^{1}

^{1}

^{1}

^{2}

^{1}

^{2}

The tooth surfaces of beveloid gears have different topography features due to machining methods, manufacturing accuracies, and surface wear, which will affect the contact state of the tooth surface, thereby affecting time-varying mesh stiffness between mating gear pairs. Therefore, a slice grouping method was proposed in this paper on the basis of potential energy to calculate the total meshing stiffness of beveloid gears with the surface topography. The method in this paper was verified by finite element method (FEM). Compared with the calculation results of this paper, the relative error is 5.9%, which demonstrated the feasibility and accuracy of the method in this paper. Then, the influence of parameters such as pressure angle, helix angle, pitch angle, tooth width, fractal dimension, and fractal roughness on meshing stiffness was investigated, of which results show that pressure angle, pitch angle, tooth width, and fractal dimension have an incremental impact on the mean value of mesh stiffness. However, the fluctuating value of mesh stiffness has also increased as the pressure angle, tooth width, and pitch cone angle increase. Both the helix angle and the fractal roughness have a depressive impact on the total stiffness. But the difference is that, with the increase of the helix angle, the fluctuation of meshing stiffness has been decreased. Conversely, with the increase of the fractal roughness, the fluctuation of meshing stiffness has been increased.

Involute beveloid gears, which were first proposed by Mettitt in 1954, have the variable profile shift modification coefficient along axes direction. Beveloid gears can be used to realize the transmission form of parallel shafts, intersected shafts, and crossed shafts, which can be used in four-wheel-drive transfer cases and marine gearboxes with a down angle. However, due to beveloid gears having different addendum circles, dedendum circles, and pressure angles on different cross-sections, it is difficult to calculate the total mesh stiffness of beveloid gears. Furthermore, due to machining methods, manufacturing accuracies, and surface wear, the tooth surface of beveloid gears is always not smooth, and the surface topography features will affect the contact state of beveloid gears, thereby affecting time-varying mesh stiffness between mating gear pairs and changing the dynamic characteristics of transmission systems. Therefore, research on the time-varying meshing parameters of the beveloid gear is beneficial to improve the meshing characteristics of the beveloid gear and enhance the transmission stability of beveloid gears.

To accurately and efficiently calculate the time-varying meshing stiffness of gears, domestic and foreign scholars have done a lot of research. The current methods for calculating the time-varying meshing stiffness of gears generally include the potential energy method and the finite element method (FEM) from the literature in recent years. For the FEM, Tang proposed the gear meshing stiffness calculation method based on the finite element numerical calculation and gave the relationship between the modification parameters and the mesh stiffness of the modification gear [

The motivation of this paper is to provide a numerical calculation method of meshing stiffness for the beveloid gear with the surface topography. The current researches on the meshing stiffness of beveloid gears were based on the assumption that the tooth surface is absolutely smooth. But in fact, due to machining methods, manufacturing accuracies, and surface wear, the tooth surface of beveloid gears is always not smooth. In view of this, the slicing idea was used in this paper to discretize the thickness of the beveloid gear teeth and comprehensively analyzed the influence of the tooth surface friction and the tooth surface topography on the meshing stiffness throughout the meshing operation. The slice grouping method was proposed on the basis of potential energy to calculate the total meshing stiffness of beveloid gears with the rough surface topography.

The sections of this paper are arranged as follows. Section

In the traditional potential energy method, the total potential energy stored in the mesh gear system was assumed to include five components: Hertzian energy _{h}, bending energy _{b}, shear energy _{s}, axial compressive energy _{a}, and base body potential energy _{f}. They can be used to calculate Hertzian mesh stiffness kh, bending mesh stiffness _{b}, shear mesh stiffness _{s}, axial compressive stiffness _{a}, and fillet-foundation stiffness _{f}, respectively. Through the knowledge of elastic mechanics, the five parts are separately calculated for the stiffness components. And then, the five stiffness components are combined to obtain the total mesh stiffness. However, because of the special tapered tooth shape of beveloid gears, the traditional potential energy method cannot calculate the mesh stiffness directly. Besides, the conventional potential energy method does not consider the influence of tooth surface friction and tooth surface topography. In summary, an improved potential energy method based on fractal theory was proposed to calculate the total meshing stiffness for beveloid gears of a rough surface in this paper.

Because the axis-direction modification coefficient of beveloid gears changes linearly, the beveloid gear can be regarded as a continuous superposition of a certain number of slices with the same thickness and different modification coefficients. As shown in Figure

Slicing diagram of beveloid gears.

To accurately calculate each slice’s meshing stiffness, the teeth of beveloid gears are simplified into a variable cross-sectional cantilever beam with variable linear along the axial direction. In Figure _{i}, and the cross-sectional modulus is _{i}, the distance between any meshing point _{b}. _{b} is the base circle radius. _{a} and _{f} are the addendum circle radius and the dedendum circle radius. _{b} is the half tooth thickness at any meshing point _{n} is the normal meshing force on the tooth surface. _{x} and _{y} are the partial force of the normal meshing force _{n} along the directions

Geometry diagram of beveloid gear teeth.

Based on the potential energy method and the slice grouping method, the five components of the potential energy of each slice under the action of normal meshing force

According to the cantilever beam model and the knowledge of elastic mechanics, the stored bending potential energy, shear potential energy, and compressive potential energy of each slice are expressed as

The component force in the

Equivalent bending moment can be expressed as_{k} is the working pressure angle at any meshing point _{0} is the pressure angle at the pitch point.

Due to the relative sliding speed, the rolling ratio and other parameters are constantly changing. The tooth surface friction coefficient has changed with time, and the direction of tooth surface friction has changed at the pitch point. Xu and Kahraman [_{ek} is the entrainment velocity. _{i}(_{k} is the rolling ratio. _{hk} is the tooth surface contact pressure, and _{k} is the combined radius of curvature at any meshing point

Moreover, due to the fact that the effective contact part of the beveloid gear teeth can be regarded as the trapezoidal section in the axial direction, the calculation formula of _{i}, _{i}, and _{ib} and _{is} are the tooth thickness of the large section and the small section of beveloid gears. _{e} is the equivalent modulus of elasticity.

The bending stiffness

Therefore, the bending stiffness of beveloid gears can be expressed as

Similarly, the shear stiffness _{s} and compressive stiffness _{a} of beveloid gears can be expressed as

The fillet-foundation stiffness _{f} of beveloid gears can be calculated by the following equation:_{f} and _{f} are shown in Figure ^{∗}, ^{∗}, ^{∗}, and ^{∗} are constants related to _{a}, _{f}, and _{f}. The calculation method is detailed in the literature [

Generally, Hertz’s theory holds that the tooth surfaces in contact with each other are frictionless, and the contact deformation of the meshing area has a linear relationship with the normal meshing force. Due to machine body error, installation error, deformation error, and other factors, the tooth surface is generally uneven and rugged, and the machined surface can be characterized by continuity, nondifferentiability, and statistically self-affinity. As shown in Figure

Rough tooth surface contact diagram.

Miao and Huang [

Similarly, the plastic critical index can be defined as

The asperities are in elastic deformation when _{min} < _{ec}, and the contact stiffness _{nec} in the elastic regime can be expressed by integrating into the whole contact surface as

The asperities are in elastic-plastic deformation when _{ec} < _{pc}, and the contact stiffness _{npc} in the elastic-plastic regimes can be expressed by integrating into the whole contact surface as

The asperities are in plastic deformation when _{pc} < _{max}, and the contact stiffness in the plastic regimes should take 0.

It follows that the total contact stiffness based on the fractal model can be given by

Therefore, the total mesh stiffness of the beveloid gear can be given by combining the five stiffness components as

According to the space meshing principles and the processing method of beveloid gears, the tooth face equations of working tooth surface and the tooth root surface of beveloid gears are derived utilizing the virtual rack cutter’s tooth face equations. As shown in Figure

In the coordinate system _{n}, the tooth face equation of the straight edges

In the coordinate system _{n}, the tooth face equation of the fillet curves

Normal cross section of a rack cutter.

Figure _{n} ⟶ _{p} ⟶ _{c}. Then, the virtual rack cutter surfaces can be expressed in coordinate system _{c} as follows:

Coordinate relationship between the normal cross section of virtual rack and the beveloid gear.

The tooth face equations of the rack cutter in the coordinate system _{1} can be derived from the tooth surface equation of the rack cutter in coordinate system _{c} by using a series of transformation matrices given by _{c} ⟶ _{0} ⟶ _{1}. The tooth face equations can be expressed in coordinate system _{1} as follows:

Through the relative movement of the rack and pinion, the tooth surface equation of the beveloid gear can be derived from the tooth surface equation of the rack cutter in the coordinate system _{1}. And the beveloid gears working tooth surfaces can be expressed as follows: _{c}, _{c}, and _{c} are the position vector of the working surface of the rack in the coordinate system _{c}. _{c}.

The rough tooth surface topography can be considered a series of randomly distributed asperities superimposed on the theoretical tooth surface’s normal direction. Therefore, the mathematical model of beveloid gears considering the surface topography characteristics can be expressed as

The height of the asperities on the rough tooth surface can be given as_{x} and _{y} are the fractal dimension of the rough surface in the directions _{x} and _{y} are the fractal roughness of the rough surface in the directions _{1} and _{2} are the number of sampling points within a finite length of the rough surface in the directions

As shown in Figures

Simulation of three-dimensional surface topography under different fractal parameters: (a) _{x} = _{y} and _{x} = _{y}, (b) _{x} = _{y} and _{x} > _{y}, (c) _{x} = _{y} and _{x}<_{y}, (d) _{x}>_{y} and _{x} = _{y}, and (e) _{x} < _{y} and _{x} = _{y}.

Parameters and material properties.

Parameter and properties | Value |
---|---|

Number of teeth | 40 |

Module (mm) | 4 |

Spiral direction | Right |

Normal pressure angle (deg) | 20 |

Helix angle (deg) | 8 |

Tooth surfaces topography characteristics of the gear specimen with uniform wear. (a) Tooth surface measurement processes of the beveloid gear. (b) The beveloid gear specimens. (c) Measurement results of tooth surfaces of the specimens.

The contour curve fluctuation value of tooth surface: (a) the tooth profile direction and (b) the tooth width direction.

It can be seen that the surfaces of beveloid gear specimens present prominent topography characteristics, which are a series of ravines and ridges in Figure

According to Qi et al. [^{2} represents the arithmetic mean of the difference.

The double logarithmic plot of log (_{s} is scale coefficient (0 < _{s} < 1).

The double logarithmic plots of structure-function method (SFM) of different surface direction: (a) the tooth profile direction and (b) the tooth width direction.

The fractal parameters of different tooth surface direction.

Fractal dimension | Fractal roughness | |
---|---|---|

The tooth profile direction | 1.45 | 6.69 |

The tooth width direction | 1.63 | 3.34 |

The tooth surfaces of beveloid gears have different topography characteristics due to different machining methods, manufacturing accuracies, and surface wear, and they are always not smooth. And the rough tooth surface topography characteristics can be considered a series of randomly distributed asperities superimposed on the theoretical tooth surface’s normal direction. Based on the fractal parameters in Table

The main geometry parameters of the beveloid gear pair.

Parameter and properties | Pinion | Gear | Parameter and properties | Pinion | Gear |
---|---|---|---|---|---|

Number of teeth | 40 | 25 | Spiral direction | Right | Left |

Module (mm) | 4 | Width (mm) | 38 | 40 | |

Pitch angle (deg) | 10 | 0 | Young’s modulus (GPa) | 209 | |

Normal pressure angle (deg) | 20 | 20 | Poisson’s ratio | 0.26 | |

Helix angle (deg) | 8 | 1.25 |

The solid models of the beveloid gear with rough surface topography. (a) The solid models of the beveloid gear. (b) The solid models of a single tooth. (c) The spatial point sets of the rough tooth surface.

The FEM was used to verify the mesh stiffness calculation model proposed in this paper. The main geometry parameters of the beveloid gear pair are shown in Table

Finite element mesh model.

Flowchart diagram for the total mesh stiffness of beveloid gears.

Figure _{l} is the transmission error.

The distribution of the contact stress.

Figure

Comparative analysis of meshing stiffness.

Time-varying mesh stiffness is one of the main reasons leading to the vibration in a gear transmission system. Analyzing the influencing factors of beveloid gears mesh stiffness has great significance for improving its meshing characteristics and transmission stability. Compared with traditional involute cylindrical gears, beveloid gears have different addendum circle, dedendum circle, and pressure angle on different cross-sections, and the meshing characteristics are more complicated. The geometry parameters of beveloid gears mainly include pressure angle, helix angle, pitch angle, and tooth width. We define fluctuating value as the max–min of time-varying mesh stiffness divided by the mean value to evaluate the fluctuation degree. In this paper, the influence of the geometry parameters and surface topography of beveloid gears on the meshing stiffness was investigated by changing the geometry parameters shown in Table

Figure

Impacts of pressure on mesh stiffness: (a) a time-varying value of mesh stiffness and (b) mean value and fluctuating value of mesh stiffness.

The impacts of different helix angles on the total mesh stiffness are shown in Figure

Impacts of helix angle on mesh stiffness: (a) a time-varying value of mesh stiffness and (b) mean value and fluctuating value of mesh stiffness.

The effects of different pitch angles on the time-varying mesh stiffness of beveloid gears are shown in Figure

Impacts of pitch angle on mesh stiffness: (a) a time-varying value of mesh stiffness and (b) mean value and fluctuating value of mesh stiffness.

The effects of different tooth widths on the time-varying mesh stiffness of beveloid gears are shown in Figure

Impacts of tooth width on mesh stiffness: (a) a time-varying value of mesh stiffness and (b) mean value and fluctuating value of mesh stiffness.

According to Mandelbrot’s fractal theory, the value of fractal dimension

Impacts of fractal dimension on mesh stiffness: (a) a time-varying value of mesh stiffness and (b) mean value and fluctuating value of mesh stiffness.

Figure

Impacts of fractal roughness on mesh stiffness: (a) a time-varying value of mesh stiffness and (b) mean value and fluctuating value of mesh stiffness.

This paper concerns the numerical calculation method of meshing stiffness for the beveloid gear with the surface topography, and the influence of the geometry parameters and surface topography parameters of beveloid gears on the meshing stiffness was investigated. The following conclusions are drawn:

In this study, an improved potential energy method based on the slice grouping method was presented to calculate the mesh stiffness for beveloid gears with the tooth surface topography. The proposed method in this paper was verified by FEM. Compared with the calculation results of this paper, the relative error is 5.9%, which demonstrated the feasibility and accuracy of the method in this paper.

Based on fractal theory and the space meshing principles, a mathematical model was derived to describe local topography features of actual tooth surfaces of beveloid gears, and the fractal parameters were calculated by the SFM. The analytical results show that fractal dimension

The analytical results show that pressure angle, pitch angle, tooth width, and fractal dimension have an incremental impact on the total stiffness of beveloid gears. However, the fluctuating value of mesh stiffness has also increased as the pressure angle, tooth width, and pitch cone angle increase. It is worth emphasizing that, with the increase of the fractal dimension, the mean value of meshing stiffness has been improved, but the fluctuation of meshing stiffness has been decreased.

Both the helix angle and the fractal roughness have a depressive impact on the total stiffness of beveloid gears. But the difference is that, with the increase of the helix angle, the fluctuation of meshing stiffness has been decreased, which means that the increase of the helix angle can improve the gear transmission’s stability. Conversely, with the increase of the fractal roughness, the fluctuation of meshing stiffness has been increased, which indicates that the roughness of the tooth surface is one of the causes of vibration and noise.

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this paper.

The authors wish to thank the Key Laboratory of Advanced Manufacturing and Intelligent Technology, Ministry of Education, Harbin University of Science and Technology, and School of Mechatronics Engineering, Harbin Institute of Technology, for providing the technical support. The research was funded by the National Key Research and Development Project of China (Grant no. 2019YFB2006400); the Major Science and Technology Projects of Heilongjiang Province (Grant no. 2019ZX03A02).