In machining processes, the deformation of shafts in C6136A lathes due to uneven force distribution can lead to dimensional and shape errors in workpieces. This study focuses on analyzing the deformation and stress conditions of the gear shaft in the feed box using finite element analysis (FEA). By leveraging FEA data, practical methods to reduce machining errors are derived. The gear shaft is a critical component, and understanding its behavior under load is essential for improving machining accuracy.
Finite element analysis involves two main stages: pre-processing and post-processing. Pre-processing includes creating a finite element model of the component, known as modeling, followed by meshing the model into discrete elements. Post-processing involves data collection and analysis from the meshed model to interpret stress and deformation results across the gear shaft. This approach allows for a detailed examination of the gear shaft’s performance under operational conditions.
The gear shaft in the C6136A lathe feed box is modeled as a stepped shaft, and its geometric model is developed using UG software. Simplifications are made, such as ignoring small relief grooves, to enhance computational efficiency during meshing. The gear shaft is subjected to constraints and forces based on a mechanical model, and tetrahedral elements are used for meshing, resulting in 4,434 elements and 8,885 nodes. This setup ensures accurate simulation of the gear shaft’s response to loads.
To establish the force model for the gear shaft, the spindle speed must be determined first, as power is transmitted from the spindle through intermediate gears to the gear shaft. The gear shaft experiences maximum torque at the lowest speeds, minimizing dynamic effects. Under constant motor power, lower feed rates result in higher loads on the gear shaft, leading to increased deformation. Thus, the low-speed branch of the gear shaft transmission is selected for analysis. The C6136A lathe has eight speed levels ranging from 42 r/min to 980 r/min in a geometric progression. The lowest speed at which the gear shaft transmits full power is identified to determine the maximum torque.
The spindle calculation speed \( n_c \) is the minimum speed at which the spindle transmits full power. The empirical formula for machine tools like lathes is given by:
$$ n_c = n_{\text{min}} \phi^{z/3 – 1} $$
where \( n_{\text{min}} \) is the minimum spindle speed (10 r/min), \( \phi \) is the common ratio of speeds (1.25), and \( z \) is the number of speed levels (8). Substituting the values:
$$ n_c = 10 \times 1.25 = 50 \, \text{r/min} $$
The gear shaft speed \( n_2 \) is derived from the transmission ratio \( i = n_c / n_2 = 8.18 \), yielding:
$$ n_2 = \frac{n_c}{i} = \frac{50}{8.18} \approx 6 \, \text{r/min} $$
The torque transmitted by the gear shaft is calculated using the formula:
$$ T = 9550 \frac{P}{n} $$
where \( T \) is the torque in N·m, \( P \) is the power in kW, and \( n \) is the speed in r/min. With a motor power of 7.5 kW and an efficiency of 0.9, the calculated power is 4.4 kW. Thus, the torque on the gear shaft is:
$$ T_1 = T_2 = 9550 \times \frac{4.4 \times 0.9}{6} = 6303 \, \text{N·m} $$
The tangential and axial forces on the gear shaft are determined based on the gear geometry. For a gear with diameter \( d = 135 \, \text{mm} \) (radius \( r = 0.0675 \, \text{m} \)) and pressure angle \( 20^\circ \):
$$ F_{t1} = \frac{T}{r} = \frac{6303}{0.0675} = 9337 \, \text{N} $$
$$ F_{r1} = F_{t1} \tan(20^\circ) = 9337 \times \tan(20^\circ) \approx 3398 \, \text{N} $$
For another gear with diameter \( d = 84 \, \text{mm} \) (radius \( r = 0.042 \, \text{m} \)):
$$ F_{t2} = \frac{6303}{0.042} = 15007 \, \text{N} $$
$$ F_{r2} = 15007 \times \tan(20^\circ) \approx 5462 \, \text{N} $$
The force model illustrates the load distribution on the gear shaft, which is essential for FEA. The gear shaft is constrained at the front support with restrictions in X-direction movement and rotation, simulating real-world conditions. The meshing process using tetrahedral elements ensures a balanced trade-off between accuracy and computational cost, critical for analyzing the gear shaft’s behavior.
In the post-processing phase, the deformation and stress results are extracted. The maximum deformation (DMX) of the gear shaft is 0.002617 mm, occurring near the gear with 90 teeth between the rear and auxiliary supports. The von Mises stress reaches a maximum of 38.6004 MPa, located between the front and auxiliary supports. These values indicate that the gear shaft maintains structural integrity under load, with minimal deformation that does not compromise machining precision.
The following tables summarize the displacement and stress results from the FEA:
| Step Name | Direction | Minimum (mm) | Maximum (mm) | Amplitude (mm) |
|---|---|---|---|---|
| SUBCASE-STATIC LOADS1 | X | 0 | 0.00209939 | 0.00261676 |
| Y | 0 | 0.000879786 | 0.00141471 | |
| Z | 0 | -0.00121431 | -0.00182357 |
| Step Name | Stress Type | Minimum (MPa) | Maximum (MPa) |
|---|---|---|---|
| SUBCASE-STATIC LOADS1 | Von Mises | 0.00175701 | 38.6004 |
| Max Shear | 0.000974704 | 20.6317 | |
| Max Principal | -49.8075 | 30.0692 |
The analysis confirms that the gear shaft exhibits high resistance to deformation and failure under high loads. The low-order modal frequencies of the gear shaft are sufficiently high, reducing the risk of resonance during normal operation. The amplitude of vibration is negligible, ensuring that the gear shaft’s rigidity supports precise machining processes. This makes the gear shaft ideal for applications requiring stability and accuracy.
In conclusion, the finite element analysis of the C6136A gear shaft provides valuable insights into its mechanical behavior. By integrating geometric modeling, force calculations, and FEA, this study demonstrates that the gear shaft can withstand operational stresses with minimal deformation. The use of advanced software like UG for modeling and meshing enhances the reliability of the results. Future work could explore dynamic analyses or material optimizations to further improve the gear shaft’s performance. Overall, the gear shaft in the C6136A lathe feed box proves to be a robust component, contributing to reduced machining errors and enhanced productivity.
The methodology applied here can be extended to other machinery components, emphasizing the importance of FEA in design and optimization. By continuously refining the gear shaft models and incorporating real-world data, engineers can achieve higher efficiency and durability in mechanical systems. The gear shaft remains a focal point in ensuring the overall performance of the lathe, underscoring the need for detailed analysis and innovation in this area.