{2} = 60^{\circ}\)

Now, calculate the drill tip length:

\(L_{tip} = \frac{D}{2} \times \cot(\alpha)\)

\(L_{tip} = \frac{20 \text{ mm}}{2} \times \cot(60^{\circ})\)

\(L_{tip} = 10 \text{ mm} \times \frac{1}{\sqrt{3}}\)

\(L_{tip} \approx 10 \text{ mm} \times 0.57735\)

\(L_{tip} \approx 5.7735 \text{ mm}\)

Calculating the Total Travel Distance

Now we sum up all the components of the total depth of cut:

• Plate Thickness = 30 mm
• Initial Gap = 3 mm
• Overtravel = 2 mm
• Drill Tip Length = 5.7735 mm

Total Depth of Cut, \(L_{total} = \text{Plate Thickness} + \text{Initial Gap} + \text{Overtravel} + L_{tip}\)

\(L_{total} = 30 \text{ mm} + 3 \text{ mm} + 2 \text{ mm} + 5.7735 \text{ mm}\)

\(L_{total} = 40.7735 \text{ mm}\)

Calculating the Feed Rate

The feed rate ($V_f$) is the speed at which the drill moves axially into the material. It is calculated by multiplying the feed per revolution by the rotational speed.

Given:

• Rotational Speed, \(N = 500 \text{ rev/min}\)
• Feed per Revolution, \(f = 0.1 \text{ mm/rev}\)

Feed Rate, \(V_f = f \times N\)

\(V_f = 0.1 \text{ mm/rev} \times 500 \text{ rev/min}\)

\(V_f = 50 \text{ mm/min}\)

Calculating the Machining Time

The machining time ($T_m$) is the total distance traveled divided by the feed rate.

\(T_m = \frac{L_{total}}{V_f}\)

\(T_m = \frac{40.7735 \text{ mm}}{50 \text{ mm/min}}\)

\(T_m \approx 0.81547 \text{ minutes}\)

The question asks for the time in seconds. To convert minutes to seconds, multiply by 60.

\(T_m (\text{in seconds}) = T_m (\text{in minutes}) \times 60\)

\(T_m (\text{in seconds}) \approx 0.81547 \times 60\)

\(T_m (\text{in seconds}) \approx 48.9282 \text{ seconds}\)

Rounding to two decimal places, the machining time is approximately 48.93 seconds, which is very close to 48.92 seconds.

Summary of Calculation

The calculated machining time is approximately 48.93 seconds.