Abstract
In this article, a finite line contact tribo-dynamics model of helical gears is established by coupling the tooth surface friction, friction moment, and vibration velocity. To obtain convergent results, a cycle iteration strategy is adopted to solve the two submodels, the dynamic model and the thermal elastohydrodynamic lubrication model. The parameters obtained from the dynamic model such as the dynamic load, dynamic transmission error, and instantaneous speed of the teeth surfaces are used as the boundary conditions for solving the lubrication model. Meanwhile, the friction force and friction moment obtained by the lubrication model are used as parametric excitations of the dynamic model. Based on the proposed model, the influence of dynamic effect, thermal effect, rotational speed, and tooth surface roughness on the tribological and dynamic properties of helical gears are discussed. The results show that the tribological and dynamic parameters deviate significantly from the quasistatic values, especially at the characteristic speed. The vibration along the off-line-of-action direction is obviously affected by the friction excitation. As the rotational speed increases, the friction coefficient and friction force decrease as well as the vibration of the gear pair along the off-line-of-action direction weakens. In addition, the surface roughness has the greatest influence on the vibration of the off-line-of-action direction in all directions of gear vibrations.
1. Introduction
Due to the stable transmission and high load-bearing, the helical gear is widely utilized in several mechanical transmission devices. In the process of gear transmission, various failure forms are accompanied, such as pitting and gluing. In general, these failures of the gear are closely related to the contact state of the tooth surface. In the past decades, the gear tribology has been widely studied from both theoretical and experimental aspects [1]. In the recent years, the coupling characteristics of tribological and dynamic of gears have been widely reported, especially for spur gears. It shows that the tribological performance of the gear interface under the dynamic load is significantly different from that under the quasistatic load.
A large number of studies have been reported on gear friction and dynamics, including theoretical and experimental studies. In the aspect of gear tribology, it has undergone the evolution from elastohydrodynamic lubrication (EHL) to mixed lubrication. In addition, the influence of dynamic characteristics on the tribological performance of the tooth surface is considered. Based on a reduced Reynolds equation, Liu et al. [2] established a mixed lubrication model for spur gears and studied the effect of operating conditions and surface roughness on the lubrication performance. Li and Kahraman [3] established a mixed thermal EHL model for spur gear and discussed the influence of thermal effect on lubrication performance. By introducing the transient effect, Bertsche and Fietkau [4] established a transient EHL model of gears. Liu and Yang [5] firstly proposed a finite line EHL model of a helical gear pair, and a specific meshing position is selected for lubrication analysis. By considering the oil supply conditions of gear lubrication, Liu et al. [6] established a starved thermal EHL model for helical gears. Liu et al. [7, 8] proposed a thermal EHL model of the spur gear under starved lubrication and discussed the influence of oil supply on lubrication performance. Based on the finite line contact theory, Liu et al. [9–12] established a finite line contact thermal EHL model for helical gears and systematically studied the influence of geometric parameters and working conditions on the lubrication performance. Liu et al. [13, 14] investigated the lubrication performance of spur gear under the dynamic load. It revealed that the lubrication performance of gears deviates from the quasistatic value obviously under the dynamic load. However, it indicated that the effect of the tooth surface friction excitation on dynamic performance is limited.
In the aspect of gear dynamic, a lot of theoretical and experimental results have been published. Based on the EHL-based friction coefficient formula proposed by Xu et al. [15], Wang [16] established a friction-dynamics model for helical gears and discussed the variation of dynamic performance of the helical gear. By adopting a fixed friction coefficient, Jiang et al. [17, 18] established a three-dimensional dynamics model of the helical gear and discussed its dynamic performance with several gear failures, such as spalling defect and tooth breakage. Han et al. [19, 20] studied the dynamic performance of helical gears considering friction and time-varying meshing stiffness. Vaishya and Singh [21] established a multidegree of a freedom dynamic model of gears considering the tooth surface friction and discussed the influence of friction on gear dynamic characteristics. Marques et al. [22] studied the influence of input torque and friction force on gear dynamic characteristics and power loss with a multidegree of the freedom dynamic model. By introducing sliding friction, He et al. [23] established a friction-dynamics model of spur gear. In all the abovementioned dynamic models, an empirical friction coefficient formula or fixed friction coefficient is adopted. This treatment strategy is generally considered to have little influence in the study of gear dynamic characteristics. However, the gear failure is closely related to the lubrication state of the tooth surface. In the recent years, the coupling analysis of gear lubrication and dynamics has become a hot topic. Based on the coupling effect of tooth surface shear force, meshing force, and velocity, Li and Kahraman [24] firstly established a tribo-dynamic model of the spur gear. It revealed the gear dynamic characteristics and lubrication performance under the tribo-dynamic model, such as oil film damping and power loss. Based on the finite line contact theory, Ouyang et al. [25, 26] established a three-dimensional tribo-dynamics model of the spur gear.
Based on the abovementioned analysis, a novel tribo-dynamics model of gears is proposed, especially for spur gears. However, most of the abovementioned models adopt the line contact assumption, which is not suitable for helical gears. The contact geometry characteristics of spur gear and helical gear are not exactly the same. So, in the article, a finite line contact tribo-dynamics of helical gear is established. First, the geometric and kinematic analyses of helical gear pairs are carried out. The corresponding parameters are obtained, such as the radius of curvature, velocity of the tooth surface, and length of the contact line. Second, two submodels are given, namely, the thermal elastohydrodynamic lubrication model and the multidegree-of-freedom dynamic model of the helical gear pair. By coupling the tooth surface friction, friction moment, and vibration velocity, these two submodels constitute the tribo-dynamics model of the helical gear pair. Then, the solution strategy of the model is given. Based on the proposed model, the influences of thermal effect, working conditions, and surface roughness on the dynamic characteristics and tribological properties of helical gears are discussed.
2. Governing Equations
2.1. The Description of Helical Gear Meshing
The contact geometry model of a single tooth of the helical gear is shown in Figure 1. The term represents the instantaneous contact line. In the process of gear meshing, the length of the contact line changes on the actual meshing region . The terms are the rotation axis of the driving gear and driven gear, respectively. The angle between the contact line and the rotation axis of the gear is equal to the helix angle of the base circle, i.e., βb. A coordinate system is established as shown in Figure 1, where the y-axis is located at the contact line direction [9–12].
In the article, the tribo-dynamic model consists of two submodels: dynamics model and the thermal EHL model. The parameters of the tooth surface geometry and kinematic are essential to solve the lubrication model. As shown in Figure 1, the radius of curvature at any point K on the contact line can be expressed aswhere in the coordinate system xyz as shown in Figure 1.
When the dynamic effect of the gear is ignored, the nominal linear velocities of the contact line along the x-axis are given aswhere ω1 and ω2 are the angular velocities of the driving and driven gears, respectively.
So, the kinematic parameters of the tooth surface can be obtained, such as the entrainment velocity ue, the sliding velocity us and the slide-roll ratio ζ. These parameters are defined as
It is well known that the contact line length of the helical gear varies during the meshing process. Figure 2 shows the motion pattern of the contact line. For a pair of meshing teeth, the paired teeth surfaces start to engage at point B1 and exit at point . The length of the contact line is gradually lengthened to the full length of the tooth width and then gradually shortened. As shown in Figure 2, the length of the contact line at a certain meshing position is set as l, and the projection length along the axial direction is equal to Bl. According to the geometric relation, the term Bl can be expressed aswhere , , rb1 is the radius of the base circle.
Then, the length of contact line can be given as
In general, in order to ensure the continuity of the gear transmission, the contact ratio of gears is greater than 1. This indicates that there are multiple pairs of gears engaged simultaneously at the same time. The variation trend of each contact line is consistent, as described in equations (4) and (5). So, the total contact line length of the helical gear pair at each meshing position can be obtained. The expression is defined aswhere li is the length of the contact line of a single contact tooth. It is determined by equation (5). N is the number of simultaneously engaged teeth.
2.2. Thermal Elastohydrodynamic Lubrication Model
In this article, a thermal elastohydrodynamic lubrication model is adopted to simulate the lubrication state of the helical gear tooth surface. In addition, the non-Newtonian characteristics of lubricants are considered [9–12]. Based on the coordinate system shown in Figure 1, the generalized Reynolds equation is given aswhere , , , , , , . h is the film thickness. p is the film pressure. ρ is the density of lubricant. is the equivalent viscosity of lubricant. t is the time.
In general, the film thickness is comparable to the surface roughness. It is composed of geometric gap, surface roughness, and elastic deformation. The film thickness is expressed aswhere h0 is the normal approach of the meshing surfaces and adjusted by the load equation. is the geometric gap under undeformed condition, which is calculated from the radius of curvature in equation (1). S1 and S2 represent the tooth surface roughness. represents the comprehensive elastic modulus of the gear material. Ω is the calculation region.
The contact pressure of the tooth surface is balanced with the dynamic load. The dynamic load is determined by a dynamic model of the helical gear, as described in the following sections. Then, the load balance equation is given aswhere Fd is the dynamic load. ω (t) is the load sharing coefficient.
In order to simulate the non-Newtonian characteristics of lubricant, the Ree–Eyring constitutive model is selected. The equation of Ree–Eyring fluid is defined aswhere τe is the comprehensive shear stress of the film, , τx, and τy are the shear stress of the oil film along the x and y directions. η is the viscosity of lubricant. τ0 is the characteristic shear stress of the lubricant.
Then, the equivalent viscosity of lubricant can be obtained.
The viscosity and density of lubricant are the functions of pressure and temperature. In the article, the functions of Roelands and Dowson–Higginson are adopted to describe the viscosity and density of the lubricant, respectively. These functions are denoted bywhere the terms η0, ρ0, and T0 represent environmental viscosity, density, and temperature, respectively. . The terms α and βT are the viscosity-pressure coefficient and viscosity-temperature coefficient, respectively. T is temperature.
In general, the sliding motion occurs at the tooth surface except for the pitch point. This phenomenon results in heat effect and power loss on the tooth surface. The temperature field of film thickness can be simulated by the energy equation. It is expressed aswhere , and uf and are the flow velocity of the fluid along the x and y directions. cf and kf are the specific heat and the heat conduction coefficient of the fluid.
Based on the assumption of no slip at the interface between the lubricant and tooth surface, the distribution of the film velocity field can be obtained. It is expressed as
In addition, the heat conduction of the gear material should be considered. The energy equation of solids is given aswhere c1 and c2 are specific heat of gear material. ρ1 and ρ2 are density of the gear material. k1 and k2 are heat conduction coefficient of the gear material. zp and are coordinate direction of the contact tooth surface, which is the same as the z direction of the film thickness.
The condition of heat flow continuity should be satisfied at the fluid-solid interface. The equation is expressed as
According to the constitutive model and the velocity field of fluid, the shear stress distribution of the tooth surface can be obtained. It is expressed as
Finally, the friction coefficient of the tooth surface can be expressed as
2.3. Dynamic Model of the Helical Gear
Generally, a gear system consists of a gear pair, bearing, shaft, and a support system. This system generates vibration under the action of internal and external excitation, including transverse and torsion vibration of each degree of freedom. In the current study, the bearing, shaft, and support system are simplified to a spring-damping system. Meanwhile, in order to simplify the model, only the transverse vibration of the gear pair in three directions and the torsional vibration around the gear shaft are considered. A multidegree of freedom lumped parameter of the helical gear is shown in Figure 3.
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As shown in Figure 3, the meshing tooth surface is simplified as the spring and damping link along the line of action direction and shear force along the off-line-of-action direction. The origin of the coordinate system x1y1z1 and x2y2z2 is located at the center of the driving and driven gear, respectively. Meanwhile, the axes y1 and y2 are built along the line of action. The axes x1 and x2 are built along the off-line of action. The axes z1 and z2 are consistent with the axial direction of the gears. The terms θ1 and θ2 represent the torsional vibration displacement of the gears around the shaft. The terms represent the stiffness and damping of support system in the x, y, and z directions. Then, the dynamic model of the helical gear is written aswhere the terms m1 and m2 are the mass of the gears. The terms are the torsional damping of gears. The terms I1 and I2 are moment of inertia of the gears. The terms T1 and T2 are the input and output torque. Fd is the total dynamic meshing force of teeth surfaces. Ftf represents the total friction force of all teeth surfaces at a certain meshing position. Mf1 and Mf2 are the friction torques on the driving and driven gears.
The total friction force is given aswhere Ff is single tooth surface friction force. It can be obtained by integrating the shear stress of tooth surface, as shown in equation (17).
The friction torques of the tooth surface are obtained bywhere the terms r1 and r2 represent the moment arm of friction, as shown in equation (1).
The total dynamic meshing force is expressed aswhere k (t) is the time-varying meshing stiffness. The term cm is the meshing damping. The term δ is the relative dynamic displacement along the normal direction of the meshing surface. It is defined as
Based on the proposed dynamic model, the reaction force of the support system can be obtained. So, the dynamic bearing forces in three directions of support systems are defined as
As described in Section 2.2, the traction velocity of the tooth surface is an important cause of interface lubrication. The equation (2) shows the expression of tooth surface velocity at nominal rotational speed. In the current study, the influence of dynamic effect on lubrication analysis should be considered. So, the velocity term of equation (2) is modified as
As stated in equation (25), the second and third terms at the right end are expressed as the effects of torsional and lateral vibration of the gear system.
3. Solution Strategy
As stated in Section 2, a tribo-dynamic model of a helical gear pair is established by coupling the dynamic meshing force, friction force, and velocities of teeth surfaces. The model is solved by the loop iteration strategy of submodels of lubrication and dynamics. To ensure the stability of the numerical solution, all the governing equations are treated dimensionless. For the lubrication submodel, the tribological variables are obtained through pressure-temperature loop iteration. In order to accelerate convergence, the discrete Fourier transform (DC-FFT) algorithm is used to calculate the elastic deformation, and the column-by-column scanning method is used to calculate the temperature field. The relative error of pressure, temperature, and load iterative convergence is set as εp = 10−5, εT = 10−5, and εF = 10−4, respectively. The dynamic model is solved by the Runge–Kutta method. The relative error of periodic convergence is set as 10−3.
The flowchart of the iterative strategy is shown in Figure 4. The friction force and friction moment of the lubrication model are used as the excitation of the dynamic model. The dynamic velocity and dynamic meshing force terms are transferred to the lubrication model as its working conditions. The iteration process is as follows: Step 1. The procedure is initialized firstly. The friction force Ff and friction moment, Mf1 and Mf2, are initially set as 0. Then, the equation (19) of the dynamic model is solved. The dynamic parameters are obtained, such as the instantaneous velocities u1 (t) and u2 (t) and dynamic meshing force Fd as stated in equations (22) and (25). Step 2. The load sharing coefficient has been obtained. It is equal to the ratio of the single contact line length to the total contact line, as stated in equations (5) and (6). Step 3. The thermal EHL model is solved and the tribological variables are obtained. The shear stress of teeth surfaces is obtained as stated in equation (17). Step 4. The friction excitations are calculated, such as friction force and frictional moment as shown in the equations (20) and (21). Step 5. The friction excitations in Step 1 are modified and steps 1–5 are repeated. The convergent solution is obtained until the variable satisfies the periodic error criterion.
In the article, the parameters of the helical gear and lubricant are shown in Table 1. The time-varying meshing stiffness of the helical gear can be calculated by using the elastic potential energy method [27]. The meshing damping is determined by meshing stiffness and mass of gears. For the selected helical gear pair, the time-varying meshing stiffness and harmonic components are shown in Figures 5(a) and 5(b), respectively. In addition, for the support system, the equivalent bearing stiffness is set as , , , , , and . The damping coefficient is set to 0.1. For the torsional vibration of gears, the torsional viscous damping is set as .
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4. Results and Discussion
4.1. Comparison of a Tribo-Dynamic Model and a Quasistatic Model
First, the influence of dynamic effects is discussed. The tribological properties of the gear surface under the tribo-dynamic are compared with that of the quasistatic model. The corresponding working condition is set as T1 = 500 N·m, ω1 = 400 r/min. Figure 6 shows the variation of the single tooth dynamic load, friction force, friction moment, entrainment velocity, and comprehensive radius of curvature under the tribo-dynamic model. In addition, Figure 6(a) also shows the tooth surface load under the quasistatic model. It shows that the tooth surface meshing dynamic load fluctuates obviously. Figures 6(b) and 6(c) show the variation of friction and friction moment, respectively. It indicates that the sign of the friction excitation on the tooth surface changed periodically. This is due to the variation of relative sliding speed of the tooth surface during the meshing process. Figure 6(c) shows that the friction moment of the driven gear is greater than that of the driving gear. This is due to the selection of a reduction gear pair, and the radius of curvature from the driven gear is larger than that of the driving gear. Figure 6(d) shows the variation entrainment velocity and comprehensive radius of curvature at the midpoint of the contact line during the meshing process. The term R is expressed as .
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Figure 7 shows the comparison of entrainment velocity and the slid-roll ratio under the two models. It indicates that the kinematic parameters of gears fluctuate around the nominal values under dynamic loads, as shown in Figures 7(a) and 7(b). It should be pointed that the kinematic parameters of gears in the quasistatic model are determined by the nominal values. As stated in equation (25), the velocity term of the coupled model contains the fluctuation terms of transverse vibration and torsional vibration.
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Figure 8 shows the comparison of tribological parameters of the tooth surface between the tribo-dynamics model and the quasistatic model. Figure 8(a) shows that the pressure at most positions is obviously higher than those in the quasistatic model. This isconsistent with the distribution of the dynamic load on the tooth surface in Figure 6(a). Meanwhile, the minimum film thickness of the tooth surface is lower than that of the quasistatic model, as shown in Figure 8(c). Figures 8(b)–8(d) show that the temperature rise and friction coefficient under the tribo-dynamic model are higher than those of the quasistatic model. For the tribo-dynamic model, lower film thickness and higher temperature rise, this phenomenon is not good for gear durability, such as gear wear and fatigue.
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4.2. Influence of Thermal Effect
It is well known that thermal effect has a significant influence on tooth surface lubrication, such as film thickness and friction coefficient. The friction excitation in the gear system is closely related to lubrication. Based on the proposed model, the influence of thermal effect on the gear dynamic characteristics is discussed. Figure 9 displays the influence of thermal effect on the dynamic characteristic of the helical gear. It can be seen that the thermal effect has a significant influence on the dynamic characteristics of gears, especially the vibration along the x direction. In the article, the x direction means the off-line of action. The fluctuation of vibration displacement and bearing force along the x direction of the isothermal model are significantly higher than those of the thermal model, as shown in Figures 9(b) and 9(d). This is due to the higher friction of the isothermal model, as shown in subsequent Figures 10(d). Figures 9(a)–9(c) indicate that the thermal effect has little influence on the dynamic meshing force and transmission error. This is because the thermal effect affects these two parameters mainly through the friction moment, as stated in equation (19). In general, the friction moment is incomparable to the external load.
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Figure 10 shows the influence of thermal effect on the lubrication performance of the helical gear. It shows that the thermal effect has a significant influence on the tribological characteristics of tooth surfaces, such as film thickness and friction coefficient, as shown in Figures 10(c) and 10(d). This is because the thermal effect leads to the reduction of lubricating oil viscosity, resulting in the reduction of film thickness and tooth surface shear stress. This conclusion is consistent with the previous gear lubrication analysis. Figure 10(b) shows the maximum temperature rise. The corresponding friction force and friction moment excitation are significantly different between the isothermal and thermal models. This is reflected in the dynamic response as shown in Figures 9(b)–9(d). In addition, the thermal effect had little effect on the pressure, as shown in Figure 10(a). In addition, due to the influence of thermal effect on the shear stress, the thermal effect should be considered in gear wear and fatigue analysis.
Figure 11 shows the influence of thermal effect on the friction excitation of tooth surface with the tribo-dynamics model. Figures 11(a)–11(c) show the distributions of friction force and the friction moment, respectively. It indicates that the friction force and friction moment of the isothermal model are obviously larger than those of the thermal model. This is consistent with the variation of the friction coefficient in Figure 10(d).
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4.3. Influence of Input Rotation Speed
In the previous articles, the influence of working conditions has been extensively studied in gear dynamics and lubrication models. It reveals that the working conditions have significant influence on the dynamic and tribological characteristics of gears. In this section, the influence of input rotation speed on the dynamic and tribological characteristics of gears is discussed based on the proposed model. Figure 12 shows the influence of input rotation speed on the dynamic characteristics of the helical gear. It can be seen that, with the increase of input rotational speed, the dynamic meshing force and transmission error increase as shown in Figures 12(a) and 12(b), while the bearing force and friction moment decrease as shown in Figures 12(c) and 12(d). This is because the friction coefficient of tooth surface decreases with the increase of input rotational speed, as shown in the subsequent Figure 13(d). It revealed that the input rotation speed has a significant effect on the vibration of the gear system at all degrees of freedom.
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Figure 13 shows the variation of pressure, temperature rise, film thickness, and friction coefficient at several input rotational speeds. In Figures 13(a) and 13(c), the pressure and film thickness represent the values in the middle of the contact line. It indicates that as the input rotation speed goes up, the film thickness and the friction coefficient go down while the temperature goes up as shown in Figures 13(b)–13(d). The effect of input rotation speed on pressure is not obvious, except for some special values.
4.4. Influence of Surface Roughness
In the previous articles, few studies have shown the effect of surface roughness on dynamic characteristics of gears. In this section, the regular sinusoidal roughness is adopted for gear lubrication analysis. The working condition is set as T1 = 500 N·m, ω1 = 400 r/min. Figure 14 shows the variation of the dynamic meshing force, dynamic transmission error, and bearing force under different roughness amplitude. It reveals that, under the current calculation conditions, the roughness amplitude has little influence on the dynamic characteristic parameters, as shown in Figures 14(a) and 14(b). Certainly, only at some local meshing moments, as the roughness amplitude increases, the bearing force increases as shown in Figures 14(c) and 14(d). This is because the lubrication state of the tooth surface belongs to full-film lubrication in all the selected roughness amplitudes. The film thickness is obviously higher than the roughness amplitudes, as shown in Figure 8(c). Meanwhile, with the increase of the surface roughness amplitude, the friction coefficient increases. It leads to the increase of bearing force.
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Figure 15 shows the variation of pressure, friction coefficient, friction force, and friction moment under different roughness amplitudes. Figure 15(a) shows that the roughness amplitude has a significant impact on the pressure, especially in the positions of engaging in and engaging out. This is due to the contact line being shorter at the engaging-in and engaging-out positions, which belongs to the point contact. Figure 15(b) shows that as the surface roughness amplitude increases, the friction coefficient increases. As shown in Figures 15(c) and 15(d), it indicates that the friction force and friction moment increase with the roughness amplitude.
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5. Conclusion
(1)In this article, based on the finite length line contact theory, a coupled tribo-dynamic model of the helical gear is established. The lubrication performance of the gear under the tribo-dynamic model deviates from that of the quasistatic model. Meanwhile, the transverse vibration of gears is determined by tooth friction force. Due to the transverse and torsional vibration, the instantaneous velocity of the gear surface under the coupled model fluctuates around the nominal values.(2)The influence of thermal effect on gear dynamic characteristics and lubrication performance cannot be ignored. The thermal effect can reduce the vibration of the gear along the off-line of action. Meanwhile, the thermal effect has little influence on the dynamic meshing force and transmission error of gears. By considering the thermal effect, the film thickness and the friction coefficient of the tooth surface are lower than those of isothermal solution.(3)The input rotation speed and surface roughness have significant influence on the dynamic characteristics and lubrication performance of gears. With the increase of input rotational speed, the dynamic meshing force and transmission error increase, while the friction force and friction moment decrease. Meanwhile, the film thickness and the friction coefficient go down while the temperature goes up with the increase of input rotation speed. By changing the lubrication state of the tooth surface, the roughness of the tooth surface has significant influence on lubrication performance. The influence of tooth surface roughness on gear dynamic characteristics cannot be ignored, especially vibration in the off-line of action.(4)In this article, the all cases belong to full-film lubrication. The results of this article can also provide valuable reference for the analysis of gear fatigue and wear performance. However, it is well known that the tooth surfaces of gears are usually in a mixed state. In future work, it can try to establish the tribo-dynamics model of gears with mixed lubrication. Then, the influence of gear wear and fatigue is considered in the model.Nomenclature
| B: | Tooth width, mm |
| Bl: | Effective meshing width, mm |
| c1, c2: | Specific heat of gear material, J·.kg−1K−1 |
| cf: | Specific heat of lubricating oil, J·kg−1K−1 |
| cm: | Meshing damping, N·s |
| ct1, ct2: | Torsional viscous damping, Nms/rad |
| cx1, cx2, cy1, cy2, cz1, cz2: | Equivalent bearing damping, Ns/m |
| E: | Equivalent elastic modulus, Gpa |
| F: | Single tooth dynamic load, N |
| Fd: | Total dynamic load, N |
| Ff: | Friction force of single tooth surface, N |
| Fbx1, Fbx2, Fby1, Fby2, Fbz1, Fbz2: | Bearing force, N |
| Ftf: | Total friction force of all surfaces, N |
| h: | Film thickness, μm |
| I1, I2: | Rotational inertia of gears, kg·m2 |
| k(t): | Mesh stiffness, N/mm |
| kf: | Thermal conductivity of lubricating oil, W/m·K |
| K1, K2: | Thermal conductivity of the gear material, W/m·K |
| kx1, kx2, ky1, ky2, kz1, kz2: | Equivalent bearing stiffness, N/mm |
| l: | Contact line of a single tooth surface, mm |
| lz: | Total length of contact line of each meshing tooth pair, mm |
| m1, m2: | Mass of gears, kg |
| mn: | Normal module, mm |
| Mf1, Mf2: | Friction torques, N·m |
| N: | Total meshing tooth pair, dimensionless |
| p: | Pressure, Gpa |
| r1, r2: | Radius of curvature, mm |
| rb1, rb2: | Radius of the base circle, mm |
| R: | Equivalent curvature radius, mm |
| s1, s2: | Surface roughness, mm |
| t: | Time, s |
| T1, T2: | Input and output torque, N·m |
| Tf: | Temperature of fluid, K |
| T0: | Environment temperature, K |
| ue: | Entrainment velocity, mm/s |
| us: | Sliding velocity, mm/s |
| ‾u1, ‾u2: | Nominal linear velocities, mm/s |
| z1, z2: | Number of teeth, dimensionless |
| α: | Viscous-pressure coefficient, Gpa−1 |
| βb: | Base helix angle, degree |
| βT: | Viscosity-temperature coefficient, K−1 |
| β: | Helix angle, degree |
| η, η′, η0: | Viscosity of the lubricant, Pa·s |
| θ1, θ2: | Torsional freedom, degree |
| ρ, ρ0: | Density of the lubricant, kg/m3 |
| τ, τx: | Surface shearing stress, Mpa |
| ω1: | Input rotational speed, rpm |
| ζ: | Slide-roll rate, dimensionless |
| δ(t): | Contact deformation of gear mesh, mm |
| μ: | Friction coefficient, dimensionless. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Hubei Key Laboratory of Modern Manufacturing Quality Engineering (Grant no. KFJJ-2022013) and the Core Technology Application of Hubei Agricultural Machinery Equipment (Grant no. HBSNYT202221).