Investigation on the Dynamic Characteristics of Non-Orthogonal Helical Face Gears with Higher-Order Tooth Surface Modification ^†

Abstract

A study on the dynamic characteristics of non-orthogonal helical face gears with higher-order tooth surface modification is presented in this article. The method of designing the non-orthogonal helical face gears with higher-order tooth surface modification is described. First, MATLAB programming that can be used for the parameterized 3D mesh calculations of non-orthogonal helical face gears with higher-order tooth surface modification are completed. Second, the calculated grid nodes from the MATLAB programming are imported into ABAQUS to generate a three-dimensional mode. The meshing stiffness of the gear pair is then estimated using finite element analysis. Ultimately, a dynamic model of a non-orthogonal helical face gear pair involving second-order and higher-order tooth surface modifications is established. One example is presented to study the dynamic characteristics of non-orthogonal helical face gear pairs with second-order and higher-order tooth surface modifications. The results show that the dynamic response from the second-order tooth surface modification has a higher peak-to-peak amplitude than that of the higher-order modification.

1. Introduction

Face gear transmission has advantages such as a compact structure, high load-bearing capacity, and large transmission ratio. Face gears are mainly used in aviation helicopter transmissions. In actual working conditions, various errors, such as manufacturing errors and installation errors, etc., can lead to an uneven load distribution and increased vibration and noise. In order to reduce the influence of various errors, it is necessary to perform tooth surface modifications on face gears [1,2,3]. In helicopters, face gears play a role in power transmission, with high rotational speeds, and their dynamic performance directly affects the noise level, stability and safety of helicopter operation [4].

For a long time, tooth modifications and the dynamic performances of gears have been hot topics in the field of gear transmission research [5,6,7,8,9,10,11]. Researchers have been exploring new tooth surface modification methods to improve the transmission performances of gears. The higher-order tooth surface modification is a relatively novel method that has attracted the attention of researchers in the industry [12,13]. Jiang et al. [14] used particle swarm optimization (PSO) to identify the coefficients of higher-order polynomials and demonstrated the viability of higher-order modification by comparing and contrasting load transmission errors before and after modification. Jia et al. [15] analyzed the load error transmission amplitude (ALTE) after a high-order transmission error correction of cylindrical gears, supporting the benefits of higher-order transmission error modification in the reduction of the vibration and noise of gears. Su et al. [16] verified that higher-order transmission error modification can reduce the bending stress of spiral bevel gears and improve the meshing performance of spiral bevel gears by conducting a loaded tooth contact simulation (LTCA) on helical bevel gears with seventh-order transmission error modification. Li et al. [17] conducted a tooth contact analysis (TCA) on a paired pinion modified with a higher-order parabolic curve to study the influence of the meshing performance of the opposite gear pair. Wei et al. [18] created a dynamic model of a single-stage gear system and analyzed the dynamic response of a gear transmission system. Shi et al. [19] proposed a dynamic model of a hypoid gear that takes into account meshing stiffness changes under dynamic situations. The model presented here offered a physics-based way for quantifying the relationship between the meshing stiffness and dynamic meshing force, which could describe the complex dynamic response of hypoid gear meshing. Peng et al. [20] presented a surface-active modification (SAM) approach to minimize face gear driving vibration. This study provides a theoretical foundation for the vibration reduction design of face gear pairs. Liu et al. [21] proposed a dynamic wear prediction approach to analyze the coupling impact between surface wear and spur gear system dynamics.

From the discussion above, it can be seen that the current research on the tooth modifications of face gear transmissions is not enough. As is well known, the speeds and loads of aviation gears are very high. High loads and high speeds will bring greater thermal deformation and elastic deformation to the face gear transmission system. Therefore, it is of great significance to study the tooth modifications of face gear transmissions and evaluate their dynamic performances after modification to ensure the quality of helicopters.

In order to further study the higher-order modification design and meshing performances of face gears, this paper presents a dynamic model of a non-orthogonal helical face gear transmission involving second-order and higher-order tooth surface modifications. Based on the established model, the dynamic performance of a higher-order-modified face gear transmission was deeply studied. The calculation results show that the dynamic response from second-order tooth surface modification has a higher peak-to-peak amplitude than that of the higher-order modification.

2. Generation of the Tooth Surfaces of Non-Orthogonal Helical Face Gears

This study proposes a methodology for machining face gears using a grinding wheel. The grinding wheel can be considered a single gear shaper cutter tooth that rotates in parallel with the direction of the tooth surface. The processing principle is depicted in Figure 1.

The grinding process involves three distinct motions, as illustrated in Figure 1: the dish-shaped grinding wheel swings around axis $Z_s$ of the virtual gear shaper cutter at an angular speed ${\omega }_s$, the face gear rotates around its own rotating $Z_2$ axis at an angular speed ${\omega }_2$, and the dish-shaped grinding wheel rapidly rotates around its own rotating $X_g$ axis to generate a cutting motion.

These motions together generate the theoretical tooth surface of non-orthogonal helical face gear; the machining coordinate system of non-orthogonal helical face gear was established according to its motion relation, where, $S_2$, $S_g$ and $S_s$ were rigidly connected with a non-orthogonal helical face gear, a dish-shaped grinding wheel and virtual gear shaper cutter, respectively, and $Z_2$, $Z_g$ and $Z_s$ were the rotation axes of the non-orthogonal helical face gear, dish-shaped grinding wheel and virtual gear shaper cutter, respectively.

The distance between the origins $O_2$ and $O_s$ of the two coordinate systems $S_2$ and $S_s$ is $L_0$. The angle $\gamma$ formed by the rotation axis of the virtual gear shaper cutter and the rotation axis of the face gear is the shaft angle of the axis, and ${\gamma }_m$ can be calculated by ${\gamma }_m={180}^{\circ }-\gamma$. The center distance between the dish-shaped grinding wheel and the virtual gear shaper cutter is $E_g$, which is defined as $E_g=r_{pg}-r_{ps}$, where $r_{ps}$ represents the pitch circle radius of the virtual gear shaper cutter, and $r_{pg}$ represents the pitch circle radius of the dish-shaped grinding wheel. $l_g$ is the moving distance of the center of the dish-shaped grinding wheel.

The grinding wheel surface group, which oscillates with the virtual gear shaper cutter based on the tooth surface equation of the dish-shaped wheel and the relationship of the machining motion, is formed. Subsequently, by combining this surface group with the meshing equation, the tooth surface of the face gear is obtained. The tooth surface of the face gear is determined by Equation (1):

$$\left\{\begin{array}{l}{\stackrel{\rightharpoonup }{r}}_g(u_2,{\theta }_g)=M_{gs}{\stackrel{\rightharpoonup }{r}}_s(u_2)\\ {\stackrel{\rightharpoonup }{r}}_2(u_2,l_g,{\phi }_g,{\theta }_g)=M_{2g}{\stackrel{\rightharpoonup }{r}}_g(u_2,{\theta }_g)\\ f_1={\stackrel{\rightharpoonup }{n}}_g(u_2,{\theta }_g){\stackrel{\rightharpoonup }{v}}_{g1}(u_2,l_g,{\phi }_g,{\theta }_g)=0\\ f_2={\stackrel{\rightharpoonup }{n}}_g(u_2,{\theta }_g){\stackrel{\rightharpoonup }{v}}_{g2}(u_2,l_g,{\phi }_g,{\theta }_g)=0\end{array}\right.$$

(1)

where ${\stackrel{\rightharpoonup }{r}}_g$ is the position vector of the dish-shaped grinding wheel surface, ${\theta }_g$ is the rotation angle of the grinding wheel around its axis, and $M_{gs}$ represents the transformation matrix from coordinate system $S_s$ to coordinate system $S_g$; ${\stackrel{\rightharpoonup }{r}}_2$ is the tooth surface equation of the non-orthogonal helical face gear, ${\phi }_g$ is the rotation angle of the grinding wheel around axis $Z_s$ of the virtual gear shaper cutter, and $M_{2g}$ represents the transformation matrix from coordinate system $S_g$ to coordinate system $S_2$; and $f_1$ and $f_2$ indicate the two meshing equations in $S_s$, and ${\stackrel{\rightharpoonup }{v}}_{g1}$ and ${\stackrel{\rightharpoonup }{v}}_{g2}$ are, respectively, the central velocity of the grinding wheel and the relative velocity between the non-orthogonal helical face gear and the grinding wheel.

3. Designation of the Tooth Modifications of Face Gears

In this study, only the tooth surface of a pinion was modified. The pinion was double-crowned through the modified tooth profile of the rack cutter and a predesigned transmission error.

The vector equation of the modified tooth profile of the rack cutter is determined by Equation (2):

$${\stackrel{\rightharpoonup }{r}}_{a2}=[\begin{array}{cccc}-a_{c2}{u_2}^2& u_2& l_g& 1\end{array}{]}^{\mathrm{T}}$$

(2)

where u₂ and l_g are the related tooth flank parameters of the rack cutter, and a_c2 is the modification parameter.

As shown in Figure 2, the transmission error was pre-designed as a sixth-order polynomial function, and its variation trend is controlled by five points. There are seven unknown parameters in the transmission error function, and seven equations are provided by five pre-designed points on the transmission error function. The coordinates of the given five points are, respectively, the rotation angle of the pinion and the corresponding transmission error.

The designation of the tooth modification of face gears is detailed in reference [22].

4. Analysis of the Dynamic Performance of Higher-Order-Modified Face Gears

4.1. Calculation of Meshing Stiffness

The general expression of the meshing stiffness of a single tooth of a gear is determined by Equation (3):

$$k_n=\frac{F_n}{{\delta }_n}$$

(3)

where $F_n$ is the normal contact force acting on the tooth profile surface, and ${\delta }_n$ is the comprehensive elastic deformation of single gear teeth pair.

The normal contact force of the face gear tooth profile surface comprises tangential, radial and axial components. The analytical calculation of this force is highly complex, often requiring numerical solutions obtained through finite element analysis. The comprehensive elastic deformation of a tooth surface generally includes the contact deformation of the tooth surface caused by Hertz contact ${\delta }_h$, the bending deformation caused by tooth bending ${\delta }_b$ and the support deformation ${\delta }_f$.

The comprehensive elastic deformation of tooth surface ${\delta }_n$ is determined by Equation (4):

$${\delta }_n={\displaystyle \sum _{i=1}^2{\delta }_{hi}}+{\displaystyle \sum _{i=1}^2{\delta }_{bi}}+{\displaystyle \sum _{i=1}^2{\delta }_{fi}}$$

(4)

The meshing stiffness of a single tooth refers to the comprehensive stiffness of a pair of gear teeth in contact, which are connected as a single tooth contact pair through series coupling. The meshing stiffness of a single tooth $k$ is determined by Equation (5):

$$k=\frac{k_{n1}{k}_{n2}}{k_{n1}+k_{n2}}$$

(5)

where $k_{n1}$ and $k_{n2}$ are the single tooth stiffnesses of the driving gear and driven gear.

In the meshing process of multiple pairs of teeth, the meshing stiffness of the gear pair engaged simultaneously at a certain instant is the superposition of the meshing stiffness of the single tooth, which is called the comprehensive meshing stiffness of the gear system at this moment. The meshing stiffness of the gear pair is determined by Equation (6):

$$K={\displaystyle \sum _{i=1}^n{k}_i}$$

(6)

where $k_i$ is the meshing stiffness of a single tooth $i$, and n is the number of teeth pairs engaged simultaneously.

Due to the simultaneous meshing of multiple teeth, it is necessary to consider the influence of the contact ratio on the comprehensive meshing stiffness of multiple teeth, and the superposition calculation of a single tooth stiffness is carried out. The contact ratio is determined by Equation (7):

$$\epsilon =\frac{\Delta T}{\Delta t}$$

(7)

where $\Delta T$ is the engagement time of a single tooth, and $\Delta t$ is the time difference when two adjacent teeth start to engage.

4.2. Establishment of Differential Equations of Face Gear Dynamics

4.2.1. Face Gear Meshing Force Analysis

Figure 3 shows the force analysis of non-orthogonal helical gear. The axial contact force of face gear is generated due to the non-vertical axis angle between face gear and cylindrical gear and the influence of helical tooth contact. Here, $F_n$ is the normal contact force of the tooth surface, and $F_{x2}$, $F_{y2}$ and $F_{z2}$ are the components in the three directions of $F_n$.

According to the force analysis of the face gear, the force balance equation of the face gear contact can be expressed by Equation (8):

$$\left\{\begin{array}{l}\begin{array}{l}{F}_{2x}=F_n{a}_{21}\\ =-F_n\cos \alpha \cos \beta \end{array}\hfill \\ \begin{array}{l}{F}_{2y}=F_n{a}_{22}\\ =-F_n(\sin \alpha \sin \gamma +\cos \alpha \sin \beta \cos \gamma )\end{array}\hfill \\ \begin{array}{l}{F}_{2z}=F_n{a}_{23}\\ =-F_n(\sin \alpha \cos \gamma +\cos \alpha \sin \beta \sin \gamma )\end{array}\hfill \end{array}\right.$$

(8)

The corresponding force balance equation of the cylindrical gear is Equation (9):

$$\left\{\begin{array}{l}{F}_{1x}=F_n{a}_{11}=F_n\cos \alpha \cos \beta \hfill \\ F_{1y}=F_n{a}_{12}=F_n\sin \alpha \hfill \\ F_{1z}=F_n{a}_{13}=F_n\cos \alpha \sin \beta \hfill \end{array}\right.$$

(9)

4.2.2. Derivation of Dynamic Differential Equations

As shown in Figure 4, the meshing force received by the non-orthogonal face gear can decompose the component forces along the X, Y and Z directions. According to the stress conditions, a nonlinear dynamic model of the transmission system of the non-orthogonal face gear under elastic support was established.

The input torque of the cylindrical gear was set to $T_1$, and the torque received by the non-orthogonal helical face gear was $T_2$. The whole transmission system had eight degrees of freedom, which are the displacement $X_1,Y_1,Z_1$ of the face gear along directions $x_1,y_1,z_1$ and displacement $X_2,Y_2,Z_2$ of the cylindrical gear along directions $x_2,y_2,z_2$, and the torsional vibration angles of the two gears around their respective axes are ${\theta }_1$ and ${\theta }_2$.

$$\left\{x_1,y_1,z_1,{\theta }_1,x_2,y_2,z_2,{\theta }_2\right\}$$

(10)

The interval function $f({\lambda }_n)$ can be determined by Equation (11):

$$f({\lambda }_n)=\left\{\begin{array}{c}{X}_n-b_m,X_n>b_m\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\left|X_n\right|\le b_m\\ X_n+b_m,X_n<-b_m.\end{array}\right.$$

(11)

where $b_m$ is half of the lateral clearance of the teeth.

According to the dynamic model of the face gear transmission system, the differential equations of the cylindrical gear and face gear can be obtained, as shown in Equation (12):

$$\left\{\begin{array}{l}{m}_1{\ddot{x}}_1+C_{1x}{\dot{x}}_1+k_{1x}{x}_1=F_{1x}\hfill \\ m_1{\ddot{y}}_1+C_{1y}{\dot{y}}_1+k_{1y}{y}_1=F_{1y}\hfill \\ m_1{\ddot{z}}_1+C_{1z}{\dot{z}}_1+k_{1z}{z}_1=F_{1z}\hfill \\ J_1{\ddot{\theta }}_1=T_1-F_{1x}{r}_1\hfill \\ m_2{\ddot{x}}_2+C_{2x}{\dot{x}}_2+k_{2x}{x}_2=F_{2x}\hfill \\ m_2{\ddot{y}}_2+C_{2y}{\dot{y}}_2+k_{2y}{y}_2=F_{2y}\hfill \\ m_2{\ddot{z}}_2+C_{2z}{\dot{z}}_2+k_{2z}{z}_2=F_{2z}\hfill \\ J_2{\ddot{\theta }}_2=-T_2+F_{2x}{r}_2\hfill \end{array}\right.$$

(12)

where $m_1$, $m_2$ and $I_1$, $I_2$ are the concentrated masses and the concentrated moments of inertia of the non-orthogonal helical face gear and cylindrical gear, respectively. $T_1$ and $T_2$ are the torques applied to the two gears.

The elastic meshing force between the two gears can be expressed by Equation (13):

$$F_n=k_m(t)f({\lambda }_n)+C_m{\dot{\lambda }}_n$$

(13)

where ${\lambda }_n$ can be expressed by Equation (14):

$$\begin{array}{l}{\lambda }_n=a_{11}{x}_1+a_{12}{y}_1+a_{13}{z}_1+a_{11}{r}_1\frac{T_1-F_{1x}{r}_1}{J_1}\\ -(a_{21}{x}_2+a_{22}{y}_2+a_{23}{z}_2+a_{21}{r}_2\frac{F_{2x}{r}_2-T_2}{J_2})\\ -e_n(t)\end{array}$$

(14)

where ${\lambda }_n$ is the normal relative displacement between the two contact tooth surfaces of cylindrical gear and face gear due to vibration and error; $k_m(t)$ is the time-varying meshing stiffness of the gear; $C_m$ is meshing damping; and $e_n(t)$ is the static transmission error of the gear.

Under the standard unit system, the values of many parameters in the dynamic differential equation of face gear are quite different, and the unit conversion is also very complicated when solving, which makes it difficult to solve the parameters in the equation set. Therefore, in the process of solving differential equations, dimensionless processing is usually used to eliminate the influence of different unit conversions.

A time nominal scale $\tau ={\omega }_{n}t$ and displacement scale $b_c$ were introduced, where ${\omega }_n$ is angular frequency:

$$\left\{\begin{array}{c}{\omega }_n=\sqrt{\frac{k_m}{m_e}}\\ m_e=\frac{J_1{J}_2}{J_1{r}_1^2+J_2{r}_2^2}\end{array}\right.$$

(15)

$$\left\{\begin{array}{c}{\overline{x}}_i(\tau )=\frac{x_i(t)}{b_c},{\overline{y}}_i(\tau )=\frac{y_i(t)}{b_c},{\overline{z}}_i(\tau )=\frac{z_i(t)}{b_c}\\ {\overline{\lambda }}_n(\tau )=\frac{{\lambda }_n(t)}{b_c}\end{array}\right.$$

(16)

According to the above formula, the variables in the differential equations can be transformed into Equation (17):

$$\left\{\begin{array}{l}{\dot{x}}_i(\tau )=\frac{dx}{d\tau }=\frac{dx}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c\overline{x})}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n{\dot{\overline{x}}}_i\hfill \\ {\dot{y}}_i(\tau )=\frac{dy}{d\tau }=\frac{dy}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c\overline{y})}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n{\dot{\overline{y}}}_i\hfill \\ {\dot{z}}_i(\tau )=\frac{dz}{d\tau }=\frac{dz}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c\overline{z})}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n{\dot{\overline{z}}}_i\hfill \\ {\dot{\lambda }}_n(\tau )=\frac{d{\lambda }_n}{d\tau }=\frac{d{\lambda }_n}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c{\overline{\lambda }}_n)}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n{\dot{\overline{\lambda }}}_n\hfill \\ {\ddot{x}}_i(\tau )=\frac{d\dot{x}}{d\tau }=\frac{d\dot{x}}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c\dot{\overline{x}})}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n^2{\ddot{\overline{x}}}_i\hfill \\ {\ddot{y}}_i(\tau )=\frac{d\dot{y}}{d\tau }=\frac{d\dot{y}}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c\dot{\overline{y}})}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n^2{\ddot{\overline{y}}}_i\hfill \\ {\ddot{z}}_i(\tau )=\frac{d\dot{z}}{d\tau }=\frac{d\dot{z}}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c\dot{\overline{z}})}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n^2{\ddot{\overline{z}}}_i\hfill \\ {\ddot{\lambda }}_n(\tau )=\frac{d{\dot{\lambda }}_n}{d\tau }=\frac{d{\dot{\lambda }}_n}{dt}\cdot \frac{dt}{d\tau }=\frac{d(b_c{\dot{\overline{\lambda }}}_n)}{dt}\cdot \frac{d({\omega }_n\tau )}{d\tau }=b_c{\omega }_n^2{\ddot{\overline{\lambda }}}_n\hfill \end{array}\right.$$

(17)

In substituting the above dimensionless conversion formula, the dimensionless differential equations of the face gear transmission system can be obtained:

$$\left\{\begin{array}{l}{\ddot{\overline{x}}}_1+\frac{c_{1x}}{m_1{\omega }_n}{\dot{\overline{x}}}_1+\frac{k_{1x}}{m_1{\omega }_n^2}{\overline{x}}_1-\frac{a_{11}{k}_{m1}(t)}{m_1{\omega }_n^2}f({\overline{\lambda }}_n)-\frac{a_{11}{c}_{m1}(t)}{m_1{\omega }_n}{\dot{\overline{\lambda }}}_n=0\hfill \\ {\ddot{\overline{y}}}_1+\frac{c_{1y}}{m_1{\omega }_n}{\dot{\overline{y}}}_1+\frac{k_{1y}}{m_1{\omega }_n^2}{\overline{y}}_1-\frac{a_{12}{k}_{m1}(t)}{m_1{\omega }_n^2}f({\overline{\lambda }}_n)-\frac{a_{12}{c}_{m1}(t)}{m_1{\omega }_n}{\dot{\overline{\lambda }}}_n=0\hfill \\ {\ddot{\overline{z}}}_1+\frac{c_{1z}}{m_1{\omega }_n}{\dot{\overline{z}}}_1+\frac{k_{1z}}{m_1{\omega }_n^2}{\overline{z}}_1-\frac{a_{13}{k}_{m1}(t)}{m_1{\omega }_n^2}f({\overline{\lambda }}_n)-\frac{a_{13}{c}_{m1}(t)}{m_1{\omega }_n}{\dot{\overline{\lambda }}}_n=0\hfill \\ I_1{\ddot{\theta }}_1=T_1-F_{x1}{r}_1\hfill \\ {\ddot{\overline{x}}}_2+\frac{c_{2x}}{m_2{\omega }_n}{\dot{\overline{x}}}_2+\frac{k_{2x}}{m_2{\omega }_n^2}{\overline{x}}_1-\frac{a_{21}{k}_{m2}(t)}{m_2{\omega }_n^2}f({\overline{\lambda }}_n)-\frac{a_{21}{c}_{m2}(t)}{m_2{\omega }_n}{\dot{\overline{\lambda }}}_n=0\hfill \\ {\ddot{\overline{y}}}_2+\frac{c_{2y}}{m_2{\omega }_n}{\dot{\overline{y}}}_2+\frac{k_{2y}}{m_2{\omega }_n^2}{\overline{y}}_2-\frac{a_{22}{k}_{m2}(t)}{m_2{\omega }_n^2}f({\overline{\lambda }}_n)-\frac{a_{22}{c}_{m2}(t)}{m_2{\omega }_n}{\dot{\overline{\lambda }}}_n=0\hfill \\ {\ddot{\overline{z}}}_2+\frac{c_{2z}}{m_2{\omega }_n}{\dot{\overline{z}}}_2+\frac{k_{1z}}{m_1{\omega }_n^2}{\overline{z}}_2-\frac{a_{23}{k}_{m2}(t)}{m_2{\omega }_n^2}f({\overline{\lambda }}_n)-\frac{a_{23}{c}_{m2}(t)}{m_2{\omega }_n}{\dot{\overline{\lambda }}}_n=0\hfill \\ I_2{\ddot{\theta }}_2=-T_2+F_{x2}r\hfill \end{array}\right.$$

(18)

4.2.3. Solutions of Dynamic Differential Equations

Due to the difficulty in solving nonlinear second-order differential equations using analytic methods, the Runge–Kutta numerical integration method is utilized in this paper. The Runge–Kutta method is a commonly used high-precision algorithm for solving differential equations. This algorithm can avoid the direct use of the Taylor formula in the algorithm, reduce the difficulty in calculation, reduce the influence of error, and ensure the accuracy of the results. The vibration response of a non-orthogonal face gear transmission system is calculated.

The general vibration equation is Equation (19):

$$M\ddot{X}+C\dot{X}+KX=F$$

(19)

In Equation (19), M, C and K are all n × n matrices; X and F are n × 1 column vectors. In order to reduce the order, the transformation can be as Equation (20):

$$\ddot{X}=-(M^{-1}KX+M^{-1}C\dot{X})+M^{-1}F$$

(20)

The second-order differential equations are converted into first-order differential equations, which is convenient for solving the dynamics equations using the ODE series functions of the Runge–Kutta method in MATLAB 2019b.

5. Results

5.1. The Establishment of the Gear Model

The basic parameters of the theoretical tooth surface of a face gear in this paper are shown in

Table 1. The basic parameters of the theoretical tooth surface of a face gear.

Design Parameter	Numerical Value
Pinion teeth number	25
Cutter tooth number	28
Face gear tooth number	160
Module (mm)	6.35
Pressure angle (degree)	25
Helix angle (degree)	15
Shaft angle (degree)	100
Inner radius (mm)	510
External radius (mm)	600
Torque (Nm)	1600
Young’s modulus (MPa)	206,800
Poisson’s ratio	0.29

.

According to the designed higher-order transmission error function curve, based on the tooth profile modification of the cutter, as well as the motion relationship between the cutter and the machined gear, the pinion tooth surface higher-order modification design was carried out.

Figure 5 shows the transmission error curves after the high-order modification design. Different from the traditional middle-convex TE, HTE is medium-concave. It can be clearly seen that the stress distribution in the middle region of the gear surface is more uniform when meshing.

5.2. Contact Force and Comprehensive Elastic Deformation of Gear Teeth

In order to ensure the meshing stiffness with full periodic fluctuations, this part will focus on the five-tooth model of a face gear. As shown in Figure 6, after processing via ABAQUS software, the normal contact force F_n of the gear tooth surface in the output is taken as a function of engagement period. When a face gear and a cylindrical gear engage, there is simultaneous contact between several teeth. The contact region on the tooth surface exhibits an elongated elliptical shape, with the contact positions on the two gears being mutually corresponding, as depicted in Figure 7. Figure 8 depicts the elastic deformation of the two gears. Apart from the elastic deformation occurring at the contact point, the adjacent region surrounding the release area also experiences deformations of different magnitudes. The distortion exhibits a progressive decrease as it extends outward from the release area. If all deformation amounts are included in the data and averaged, a small comprehensive deformation result will be obtained, and the derived meshing stiffness will also contain substantial errors. Therefore, when extracting the total amount of elastic deformation during data processing, it is necessary to compare the contact areas of different tooth surfaces, as shown in Figure 7.

The curve of the comprehensive deformation of all nodes in the contact area of the tooth surface over time is output through the post-processing of ABAQUS.

As shown in Figure 9, The comprehensive deformation of each contact point in the contact area of the tooth surface is different, and the average value of the comprehensive deformation should be taken.

5.3. Calculation of Multi-Tooth Meshing Stiffness

After the normal contact force and the comprehensive elastic deformation of gear teeth were obtained through a finite element calculation, the meshing stiffnesses of single teeth at each contact position were calculated using Equation (3), as shown in Figure 10.

The gear pair generate normal contact force in contact; as shown in Figure 11, the actual engagement time of the gear teeth can be estimated using the variation trend of the gear meshing force. The contact ratio is determined by Equation (7).

During the meshing process, the meshing stiffness also changes due to the change in the number of meshing teeth. The meshing stiffness of multiple teeth can be obtained from the superposition of the meshing stiffness of single teeth according to the change in the contact ratio. Figure 12, Figure 13 and Figure 14 show the calculation results of the meshing stiffnesses of three meshing pairs: standard tooth surface, second-order parabolic modified tooth surface and higher-order parabolic modified tooth surface.

By comparison, it can be seen that the contact ratio of the face gear changes obviously after the higher-order modification, and the corresponding multi-tooth comprehensive meshing stiffness of the gear also changes accordingly, which provides data support for the subsequent analysis of gear dynamic characteristics.

5.4. Analysis of Dynamic Characteristics

Through MATLAB programming, the vibration displacement of the face gear at each degree of freedom could be obtained, and its 10 periods under steady vibration were taken, as shown in Figure 15.

As shown in

Table 2. Comparison of vibration displacement amplitudes.

	X Direction		Y Direction		Z Direction
	Result (μm)	Amplitude	Result (μm)	Amplitude	Result (μm)	Amplitude
Standard	[7.83, 9.79]	1.96	[4.08, 5.11]	1.03	[2.72, 3.40]	0.68
Second-order	[7.79, 9.74]	1.95	[4.07, 5.09]	1.02	[2.70, 3.39]	0.69
Higher-order	[7.67, 9.32]	1.65	[4.00, 4.86]	0.86	[2.66, 3.24]	0.58

, the standard face gear’s amplitude of vibration displacement in the X direction was 1.96 μm, while the modified face gear’s amplitude was 1.95 μm, which is a 0.5% decrease. The higher-order modified face gear’s vibration displacement amplitude was 1.65 μm, which is a decrease of 15.8%. The vibration displacement amplitude of the standard face gear was 1.03 μm, while that of the second-order modified face gear was 1.02 μm, which is a reduction of 0.97%. The higher-order modified face gear’s vibration displacement amplitude was 0.86 μm, which is a 16.5% decrease. In comparison to the vibration displacement in the Z direction, the vibration displacement amplitude of the standard face gear was 0.68 μm, and that of second-order modified face gear was 0.69 μm, representing an increase of 1.47%. The higher-order modified face gear had a vibration displacement amplitude of 0.58 μm, which is a 14.7% decrease. These findings indicate that the amplitudes of vibration displacement in both directions are reduced for the higher-order modified face gear compared to both the standard and second-order modified face gear. Additionally, the higher-order modified face gear exhibited clear advantages in terms of meshing performance.

The dynamic meshing forces of the three face gear pairs can be calculated using the formula for dynamic meshing force. In Figure 16, the dynamic meshing force of the standard face gear pair fluctuates above and below 2306 N with a fluctuation interval of [2028, 2696] and a fluctuation amplitude of 668 N. Figure 17 shows the trend of the dynamic meshing force of the second-order modified surface gear pair as a function of the periodicity of the gear meshing. It can be seen from the figure that the dynamic meshing force of the second-order modified surface gear pair fluctuates above and below 2380 N, with a fluctuation interval of [2148, 2885] and a fluctuation amplitude of 737 N. Figure 18 shows the trend of the dynamic meshing force of the higher-order profile gear pair as a function of the periodicity of the gear meshing. The results show that the dynamic meshing force of the higher-order profile gear pair fluctuates and remains cyclical above and below 2355 N, with a fluctuation interval of [2102, 2662] and a fluctuation amplitude of 560 N. As the input torque remains constant for all three gears and the theoretical meshing force remains the same, the average value of the dynamic meshing force for all three gears is almost equal once the time-varying meshing stiffness and tooth clearance are accounted for. In terms of the fluctuating amplitude, the higher-order modified surface gear pair exhibits the smallest amplitude, significantly lower than those of the second-order modified surface gear pair and the standard surface gear pair.

Figure 19, Figure 20 and Figure 21 show the dynamic transmission errors for the three face gear pairs. Figure 19 shows the trend of the dynamic transfer error of the standard face gear pair as a function of the periodicity of the gear meshing. It can be seen from the figure that the dynamic transfer error of the standard face gear pair fluctuates above and below 15.6 μm, with a fluctuation interval of [14.2, 17.8] and a fluctuation amplitude of 3.6 μm. Figure 20 shows the trend of the dynamic transfer error of the second-order modified surface gear pair with the periodicity of the gear meshing. It can be seen from the graph that the dynamic transfer error of the second-order modified surface gear pair fluctuates above and below 13.7 μm, with a fluctuation interval of [11.7, 15.6] and a fluctuation amplitude of 3.9 μm. Figure 21 shows the trend of the dynamic transfer error of the higher-order modified surface gear pair with the periodicity of the gear meshing. It can be seen from the graph that the dynamic transmission error of the higher-order modified surface gear pair fluctuates above and below 13.2 μm, with a fluctuation interval of [10.8, 14.5] and a fluctuation amplitude of 3.7 μm. When comparing the three face gear pairs, the amplitudes of the dynamic transmission error fluctuations are similar in magnitude. However, in terms of the mean values, the standard face gear exhibits the largest mean dynamic transmission error, followed by the second-order modified face gear with a smaller mean dynamic transmission error, and the higher-order modified face gear with the smallest mean dynamic transmission error.

The results show that the amplitude of vibration displacement, the amplitude of fluctuation of the dynamic meshing force and the dynamic transmission error of the higher-order modified face gear are smaller than those of the standard and second-order modified face gears, and the higher-order modified face gear has a better meshing performance.

6. Conclusions

In order to investigate the dynamic characteristics of non-orthogonal helical face gears with higher-order tooth surface modification, a dynamic model of face gear transmission involving second-order and higher-order tooth surface modifications is presented, and the dynamic performance was deeply studied. From the present study, the following conclusions are drawn:

(1)

The three-dimensional grid model of a modified pinion was obtained based on the modified tooth profile of the rack cutter and a predesigned transmission error. The three-dimensional grid model of a face gears can achieve parameterized and fast modeling, and the grid density can be set autonomously.

(2)

The meshing stiffnesses of the standard, the second-order modified and the higher-order modified face gears were obtained based on finite element calculations. Through a comparison, it was found that the meshing stiffness fluctuation of higher-order modified face gears is the smallest under a given load.

(3)

A dynamic model of a non-orthogonal helical face gear pair involving second-order and higher-order tooth surface modifications was established. The dynamic performance of the high-order modified face gear transmission was deeply studied.

(4)

The advantage of higher-order modification is demonstrated through a comparison of results. Compared with second-order modification, the higher-order modification has a smaller vibration displacement, dynamic meshing force and dynamic transmission error.