1. Introduction
The spur gear transmission system is widely employed in automotive, aerospace, marine and precision machinery fields due to its high transmission efficiency, long service life and stable operation. The dynamic performance of a spur gear pair directly determines the reliability and accuracy of the entire mechanical equipment. Over the past two decades, nonlinear dynamics of gear systems has attracted extensive attention because of the strong nonlinearities originating from time-varying mesh stiffness, backlash, friction, transmission errors, and bearing clearances. However, most existing studies focus on healthy gear pairs or gear systems with wear and crack faults. In practice, gear tooth breakage, which often occurs as a result of severe overload, fatigue crack propagation, or unexpected impact, represents one of the most destructive failure modes of the spur gear transmission system. When a gear tooth is broken or damaged, the gear pair may still operate temporarily, especially in special working conditions where immediate replacement is impossible. The presence of a broken tooth changes the local meshing geometry, reduces the load-carrying capacity, and induces severe vibration and noise. Moreover, the backlash-induced tooth separation and tooth back-side collision may occur, causing strongly non-smooth dynamic behaviors.
Aiming at the transient nature of tooth back-side contact during high-speed operation, I introduce a dissipative contact force model to characterize the instantaneous collision between the back-side tooth profiles of the driving and driven gears. The research work in this dissertation focuses on the multi-state meshing-impacting nonlinear dynamics of a spur gear transmission system with tooth breakage. I investigate two types of tooth breakage faults: local tooth breakage and tooth root breakage. For local tooth breakage, a standard involute spur gear pair is studied. For tooth root breakage, a high-contact-ratio spur gear pair is analyzed. The key research tasks include: (1) establishing a simplified physical model of gear meshing-impacting based on gear meshing theory and continuous collision contact theory; (2) analyzing the influence of tooth breakage on the time-varying contact ratio, time-varying mesh stiffness and load-sharing ratio; (3) building the multi-state meshing-impacting nonlinear dynamic models of the spur gear systems under tooth breakage; (4) exploring the contact force evolution, bifurcation, chaos and collision characteristics of the gear systems under different system parameters; (5) verifying the local load-carrying capacity degradation of the broken gear systems by means of finite element analysis.
The remainder of this article is organized as follows. Section 2 describes the influence of tooth breakage on the multi-state meshing-impacting behavior of the spur gear pair. Section 3 presents the nonlinear dynamic modeling procedures for both local tooth breakage and tooth root breakage. Section 4 discusses the numerical results including contact force evolution, bifurcation diagrams, phase portraits, Poincaré maps and finite element verification. Finally, Section 5 provides the concluding remarks.
2. Influence of Tooth Breakage on Multi-State Meshing-Impacting Behavior
2.1 Physical Model of Gear Pair Meshing-Impacting
In this research, I establish a simplified torsional vibration model of the spur gear pair as shown in the conceptual framework. The model consists of a driving gear (subscript p) and a driven gear (subscript g), connected through a time-varying mesh stiffness \(k_{m}(\tau)\), a mesh damping \(c_{g}\), and a backlash of half size \(D\). The relative displacement along the line of action is defined as \(x = R_{bp}\theta_{p} – R_{bg}\theta_{g} – e(\tau)\), where \(R_{bp}\) and \(R_{bg}\) are the base circle radii of the driving and driven gears, respectively, and \(e(\tau)\) is the comprehensive transmission error. The gear backlash causes three possible motion states: tooth face meshing (\(x \ge D\)), tooth separation (\(-D < x < D\)), and tooth back-side collision (\(x = -D\)). Since the tooth back-side contact occurs within an extremely short time interval, I treat this contact as an impact process rather than a continuous meshing state. The collision recovery coefficient \(R\) is introduced to characterize the energy dissipation during the back-side impact.
The physical model is conceptualized with three types of contacts: the driving tooth flank pushing the driven tooth flank (face meshing), separation with no contact, and the driven tooth back flank striking the driving tooth back flank (back-side collision). The torque \(T_{p}\) and \(T_{g}\) act on the driving and driven gears, respectively. The symbols \(\theta_{p}\) and \(\theta_{g}\) denote the angular displacements of the two gears. The mesh damping is assumed to be viscous. Based on the continuous contact force model, the collision force during the back-side impact is described through a spring-damper contact law combined with the restitution coefficient.
2.2 Local Tooth Breakage and Time-Varying Contact Ratio
For the standard spur gear pair with local tooth breakage, I consider a case where the tip of one driving gear tooth is locally broken. The broken tooth loses partial contact with the driven gear tooth, which reduces the effective contact ratio. In the healthy gear pair, the actual meshing line is AD, where AB and CD are double-tooth contact zones and BC is the single-tooth contact zone. After the local breakage, the broken tooth disengages earlier at point D1, so the effective meshing line becomes AD1. The time-varying contact ratio is expressed as:
\[
\varepsilon_{b} = \frac{l_{AD1}}{p_{bt}} < \varepsilon_{m}
\]
where \(l_{AD1}\) is the length of the new meshing line AD1, \(p_{bt}\) is the base pitch, and \(\varepsilon_{m}\) is the contact ratio of the healthy gear pair. The length \(l_{AD1}\) is computed from:
\[
l_{AD1} = \sqrt{R_{ab}^{2} – R_{bp}^{2}} + \sqrt{R_{ag}^{2} – R_{bg}^{2}} – (R_{bp} + R_{bg})\tan\alpha_{0}
\]
where \(R_{ab}\) is the radius of the broken tooth tip circle and \(\alpha_{0}\) is the pressure angle.
Under the assumption that the local breakage affects the \(n_{b}\)-th meshing period, I classify the meshing-impacting states of the gear pair into five categories:
1. Double-tooth face meshing with \(x \ge D\), occurring in the zones AB1 or CD1 for the broken period or AB and CD for the healthy period.
2. Single-tooth face meshing (zone BC) with \(x \ge D\).
3. Single-tooth face meshing caused by the local breakage (zone B1B or D1D) with \(x \ge D\).
4. Tooth separation with \(-D < x < D\).
5. Tooth back-side collision with \(x = -D\).
2.3 Tooth Root Breakage of High-Contact-Ratio Spur Gear
For the high-contact-ratio spur gear pair, the healthy gear pair exhibits a “three-tooth–two-tooth–three-tooth–two-tooth–three-tooth” alternating meshing pattern along the line of action AF. The zones AB, CD and EF correspond to three-tooth meshing, while BC and DE correspond to two-tooth meshing. When a tooth root breakage occurs at the root of one driving gear tooth, the broken tooth loses its load-carrying capability along the entire meshing line. Consequently, the three-tooth meshing zones become two-tooth meshing zones, and the two-tooth meshing zones become single-tooth meshing zones. The meshing pattern becomes “two-tooth–single-tooth–two-tooth–single-tooth–two-tooth”.
The multi-state meshing-impacting behavior of the high-contact-ratio spur gear pair under tooth root breakage is classified into six categories:
1. Three-tooth face meshing: \(x \ge D\), with \(t \in [n T_{0}, (n + \varepsilon_{m} – 2) T_{0}]\) for healthy periods.
2. Normal two-tooth face meshing I: \(x \ge D\), with \(t \in ((n + \varepsilon_{m} – 2) T_{0}, (n+1) T_{0})\).
3. Two-tooth face meshing II caused by root breakage: \(x \ge D\), with \(t \in [n T_{0}, (n + \varepsilon_{m} – 2) T_{0}]\) for broken periods.
4. Single-tooth face meshing caused by root breakage: \(x \ge D\), with \(t \in ((n + \varepsilon_{m} – 2) T_{0}, (n+1) T_{0})\) for broken periods.
5. Tooth separation: \(-D < x < D\).
6. Tooth back-side collision: \(x = -D\).
The geometric parameters of the two gear pairs studied in this work are listed in Table 1 and Table 2.
| Symbol | Quantity | Value |
|---|---|---|
| \(z_p\) | Teeth number of driving gear | 40 |
| \(z_g\) | Teeth number of driven gear | 40 |
| \(m\) | Module (mm) | 3 |
| \(h_a^*\) | Addendum coefficient | 1 |
| \(c^*\) | Tip clearance coefficient | 0.25 |
| \(\alpha_0\) | Pressure angle (°) | 20 |
| \(B\) | Tooth width (mm) | 20 |
| \(E\) | Elastic modulus (N/mm²) | 206000 |
| \(I_p, I_g\) | Moments of inertia (kg·m²) | 3.52×10⁻³ |
| \(\nu\) | Poisson ratio | 0.3 |
Table 1. Geometric parameters of the standard spur gear pair with local tooth breakage.
| Symbol | Quantity | Value |
|---|---|---|
| \(z_p\) | Teeth number of driving gear | 40 |
| \(z_g\) | Teeth number of driven gear | 55 |
| \(m\) | Module (mm) | 3 |
| \(h_a^*\) | Addendum coefficient | 1.2 |
| \(c^*\) | Tip clearance coefficient | 0.25 |
| \(\alpha_0\) | Pressure angle (°) | 20 |
| \(B\) | Tooth width (mm) | 40 |
| \(E\) | Elastic modulus (N/mm²) | 206000 |
| \(I_p\) | Moment of inertia of driving gear (kg·m²) | 5.25×10⁻³ |
| \(I_g\) | Moment of inertia of driven gear (kg·m²) | 17.83×10⁻³ |
| \(\nu\) | Poisson ratio | 0.3 |
Table 2. Geometric parameters of the high-contact-ratio spur gear pair with tooth root breakage.
2.4 Time-Varying Mesh Stiffness and Load-Sharing Ratio
For the spur gear pair, the comprehensive time-varying mesh stiffness \(k_{m}(\tau)\) is computed by considering the Hertzian contact stiffness \(k_{h}\), the bending stiffness \(k_{bji}\), the axial compressive stiffness \(k_{aji}\), the shear stiffness \(k_{sji}\) and the gear body stiffness \(k_{fji}\), where \(j = p, g\) denotes the driving and driven gears, and \(i\) denotes the meshing tooth pair index. The mesh stiffness is expressed as:
\[
\frac{1}{k_{m}(\tau)} = \frac{1}{k_{h}} + \sum_{i} \left( \frac{1}{k_{bpi}} + \frac{1}{k_{api}} + \frac{1}{k_{spi}} + \frac{1}{k_{fpi}} \right) + \sum_{i} \left( \frac{1}{k_{bgi}} + \frac{1}{k_{agi}} + \frac{1}{k_{sgi}} + \frac{1}{k_{fgi}} \right)
\]
The local tooth breakage reduces the time-varying mesh stiffness in the influenced meshing period because the number of simultaneously meshing tooth pairs decreases locally. The double-tooth contact zone shrinks, and the single-tooth contact zone expands. This causes a local reduction of mesh stiffness and thus degrades the load-carrying capacity of the damaged spur gear.
For the high-contact-ratio spur gear pair operating in a steady temperature field, I further consider the thermal stiffness \(k_{tji}\) caused by the flash temperature on the tooth surface. The comprehensive mesh stiffness is then:
\[
\frac{1}{k_{m}(\tau)} = \frac{1}{k_{h}} + \sum_{i} \left( \frac{1}{k_{bpi}} + \frac{1}{k_{api}} + \frac{1}{k_{spi}} + \frac{1}{k_{fpi}} + \frac{1}{k_{tpi}} \right) + \sum_{i} \left( \frac{1}{k_{bgi}} + \frac{1}{k_{agi}} + \frac{1}{k_{sgi}} + \frac{1}{k_{fgi}} + \frac{1}{k_{tgi}} \right)
\]
The load-sharing ratio \(L_{i}(\tau)\) of the \(i\)-th meshing tooth pair is determined based on the minimum elastic potential energy theory:
\[
L_{i}(\tau) = \frac{u_{i} v_{i}}{\sum_{i} u_{i} v_{i}} = \frac{1 / k_{mi}(\tau)}{\sum_{i} 1 / k_{mi}(\tau)}
\]
where \(u_{i} = v_{i}^{-1} = k_{mi}(\tau)\), and \(k_{mi}(\tau)\) is the stiffness of the \(i\)-th tooth pair. Under tooth root breakage, the load-sharing ratio of the adjacent tooth pairs increases, causing stress concentration and aggravating the fatigue damage of the spur gear system.
3. Multi-State Meshing-Impacting Dynamic Modeling of the Spur Gear System
3.1 Dynamic Model of the Spur Gear Pair with Local Tooth Breakage
Based on the multi-state meshing-impacting classification and the time-varying meshing parameters, I establish the torsional vibration equations of the spur gear system under local tooth breakage. The governing equations are derived from Newton’s second law for each motion state.
For double-tooth face meshing, the equations of motion of the driving and driven gears are:
\[
\begin{cases}
I_{p}\ddot{\theta}_{p} = T_{p} – R_{bp}F_{Np1} – S_{dp1}F_{fp1} – R_{bp}F_{Np2} – S_{dp2}F_{fp2} \\
I_{g}\ddot{\theta}_{g} = -T_{g} + R_{bg}F_{Ng1} + S_{dg1}F_{fg1} + R_{bg}F_{Ng2} + S_{dg2}F_{fg2}
\end{cases}
\]
where \(F_{Npi} = F_{Ngi} = L_{i}(\tau) F_{m}\), and \(F_{m}\) is the total dynamic meshing force:
\[
F_{m} = k_{md}(\tau)(x – D) + c_{g}\dot{x}
\]
The friction forces are given by:
\[
F_{fpi} = F_{fgi} = \lambda_{di}(\tau) \mu_{di}(\tau) L_{i}(\tau) F_{m}
\]
where \(\lambda_{di}(\tau)\) is the friction direction coefficient and \(\mu_{di}\) is the dry friction coefficient. After rearranging, the relative motion equation for double-tooth meshing is obtained as:
\[
m_{e}\ddot{x} + \left[ c_{g} + \lambda_{d1}\mu_{d1}g_{d1}L_{1} + \lambda_{d2}\mu_{d2}g_{d2}L_{2} \right]_{1} \dot{x} + k_{md}(\tau)(x – D) = F + F_{h}(\tau)
\]
where \(m_{e}\) is the equivalent mass, \(g_{di}\) is the equivalent friction arm of the \(i\)-th tooth pair, \(F\) is the external load, and \(F_{h}(\tau)\) is the internal excitation due to transmission error.
For single-tooth face meshing I, the equation is:
\[
m_{e}\ddot{x} + \left[ c_{g} + \lambda_{d1}\mu_{d1}g_{d1} \right]_{1} \dot{x} + k_{ms1}(\tau)(x – D) = F + F_{h}(\tau)
\]
For single-tooth face meshing II, the equation has the same form but with the stiffness \(k_{ms2}(\tau)\). During tooth separation, the meshing force and friction force vanish, and the relative motion equation becomes:
\[
m_{e}\ddot{x} = F + F_{h}(\tau)
\]
The tooth back-side collision is described by the impact model with the recovery coefficient \(R\):
\[
\dot{x}^{+} = -R \dot{x}^{-}
\]
The collision force is characterized at the maximum deformation point:
\[
F_{c} = \frac{5}{4} k_{c}(\tau) \delta_{m} \left[ \frac{5 R^{2} m_{e} \dot{x}^{-2}}{4 k_{c}(\tau)} \right]^{2/5}
\]
where \(\delta_{m}\) is the maximum deformation and \(k_{c}(\tau)\) is the tooth back-side contact stiffness.
By introducing the dimensionless time \(t = \omega_{n} \tau\) and the dimensionless relative displacement \(x = x / D\), the dimensionless governing equation of the spur gear system under local tooth breakage is expressed as:
\[
\ddot{x} – \eta_{1}(t, x) f_{1}(x, \dot{x}) = F + \varepsilon \omega^{2} \cos(\omega t)
\]
with the state-dependent function \(\eta_{1}(t, x)\) and the meshing force function \(f_{1}(x, \dot{x})\). The dimensionless dynamic contact force is:
\[
F_{m} = k_{d}(t)(x – D) + c \dot{x}, \quad x > D
\]
3.2 Dynamic Model of the High-Contact-Ratio Spur Gear with Tooth Root Breakage
For the high-contact-ratio spur gear pair with tooth root breakage, I establish six different groups of equations according to the motion states. For three-tooth face meshing:
\[
m_{e}\ddot{x} + \left[ c_{g} + \lambda_{d1}\mu_{d1}g_{d1}L_{1} + \lambda_{d2}\mu_{d2}g_{d2}L_{2} + \lambda_{d3}\mu_{d3}g_{d3}L_{3} \right] \dot{x} + k_{mt}(\tau)(x – D) = F + F_{h}(\tau)
\]
For two-tooth face meshing I (healthy two-tooth meshing):
\[
m_{e}\ddot{x} + \left[ c_{g} + \lambda_{d1}\mu_{d1}g_{d1}L_{1} + \lambda_{d2}\mu_{d2}g_{d2}L_{2} \right] \dot{x} + k_{md1}(\tau)(x – D) = F + F_{h}(\tau)
\]
For two-tooth face meshing II (caused by root breakage), the equation has the same form but with the equivalent friction arm \(g_{di}^{\ast}\) and the stiffness \(k_{md2}(\tau)\). For single-tooth meshing:
\[
m_{e}\ddot{x} + \left[ c_{g} + \lambda_{d1}\mu_{d1}g_{d1}^{\ast} \right] \dot{x} + k_{ms}(\tau)(x – D) = F + F_{h}(\tau)
\]
During tooth separation:
\[
m_{e}\ddot{x} = F + F_{h}(\tau)
\]
The dimensionless normalized equation for the high-contact-ratio spur gear system under tooth root breakage is:
\[
\ddot{x} – \eta_{2}(t, x) f_{2}(x, \dot{x}) = F + \varepsilon \omega^{2} \cos(\omega t)
\]
where \(\eta_{2}(t, x)\) represents the state function distinguishing the three-tooth, two-tooth, single-tooth, separation and collision states.
4. Nonlinear Dynamic Characteristics of the Spur Gear System with Tooth Breakage
4.1 Numerical Methods and Poincaré Maps
The dynamic equations are solved numerically using the fourth-order Runge-Kutta method with variable step size. I define three Poincaré sections to characterize the meshing, separation and collision behaviors:
– Stroboscopic section: \(\beta_{t} = \{(x, \dot{x}, t) \in \mathbb{R}^{2} \times \mathbb{T} \mid \omega t \bmod 2\pi = 0\}\)
– Tooth face contact section: \(\beta_{d} = \{(x, \dot{x}, t) \in \mathbb{R}^{2} \times \mathbb{T} \mid x = D\}\)
– Tooth back-side collision section: \(\beta_{b} = \{(x, \dot{x}, t) \in \mathbb{R}^{2} \times \mathbb{T} \mid x = -D\}\)
The phase portrait, time history, Poincaré map and frequency spectrum are combined to identify the motion types and meshing-impacting characteristics of the spur gear system.
4.2 Contact Force Evolution under Local Tooth Breakage
With the system parameters \(D=1\), \(c=0.01\), \(\varepsilon=0.05\), \(F=0.05\), \(\mu_{d1}=\mu_{d2}=0.15\), \(\omega=1.1\), and \(R=0.85\), the time history of the dimensionless dynamic contact force is shown in Fig. 4(a). The dynamic contact force \(F_{m}\) alternates among positive values, zero and negative values periodically, indicating that the spur gear system experiences tooth face meshing, tooth separation and tooth back-side collision in one motion period. The local tooth breakage causes a local reduction of \(F_{m}\), while the adjacent tooth pairs suffer increased forces. This means that the broken spur gear has a lower local power-transmission capability, and the neighboring teeth are subjected to higher loads.
The comparison between the broken and healthy periods shows that:
\[
T_{b1} > T_{1}, \quad T_{b2} < T_{2}, \quad T_{b1} + T_{b2} = T_{1} + T_{2}
\]
where \(T_{1}\) and \(T_{2}\) are the single-tooth and double-tooth meshing periods of the healthy spur gear, while \(T_{b1}\) and \(T_{b2}\) correspond to the broken cases. The single-tooth meshing period increases and the double-tooth meshing period decreases after the breakage.
4.3 Influence of Load Coefficient on the Spur Gear System with Local Tooth Breakage
The multi-initial bifurcation diagrams and the collision force period diagrams of the locally broken spur gear system are illustrated in Fig. 5. When the load coefficient is large, the system exhibits stable period-1 motion without tooth back-side collision. With decreasing load, the system first enters chaos and then develops into period-4 motion, accompanied by the appearance of back-side collision. A period-2 attractor coexists with period-4 at moderate loads, which significantly increases the complexity of the system dynamics. The coexistence of multiple attractors indicates that the system is sensitive to initial conditions. Further decreasing the load, the system enters chaos again, and the chaotic motion induces tooth back-side collision. When the load becomes sufficiently small, the multi-attractor coexistence disappears, and the system evolves into a relatively stable periodic motion.
The main conclusions can be summarized as:
1. Large load suppresses tooth back-side collision, so a spur gear system is more stable under heavy load.
2. Chaos induces tooth back-side collision, which aggravates the vibration and noise.
3. The multi-attractor coexistence phenomenon appears in a moderate load range, which may lead to sudden jumps between different motion states.
4.4 Influence of Meshing Frequency on the Spur Gear System with Local Tooth Breakage
When the meshing frequency is used as the bifurcation parameter, the locally broken spur gear system displays rich dynamic behaviors. At relatively low meshing frequencies, the system is chaotic and back-side collision occurs. With increasing meshing frequency, the chaotic motion degenerates into period-1 motion and the back-side collision disappears. Period-1 and period-2 attractors coexist in a certain frequency range. Later, the period-1 motion jumps to period-3 motion, and period-3 and period-8 attractors coexist with period-2 attractors. When the frequency becomes relatively high, the system enters chaos again through boundary crisis, and back-side collision reappears. The coexistence of chaotic and periodic attractors makes the spur gear system highly sensitive to initial conditions. At the end of the frequency range, the system degenerates into a periodic motion with tooth face meshing, tooth separation and back-side collision alternating periodically.
Therefore, both too low and too high meshing frequencies may induce chaotic motion and back-side collision in the broken spur gear system. Choosing a reasonable meshing frequency interval can reduce the probability of back-side collision and improve the smoothness of the spur gear system.
4.5 Bifurcation Comparison between the Broken and Healthy Spur Gear Systems
The bifurcation diagram of the healthy spur gear system with load variation is compared with that of the locally broken spur gear system. For the healthy system, when the load is large, the system exhibits period-1 and period-2 motion coexistence without back-side collision. With decreasing load, the system enters chaos through crisis and then degenerates into period-6 motion. The local tooth breakage changes not only the dynamic behavior types but also the bifurcation characteristics. Under moderate load, the local breakage may induce multi-attractor coexistence and cause back-side collision. Under small load, the influence of the breakage becomes weak, and the chaotic load range is similar to that of the healthy system. Under very small load, the breakage may induce chaotic motion and aggravate the vibration.
4.6 Finite Element Verification of the Locally Broken Spur Gear
A three-dimensional finite element model of the locally broken spur gear pair is established in Workbench. The material is alloy steel with a friction coefficient of 0.15 and an applied torque of 500 N·m. The equivalent stress distributions of the healthy and broken spur gears at the same meshing position are compared. For the healthy spur gear in double-tooth meshing, the maximum contact stress is 538.29 MPa. For the broken spur gear, the double-tooth meshing becomes single-tooth meshing, and the maximum contact stress increases to 1045.4 MPa. Thus, the local tooth breakage of the spur gear leads to a significant stress increase at the tooth root and the contact area, which may further cause crack propagation, wear or plastic deformation.
4.7 Dynamic Behaviors of the High-Contact-Ratio Spur Gear with Tooth Root Breakage
In this subsection, I investigate the influence of tooth root breakage on the high-contact-ratio spur gear system. Four typical motion behaviors are compared: tooth face meshing, tooth face meshing–tooth separation, tooth face meshing–tooth separation–tooth back-side collision, and the transition of motion behavior.
When the system operates in pure tooth face meshing, the tooth root breakage causes an increase in the peak value of the dynamic contact force and changes the motion period from period-1 to near period-3. The topology of the phase trajectory is enlarged. When the system operates in tooth face meshing–tooth separation, the tooth root breakage causes a local reduction of the contact force and leads to an earlier separation. If the broken tooth meshes during the separation state, the influence is negligible. When the system operates in tooth face meshing–tooth separation–tooth back-side collision, the tooth root breakage slightly increases the peak force and the collision force, and the separation ratio decreases slightly. When the tooth root breakage changes the motion behavior, the peak force changes significantly, and the system may develop near-period motion with a larger probability of tooth separation.
The separation ratio is defined as the proportion of the separation time in one period. I analyze the influence of the backlash \(D\) and the load \(F\) on the separation ratio of the broken high-contact-ratio spur gear system. The results show that smaller \(D\) leads to smaller contact force, smaller collision force, smaller topology and smaller separation ratio. Larger \(F\) leads to smaller peak contact force but has little influence on the collision force. The topology of the phase trajectory and the separation ratio decrease with increasing load \(F\).
4.8 Bifurcation and Chaos of the Broken High-Contact-Ratio Spur Gear with Meshing Frequency
The bifurcation diagrams of the broken high-contact-ratio spur gear system with respect to the meshing frequency are shown in Fig. 14. When the meshing frequency is very small, the system is chaotic with back-side collision. With increasing frequency, the chaotic motion degenerates into period-1 motion, and the back-side collision disappears. The system then experiences period-2, period-4, chaos, period-6, period-3, chaos and finally period-1 motion as the frequency increases. The system is a near-period-3 motion in certain ranges, meaning that within one meshing period there are three sub-periods. The back-side collision appears only in the chaotic region and in some periodic regions with very small frequency. Therefore, choosing a relatively large meshing frequency can effectively suppress the back-side collision of the broken spur gear system.
4.9 Bifurcation and Chaos of the Broken High-Contact-Ratio Spur Gear with Error Coefficient
The error coefficient \(\varepsilon\) is used as the bifurcation parameter. For very small \(\varepsilon\), the system is in period-1 motion with pure tooth face meshing. When \(\varepsilon\) increases beyond the first critical value, tooth separation appears. The system then evolves into period-2 motion through jump. A further increase of \(\varepsilon\) causes the system to enter chaos at the boundary crisis, accompanied by tooth back-side collision. After that, the system experiences the following evolution: chaos \(\rightarrow\) period-3 \(\rightarrow\) period-6 \(\rightarrow\) period-12 \(\rightarrow\) period-6 \(\rightarrow\) period-3 \(\rightarrow\) chaos \(\rightarrow\) period-3 \(\rightarrow\) chaos \(\rightarrow\) period-4 \(\rightarrow\) chaos. The tooth back-side collision occurs in the chaotic regimes and in most periodic regimes when the error coefficient is relatively large. For very small error coefficients, the meshing-impacting behavior disappears, and the spur gear system operates smoothly. Therefore, choosing a small error coefficient can suppress back-side collision, but it may not reveal the complete nonlinear dynamic characteristics of the gear system. A moderate error coefficient is preferred for the investigation of the meshing-impacting mechanism.
4.10 Comparison of the Bifurcation Characteristics between the Broken and Healthy High-Contact-Ratio Spur Gears
The bifurcation diagram of the healthy high-contact-ratio spur gear system with respect to the error coefficient is compared with that of the broken system. For the healthy system, the evolution is: period-1 (pure meshing) \(\rightarrow\) period-1 (meshing-separation) \(\rightarrow\) period-2 (meshing-separation) \(\rightarrow\) chaos \(\rightarrow\) period-3 \(\rightarrow\) period-6 \(\rightarrow\) chaos. For the broken system, the evolution is more complicated: the system undergoes an inverse period-doubling bifurcation (period-12 \(\rightarrow\) period-6 \(\rightarrow\) period-3) before entering the final chaotic regime. The tooth root breakage significantly increases the dynamic complexity of the spur gear system in the error coefficient interval from 0.03 to 0.083. In this interval, the broken system exhibits additional periodic windows, inverse period-doubling bifurcations and larger chaotic attractors, which may cause severe vibration and noise. Outside this interval, the influence of the tooth root breakage on the system motion behavior is relatively small.
4.11 Finite Element Verification of the Broken High-Contact-Ratio Spur Gear
A finite element model of the high-contact-ratio spur gear pair with tooth root breakage is established. The material is alloy steel, the friction coefficient is 0.15, and the applied torque is 1200 N·m. The equivalent stress distributions of the healthy and broken gears at the same meshing positions are compared. For the healthy three-tooth meshing state, the maximum contact stress is 396.13 MPa. After the tooth root breakage, the three-tooth meshing becomes two-tooth meshing, and the maximum stress increases to 494.71 MPa. For the healthy two-tooth meshing state, the maximum stress is 399.02 MPa. After the breakage, the two-tooth meshing becomes single-tooth meshing, and the maximum stress increases to 614.94 MPa. The finite element analysis verifies that the tooth root breakage reduces the number of simultaneously meshing tooth pairs and causes stress concentration at the adjacent tooth roots. This is the primary reason for the increased vibration and noise of the broken spur gear system.
5. Conclusion and Prospects
In this research, I have systematically investigated the multi-state meshing-impacting nonlinear dynamics of spur gear transmission systems under two types of tooth breakage faults: local tooth breakage and tooth root breakage. The main contributions and conclusions are summarized as follows:
(1) A spur gear meshing-impacting physical model was established by introducing a dissipative contact force model with the collision recovery coefficient. The instantaneous back-side contact was treated as a collision process rather than a continuous meshing state, which captures the energy dissipation during the impact. The tooth breakage alters the local contact ratio and the meshing pattern of the spur gear pair. Local tooth breakage reduces the effective meshing line and changes the double-tooth and single-tooth meshing zones. Tooth root breakage of the high-contact-ratio spur gear changes the meshing pattern from “three-tooth–two-tooth–three-tooth–two-tooth–three-tooth” to “two-tooth–single-tooth–two-tooth–single-tooth–two-tooth”. The time-varying mesh stiffness and load-sharing ratio are locally reduced and increased, respectively, which degrades the local load-carrying capacity.
(2) The multi-state meshing-impacting dynamic models of the spur gear systems under tooth breakage were established based on Newton’s second law, gear meshing theory and contact impact theory. For the locally broken standard spur gear, five motion states were classified: double-tooth face meshing, single-tooth face meshing I, single-tooth face meshing II, tooth separation and tooth back-side collision. For the root-broken high-contact-ratio spur gear, six motion states were classified: three-tooth face meshing, two-tooth face meshing I, two-tooth face meshing II, single-tooth face meshing, tooth separation and tooth back-side collision. The dimensionless normalized equations were derived for both cases.
(3) The numerical results reveal that the tooth breakage causes a local reduction of the dynamic contact force and an increase of the adjacent tooth contact forces. The dynamic complexity of the broken spur gear system is much richer than that of the healthy system. The system exhibits periodic motions, multi-attractor coexistence, chaos, tooth separation and tooth back-side collision with the variation of the load coefficient, meshing frequency and error coefficient. Chaos can induce tooth back-side collision, and the multi-attractor coexistence increases the sensitivity of the spur gear system to initial conditions. Larger load suppresses back-side collision, while smaller load induces it. Larger error coefficients induce chaos and back-side collision, while smaller error coefficients suppress them. Choosing reasonable parameter ranges can reduce the probability of back-side collision and improve the stability of the broken spur gear system.
(4) The finite element analysis of the broken spur gear pairs verifies that the tooth breakage causes stress concentration at the adjacent tooth roots and contact areas. The maximum equivalent stress increases significantly compared with the healthy gear pair, which increases the risk of subsequent failures such as crack propagation, pitting and even tooth fracture.
Although the present work has provided a comprehensive framework for the nonlinear dynamics of broken spur gear systems, several aspects deserve further investigation. First, the current model simplifies the friction effect on the tooth back-side collision. A more accurate impact model considering the tangential friction and the oblique impact angle should be developed. Second, the present study focuses on the deterministic excitation. The influence of random excitation on the meshing-impacting behavior of the broken spur gear system is still an open issue. Third, the global dynamics, including the erosion of safe basins and the evolution of attraction basins, should be investigated for the broken spur gear system to provide a more complete understanding of the safety characteristics. Fourth, the experimental validation of the theoretical and numerical results is necessary to confirm the feasibility and accuracy of the proposed models. These issues will be addressed in the future research.