Research on the calculation method of grinding temperature field of spiral bevel gear based on generative machining

At present, the process parameter setting of spiral bevel gear grinding is mainly based on an empirical method, and the tooth surface temperature is difficult to estimate effectively during grinding, which may lead to excessive tooth surface temperature and surface burn caused by improper selection of process parameters. Aiming at the above problems, this paper presents a method for calculating the temperature field of spiral bevel gear forming based on a combination of analytical and numerical methods. The motion equation of spiral bevel gear grinding is derived based on the method of tooth surface development. The time-varying heat flow calculation method in tooth surface grinding was developed through the discrete and two-dimensional contact zone. The time-varying temperature field analysis method of spiral bevel gear forming is developed using the time-varying heat flow and finite element calculation method. The related research can realize the temperature prediction of grinding tooth surfaces and provide the basis for improving grinding technology.


Introduction
The research on the grinding theory and technology of spiral bevel gear has always been the key research problem in the machining process.The thermal damage in the grinding process seriously affects the precision of the tooth surface and the smoothness of the gear transmission system.
For the above problems, Ramanath and Shaw [1] predicted for the first time the heat distribution ratio between the abrasive particle and the workpiece in the grinding contact zone in the case of shallow grinding.Still, their research results were rarely applied due to the small consideration factor.On this basis, Chen et al. [2] comprehensively considered the influence of the workpiece, grinding chips, grinding fluid, and grinding wheel in the grinding contact zone and calculated the respective heat distribution ratio of all parts in the grinding contact zone.Ming [3] systematically studied the heat generation source of spiral bevel gear during grinding on CNC machine tools and revealed the rules of the grinding interface.Tang et al. [4] established a mathematical geometric model and deduced the calculation formula of grinding force according to the generative motion grinding process of the cone of the grinding wheel.Zhao [5] used the finite element analysis theory to analyze the grinding process of face gear with a rectangular heat source and verified the feasibility of the model through experiments.Kizaki [6] developed a system aiming at the heat generation mechanism in the generating gear grinding process that can observe transient phenomena in the removal area, which has the potential to improve the cooling efficiency of continuous generating grinding significantly.Based on the grinding theory, Zhao et al. [7] introduced the grinding wheel's grinding area to establish a calculation model of grinding force, conducted an experimental study on the relationship between grinding force and grinding wheel's grinding area, and verified the accuracy of the model in reverse with the results.Based on the thermal model of earlier studies, Jin et al. [8] developed a theoretical grinding temperature field model for the direct grinding process to predict the grinding process temperature field.To accurately predict the temperature change of the workpiece surface, Lan [9] collected data signals during processing through tests, converted them into specific values of ground temperature, and calculated the distribution of the grinding temperature field with the finite element method.Kim et al. [10] further developed the unequal triangular heat source distribution model based on the triangular heat source distribution, and the results showed that this heat source model was more consistent with the actual situation for the grinding process with slow feed.
There is little research on the calculation method of grinding temperature field for spiral bevel gear, which has a complex surface and forming motion.Therefore, this paper proposes a temperature field method for spiral bevel gears by combining the motion geometry of spiral bevel gears, metal cutting theory, and computational heat transfer method.Firstly, based on the relative motion of the workpiece in the generative method, the formula for calculating the local second-order contact characteristics in the spiral bevel gear grinding process is derived.Secondly, using the surface grinding heat generation formula, the heat flow calculation grinding of spiral bevel gear is established.Finally, the temperature field distribution of the spiral bevel gear in the forming process is calculated using the finite element heat transfer analysis method.

Spiral bevel gear machining movement
In the theoretical cutting mathematical model of spiral bevel gear grinding, As shown in Figure 1, L and R, respectively, represent the transverse and longitudinal lengths of the conjugate contact points of r q and  from the interleaving point of the shaft when the gear pair engages and br R represent the radius of the same point to its axis.
In the above Equation (2), x  is the direction angle of the instantaneous contact line of the grinding process, and , , , ,  a a a a .

Developed into grinding contact length
According to the actual grinding conditions of the spiral bevel gear generating method, the local area of any point on the instantaneous contact line is the contact of two spatial surfaces here, as shown in Figure 2. Wherein the grinding speed m V at any point on the instantaneous contact line meets the vector triangle rule with the grinding wheel linear speed gx V and the feed speed s V .According to the actual situation, the surface contact theory is used to analyze the grinding contact zone.and point 0 y  , whose expression is as follows: The fitting surface S was transformed symmetrically by taking the plane as the mirror symmetry plane.Assuming the grinding depth was h z , the micro-element grinding removal area was constructed, as shown in Figure 3.According to the fitting surface expression presented by Equation (3), it can be seen that the grinding area g S is: The conic curve   , L x y obtained in Figure 4 is the grinding shape of the machined workpiece involved in the production process and located in the 1 O coordinate system.In order to obtain the grinding arc length at this time, it is converted to the 2 O coordinate system and finally represented by the plane curve 2 L .Suppose , the grinding length z l is:

Generative grinding force
The derivation of grinding forces in grinding micro-elements is shown in Equation ( 6 Here, ds is the instantaneous contact line element. When the grinding force generated by the grinding element is known, the grinding force generated by grinding at this moment is obtained by curve integration along the instantaneous contact line, as shown in Equation ( 7): ( ) Here, L is the instantaneous contact line.

Instantaneous grinding zone heat flow
The total heat flux of the grinding zone is calculated according to grinding force F , grinding speed s v and grinding contact area S .
During grinding, the moving direction of the grinding heat source of the grinding wheel is parallel to the plane of the workpiece, and the grinding contact arc length is small.Assuming that the heat source is the moving line heat source with triangular distribution, the heat flow rate 1 Q of the grinding method of spiral bevel gear is as follows: Here, L represents the grinding contact length of the instantaneous contact line.

Gear parameter
The gear is a pair of spiral bevel gear.The specific parameters of the gear are shown in Table 1. in the finite element model: The grinding wheel linear speed is 31.4m/s, the grinding wheel feed speed is 1000 mm/min, and the grinding depth is 0.05 mm.The grinding fluid is water based grinding fluid.

Modeling calculation by the generative method
The finite element model of the spiral bevel gear is shown in Figure 5.The temperature field calculation results are calculated according to the second and third boundary conditions of heat transfer theory as the finite element simulation conditions.

Calculation result of temperature field of gear grinding
Figure 6 shows the distribution state of the transient temperature field on the tooth surface during the generation method.The temperature field is roughly distributed around the grinding area, showing an irregular shape similar to an ellipse, and the highest temperature appears in the center of the grinding area.According to Ming [3] , the critical condition value of the adopted material 20CrMnTi is 600 degrees.According to the comparison with the experimental results proposed by Xu et al. [11] , the location and shape of the burn area are relatively close.Hence, the calculated results in this paper agree with the experimental results.According to Figures 7, 8 and 9, when the grinding depth exceeds 0.13mm , the maximum temperature during grinding will exceed the burn temperature limit.When the temperature field does not exceed the burn critical value, the feed speed of the grinding wheel can be increased.

Conclusion
(1) The shape of the grinding zone obtained by the fitting surface method is the intersection area of the conic curve and the plane in three-dimensional space.On the basis of this, the grinding force in the grinding process can be obtained through the single grain-plane grinding force model.
(2) The heat source model of spiral bevel gear grinding based on gear meshing theory and metal cutting theory can calculate the heat source size and heat flux distribution to provide a basis for the calculation and further analysis of the coefficient of convective heat transfer between the workpiece and other parts.

Figure 1 .
Figure 1.Coordinate points of gear shaft section.

Figure 2 .
Figure 2. Grinding contact area by generating method.

Figure 3 .
Figure 3. Grinding removal area by generating method.

Figure 4 .
Figure 4. Grinding section line by generating method.

Figure 6 .
Figure 6.Grinding temperature field by generating method.

4 Figure 7 .
Figure 7. Temperature field changing with feed speed.

Figure 8 .
Figure 8. Temperature field changing with grinding depth.

Figure 9 .
Figure 9. Temperature field changing with a linear speed of the grinding wheel. )
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