Basic Cutting Theory

Orthogonal Cutting

Orthogonal cutting is a convention used in the study of cutting in order to simplify it. This convention describes a case where the cutting tool has a wedge shape, a greater width than the workpiece, and is positioned perpendicular to the cutting direction, as shown in the figure. As the tool moves relative to the workpiece, it begins to remove material, creating a chip (burr). This chip is produced because the hard cutting tool presses and cuts the workpiece material during their contact.

During orthogonal cutting, 3 areas of interest are created, as shown below:

• Area 1: It is located between the deformed and the undeformed material of the workpiece. Of interest here is the plastic deformation of the machined material
• Area 2: It is the contact surface between the chip and the cutting tool. Here we are interested in the friction between the cutting tool and the machined material, which results in an increase in temperature and wear of the cutting tool.
• Area 3: It concerns the machined surface of the workpiece, i.e., the point that has already been cut. At this point we examine the surface quality characteristics

Cutting Angles

During orthogonal cutting, the combination of the cutting tool with the workpiece creates three basic angles, which play an important role in the cutting process. These angles determine the way material is removed and affect the efficiency and quality of the machining.

• Clearance angle - clearance angle (α): is the angle formed between the clearance surface of the cutting tool that faces the already machined surface. This angle helps prevent the tool from rubbing on the workpiece and reduces resistance during cutting. It directly affects wear and the tool’s strength.
• Wedge angle - wedge angle (β): is formed between the surface of the cutting tool that is in contact with the chip and the clearance surface of the cutting tool. It reflects how “sharp” the tool is.
• Chip angle – rake angle (γ): It is the angle of the surface on which the chip moves as the material is removed. It affects chip flow and cutting forces.

These angles play an important role in cutting, as they directly affect tool performance and the quality of the final result. They are selected based on the type of cutting tool, the workpiece material, and the cutting conditions. Between these three angles, the following relationship applies:

$$α+β+γ=90°$$

Cutting conditions

The cutting conditions are the main parameters that determine how machining is performed. They include cutting speed, feed, and depth of cut. Correct selection of these values significantly affects cutting quality, tool wear, and machining efficiency.

Cutting Speed (Cutting Speed - Vc)

• Definition:
• Cutting speed is the speed at which a point on the circumference of a rotating object (such as a cutting tool or a workpiece) moves through space.

• In practice, it is the circumference of the rotating object multiplied by the revolutions. It is usually measured in meters per minute (m/min).

• Basic Formula:

$$V_c = \frac{\pi \cdot D \cdot n}{1000}$$

where:

• Vc → Cutting speed (m/min)
• D → Diameter (mm)
• n → Revolutions (RPM)
• π → 3,14

• The 1000 is used for unit conversion from mm/min to m/min

To calculate revolutions with a given peripheral speed, we solve as follows:

$$n = \frac{V_c \cdot 1000}{\pi \cdot D}$$

Feed (Feed Rate - fn )

• Definition: (lathe - fn)
• The distance the cutting tool travels per revolution of the workpiece. It is measured in millimeters per revolution (mm/rev)
• It can also be measured in (mm/min) by multiplying by the revolutions

• Definition: (milling machine - FEED)
• The distance the cutting tool travels along the workpiece. It is usually expressed as mm/min and depends on the tool rotational speed (RPM) and the feed per tooth (fz).

$$f = f_z \cdot Z \cdot n$$

• where:
• f = total feed (mm/min)
• fz = feed per tooth (mm/tooth)
• Z = number of tool teeth
• n = number of revolutions (RPM)

Depth of cut Ap - Ae (mm)

• Definition:

• On a lathe, the depth of cut is only the distance between the new and the old surface of the workpiece.

  

  
  * In the case of a milling machine, and specifically side milling, there are 2 different values for the depth of cut
  * Ap (Depth of cut - Depth of cut): The vertical depth removed in each pass.
  * Ae (Radial depth of cut - Width of cut): The horizontal width removed in each pass.

Material Removal Rate (MRR) – Material Removal Rate

• Definition:

• The material removal rate is defined as the volume of chips produced by a machining operation in one minute and is the result of all the previous conditions. It is measured in (mm³/min)

  
  * Basic formula:

$$MRR = V_c \cdot f \cdot a_p$$