Research on Time-Varying Meshing Stiffness of Marine Beveloid Gear System

Abstract

Beveloid gears have the advantages of compensating for axial error, providing smooth transmission, and eliminating turning error. Therefore, they are widely used in applications that require high transmission accuracy and stability. However, research on calculating the time-varying meshing stiffness of beveloid gears is still limited, and there is an urgent need to propose a method that can calculate the meshing stiffness of beveloid gears quickly and accurately. We first established the tooth profile expressions, assuming a pair of beveloid gears meshing with the same rack, and the contact line equations of parallel axis beveloid gear pairs were derived. Next, we analyzed the contact process of beveloid gears. We propose an analytical algorithm based on the slicing method to calculate the meshing stiffness of helical gears, straight beveloid gears, and helical beveloid gears. Then, the influence of different parameters on the meshing stiffness of helical beveloid gears was analyzed by changing the respective parameters. Finally, the finite element method (FEM) was used to verify the correctness of the analytical results, and then the errors were analyzed. The study demonstrates that the results obtained from the analytical algorithm we proposed have the same magnitude as those obtained by the FEM for the time-varying meshing stiffness calculation of beveloid gears.

1. Introduction

The power systems for ships are core components, and their functionality has a direct impact on the ship’s speed and safety stability [1,2]. The development of power systems has become a major focus in the global shipbuilding industry [3]. The primary transmission component for high-speed waterborne transportation is the high-speed ship gearbox [4,5,6]. For its specialized application, the high-speed ship gearbox is characterized by compact size, lightweight, and rigorous performance demands [7]. As the core component of the gearbox, the gear system is susceptible to malfunctions which can lead to the ship’s failure to operate normally, resulting in significant economic losses [8,9]. Whether the gear system can operate reliably or not becomes a bottleneck restricting the further development of smart ships [10]. Time-varying meshing stiffness stands as a fundamental aspect of mechanical system dynamics research, and it is also important in areas such as trains, wind turbines, bearings, and aircraft [11,12,13,14,15]. Beam A S. proposed the concept of beveloid gears in 1954 [16]. Nowadays, beveloid gears have been widely used in engineering practices, such as the gearbox of the Audi Q5, the ZF220A marine gearbox, and the planetary gear train of the RV reducer for robots [17,18].

Compared with traditional gears, the addendum modification factor varies linearly along the beveloid gears’ tooth-width direction. Compensating for axial error, eliminating turning errors, and providing smooth transmission can be accomplished by adjusting the beveloid gears’ axial position so that normal meshing can be ensured when gears’ wear or mesh clearance occurs. Furthermore, beveloid gears can meet a higher transmission ratio, helping to minimize system volume, and make the structure more compact. they also have better adaptation to complex operating conditions and can reduce system vibration [19,20].

The time-varying meshing stiffness of the beveloid gear is widely accepted as its inherent property that results exclusively from its design parameters and consequently exhibits periodic variations, enabling stiffness excitation of the gear system during transmission. Such an excitation affects the dynamic features of the system directly, such as noise and vibration, leading to a strong nonlinearity in the instant characterization of the system. The analytical study of time-varying meshing stiffness of beveloid gears not only helps in the development of a ship’s kinematic and dynamic properties which are critical for ship navigation and control, but also optimizes the performance of the drivetrain and provides a reference for the design of subsequent transmission systems. Therefore, there is a high demand for this research.

To analyze the time-varying meshing stiffness of beveloid gears, it is necessary to analyze the meshing contact process first. For the parametric design of beveloid gears, Li et al. [21] provided the entire design process of the beveloid gear as well as the procedure of calculating two lateral face parameters. Ni et al. [22] completed the geometric design of beveloid gears based on meshing theory and also investigated beveloid gear meshing properties. For the derivation of the tooth profile and the meshing equation of beveloid gears, K. Mitome [23] defined the ideal gear from the roll-cutting process of a beveloid gear and generated mathematical formulas for beveloid gear tooth profiles. Based on the basic principle of gear meshing, Yu et al. [24] derived the mathematical expressions for the tooth profile and mesh line equation for parallel-axis beveloid gear pairs. The time-varying meshing stiffness analysis also needs to consider the influence of machining methods, which directly affect the tooth geometry parameters [25,26]. From the above, the meshing contact process can be systematically analyzed and the relevant expressions can be obtained.

In terms of the analytical solution of traditional gear meshing stiffness, it is generally simplified into a two-dimensional plane problem. The solution is then based on the Ishikawa formula [27,28] and the Weber energy method [29,30]. However, the calculation of beveloid gears’ meshing stiffness cannot be solved as a two-dimensional problem. Therefore, Smith [31] proposed the slice theory. However, the influence of adjacent slices, inter-tooth gaps and other factors cannot be taken into account in the calculation process, and the results still have large errors. In addition, because the beveloid gear has a variable cross section along the axial and radial directions, the elastic deformation of the beveloid gear should be considered in the calculation and the method needs to be improved. Therefore, Wu et al. [32] proposed an improvement to the elastic deformation formula for calculating the meshing stiffness of cylindrical gears. Mao et al. [33] improved the Weber method based on the slicing principle. However, these methods can only accomplish the calculation of the meshing stiffness of variable beveloid gears with specific parameters and the systematic and generalized approach still presents a gap.

The finite element method (FEM) is the widely-chosen method among the various numerical methods for calculating the meshing stiffness of beveloid gears [34,35,36,37]. However, these researchers only analyzed models with specific parameters. Compared with FEM, the boundary element method (BEM) is typically specialized in solving a specific problem [38,39], and its pre- and post-processing workload is substantial, posing a significant undertaking in solving multiple types of gears. To calculate the meshing stiffness of beveloid gears, it is necessary to consider the forces and deformations of the entire tooth. In doing so, the advantages of BEM for dimension reduction disappear [40]. What is more, FEM can adapt to complex geometry and boundary conditions due to its versatility [41,42]. Although the fractal method is advantageous in analyzing nonlinear systems and complex dynamic phenomena, its computational complexity is high, and parameter selection has a considerable impact on the results [39,43,44]. Additionally, compared to the finite difference method (FDM), FEM is highly adaptable, and adaptive meshing can be utilized to balance accuracy and computational cost [45,46,47]. Therefore, in this study, we chose FEM as our numerical method to calculate the meshing stiffness of beveloid gears.

In conclusion, much research on the design of beveloid gear parameters, contact characteristics, and machining procedures is being conducted. From the above, designing specific beveloid gear pairs and completing machining operations according to the real application requirements can be carried out. However, existing research on the meshing stiffness of beveloid gears is limited, and the calculation process is only for beveloid gears with specific parameters. Therefore, research on the systematic and generalized calculation method of time-varying meshing stiffness of beveloid gears is required.

Given the importance of research on the time-varying meshing stiffness of beveloid gears for intelligent ships, we propose a method that can calculate the meshing stiffness of beveloid gears quickly and accurately. In this paper, we took involute beveloid gears as the primary object. In Section 2, the tooth profile expression of involute beveloid gears is obtained and the meshing contact position is analyzed. In Section 3, the analytical solution of the time-varying meshing stiffness of beveloid gears for different contact positions is calculated, and the influence of different gear parameters on the time-varying meshing stiffness is analyzed. In Section 4, the numerical solution of the time-varying meshing stiffness of a single tooth is calculated as well as of the multi-tooth, and the results are compared to ensure the accuracy of the analytical solution. In Section 5, the main conclusions are summarized.

2. Analysis of Meshing Contact Position of Beveloid Gears

To calculate the time-varying meshing stiffness of the beveloid gear, the force situation and normal deformation of the beveloid gear during the meshing process must be obtained. The force and direction of the meshing point at different contact positions are different which will affect the calculation of the meshing stiffness. To solve the time-varying meshing stiffness of beveloid gears accurately, the accurate tooth profile expression must be calculated first, then the meshing contact location of the beveloid gear pair is obtained. This section’s research strategy is as follows:

(1)

Starting from the normal tooth profile equation of the rack, combined with the meshing principle, the contact line equation of the gear and rack in the comoving coordinate system of the beveloid gear is obtained based on coordinate transformation.

(2)

Changing the size of the corner, the contact line equation at different contact positions is obtained, and the beveloid gear tooth profile is enveloped.

(3)

The left and right teeth-surface images of the beveloid gears are drawn as parametric equations and then verified for the correctness of the solution procedure of the beveloid-gear tooth-profile equation.

(4)

Letting a pair of beveloid gears mesh with the same rack, the beveloid gear is in line contact when meshing is demonstrated, and then the contact line equation at various corners is obtained.

2.1. Derivation of the Beveloid Gear’s Tooth-Surface Equation

Due to the complexity of the beveloid gear structure, it is difficult to calculate the tooth-profile expression for beveloid gears in each cross section directly. However, the expression is easier to obtain because the tooth profile of the gear teeth meshes with the beveloid gear in a straight line on the normal surface. Therefore, the tooth profile of beveloid gears can be calculated by first solving the tooth profile equation of the rack meshing with the beveloid gear. Then, calculating the tooth profile of the beveloid gear is indirectly based on the meshing principle.

For the left and right tooth surfaces of individual gear teeth to mesh on both sides of the tooth groove of the rack, the rack coordinate system must be established that takes the intersection of the normal plane of the rack tooth groove and the index line as the origin, as shown in Figure 1:

The basic normal parameters of the beveloid gear are shown in

Table 1. The basic normal parameters of the beveloid gear.

${\mathit{m}}_{\mathit{n}}$	$\mathit{z}$	${\mathit{\alpha }}_{\mathit{n}}$	${\mathit{h}}_{\mathit{a}}^{\ast }$	${\mathit{c}}^{\ast }$	$\mathit{\beta }$	$\mathit{\delta }$	$\mathit{B}$
Module	Number of teeth	Pressure angle	Addendumcoefficient	Tip clearancecoefficient	Helix angle	Taper angle	Tooth width

, below:

In the figure, the coordinate system $O_{T-XYZ}$ is the end coordinate system of the rack, and the coordinate system $O_{n-XYZ}$ is the normal coordinate system of the rack. The conversion matrix $R_{nT}$ from the normal to the end coordinate system can be expressed as:

$$R_{nT}=\left[\begin{array}{cccc}\cos \delta & -\sin \delta \sin \beta & \sin \delta \cos \beta & 0\\ 0& \cos \beta & \sin \beta & 0\\ -\sin \delta & -\sin \beta \cos \delta & \cos \beta \cos \delta & 0\\ 0& 0& 0& 1\end{array}\right]$$

(1)

Then, the tooth profile of the rack on the normal plane is two straight lines, and the shape of the rack tooth in the normal coordinate system is shown in Figure 2:

Taking the left tooth profile of the rack as an example, the equation of the tooth profile of the rack in the normal coordinate system $O_{n-XYZ}$ can be derived based on the tooth shape of the rack in the normal section:

$$x_n\tan {\alpha }_n+y_n+\frac{p}{4}=0$$

(2)

$p$—indexing rounded pitch of beveloid gears (mm).

Then, the left profile of the rack can be expressed in the coordinate system of the rack end face as:

$$x_T\left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)+y_T\cos \beta +z_T\left(-\sin \delta \tan {\alpha }_n-\sin \beta \cos \delta \right)+\frac{p}{4}=0$$

(3)

A coordinate system is established for the meshed beveloid gear rack, as shown in Figure 3. Where the coordinate system $O_{-XYZ}$ is the fixed coordinate system, the origin is located at the center of the circle of the large end face of the beveloid gear, and the y-axis points to the direction of the rack movement. The coordinate system $O_{T-XYZ}$ is the end-face coordinate system of the rack, which is the follower coordinate system of the rack. The distance from the fixed coordinate system in the x-direction is the gear pitch circle radius $r$. Coordinate system $O_{O-XYZ}$ is the follower coordinate system of the beveloid gear and the angle with the coordinate system $O_{-XYZ}$ is $\phi$.

According to the coordinate system established in Figure 3, the transformation matrix $R_{T0}$ from the coordinate system to the coordinate system is:

$$R_{T0}=\left[\begin{array}{cccc}\cos \phi & -\sin \phi & 0& -r\cos \phi -r\phi \sin \phi \\ \sin \phi & \cos \phi & 0& -r\sin \phi +r\phi \cos \phi \\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(4)

Then, in the coordinate system $O_{O-XYZ}$, the equation of the left tooth surface of the rack is expressed as:

$$\begin{array}{l}\left[x_0-\left(\frac{p}{4}\sin \delta \sin \beta \cos \phi +\frac{p}{4}\cos \beta \sin \phi -r\cos \phi -r\phi \sin \phi \right)\right]\times \\ \left[\cos \phi \left(\tan {\alpha }_n\cos \beta -\sin \beta \sin \delta \right)-\cos \beta \sin \phi \right]+\\ \left[y_0-\left(\frac{p}{4}\sin \delta \sin \beta \sin \phi -\frac{p}{4}\cos \beta \cos \phi -r\sin \phi +r\phi \cos \phi \right)\right]\times \\ \left[\sin \phi \left(\tan {\alpha }_n\cos \beta -\sin \beta \sin \delta \right)+\cos \beta \cos \phi \right]+\\ \left[z_0-\frac{p}{4}\sin \beta \cos \delta \right]\left(\tan {\alpha }_n\sin \delta +\sin \beta \cos \delta \right)=0\end{array}$$

(5)

Based on the principle of gear meshing, the tangential velocity of the gear rack should be perpendicular to the normal vector of the profile of the rack to make the gear rack mesh continuously. The coordinates of the meshing point satisfy the equation:

$$x_T\cos \beta -\left(y_T+r\phi \right)\left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)=0$$

(6)

Due to the meshing point being located on the tooth profile of the rack, the spatial equation of the mesh line of the gear and rack in the coordinate system $O_{T-XYZ}$ can be expressed as:

$$\left\{\begin{array}{l}\left[x_0-\left(\frac{p}{4}\sin \delta \sin \beta \cos \phi +\frac{p}{4}\cos \beta \sin \phi -r\cos \phi -r\phi \sin \phi \right)\right]\times \\ \left[\cos \phi \left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)-\cos \beta \sin \phi \right]+\\ \left[y_0-\left(\frac{p}{4}\sin \delta \sin \beta \sin \phi -\frac{p}{4}\cos \beta \cos \phi -r\sin \phi +r\phi \cos \phi \right)\right]\times \\ \left[\sin \phi \left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)+\cos \beta \cos \phi \right]+\\ \left[z_0-\frac{p}{4}\sin \beta \cos \delta \right]\left(\tan {\alpha }_n\sin \delta +\sin \beta \cos \delta \right)=0\\ \\ \left[x_0-\left(-r\cos \phi \right)\right]\times \left[\cos \beta \cos \phi +\left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)\sin \phi \right]+\\ \left[y_0-\left(-r\sin \phi \right)\right]\times \left[\cos \beta \sin \phi -\left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)\cos \phi \right]=0\end{array}\right.$$

(7)

Therefore, it can be verified that the mesh line of the beveloid gear and the rack is straight. Similarly, the equation of the mesh line between the right tooth profile of the beveloid gear and the rack can be obtained. Changing the corner $\phi$, the left and right tooth profiles of the beveloid gear can be enveloped. By ensembling the left and right tooth profile equations, the mathematical expression of the equation for the single-tooth profile equation of the beveloid gear can be obtained.

2.2. Beveloid Gears’ Tooth-Surface Drawing

To verify the correctness of the derived tooth-surface profile equation, data processing software is used to obtain the point set on the tooth surface of the beveloid gear. After Excel 2016 processing, the point is imported into the drawing software Origin 2018 to draw a three-dimensional tooth-surface image. Whether the tooth-profile-surface equation is correct is judged by the definition of the beveloid gear. The parameters selected during this verification are described in

Table 2. The basic parameters of the beveloid gear.

${\mathit{m}}_{\mathit{n}}$	$\mathit{z}$	${\mathit{\alpha }}_{\mathit{n}}$	${\mathit{h}}_{\mathit{a}}^{\ast }$	${\mathit{c}}^{\ast }$	$\mathit{\beta }$	$\mathit{\delta }$	$\mathit{B}$
4 mm	40	20°	1	0.25	10°	6°	40 mm

:

To facilitate the determination of the tooth-surface area of the beveloid gear, the coordinates $x_n$, $y_n$ and $z_n$ on the rack normal sections are used as parameters. After the coordinate transformation, the rack profile equation in the coordinate system $O_{O-XYZ}$ can be represented by the parameters $x_T$, $y_T$ and $z_T$:

$$\left\{\begin{array}{c}{x}_0=x_T\cos \phi -y_T\sin \phi -r\cos \phi -r\phi \sin \phi \\ y_0=x_T\sin \phi -y_T\cos \phi -r\sin \phi +r\phi \cos \phi \\ z_0=z_T\end{array}\right.$$

(8)

Based on the above, the corresponding value $\phi$ is obtained:

$$\phi =\frac{\frac{x_T\cos \beta }{\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta }-y_T}{r}$$

(9)

The value obtained in the formula is brought into the parametric Equation (8), and the dataset of the right tooth surface of the beveloid gear is obtained. The dataset of the left tooth surface is obtained by the same method for image drawing.

Drawing, as shown in Figure 4:

From the single-tooth-surface diagram of the beveloid gear plotted in Figure 4, it can be seen that the bottom is the large end of the beveloid gear. It is verified that the tooth profile of the variable beveloid gear teeth obtained by this method is involute in each section parallel to the x-y plane as well as meeting the definition of beveloid gears. So, the correctness of the tooth-profile-surface expression can be verified. According to the tooth-surface diagram of the graduated beveloid gear, it can be seen that the shapes of the left and right tooth surfaces of the helical beveloid gear are different. Therefore, it is necessary to calculate the deformation separately for the left and right tooth surfaces involved in meshing when calculating the stiffness.

2.3. Analysis of Parallel-Axis Meshing Contact Position of Beveloid Gears

According to the above conclusion, the straight line obtained by the intersection of the two planes of the contact line between the left and right tooth profiles and the rack of the beveloid gear can be obtained.

For parallel-axis meshing beveloid gears, a pair of gears meshing with each other can be considered to be meshing with the same rack at the same time, as shown in Figure 5.

For the pair of beveloid gears and racks in Figure 5, the coordinate system is established in the same way as shown in Figure 3. Assuming the right gear is the active wheel with rotates counterclockwise, regard the right gear as gear 1, and the left gear as gear 2, while the relevant parameters are distinguished by subscripts. When meshing, the left tooth surface of gear 1 and the right tooth surface of gear 2 mesh with each other, and the helix angle of the two has the same magnitude and the direction is opposite.

Based on a series of coordinate transformations, the meshing line equation of the right tooth surface of gear 2 and the rack is converted from the coordinate system $O_{O2-XYZ}$ to the coordinate system $O_{O1-XYZ}$, which can be expressed as:

$$\left\{\begin{array}{l}\left[x_0-\left(\frac{p}{4}\sin \delta \sin \beta \cos \phi +\frac{p}{4}\cos \beta \sin \phi -r\cos \phi -r\phi \sin \phi \right)\right]\times \\ \left[\cos \phi \left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)-\cos \beta \sin \phi \right]+\\ \left[y_0-\left(\frac{p}{4}\sin \delta \sin \beta \sin \phi -\frac{p}{4}\cos \beta \cos \phi -r\sin \phi +r\phi \cos \phi \right)\right]\times \\ \left[\sin \phi \left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)+\cos \beta \cos \phi \right]+\\ \left[z_0-\frac{p}{4}\sin \beta \cos \delta \right]\left(\tan {\alpha }_n\sin \delta +\sin \beta \cos \delta \right)=0\\ \\ \left[x_0-\left(-r\cos \phi \right)\right]\times \left[\cos \beta \cos \phi +\left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)\sin \phi \right]+\\ \left[y_0-\left(-r\sin \phi \right)\right]\times \left[\cos \beta \sin \phi -\left(\tan {\alpha }_n\cos \delta -\sin \beta \sin \delta \right)\cos \phi \right]=0\end{array}\right.$$

(10)

Equations (10) and (7) are the same. Therefore, the contact form of a pair of parallel-axis beveloid gears meshing with each other is line contact and the meshing line of the two gears is the same as the straight line are verified. At the same time, for $\phi$ in different values, it is also possible to solve for their real-time contact lines.

3. Analytical Solution of Time-Varying Meshing Stiffness of Beveloid Gears

3.1. Calculation of Time-Varying Meshing Stiffness of Helical Gears

An approximate model of gear tooth slices should be established first. Then, the time-varying meshing stiffness of helical gears can be calculated based on Ishikawa’s formula (Ishikawa’s formula is to simplify the gear teeth into a rectangular and trapezoidal combination of the cantilever beam. Then, the 30° section method is used to determine the dangerous section of the gear. Next, the deformation along the meshing line is divided into four parts: the bending deformation of the trapezoidal part, the bending deformation of the rectangular part, the deformation caused by shear, and the deformation of the basic part. The deformation of the gear tooth is obtained by superposing them, and then combining the contact deformation to obtain the single-tooth meshing stiffness through the stiffness calculation formula). The basic parameters of helical gears and their representative symbols are shown in

Table 3. The normal parameters of the helical gear and their representative symbols.

${\mathit{m}}_{\mathit{n}}$	$\mathit{z}$	${\mathit{\alpha }}_{\mathit{n}}$	${\mathit{h}}_{\mathit{a}}^{\ast }$	${\mathit{c}}^{\ast }$	$\mathit{\beta }$	$\mathit{x}$	$\mathit{B}$
Module	Number of teeth	Pressure angle	Addendumcoefficient	Tip clearance coefficient	Helix angle	Modification coefficient	Tooth width

.

According to the idea of the slice method, one tooth of the helical gear is equally divided into N slices along the tooth width direction, as shown in Figure 6:

Considering the gear tooth as a cantilever beam (the displacement and rotation angle at the tooth root position are always 0 during the meshing process), each gear tooth slice is simplified as a combination of a rectangle and an isosceles trapezoid, as shown in Figure 7. Based on the above, calculating the deformation of a single tooth translates into calculating the deformation of this approximation.

Calculation of each parameter:

The tooth thickness of the end faces the tooth apex circle of the gear tooth slice:

$$S_a=2r_{at}\sin \left(\frac{\pi +4x_t\tan {\alpha }_t}{2z}+inv{\alpha }_t-inv{\alpha }_{at}\right)$$

(11)

$r_{at}$—the radius of the end-face tooth top circle of the helical gear.

$$r_{at}=m_t\left(z+2h_t+2x_t\right)/2$$

(12)

• $m_t$—the end-face modulus;
• $h_t$—the end-face tooth-top height coefficient;
• $x_t$—the end-face displacement coefficient of the helical gear.
• ${\alpha }_t$—the pressure angle of the end-face indexing circle of the helical gear.
• ${\alpha }_{at}$—the pressure angle of the top circle of the end face of the helical gear.
• $r_{bt}$—base radius of the end face of the helical gear.
• $r_F$—radius of effective root circle of the end face.

  $$r_F=r_{at}-2m_t{h}_t^{\ast }$$

  (13)
• $s_f$—the tooth thickness of the end-face tooth root circle of the gear tooth slice:
• $h_r$—the height of the approximate rectangle:

When $r_F\ge r_{bt}$:

$$s_f=2r_F\sin \left(\frac{\pi +4x_t\tan {\alpha }_t}{2z}+inv{\alpha }_t-inv{\alpha }_F\right)$$

(14)

$$h_r=\sqrt{r_{bt}^2-\frac{s_f^2}{4}}-\sqrt{r_{ft}^2-\frac{s_f^2}{4}}$$

(15)

When $r_F<r_{bt}$:

$$s_f=2r_F\sin \left(\frac{\pi +4x_t\tan {\alpha }_t}{2z}+inv{\alpha }_t\right)$$

(16)

$$h_r=\sqrt{r_F^2-\frac{s_f^2}{4}}-\sqrt{r_{ft}^2-\frac{s_f^2}{4}}$$

(17)

• ${\alpha }_F$—effective root circle pressure angle of the helical gear end face.
• $r_{ft}$—radius of end-face root circle of the helical gear:

  $$r_{ft}=m_t\left(z-2h_t-2c_t+2x_t\right)/2$$

  (18)
• $c_t$—end-face top clearance coefficient of the helical gear.

Full tooth height of the tooth slice:

$$h=\sqrt{r_{at}^2-\frac{s_f^2}{4}}-\sqrt{r_{ft}^2-\frac{s_f^2}{4}}$$

(19)

Height of the intersection of the two waist extensions of the trapezoid:

$$h_i=\frac{s_{f}h-s_{a}h}{s_f-s_a}$$

(20)

For a pair of gears involved in meshing, let subscript 1 denote the active wheel and subscript 2 denote the driven wheel. Let the angle of rotation $\theta$ of the nth tooth slice just entering the mesh be 0. Then when the tooth slice of the wheel is rotated by an angle $\theta$, the radius at the position of the mesh point:

$$r_{x1}=\sqrt{r_{bt1}^2+{\left(r_{1B_1}\sin {\alpha }_{1B_1}+r_{bt1}\theta \right)}^2}$$

(21)

• ${\alpha }_{1B_1}$—pressure angle of the gear tooth slice entering the contact position.
• $r_{1B_1}$—radius of entering the contact position.

Radius at the contact position of the driven wheel:

$$r_{x2}=\sqrt{r_{bt2}^2+{\left[\left(r_{bt1}+r_{bt2}\right)\tan {\alpha }_t-r_{x1}\sin {\alpha }_{x1}\right]}^2}$$

(22)

${\alpha }_{x1}$—the meshing angle at the active wheel contact position.

The angle between the direction of force and the horizontal axis:

$${\omega }_x=\arccos \left(\frac{r_{bt}}{r_x}\right)-\left(\frac{\pi +4x_t\tan {\alpha }_t}{2z}+inv{\alpha }_t-inv{\alpha }_x\right)$$

(23)

Height of the force position:

$$h_x=r_x\cos \left({\alpha }_x-{\omega }_x\right)-\sqrt{r_{ft}^2-\frac{s_f^2}{4}}$$

(24)

The values of ${\omega }_x$ and $h_x$ for the active wheel and driven wheel are $r_{x1}$, ${\alpha }_{x1}$ and $r_{x2}$, ${\alpha }_{x2}$, respectively.

According to the Ishikawa formula, the bending deformation of the trapezoid:

$${\delta }_{Bt}=\frac{6h_i^3{F}_N{\cos }^2{\omega }_x}{Eb{s}_f^3}\left\{\frac{h_i-h_x}{h_i-h_r}\left[4-\frac{h_i-h_x}{h_i-h_r}\right]-2\ln \frac{h_i-h_x}{h_i-h_r}-3\right\}$$

(25)

E—modulus of elasticity of the gear.

The bending deformation of the rectangle:

$${\delta }_{Br}=\frac{12F_N{\cos }^2{\omega }_x}{Eb{s}_f^3}\left\{\left(h_x-h_r\right)h_x{h}_r+\frac{h_r^3}{3}\right\}$$

(26)

Shear deformation:

$${\delta }_S=\frac{2\left(1+v\right)F_N{\cos }^2{\omega }_x}{Eb{s}_f}\left\{h_r+\left(h_i-h_r\right)\ln \frac{h_i-h_r}{h_i-h_x}\right\}$$

(27)

$v$—Poisson’s ratio of the gear tooth section.

Deformation of the substrate:

$${\delta }_G=\frac{24F_N{h}_x^2{\cos }^2{\omega }_x}{\pi Eb{s}_f^2}$$

(28)

The total deformation of the nth gear tooth slice:

$$\delta ={\delta }_{Bt}+{\delta }_{Br}+{\delta }_S+{\delta }_G$$

(29)

The total deformation of the active and driven wheel tooth slices is calculated, and the contact deformation of the two slices is obtained according to the Hertz formula:

$${\delta }_{pV}=\frac{4F_N\left(1-v^2\right)}{\pi Eb}$$

(30)

The total deformation of the two corresponding gear slices in the process of meshing:

$${\delta }_n={\delta }_{n1}+{\delta }_{n2}+{\delta }_{npV}$$

(31)

The stiffness of the gear tooth slices:

$$K_n=\frac{F_N}{{\delta }_n}$$

(32)

Let the angle of rotation of the gear $\Theta =0$ when the first tooth slice starts to participate in the meshing process.

The rotation angle of the nth gear segment:

$$\theta =\Theta -\frac{\left(n-1\right)B\tan \beta }{N{r}_t}$$

(33)

$r_t$—radius of the indexing circle of the end face of the helical gear.

The angle of rotation of the gear when a single tooth slice is always meshed in a meshing cycle:

$$\phi =\frac{p_{bt}\epsilon }{r_{bt}}$$

(34)

$\epsilon$—end overlap of helical gears:

$$\epsilon =\left[z_1\left(\tan {\alpha }_{at1}-\tan {\alpha }_t\right)+z_2\left(\tan {\alpha }_{at2}-\tan {\alpha }_t\right)\right]/\left(2\pi \right)$$

(35)

$p_{bt}$—thickness of the end indexing circle of the helical gear.

The range of angles in which the nth tooth slice is involved in the mesh:

$$\left[\left(n-1\right)B\tan \beta \right]/N{r}_t\le \Theta \le \left[\left(n-1\right)B\tan \beta \right]/N{r}_t+\phi$$

(36)

Total stiffness of a single tooth mesh:

$$K\left(\Theta \right)={\displaystyle \sum _{n=m}^{m+i}{k}_n\left({\theta }_n\right)}$$

(37)

$i$—number of gear slices involved in simultaneous meshing.

The helical gear parameters in

Table 4. The basic parameters of the helical gear.

${\mathit{m}}_{\mathit{n}}$	$\mathit{z}$	${\mathit{\alpha }}_{\mathit{n}}$	${\mathit{h}}_{\mathit{a}}^{\ast }$	${\mathit{c}}^{\ast }$	$\mathit{\beta }$	$\mathit{x}$	$\mathit{B}$
4 mm	40	20°	1	0.25	5°	0	40 mm

are used to calculate the single-tooth stiffness at different corners. The calculated data is imported into Origin to plot the single-tooth time-varying meshing stiffness image of the helical gear as shown in Figure 8.

The multi-tooth meshing stiffness of the helical gear can be obtained by adding the meshing stiffnesses of multiple gear teeth, as shown in Figure 9. It can be seen that the helical gear is divided into a two-tooth meshing zone and a three-tooth meshing zone during the meshing process.

3.2. Calculation of Time-Varying Meshing Stiffness of Straight Beveloid Gears

While the helix angle is 0, the time-varying meshing stiffness of straight beveloid gears is calculated by using the Ishikawa formula and the slice method. The end-face modulus of the straight tooth beveloid gear is fixed, but ${\alpha }_n$, $h_t^{*}$, $c_t^{*}$ will change:

$${\alpha }_t=\arctan \left(\tan {\alpha }_n\cos \delta \right)$$

(38)

$$h_t^{\ast }=\frac{h_n^{\ast }}{\cos \delta }$$

(39)

$$c_t^{\ast }=\frac{c^{\ast }}{\cos \delta }$$

(40)

The displacement coefficient of the beveloid gear at different tooth-width positions is different. For the beveloid gear with the intermediate end-face displacement coefficient of 0, the displacement coefficient of the nth tooth slice satisfies:

$$x=\frac{\left[\frac{\left(n-1\right)B}{N}-\frac{B}{2}\right]\tan \delta }{m_t}$$

(41)

The meshing stiffness $K_n$ of individual gear slices can be calculated by substituting the parameters of straight beveloid gears into the formula above. Due to the different displacement coefficients of the gear slices at different tooth widths of the straight beveloid gear, the coincidence degree is different for each group of gear tooth slices when meshing.

After calculation, the slices at the middle-end face have the largest coincidence degree during meshing and are involved in the longest meshing process during the gear rotation process.

The stiffnesses of the gear tooth slices involved in simultaneous meshing were superimposed to obtain the total stiffness of single tooth meshing. The single-tooth stiffness at different corners was calculated by using the variable beveloid gear parameters except for the helix angle in Table 2. The calculated data was imported into Origin and then the image was produced of a single tooth time-varying meshing stiffness of beveloid gears at different angles of rotation and is shown in Figure 10. The image of single tooth time-varying meshing stiffness of beveloid gears at different angles of rotation is shown in Figure 11.

3.3. Calculation of Time-Varying Meshing Stiffness of Helical Beveloid Gears

For helical beveloid gears, the end-indexing circle pressure angles on the left and right tooth profiles are different and can be expressed as:

$${\alpha }_{tL}=\arctan \left(\frac{\tan {\alpha }_n\cos \delta }{\cos \beta }-\sin \delta \tan \beta \right)$$

(42)

$${\alpha }_{tR}=\arctan \left(\frac{\tan {\alpha }_n\cos \delta }{\cos \beta }+\sin \delta \tan \beta \right)$$

(43)

The size of the corresponding base circle on the left and right tooth surfaces:

$$r_{btL\left(R\right)}=\frac{1}{2}{m}_{t}z\cos {\alpha }_{L\left(R\right)}$$

(44)

Due to the tooth shape obtained by the slicing method being asymmetric, the meshing deformation cannot be calculated directly by Ishikawa’s simplification, so the tooth cross section of the helical beveloid gear is simplified into a combination of an oblique trapezoid and a rectangle, as shown in Figure 12.

From the calculation of the basic gear parameters, it was found that the top circular pressure angle and the root circular pressure angle on the left and right tooth surfaces were also different, so they should be calculated separately from the left and right tooth surfaces in the calculation of $s_{fL\left(R\right)}$, $s_{aL\left(R\right)}$, $h_{rL\left(R\right)}$. For the approximate tooth shape to be asymmetrical, its deformation needs to be recalculated using the formula of material mechanics. The parameters to be changed are calculated as follows:

When $r_F\ge r_{btL\left(R\right)}$:

$$s_{fL\left(R\right)}=r_F\sin \left(\frac{\pi +4x_t\tan {\alpha }_{tL\left(R\right)}}{2z}+inv{\alpha }_{tL\left(R\right)}-inv{\alpha }_{FtL\left(R\right)}\right)$$

(45)

$$h_{rL\left(R\right)}=\sqrt{r_{btL\left(R\right)}^2-\frac{s_{fL\left(R\right)}^2}{4}}-\sqrt{r_{ftL\left(R\right)}^2-\frac{s_{fL\left(R\right)}^2}{4}}$$

(46)

When $r_F<r_{btL\left(R\right)}$:

$$s_{fL\left(R\right)}=r_F\sin \left(\frac{\pi +4x_t\tan {\alpha }_{tL\left(R\right)}}{2z}+inv{\alpha }_{tL\left(R\right)}\right)$$

(47)

$$h_{rL\left(R\right)}=\sqrt{r_F^2-\frac{s_{fL\left(R\right)}^2}{4}}-\sqrt{r_{ft}^2-\frac{s_{fL\left(R\right)}^2}{4}}$$

(48)

Then:

$$S_{aL\left(R\right)}=r_{at}\sin \left(\frac{\pi +4x_t\tan {\alpha }_{tL\left(R\right)}}{2z}+inv{\alpha }_{tL\left(R\right)}-inv{\alpha }_{atL\left(R\right)}\right)$$

(49)

$$s_f=s_{fL}+s_{fR}$$

(50)

$$s_a=s_{aL}+s_{aR}$$

(51)

$$h_r=\frac{1}{2}\left(h_{rL}+h_{rR}\right)$$

(52)

Observe from the large end of the active wheel. Let the active wheel rotate counterclockwise, the left tooth surface of the active wheel is in contact with the right tooth surface of the driven wheel under force. Then, the pressure angle at the position where the slices of the wheel teeth start to mesh:

$${\alpha }_{1B_1}=\arctan \left[\frac{\left(r_{btL1}+r_{btR2}\right)\tan \left({\alpha }_{tL}\right)-r_{at2}\sin \left({\alpha }_{atR2}\right)}{r_{btL1}}\right]$$

(53)

The radius equation at the contact position corresponding to different angles:

$$r_{x1}=\sqrt{r_{btL1}^2+{\left(r_{1B_1}\sin {\alpha }_{1B_1}+r_{btL1}\theta \right)}^2}$$

(54)

The radius at the contact position of the driven wheel:

$$r_{x2}=\sqrt{r_{btR2}^2+{\left[\left(r_{btL1}+r_{btR2}\right)\tan {\alpha }_{tR}-r_{x1}\sin {\alpha }_{x1}\right]}^2}$$

(55)

The distance between the line of symmetry of the tooth thickness at the top of the tooth and $l$:

$$\Delta =\frac{s_a}{2}-s_{aR}-\frac{s_f}{2}+s_{fR}$$

(56)

According to the energy method, the bending deformation can be expressed as:

$${\delta }_B={\displaystyle {\int }_0^{h_x}\frac{F_N{\cos }^2{\omega }_x{\left(h_x-x\right)}^2}{E{I}_x}}dx$$

(57)

$I_x$—the moment of inertia of the approximate toothed section.

Moment of inertia of rectangular section:

$$I_{Br}=\frac{b{s}_f^3}{12}$$

(58)

Moment of inertia of trapezoidal section:

$$I_{Bt}=\frac{b{\left(h_i-x\right)}^3{s}_f^3}{12{\left(h_i-h_r\right)}^3}+{\left(\frac{\Delta x}{h-h_r}\right)}^{2}b\frac{\left(h_i-x\right)s_f}{h_i-h_r}$$

(59)

Then:

$${\delta }_B={\displaystyle {\int }_0^{h_x}\frac{F_N{\cos }^2{\omega }_x{\left(h_x-x\right)}^2}{E{I}_{Br}}}dx+{\displaystyle {\int }_0^{h_x}\frac{F_N{\cos }^2{\omega }_x{\left(h_x-x\right)}^2}{E{I}_{Bt}}}dx$$

(60)

Similarly, the energy method can be used to calculate the shear deformation of the approximate tooth shape:

$${\delta }_S={\displaystyle {\int }_0^{h_x}\frac{{\alpha }_S{F}_N{\cos }^2{\omega }_x}{2GA}}dx$$

(61)

• ${\alpha }_S$—the coefficients corresponding to the cross-sectional shape.
• $G$—shear modulus of the material.
• $A$—cross-sectional area of different sections.

The substrate deformation and contact deformation can be calculated by Ishikawa’s formula. For an intermeshing pair of tooth slices, the calculated deformations are summed to obtain the total deformation of the tooth slices. Then, the stiffness of the tooth slices is calculated by dividing the total deformation with the normal force. As shown in Figure 13, the image of the single tooth meshing stiffness in forward rotation is obtained by superposition. The image of the multi-tooth meshing stiffness in the forward rotation is shown in Figure 14:

Similarly, the images of the single-tooth and the multi-tooth meshing stiffness in reverse rotation are shown in Figure 15 and Figure 16.

After comparison, the change in single tooth meshing stiffness was the same for both gears in forward and reverse rotation, while the total overlap was relatively small in reverse rotation, resulting in the difference in the variable meshing stiffness image in the multi-tooth case.

The differences in the multi-tooth time-varying meshing stiffness of helical beveloid gears, helical gears, straight beveloid gears, and straight gears were compared.

The results are shown in Figure 17.

After comparison, it was found that the multi-tooth time-varying meshing stiffness of beveloid straight gears was smoother than that of straight gears. While the improvement was not large, and the stiffness image was similar to that of straight gears. Helical and helical beveloid gears changed more smoothly, the meshing was more stable and provided greater stiffness when meshing. Helical beveloid gears had a smaller stiffness variation range and were more stable during transmission under the premise of providing stiffness similar to helical gear size.

3.4. Analysis of Stiffness Influencing Factors

For the helical beveloid gears under forward rotation conditions, the basic parameters were changed. The influence of the module, number of teeth, pressure angle, helix angle, taper angle, and tooth width on the multi-tooth time-varying meshing stiffness was analyzed with the average meshing stiffness and fluctuation degree as indicators.

From Figure 18, it could be seen that as the normal modulus increased, the single-tooth meshing stiffness increased, while the total coincidence degrees decreased, which increased the degree of gear meshing fluctuation.

From Figure 19, it was found that the number of teeth had little effect on the degree of fluctuation.

From Figure 20, it was found that as the tooth thickness increased, the single-tooth meshing stiffness and the multi-tooth meshing stiffness increased. At the same time, the total coincidence degrees increased.

From Figure 21, it was found that as the helix angle increased, the single-tooth meshing stiffness and the multi-tooth meshing stiffness decreased, and the total coincidence degrees increased.

From Figure 22, it was found that as the taper angle increased, the single-tooth meshing stiffness and the multi-tooth meshing stiffness decreased, and the total coincidence degrees decreased.

4. Numerical Solution of Time-Varying Meshing Stiffness of Beveloid Gears

To verify the correctness and accuracy of the analytical method, three-dimensional models of helical gears and helical beveloid gears were established and then imported into ABAQUS 2018 for finite element simulation. The contact forces and contact displacements were read, and the respective time-varying meshing stiffnesses of single tooth and multi-tooth gears were calculated.

4.1. Finite Element Contact Analysis of Helical Gears

According to the definition of a helical gear, using the parameters in Table 4, a three-dimensional diagram of the helical gear was drawn, and after assembly, it was imported into ABAQUS for analysis. Firstly, the material of both gears was set as $45\#$ steel, with a density of $7.8\times {10}^{-9}T$/mm, the modulus of elasticity of 208,000 MPa, and Poisson’s ratio set to 0.27. Next, the inner surface of the gear was coupled about the gear center point, and the contact situation set for both gears: the friction coefficient was set to 0.1 in the normal behavior, and the tangential behavior was set to “hard contact” in the tangential behavior. Finally, the boundary conditions and loads of the two helical gears were set, and all degrees of freedom except axial rotation were restricted for the active wheel, and all degrees of freedom were fixed for the driven wheel, and a torque of $100,000 \mathrm{N}\cdot \mathrm{mm}$ was applied to the active wheel. To ensure that the result was easy to converge, a small rotation angle in the same direction as the torque was applied to the driving wheel in the analysis step. When meshing, a hexahedral-shaped mesh is used for helical gears. Helical gears are partitioned with the aim of reducing the calculation time and the active and driven wheels are divided into two blocks, with a circular column in the middle part, and the approximate global size of the layout points set to 5, and the approximate global size of the layout points at the tooth positions is 0.8. The mesh for the helical gears is shown in Figure 23:

After finite element analysis, the deformation and force at the contact position of the helical gear during meshing was obtained, as shown in Figure 24:

The contact displacement and contact force at the contact position were read by setting different rotation angles for the gear. The single-tooth meshing stiffness obtained from the finite element simulation was plotted simultaneously with the multi-tooth meshing stiffness and the analytical calculation results, as shown in Figure 25:

From Figure 25a, it can be seen that the trend of the single tooth meshing stiffness of the finite element calculation was consistent with the analytical calculation, but the single tooth meshing stiffness size was smaller than the analytical calculation. The maximum value of the result of the analytic calculation was 6.95 × 10⁸ N/m, and the difference between the two was about 27.5%. The possible reason for the above error is that the finite element calculation has a large span of the angle of rotation and does not take the maximum contact position of the gear tooth stiffness.

From Figure 25b, the trend of the multi-tooth time-varying meshing stiffness from the finite element calculation was approximately the same as that from the analytical calculation. However, the finite element calculation results were relatively larger and less volatile. By comparing the finite element calculation results and the analytical results, it was found that the multi-tooth meshing stiffness obtained by the two methods was consistent by the order of magnitude, but the finite element results were 13.6% larger than the analytical results. The difference in fluctuation degree was about 2.7%.

The finite element analysis results prove the correctness of the analytical calculation results as the difference between the two calculation methods for helical gears is considered to be within the acceptable range.

4.2. Finite Element Contact Analysis of Helical Beveloid Gears

The helical beveloid gear was simplified by removing the intermediate base part and establishing a model with only six complete gear teeth at the contact position. The same material, force, and boundary conditions were applied to the helical gear. In drawing the mesh, due to the complexity of the helical beveloid gear, a tetrahedral shape mesh was used at the boundary position of the gear teeth. The approximate global size of the cloth points at the gear teeth position was set to 0.8, and the approximate global size at the inner ring position was set to 5. The mesh for the helical beveloid gear is shown in Figure 26:

Where the black arrow points to is the meshing area of the tooth.

To obtain the time-varying meshing stiffness of the beveloid gear, it is necessary to make a pair of beveloid gear pairs mesh at different angles of rotation to derive the contact displacement and contact force at different angles of rotation. When the helical beveloid gears mesh, the magnitude of the forces at both ends of the contact line is relatively small, and this part of the data can be considered to be removed when calculating the stiffness. To ensure that the sampling can be carried out at the positions with the largest force, a point was selected for every 6 meshes along the tooth profile direction to complete the path creation. The contact data of all the teeth involved in meshing at the same angle of rotation were extracted. The average values of force and displacement were obtained by averaging, and then the meshing stiffness was calculated.

After finite element analysis, the deformation and force at the contact position of the helical beveloid gear during meshing could be obtained, as shown in Figure 27:

The single-tooth meshing stiffness obtained from the finite element simulation was plotted together with the multi-tooth meshing stiffness and the analytical calculation results, as shown in Figure 28:

From Figure 28, it can be seen that the finite element calculation of the single-tooth meshing stiffness and the multi-tooth meshing stiffness was greater than the analytical calculation results. For the single-tooth meshing stiffness, the maximum stiffness calculated by finite element was about 7.73 × 10⁸ N/m, while the maximum stiffness obtained by analytical calculation was 6.06 × 10⁸ N/m, with a difference of about 21.6%. The average multi-tooth meshing stiffness calculated by finite element was about 14.38 × 10⁸ N/m and the peak stiffness was 1.92 × 10⁸ N/m. Compared with the analytical results, it was found that the difference between the average stiffness and the peak stiffness was about 13.5%.

After comparison, the difference between the finite element calculation results and the analytical error results was found to be large. For helical beveloid gears, due to model complexity and insufficient modeling accuracy, there are problems with the process of importing the finite element software that result in large errors.

5. Conclusions

To develop the study of the dynamic performance of beveloid gears, the time-varying meshing stiffness of helical gears, straight beveloid gears, and helical beveloid gears were calculated based on the relevant theory of slicing method and Ishikawa formula. Then, the correctness of the calculation results was verified. The following are the main conclusions:

• Starting with the rack’s normal coordinate system, the equation for the rack tooth shape in the rack end coordinate system was obtained via a coordinate transformation. The meshing line equation in the rack-end coordinate system was then obtained by combining it with the meshing principle. The meshing line equation was then transferred into the beveloid gear’s follow-up coordinate system through coordinate transformation. By adjusting the gear rotation angle, the beveloid gear’s tooth-surface equation was enveloped. The equation of the rack in the normal coordinate system was then expressed as a parameter, and the parameter equation of the tooth surface of the beveloid gear was obtained. The value of parameters can be changed to draw the left and right tooth surface images of the beveloid gear to confirm the correctness of the equation derivation process. For a pair of beveloid gears that mesh with the same rack, assuming the meshing line of the parallel-axis beveloid gear is a straight line, the equation expression of the meshing line in space at different moments is given.
• Starting with the helical gear, the gear tooth slice was evenly divided in the direction of tooth width. To calculate the deformation of each gear tooth slice, the stiffness of a single gear tooth slice was obtained by using the Ishikawa formula, and then the single tooth meshing stiffness was obtained by superposition. Unfolding the helical gear’s working plane, the contact of the adjacent gear teeth was analyzed, and the meshing stiffness of the multi-tooth was obtained by superimposing the stiffness of each distinct gear tooth at the same angle. For straight beveloid gears, the characteristics of different coincidence degrees of meshing between different layer gear tooth slices were highlighted, and the time-varying meshing stiffness was determined. Due to the asymmetry of the end face’s left and right tooth faces, the end tooth shape was approximated as a combination of a rectangle and an oblique trapezoid. Then the deformation of the contact position was calculated based on the energy principle, and the meshing stiffness of the beveloid gear was calculated. The time-varying meshing stiffness of the beveloid gear pair was calculated separately under forward and reverse rotation, from which it was found that the changing trend of single-tooth meshing stiffness in both forward and reverse rotation was consistent. However, the total coincidence degrees during a reversal was relatively small, resulting in a difference in the multi-tooth meshing stiffness image. Comparing the meshing stiffness of straight gear, helical gear, straight beveloid gear, and helical beveloid gear when multi-tooth meshing, it was found that the meshing stiffness of the straight gear and straight beveloid gear was not significantly different, whereas the average meshing stiffness of helical gear and helical beveloid gear was larger and the degree of fluctuation was smaller. The basic parameters of the helical beveloid gear were then altered, and the effects of Module, Number of teeth, Pressure angle, Helix angle, Taper angle, and Tooth width on the meshing stiffness of the multi-tooth were investigated using the average meshing stiffness and fluctuation degree as indicators.
• The static analysis of helical gears and helical beveloid gears was performed using ABAQUS. Setting different gear corners and reading the contact displacement and contact force at the contact position, the single-tooth and multi-tooth meshing stiffnesses were calculated. The maximum stiffness was utilized as the standard for single-tooth meshing stiffness. For multi-tooth meshing stiffness, the difference between the analytical and finite element results was determined using the average stiffness and fluctuation degree. After calculation and comparison, it was determined that the difference between the results obtained by the two calculation methods was within an acceptable range. In other words, the finite element analysis results verify the correctness of the analytical calculation results. For helical beveloid gears, due to model complexity and insufficient modeling accuracy, there are problems in the process of importing the finite element software that result in large errors.