Abstract
Localized tooth breakage is one of the common failures of spur gears, which affects the smooth and safe operation of spur gear transmission systems. The tooth collision cannot be neglected, and it is especially important to reveal the meshing-impacting dynamic characteristics of the gear system under localized tooth breakage to improve the safe and stable operation of the gear system. Based on the gear meshing principle and the dissipative collision contact force model, the drive-side tooth meshing model and back-side tooth impacting model are established with considering the transient nature of tooth back-side contact. According to the contact state and force environment of the gear pair, the multi-state meshing-impacting behavior under local breakage is classified, and the discrete meshing-impact dynamics model of an involute spur gear system under local breakage is established to explore the influence of local breakage on the meshing stiffness and load distribution. The mechanism of contact force under partial tooth breakage is revealed, and the influence of load coefficient and meshing frequency on the nonlinear dynamics is studied by defining two Poincaré maps. It is found that local tooth breakage affects the contact force of single-and double-tooth meshes and reduces gear load carrying capacity. Larger loads inhibit back-side impact, and smaller loads induce the coexistence behavior and back-side impact. Larger or smaller meshing frequency induces back-side impact behavior. The coexistence of chaotic and periodic motions induces back-side impact behavior, and localized tooth breakage affects the coexistence phenomenon and aggravates the complexity of the dynamic behavior. The dynamic model of gear system considering energy dissipation and nonlinear vibration under the presence of local tooth breakage of the pinion is explored, and the conditions of back-side impact are studied. This research provides new methods and ideas for nonlinear dynamic modeling and analysis of faulty gear systems.
1. Introduction
Spur gears are widely used in various mechanical equipment due to their ability to efficiently transmit power and motion. However, localized tooth breakage is a common fault in spur gears, which can significantly affect the smooth operation and service life of gear systems. The phenomenon of spur gears operating with minor localized tooth breakage is prevalent and can lead to issues such as gear rattling, particularly under high-speed and light-load conditions. Therefore, investigating the meshing-impacting characteristics of spur gear systems under localized tooth breakage is crucial for enhancing the transmission performance and ensuring the healthy and stable operation of gear systems.
Previous studies have primarily focused on the effects of cracks and wear on gear systems. For instance, Thirumurugan et al. used the finite element method to study the influence of load on crack propagation paths in gears. Liu et al. established a dynamic model for planetary gear systems with sun gear crack faults. Zhou et al. developed a gear wear model based on Hertz contact theory and the Archard formula. Ma et al. considered tooth breakage and established a dynamic model for gear systems. Yang et al. established a time-varying mesh stiffness model for gears with tip chipping. Park developed an improved model for spur gear systems with tip crack faults. However, these models did not consider the influence of multi-state meshing and localized tooth breakage on the dynamic characteristics of gear systems.
In addition, tooth backlash can induce multi-state meshing behaviors such as gear disengagement and back-side contact . Yin et al. established a dynamic model for gear systems considering tooth surface impact and friction. Shi et al. developed multi-state meshing models for gear systems considering time-varying backlash. Huang et al. proposed a dynamic model for asymmetric gear systems. Gao et al. discussed the influence of grazing impacts on the dynamic performance of gear-bearing systems. Jin et al. analyzed the evolution of coexisting attractors in gear systems. However, most of these studies focused on healthy or normal gears and neglected the dynamic characteristics and transient nature of back-side contact under localized tooth breakage.
This paper introduces a continuous contact force model established by Hu et al. and defines back-side contact as back-side impact. Focusing on localized tooth breakage in a single pinion tooth, the changes in local contact ratio are discussed, and the multi-state meshing-impacting behaviors under localized tooth breakage are classified. A dynamic model for a single-stage gear system with localized tooth breakage is established, and the influence of localized tooth breakage on time-varying factors and contact forces is discussed. Two Poincaré maps are defined, and the fourth-order Runge-Kutta method with variable step size is used to solve the gear system dynamic model. By analyzing the bifurcation diagram, impact force periodogram, phase diagram, and Poincaré section diagram of the system, the meshing-impacting characteristics of spur gear systems under localized tooth breakage with varying loads and meshing frequencies are studied and compared with the motion transition process of healthy gear systems with varying loads. The results provide a theoretical reference for gear system parameter design and localized tooth breakage fault prediction.
2. Dynamic Modeling of Spur Gear Systems with Localized Tooth Breakage
2.1 Physical Model of Meshing-Impacting
Assuming that the gear pair is rigidly supported and only considering back-side impact, the simplified meshing-impacting physical model is shown in Figure 1.
Figure 1. Physical model of meshing-impacting in gear systems
In this model:
- cg: Meshing damping
- D: Half of the tooth backlash
- μ: Friction coefficient
- e(t)=Eaωhcos(ωht): Dynamic transmission error, where Ea is the error fluctuation coefficient, and ωh is the meshing frequency
- km(t): Time-varying meshing stiffness
- R: Coefficient of restitution
- Tj, θj, Ii, and Rbj: Torque, angular displacement, moment of inertia, and base circle radius, respectively (where j=p,g represents the pinion and gear, respectively; relevant geometric parameters are shown in Table 1)
Table 1. Geometric parameters of gears
| Parameter | Pinion (p) | Gear (g) |
|---|---|---|
| Number of teeth (z) | 40 | 40 |
| Module (m/mm) | 3 | 3 |
| Addendum coefficient (ha) | 1 | 1 |
| Top clearance coefficient (c) | 0.25 | 0.25 |
| Pressure angle (α0/°) | 20 | 20 |
2.2 Characterization of Meshing and Calculation of Time-varying Contact Ratio
Figure 2 shows the detail of the gear meshing line. AD represents the actual meshing line of a healthy gear pair, with AB or CD being the double-tooth meshing region and BC being the single-tooth meshing region. The locally broken tooth exits the meshing at point D1 earlier, turning the original double-tooth meshing regions AB or CD into single-tooth meshing regions AB1 or CD1, respectively. ϵm is the contact ratio of the healthy gear pair, and ϵb is the contact ratio under localized tooth breakage, with 1<ϵb<ϵm<2. ϵb can be derived from Equation (1).
epsilonb=pblAD1(1)
Where:
- pb: Base pitch
- lAD1: Length of meshing line AD1, which can be calculated from Equation (2)
lAD1=Rbp(tanαab−tanα)−Rbgtanα+Rag2−Rbg2(2)
Where:
- Rab and αab: Tip radius and pressure angle of the broken tooth
- Rag: Tip radius of the driven gear
Figure 2. Detail of meshing line
2.3 Classification of Multi-state Meshing-Impacting Behaviors
Let the relative displacement of gear teeth be x=Rbpθp−Rbgθg−e(t), with motion time t=nT0 (n∈N), and meshing period T0=ωpzp2π. Assuming that localized tooth breakage affects the nb-th meshing period, while the remaining nf periods are healthy meshing periods, we have S={nbmod(n,zp)=1} and P={nf}, with S∪P=N. Based on the relationship between x and D, the meshing-impacting behaviors of the gear pair are classified into the following five types:
- Double-tooth Surface Meshing:
- Boundary condition: x≥D, nT0≤t≤(n+ϵs−1)T0
- When the meshing point is in region AB1 or CD1, ϵs=ϵb, n=nb (broken double-tooth surface meshing)
- When the meshing point is in region AB or CD, ϵs=ϵm, n=nf (healthy double-tooth surface meshing)
- Single-tooth Surface Meshing I (Region BC):
- Boundary condition: x≥D, (n+ϵm−1)T0<t<(n+1)T0
- Single-tooth Surface Meshing II:
- Localized tooth breakage causes the double-tooth meshing region to become a single-tooth meshing region (regions B1B or D1D)
- Boundary condition: x≥D, (nb+ϵb−1)T0<t≤(nb+ϵm−1)T0
- Gear Disengagement (Region AD):
- Boundary condition: −D<x<D, nT0≤t≤(n+1)T0
- Back-side Impact (Theoretically at any position in region AD):
- Boundary condition: x=−D, nT0≤t≤(n+1)T0
2.4 Meshing-Impacting Dynamic Modeling
2.4.1 Double-tooth Surface Meshing
Figure 3 shows the force analysis for double-tooth surface meshing. In this state, two gear teeth pairs mesh simultaneously, corresponding to regions AB and CD or AB1 and CD1 in Figure 2. According to Newton’s second law, the absolute rotation equations for the pinion and gear are given by Equation (3), with different boundary conditions for healthy and broken gear pairs, characterized by ϵm and ϵb, respectively.
Figure 3. Force analysis of double-tooth surface meshing
begincasesIpθ¨p+RbpFNp1+Sdp1Ffp1+RbpFNp2+Sdp2Ffp2=TpIgθ¨g−RbgFNg1−Sdg1Ffg1−RbgFNg2−Sdg2Ffg2=−Tgendcases(3)
Where:
- FNp1=FNg1=L1(t)Fm, FNp2=FNg2=L2(t)Fm
- Total dynamic meshing force Fm(t)=kmd(t)(x−D)+cgx˙
- Li(t): Load sharing coefficient of the i-th meshing gear tooth pair
- kmd(t): Double-tooth surface meshing stiffness
- Frictional forces Ffpi,Ffgi (see Equation (4))
begincasesFfp1=Ffg1=λd1(t)μd1L1(t)Fm(t)Ffp2=Ffg2=λd2(t)μd2L2(t)Fm(t)endcases(4)
Where:
- μdi and λdi(t): Dry friction coefficient and friction direction coefficient of the i-th meshing gear tooth pair
The friction arms Sdpi(t) and Sdgi(t) (see Equation (5)) can be calculated, with Sdp2(t)=Sdp1(t+T0) and Sdg2(t)=Sdg1(t+T0).
begincasesSdp1(t)=(Rbp+Rbg)tanα−Rap2−Rbg2+RbpωptSdg1(t)=Rap2−Rbg2−Rbpωptendcases(5)
Considering x=Rbpθp−Rbgθg−e(t), Equation (3) can be simplified to Equation (6).
mex¨+[1+λd1μd1gd1(t)L1(t)+λd2μd2gd2(t)L2(t)][kmd(t)(x−D)+cgx˙]=F+Fh(t)(6)
Where:
- me: Equivalent mass
- gdi(t): Equivalent friction arm of the i-th meshing gear tooth pair
- F: Total load
- Fh(t): Internal error excitation of the gear pair
2.4.2 Single-tooth Surface Meshing I
Single-tooth surface meshing I corresponds to healthy single-tooth meshing, with only one gear tooth pair participating in the meshing (region BC in Figure 2). The dynamic equation is given by Equation (7), where kms(t) is the single-tooth surface meshing stiffness, with kms1(t) for this case.
mex¨+[1+λd2(t)μd2gd2(t)][kms(t)(x−D)+cgx˙]=F+Fh(t)(7)
2.4.3 Single-tooth Surface Meshing II
Figure 4 shows the force analysis for single-tooth surface meshing II. Localized tooth breakage in the pinion leads to a reduction in the number of simultaneously meshing gear teeth pairs, corresponding to regions B1B or D1D in Figure 2. The dynamic equation is the same as Equation (7), with differences in boundary conditions and kms(t), specifically kms2(t) for this case (detailed calculations in Section 2.1).
Figure 4. Force analysis of single-tooth surface meshing II
2.4.4 Gear Disengagement
During gear disengagement, the two gears are separated, and both the normal meshing force and frictional force on the tooth surface are zero. The equation of motion is given by Equation (8).
mex¨=F+Fh(t)(8)
2.4.5 Back-side Impact
When x=−D, back-side contact occurs, and the impact equation is given by Equation (9).
dotx(+)=−Rx˙(−)(9)
Where:
- x˙(−) and x˙(+): Relative velocities before and after impact
According to Hu et al. [15], the maximum impact force Fc can be calculated using Equation (10).
Fc=−4kc(t)kc(t)5mRδ53(−)(10)