1. Introduction
Spur gears are widely used in various mechanical equipment for efficient power and motion transmission. However, the phenomenon of slightly damaged teeth running with faults is common, which affects the smoothness and lifespan of the gear system. Localized tooth breakage is one of the common failures of spur gears, and it is crucial to study the meshing – impacting characteristics of spur gear systems under such conditions to improve the transmission performance and ensure the healthy and stable operation of the gear system.
1.1 Research Background
Most scholars mainly focus on the effects of cracks and wear on gear systems. For example, Thirumurugan et al. studied the influence of load on the crack propagation path of gears using the finite element method. Liu Jie et al. established a dynamic model of a planetary gear system with a sun gear crack fault. Zhou Yatian et al. established a gear system wear model based on Hertz contact theory and Archard formula. However, these models do not consider the influence of multi – state meshing and localized breakage faults on the dynamic characteristics of the gear system. Although some studies have considered factors such as tooth surface impact and friction, most of them focus on normal or healthy gears and ignore the dynamic characteristics and the transient nature of tooth back – side contact under localized breakage.
1.2 Research Objectives and Significance
This study aims to establish a dynamic model of a spur gear system considering friction and multi – state meshing under localized tooth breakage, classify the multi – state meshing – impacting behavior, analyze the influence of localized breakage on the system’s dynamic characteristics, and explore the conditions of tooth back – side impact. The research results can provide new methods and ideas for the nonlinear dynamic modeling and analysis of faulty gear systems.
2. Dynamic Modeling of Spur Gear System Under Localized Tooth Breakage
2.1 Physical Model and Assumptions
The simplified meshing – impacting physical model of a gear pair with rigid support and considering only tooth back – side impact is shown in Figure 1. The relevant parameters include meshing damping , half of the tooth side clearance , friction factor , dynamic transmission error , error fluctuation coefficient , meshing frequency , time – varying meshing stiffness , collision recovery coefficient , torque , rotation angle displacement , moment of inertia and base circle radius ( representing the driving and driven wheels respectively).
2.2 Meshing Characterization and Calculation of Time – varying Contact Ratio of Localized Broken Gears
The normal gear pair’s actual meshing line is , with or as the double – tooth meshing area and as the single – tooth meshing area. For a localized broken tooth, it exits meshing early at point . The contact ratio for a healthy gear pair and for a localized broken gear pair () can be calculated as follows:
2.3 Classification of Multi – state Meshing – Impacting Behavior of Localized Broken Gears
Let the relative displacement of the teeth be and the motion time be with the meshing period . The gear pair’s meshing – impacting behavior can be classified into the following 5 types according to the relationship between and :
| Meshing – Impacting Type | Boundary Conditions |
|---|---|
| Double – tooth tooth surface meshing (healthy) | , , when in region or , , |
| Double – tooth tooth surface meshing (broken) | , , when in region or , , |
| Single – tooth tooth surface meshing I (region ) | |
| Single – tooth tooth surface meshing II (region or ) | |
| Tooth disengagement (region ) | |
| Tooth back – side impact (region ) | , |
2.4 Meshing – Impacting Dynamics Modeling
- Double – tooth tooth surface meshing: The dynamics equations for the driving and driven wheels are as follows:
where , ; ; is the load distribution coefficient of the -th pair of meshing teeth; is the double – tooth tooth surface meshing stiffness. The friction forces on the driving and driven wheels of the -th pair of meshing teeth are given by:
The main and driven wheel friction arm of the -th pair of meshing teeth , can be calculated as:
After simplification, the equation becomes:
where is the equivalent mass, is the equivalent friction arm of the -th pair of meshing teeth, is the total load, and is the internal error excitation of the gear pair. - Single – tooth tooth surface meshing I: The dynamics equation is:
where is the single – tooth tooth surface meshing stiffness corresponding to single – tooth tooth surface meshing I. - Single – tooth tooth surface meshing II: The dynamics equation is the same as that of single – tooth tooth surface meshing I, but with different boundary conditions and . The stiffness for single – tooth tooth surface meshing II is .
- Tooth disengagement: The motion equation is .
- Tooth back – side impact: When , the impact equation is . The maximum collision force can be calculated as , where is the equivalent mass and is the time – varying tooth back – side contact stiffness.
2.5 Normalization Processing
The dimensionless parameters are introduced as follows: , , , , , . The dimensionless dynamic equation of the gear system is:
where is the dimensionless dynamic meshing force function and is the dimensionless state function. The comprehensive dimensionless dynamic contact force is:
3. Calculation of Time – varying Parameters
3.1 Time – varying Stiffness Calculation
Assume that the dimensionless tooth surface meshing stiffness and the tooth back – side contact stiffness are equal. The stiffness can be calculated as:
where is the time – varying Hertz contact stiffness, is the bending stiffness, is the axial compression stiffness, is the shear stiffness, and is the matrix stiffness. The local breakage of the gear affects the meshing period, reducing the double – tooth meshing area and increasing the single – tooth meshing area, resulting in a decrease in the time – varying meshing stiffness and a weakening of the local load – carrying capacity of the teeth.
3.2 Time – varying Load Distribution Coefficient Calculation
The load distribution coefficient under localized breakage can be calculated as: