Understanding Material Removal Rate (MRR) in Turning

This question asks us to calculate the Material Removal Rate (MRR) for a copper bar being turned on a lathe. We are given the bar's diameter, the feed rate, the depth of cut, and the spindle speed.

Parameters Provided

Let's list the given parameters for clarity:

Calculating Material Removal Rate (MRR)

The Material Removal Rate (MRR) represents the volume of material removed per unit time during a machining operation. For turning operations, MRR can be calculated using the following formula:

MRR (in mm³/min) = Feed Rate ($f$) × Depth of Cut ($d$) × Cutting Speed ($v$)

Where:

• $f$ is the feed rate in mm/rev.
• $d$ is the depth of cut in mm.
• $v$ is the cutting speed in mm/min.

First, we need to calculate the cutting speed ($v$). The cutting speed is the peripheral speed of the workpiece surface relative to the cutting tool.

Step 1: Calculate Cutting Speed ($v$)

The formula for cutting speed is:

$v = \pi \times D \times N$

Substituting the given values:

$v = \pi \times 200 \text{ mm} \times 160 \text{ rpm}$

$v = 32000\pi \text{ mm/min}$

Step 2: Calculate MRR in mm³/min

Now, we can use the MRR formula. A commonly used formula that accounts for the given units is:

MRR (mm³/min) = $f \times d \times \pi \times D \times N$

This formula effectively combines the chip area concept with the rotational speed.

Substituting the values:

MRR (mm³/min) = $(0.25 \text{ mm/rev}) \times (4 \text{ mm}) \times \pi \times (200 \text{ mm}) \times (160 \text{ rpm})$

MRR (mm³/min) = $(1 \text{ mm}^2/\text{rev}) \times (\pi \times 32000 \text{ mm/min})$

MRR (mm³/min) = $32000\pi \text{ mm³/min}$

Step 3: Convert MRR to mm³/s

The question asks for the MRR in mm³/s. To convert from mm³/min to mm³/s, we divide by 60 (since there are 60 seconds in a minute).

MRR (mm³/s) = $\frac{\text{MRR (mm³/min)}}{60}$

MRR (mm³/s) = $\frac{32000\pi \text{ mm³/min}}{60 \text{ s/min}}$

MRR (mm³/s) = $\frac{3200\pi}{6} \text{ mm³/s}$

MRR (mm³/s) = $\frac{1600\pi}{3} \text{ mm³/s}$

Step 4: Numerical Calculation

Now, let's calculate the numerical value using $\pi \approx 3.14159$:

MRR (mm³/s) $\approx \frac{1600 \times 3.14159}{3}$

MRR (mm³/s) $\approx \frac{5026.544}{3}$

MRR (mm³/s) $\approx 1675.51 \text{ mm³/s}$

Therefore, the Material Removal Rate is approximately 1675.5 mm³/s.