Abstract
The accurate explanation of the tooth deregulation or back meshing induced by the backlash can provide theoretical guidance for the optimal design of the gear structure. Based on the Johnson contact force model, the dynamic meshing force calculation model of an involute internal meshing spur gear transmission system was constructed, taking into account the time-varying backlash and energy dissipation in the meshing process. According to the meshing principle of internal meshing spur gear and the force condition of gear teeth, the meshing process was decomposed, and the nonlinear dynamic model of the involute internal meshing spur gear system with five meshing states under time-varying backlash was established. Five different Poincaré maps were constructed to clarify the characteristics of five-state meshing behaviors, and the generation mechanism of multi-state meshing behaviors of the system was revealed by the phase diagram and dynamic meshing force time history diagram. The results show that with the gradual increase of the meshing frequency, the single-tooth and double-tooth surface meshing occurs in the system, and the sudden directional change of the dynamic meshing force leads to the meshing between the gear teeth and the back of the single-tooth and double-tooth, resulting in the impact vibration behavior between the gear teeth. Even if the system shows a stable meshing state, the single and double teeth dismeshing still occurs. The previous dynamic models of internal meshing gear systems can not reveal the multi-state meshing characteristics, so that, aiming at the multi-state meshing behavior, the paper analyzes the nonlinear vibration characteristics in the internal meshing gear transmission cycle.
1. Introduction
Internal meshing spur gear possess several advantages such as small center distance, light wear, smooth transmission, and long service life . Due to the periodic alternating meshing between single and double teeth, as well as the presence of backlash, spur gear meshing exhibits various states such as tooth surface meshing, tooth separation, and tooth back meshing as the rotational speed and load change. Accurately explaining the multi-state meshing behavior induced by backlash can provide theoretical guidance for optimizing the gear structure design.
Currently, the meshing force between gear is primarily calculated using the linear spring-damping model, which consists of mass, spring, and damping elements When long cycles and impacts are involved, the variation in normal meshing force can provide a basis for the dynamic impact analysis of the system. Based on the early work of Hunt et al. , Lankarani et al. proposed an improved meshing force model that accounts for energy dissipation during transmission. Kahraman et al. previously used this model to analyze the dynamic characteristics of a single-degree-of-freedom gear system combined with error excitation. Nevzat et al. provided a comprehensive review of nonlinear dynamic models of spur gear, laying the foundation for analyzing the impact of backlash on spur gear meshing process. Liu et al. obtained a damping term in the linear model used to describe energy loss during meshing based on the oscillation decay period of the linear spring-damping system. Chen et al. conducted an analysis of the nonlinear dynamic characteristics of spur gear by combining backlash and tooth surface friction. Research on the alternating meshing characteristics of single and double teeth has primarily focused on the time-varying meshing stiffness of spur gear. Huang et al. derived an analytical model for the time-varying meshing stiffness of spur gear systems considering the influence of single and double tooth meshing characteristics based on the energy method. Yuan et al. calculated the meshing characteristics of cylindrical gear under elastohydrodynamic lubrication conditions and analyzed the impact of different operating parameters on friction and meshing forces. Xiang et al. established a bending-torsional coupling nonlinear dynamic model of a gear-rotor-rolling bearing system considering dynamic backlash, tooth surface friction, spur gear eccentricity, and time-varying meshing stiffness. Shi et al. derived three dynamic models for tooth surface meshing, disengagement, and tooth back meshing under the influence of backlash. Constant backlash can clearly describe tooth surface meshing, tooth separation, and tooth back meshing in spur gear system, but it cannot effectively distinguish between single and double tooth meshing on the tooth surface or back. However, this can be accurately identified through time-varying backlash. Li et al. established a nonlinear dynamic model of spur gear system with dynamic backlash under combined periodic excitation from both internal and external sources. Wang et al. developed a time-varying backlash model considering gear geometric errors and center distance deviations based on the finite element method. In previous gear systems, the dynamic meshing force model was represented as an implicit function. Therefore, in nonlinear dynamic analysis, the meshing force at each time step needed to be calculated through numerical iteration. This not only increases the computational cost but also poses challenges to the performance of the computational program. Additionally, since the normal meshing force of spur gears is an important source of damping in practical engineering applications, previous models did not account for energy dissipation during transmission. In the current literature on nonlinear dynamic research of spur gear systems, there have been no reports considering the impact of multi-state meshing behavior on involute internal meshing spur gear systems. As the factors considered become more comprehensive and operating conditions become more complex, it is necessary to establish dynamic models of spur gear systems for each meshing state based on the spur gear tooth profile, meshing line, and force conditions under different meshing states.
Considering the impact of tooth surface friction on the gear system, this paper proposes an improved calculation model for the dynamic meshing force of an involute internal meshing spur gear transmission system that incorporates energy dissipation based on the Johnson contact force model. In this meshing force model, the limitations of existing models are overcome, and the dynamic meshing force is defined as an explicit function considering energy dissipation during impact. Simultaneously, based on the transmission principle of involute internal meshing cylindrical spur gear, time-varying backlash, and the force conditions between gear teeth, dynamic models of the gear pair system under multiple states, including single-tooth surface meshing, double-tooth surface meshing, single-tooth back meshing, double-tooth back meshing, and tooth separation, are established. This research provides a useful reference for the dynamic performance optimization, parameter design, and further study of gear transmission systems.
2. Calculation of Time-varying Parameters
2.1 Dynamic Meshing Force Based on the Johnson Contact Model
Accurately determining the spur gear meshing force is crucial for predicting the dynamic performance and load-bearing capacity of gear transmission systems. When analyzing the dynamic characteristics of spur gear systems, most meshing force models are based on Hertzian pressure, which has the limitations of Hertz contact theory, restricting their use to meshing conditions with low backlash. Traditional models represent the meshing force as a mathematical form of an implicit function, making them computationally expensive for practical applications.
The dynamic meshing force calculation model in this paper is improved based on the Johnson contact force model. During spur gear meshing, collisions or impacts between meshing teeth lead to energy dissipation. Lankarani et al. established a meshing force model that accounts for energy dissipation, proposed modifications using the restitution coefficient c_e, and verified its effectiveness. Simultaneously, the meshing force is defined as an explicit function, eliminating the usage limitations of low backlash and simplifying calculations, as calculated by Equation (1).
Fm=[aD(τ)+b]D(τ)n−1LE∗⋅x⋅(1+43(1−ce2)⋅x˙(−)x˙)
Where:
- D(τ) is the time-varying backlash.
- L is the center distance between gears.
- E∗=2(1−μ2)E is the combined elastic modulus, where E is the elastic modulus and μ is the Poisson’s ratio.
- x is the relative displacement of the two gears in the direction of the meshing line, and x˙ is its first derivative, representing the relative velocity between the two gear teeth in the direction of the meshing line.
- x˙(−) is the theoretical relative velocity, with the restitution coefficient ce=0.8.
- a, b, and n are constants related to the backlash ΔR, as shown in Equations (2) to (4).
a = \begin{cases} 0.965, & 50\mu m < \Delta R \leq 10mm \\ 0.39, & 10mm < \Delta R < 500mm end{cases}
b = \begin{cases} 0.0965, & 50\mu m < \Delta R \leq 10mm \\ 0.85, & 10mm < \Delta R < 500mm end{cases}
n = \begin{cases} Y\Delta R – 0.005, & 50\mu m < \Delta R \leq 10mm \\ 1.094, & 10mm < \Delta R < 500mm end{cases}
Where Y is a constant that is difficult to obtain a good fit with a single expression for internal contacts, as shown in Equation (5).
Y = \begin{cases} 1.51\left[ \ln(1000\Delta R) \right] – 0.151, & 0.005mm < \Delta R \leq 0.34954mm \\ 0.151\Delta R + 1.15, & 0.3954mm < \Delta R < 10mm end{cases}
Combining the above, the mathematical expression for the gear pair meshing force can be obtained, as shown in Equation (6).
Fm=Kix+cix˙,i=k,d
Where:
Ki=[aD(τ)+b]ΔRLE∗
ci=[aD(τ)+b]D(τ)LE∗⋅4x˙(−)3(1−ce2)
Here, i=k,d represent the tooth surface and tooth back, respectively.
2.2 Friction Coefficient Based on Elastohydrodynamic Lubrication Theory
The friction coefficient on the tooth surface is primarily influenced by factors such as gear tooth geometry, surface hardness, contact pressure, and relative sliding speed of the tooth surface. This paper considers the friction coefficient under elastohydrodynamic lubrication. According to the research of He et al. [18], the time-varying friction formula for the i-th meshing gear pair is expressed as Equation (6).
mu_i(t) = \lambda_i(t) e^{f\left[ SR_i{b_6} \eta^{b_7} M \rho^{b_8} h_i(t)}
f[SRi(t),Phi(t),ηM,Raavg]=b1+b9eRaavg+b4SRi(t)Phi(t)lgηM+b5e−SRi(t)Phi(t)lgηM
Where:
- Raavg=2Ra1+Ra2 is the average surface roughness.
- ηM=0.058 is the dynamic viscosity.
- bi (i = 1, 2, …, 9) are empirical factors, and their values can be found in the research of Chen et al.
- The maximum Hertzian pressure Phi(t) for the i-th meshing gear pair is expressed as Equation (10).
Phi(t)=πρri(t)1−υ2feE
Where:
- fe=bRbpcosαTp is the unit normal load.
- ρri(t) is the relative curvature radius of the i-th meshing gear pair.
In Equations (8) and (9), the dimensionless roll-slide ratio SRi(t)=vei(t)2vs(t), where vs(t) is the tooth surface slip speed and vei(t) is the entrainment speed. λi(t) is the direction coefficient of friction force, which can be obtained from Equation (11).
lambdai(t)=sgn[vs(t)]
Where sgn(⋅) is the sign function used to determine the direction of friction force.
3. Multi-state Meshing Dynamic Model of Internal Meshing Gear System
Under low-speed and heavy-load conditions, the spur gear system is in tooth surface meshing state. Under high-speed and light-load conditions, backlash can cause periodic tooth separation or tooth back meshing behavior. To better understand the multi-state meshing characteristics of gear transmission systems, it is assumed that the gear pair is rigidly supported, and only the torsional vibration of the spur gear is considered. The simplified physical model of the involute internal meshing cylindrical spur gear in tooth surface and tooth back meshing states.
In this model, the rotational angular displacements of the driving and driven gears are θp and θg, respectively, with moments of inertia Ip and Ig. The backlash is D(τ), and the meshing stiffness and damping on the tooth surface are Kk and Ck, respectively, with a friction coefficient μk. The meshing stiffness and damping on the tooth back are Kd and Cd, respectively, with a friction coefficient μd. The dynamic transmission error of the spur gear pair is e(τ), where the tooth surface error excitation is ek(τ) and the tooth back error excitation is ed(τ). The remaining case parameters are shown in Table 1.
Table 1. Gear parameter table
| Geometric parameter | Active gear 1 | Driven gear 2 |
|---|---|---|
| Number of teeth (z) | 21 | 84 |
| Module (m) / mm | 5 | 5 |
| Pressure angle (α) / rad | 0.35 | 0.35 |
| Tooth width (b) / mm | 50 | 50 |
| Elastic modulus (E) / GPa | 210 | 210 |
| Poisson’s ratio (μ) | 0.3 | 0.3 |
3.1 Classification of Multi-state Meshing of Internal Meshing Spur Gear
The schematic diagram of tooth surface and tooth back meshing of internal meshing spur gear. Rbp, Rp, and Rap are the base circle radius, pitch circle radius, and tip circle radius of gear p, respectively. Rbg, Rg, and Rag are the base circle radius, pitch circle radius, and tip circle radius of gear g, respectively. εm is the contact ratio of the gear, and according to the research of Zhang et al., the condition for continuous transmission of internal meshing is εm>1. When 1<εm<2, there is periodic alternating meshing between single and double teeth. Based on the properties of involute gears, gear p serves as the driving gear. Due to the determined sizes and positions of the base circles of the internal and external gears, there is only one common tangent (meshing line) in one direction, namely the tooth surface meshing line (light solid line) D→A, where A and D are the actual meshing points when the two gears mesh, entering at point D and exiting at point A to transmit motion and power. The dark solid line represents the tooth back meshing line, entering at point A’ and exiting at point D’. For ease of analysis, the direction along the meshing line D→A is set as positive, and the relative displacement of the two gears in the direction of the meshing line can be expressed as x−=Rbpθp−Rbgθg−e(τ).
Based on the position of the meshing point during meshing and the geometric relationship with the backlash, the spur gear meshing states and their boundary conditions can be classified as follows:
Double-tooth surface meshing state: The boundary conditions are x−≥D−(τ) and nTm≤τ≤(εm−1)nTm (where n=0,1,2,…).
Single-tooth surface meshing state: The boundary conditions are x−≥D−(τ) and (εm−1)nTm≤τ≤(n+1)Tm.
Double-tooth back-side meshing state: The boundary conditions are x−≤−D−(τ) and nTm≤τ≤(εm−1)nTm.
Single-tooth back-side meshing state: The boundary conditions are x−≤−D−(τ) and (εm−1)nTm≤τ≤(n+1)Tm.
Tooth disengagement state: The boundary condition is x−<D−(τ) and nTm≤τ≤(n+1)Tm.
In the above classifications, x−=Rbpθp−Rbgθg−e(τ) represents the relative displacement of the two gears along the meshing line, D−(τ) is the time-varying backlash, Tm=zpωp2π is one complete meshing cycle, including both single-tooth and double-tooth meshing periods (AC or A’C’), zp and ωp are the number of teeth and angular velocity of the driving gear, respectively. εm is the spur gear meshing overlap coefficient, and for continuous internal meshing, the condition εm>1 must be satisfied. When 1<εm<2, there exists periodic alternation between single-tooth and double-tooth meshing.
Based on these classifications and their boundary conditions, the corresponding dynamic models for each meshing state can be established. These models consider the forces acting on spur gear teeth, including normal forces and frictional forces, as well as the effects of backlash, meshing stiffness, damping, and friction coefficients. The dynamic equations of motion for each state are then derived using Newton’s second law of motion, and these equations form the basis for analyzing the multi-state meshing dynamics of the internal meshing spur gear system.