To solve the problem you've presented, we need to break it down into several parts. We will calculate the coefficient of friction, chip thickness ratio, shear plane angle, velocity of the chip along the tool face, and the velocity of shear along the shear plane using the given parameters. Let's go through each calculation step by step.
Given Data
- Outside diameter of the tube: 3 cm
- Cutting speed (V): 40 m/min
- Depth of cut (d): 0.125 mm = 0.0125 cm
- Length of continuous chip for one revolution (L): 17.77 cm
- Cutting force (F_c): 200 kg
- Feed force (F_f): 75 kg
- Rake angle (α): 35°
Calculating Coefficient of Friction
The coefficient of friction (µ) can be calculated using the formula:
µ = F_f / F_c
Substituting the values:
µ = 75 kg / 200 kg = 0.375
Determining Chip Thickness Ratio
The chip thickness ratio (r) is given by the formula:
r = t_0 / t
Where:
- t_0 = depth of cut = 0.0125 cm
- t = chip thickness
To find the chip thickness (t), we can use the relationship:
t = L / (π * D)
Where D is the diameter of the tube. Since the diameter is 3 cm, we have:
t = 17.77 cm / (π * 3 cm) ≈ 1.88 cm
Now, substituting into the chip thickness ratio formula:
r = 0.0125 cm / 1.88 cm ≈ 0.00664
Calculating Shear Plane Angle
The shear plane angle (φ) can be calculated using the formula:
tan(φ) = (r * cos(α)) / (1 - r * sin(α))
Substituting the values:
tan(φ) = (0.00664 * cos(35°)) / (1 - 0.00664 * sin(35°)
Calculating cos(35°) and sin(35°):
- cos(35°) ≈ 0.8192
- sin(35°) ≈ 0.5736
Now substituting these values:
tan(φ) = (0.00664 * 0.8192) / (1 - 0.00664 * 0.5736) ≈ 0.00544
Thus, φ = arctan(0.00544) ≈ 0.312°
Velocity of the Chip Along the Tool Face
The velocity of the chip (V_c) along the tool face can be calculated using the formula:
V_c = V * (1 + r)
Substituting the values:
V_c = 40 m/min * (1 + 0.00664) ≈ 40.2656 m/min
Velocity of Shear Along Shear Plane
The velocity of shear (V_s) can be calculated using the formula:
V_s = V * cos(φ)
Substituting the values:
V_s = 40 m/min * cos(0.312°) ≈ 40 m/min * 0.9998 ≈ 39.992 m/min
Summary of Results
- Coefficient of Friction (µ): 0.375
- Chip Thickness Ratio (r): 0.00664
- Shear Plane Angle (φ): 0.312°
- Velocity of the Chip Along the Tool Face (V_c): 40.2656 m/min
- Velocity of Shear Along Shear Plane (V_s): 39.992 m/min
These calculations provide a comprehensive understanding of the machining process for the seamless tube. Each parameter plays a critical role in determining the efficiency and effectiveness of the cutting operation.
