A straight-teeth horizontal slab milling cutter is shown in the figure. It has 4 teeth and diameter (D) of 200 mm. The rotational speed of the cutter is 100 rpm and the linear feed given to the workpiece is 1000 mm/minute. The width of the workpiece (w) is 100 mm, and the entire width is milled in a single pass of the cutter. The cutting force/tooth is given by F = Kt_cw, where specific cutting force K = 10 N/mm², w is the width of cut, and t_c is the uncut chip thickness. The depth of cut (d) is D/2, and hence the assumption of $\frac{d}{D}<<1$ is invalid. The maximum cutting force required is ______ KN (round off to one decimal place).

The correct answer is 2.5.

1. Identified Parameters from the Question and Schematic:
From the given description and the schematic diagram showing the milling cutter of diameter D and the workpiece with a depth of cut d = D/2, we have the following parameters:
• Cutter diameter, D = 200 mm
• Number of teeth, z = 4
• Rotational speed of the cutter, N = 100 rpm
• Linear feed rate of the workpiece, f = 1000 mm/minute
• Workpiece width, w = 100 mm
• Specific cutting force, K = 10 N/mm²
• Depth of cut, d = D/2 = 100 mm

2. Calculating Feed per Tooth:
The feed per tooth, which represents the advance of the workpiece per tooth of the cutter, is calculated as:

$$f_z=\frac{f}{N\times z}$$

Substituting the given values:

$$f_z=\frac{1000}{100\times 4}=2.5\text{mm/tooth}$$

3. Determining the Maximum Uncut Chip Thickness:
The uncut chip thickness $t_c$ at any point along the cutting arc is given by:

$t_c=f_z\sin \left(\theta \right)$

where $\theta$ is the angle of rotation of the cutter tooth. The maximum uncut chip thickness occurs at the maximum angle of immersion.

Without approximating with the assumption that $\frac{d}{D}\ll 1$ (which is stated to be invalid because the depth of cut is half the cutter diameter, $d=\frac{D}{2}$), the maximum uncut chip thickness is determined using the exact relation:

$$t_{c,max}=f_z\sqrt{1-{(1-\frac{2d}{D})}^2}$$

Given that $d=\frac{D}{2}$, we substitute this relationship into the equation:

$$t_{c,max}=f_z\sqrt{1-{(1-\frac{D}{D})}^2}=f_z\sqrt{1-0}=f_z$$

Thus, the maximum uncut chip thickness is:

$$t_{c,max}=2.5\text{mm}$$

4. Calculating the Maximum Cutting Force:
The cutting force per tooth is given by:

$$F=K\times t_c\times w$$

Therefore, the maximum cutting force required is:

$$F_{max}=K\times t_{c,max}\times w$$

Substituting the parameters:

Converting the force from Newtons (N) to kilonewtons (kN):