Abstract
Gear transmission systems serve as core components in mechanical equipment, and their dynamic characteristics directly influence the reliability and stability of the entire machine operation. Tooth meshing impact, as one of the primary sources of dynamic excitation in gear systems, has been identified as a key scientific problem in the field of mechanical transmission. In this study, I focus on the single-stage involute spur gear pair and systematically investigate the dynamic behavior of spur gears considering tooth meshing impacts. I categorize tooth meshing impacts into tooth switching impacts and tooth boundary impacts based on the multi-state meshing characteristics of spur gear pairs. A tooth meshing impact model is established based on the kinetic energy theorem and Hertz contact theory, and the results are compared with those obtained from the traditional impulse method. The comparison demonstrates that the energy method can more accurately characterize the transient impact characteristics induced by extremely short-duration force mutations during spur gear meshing processes. Furthermore, I construct a nonlinear dynamic model of spur gear pairs considering meshing impacts by integrating time-varying parameters and multi-state meshing characteristics. Through phase portraits, Poincaré maps, force-time diagrams, bifurcation diagrams, and top Lyapunov exponent diagrams, I systematically investigate the effects of meshing frequency, load, damping, and comprehensive transmission error on the nonlinear dynamic characteristics of spur gear systems with and without tooth meshing impacts.
1. Introduction
The dynamic performance of spur gear transmission systems significantly affects the stability, transmission efficiency, and service life of the entire system. During the gear meshing process, tooth meshing impact phenomena are frequently observed, particularly under high-speed and heavy-load conditions. These impacts affect the smoothness of tooth meshing and further induce vibration, noise, wear, and fatigue damage in spur gear pairs. Therefore, an in-depth investigation into the generation mechanisms of tooth impacts and their influence on system dynamic characteristics holds significant theoretical and engineering importance for improving the reliability and service performance of spur gear transmission systems.
Spur gear pairs exhibit complex multi-state meshing characteristics due to the presence of backlash and contact ratio. Zhou and Chen proposed an off-line meshing impact model that quantitatively described the impact force and friction coefficient based on the equivalent error-tooth comprehensive deformation method. Zhu et al. established a tooth meshing impact model based on the impulse method; however, the calculated impact force amplitudes were notably underestimated, and the predicted impact durations were excessively long, limiting the accuracy of their results. To address these deficiencies, I propose an improved modeling approach based on the energy method and Hertz contact theory in this study.
In this paper, I first analyze the multi-state meshing characteristics and calculate key time-varying parameters including load distribution ratio, mesh stiffness, and friction coefficient for spur gears. Subsequently, I establish a tooth meshing impact model using the kinetic energy theorem and Hertz contact theory, enabling precise calculation of both impact forces and impact durations. The proposed energy-based approach is systematically compared with the traditional impulse method to validate its superiority. Finally, I construct a comprehensive nonlinear dynamic model of spur gear pairs incorporating both tooth switching impacts and tooth boundary impacts, and investigate the influence of critical parameters on system dynamic behavior.
2. Multi-State Meshing Characteristics and Time-Varying Parameters of Spur Gear Pairs
2.1 Multi-State Meshing Characteristics
Based on the relative displacement $x$ along the line of action and the half-backlash $D$, spur gear pairs can be classified into three fundamental meshing states: tooth surface meshing ($x > D$), tooth back contact ($x < -D$), and disengagement ($-D \le x \le D$). Due to the contact ratio $\varepsilon_m = 1.595$, both tooth surface meshing and tooth back contact contain double-tooth and single-tooth sub-states. The physical model of the spur gear pair is shown in Table 1 with the basic parameters.
| Parameter | Driving gear, $p$ | Driven gear, $g$ |
|---|---|---|
| Number of teeth | 21 | 26 |
| Module $m$ (mm) | 5 | 5 |
| Moment of inertia $I$ (kg·mm²) | 0.21 | 0.26 |
| Face width $B$ (mm) | 50 | 45 |
| Elastic modulus $E$ (GPa) | 163 | 211 |
| Poisson’s ratio $\nu$ | 0.277 | 0.33 |
| Material | Constantan | 40Cr steel |
| Pressure angle $\alpha_0$ (°) | 20 | 20 |
| Contact ratio $\varepsilon_m$ | 1.595 | 1.595 |
The $i$-th pair of teeth experiences the following sequence during the meshing process: double-tooth meshing (from mesh-in point $M_i$ to point $A$), single-tooth meshing (from point $A$ to point $C$), and double-tooth meshing (from point $C$ to mesh-out point $M_o$). The transition points $t_{M_i}$, $t_A$, $t_C$, and $t_{M_o}$ are critical instants where sudden changes in meshing force occur, leading to tooth meshing impacts in spur gear systems.
2.2 Time-Varying Load Distribution Ratio
For the spur gear pair with contact ratio $\varepsilon_m = 1.595$, the load distribution ratio $L_{jc}(t)$ for the $c$-th pair of teeth in meshing state $j$ (where $j = d$ for tooth surface meshing and $j = k$ for tooth back contact) is calculated as follows:
$$L_{jc}(t) = \begin{cases} \frac{1}{3} + \frac{1}{3}\frac{\zeta_{j}^{i} – \zeta_{jc}(t)}{\zeta_{Mi} – \zeta_{A}} & \zeta_{Mi} \le \zeta < \zeta_A \\ 1 & \zeta_A \le \zeta < \zeta_C \\ \frac{1}{3} + \frac{1}{3}\frac{\zeta_{j}^{i+1} – \zeta_{jc}(t)}{\zeta_C – \zeta_{M_o}} & \zeta_C \le \zeta < \zeta_{M_o} \end{cases}$$
This equation demonstrates that the load distribution ratio exhibits abrupt changes at the four critical time instants: $t_{Mi}$, $t_A$, $t_C$, and $t_{M_o}$. Consequently, the corresponding meshing forces also experience sudden discontinuities at these moments, which is the root cause of tooth meshing impact in spur gear systems.
2.3 Time-Varying Mesh Stiffness
Using the potential energy method, the mesh stiffness components for spur gear pairs include Hertzian contact stiffness $k_h$, bending stiffness $k_{blc}$, axial compressive stiffness $k_{alc}$, shear stiffness $k_{slc}$, and fillet foundation stiffness $k_{flc}$. The Hertzian contact stiffness is given by:
$$k_h = \frac{\pi EB}{4(1 – \nu^2)}$$
where $E$ is the elastic modulus, $B$ is the face width, and $\nu$ is the Poisson’s ratio. Considering the coupling effect between adjacent teeth on the foundation stiffness, the improved analytical expressions are:
$$k_{flc} = \frac{EB}{\cos^2\alpha_l}\left[\frac{u_{fc1}^2}{S_{fl}^2}L_{fc1}^* + \frac{u_{fc1}}{S_{fl}}M_{fc1}^* + P_{fc1}^* + \frac{u_{fc1}}{S_{fl}}Q_{fc1}^*\right]$$
The total time-varying mesh stiffness within one meshing period $T_0$ can be expressed as:
$$k(t) = \begin{cases} 1/\sum_{c=1}^{2}\left(\frac{1}{k_{h}} + \frac{1}{k_{bpc}} + \frac{1}{k_{apc}} + \frac{1}{k_{spc}} + \frac{1}{k_{fpc}} + \frac{1}{k_{bgc}} + \frac{1}{k_{agc}} + \frac{1}{k_{sgc}} + \frac{1}{k_{fgc}}\right) & 0 \le t < (\varepsilon_m – 1)T_0 \\ 1/\left(\frac{1}{k_h} + \frac{1}{k_{bp1}} + \frac{1}{k_{ap1}} + \frac{1}{k_{sp1}} + \frac{1}{k_{fp1}} + \frac{1}{k_{bg1}} + \frac{1}{k_{ag1}} + \frac{1}{k_{sg1}} + \frac{1}{k_{fg1}}\right) & (\varepsilon_m – 1)T_0 \le t < T_0 \end{cases}$$
The comparison between considering and not considering the coupling effect of adjacent teeth shows that the stiffness in the double-tooth meshing region decreases noticeably when the coupling effect is considered, while the stiffness in the single-tooth meshing region remains unchanged. Both double-tooth meshing regions exhibit identical stiffness characteristics; only the engaged tooth pairs differ.
2.4 Time-Varying Friction Coefficient
The time-varying friction coefficient for spur gear pairs is determined by:
$$\mu_{jc}(t) = e^{f( SR_{jc}(t), P_{jc}(t), R_{avgi}, \eta_{Mi} )} \cdot b_2 \cdot P_{jc}^{b_3}(t) \cdot |SR_{jc}(t)|^{b_6} \cdot v_e^{b_7}(t) \cdot \eta_{Mi}^{b_8} \cdot R_{avgi}^{b_9}$$
where $SR_{jc}(t)$ is the slide-to-roll ratio and $P_{jc}(t)$ is the contact pressure. After considering the load distribution ratio, the time-varying friction coefficient exhibits abrupt changes at the tooth switching instants $t_A$ and $t_C$, and becomes zero at the pitch point $t_p$. Additionally, the friction coefficient at mesh-in ($t_{Mi}$) is greater than that at mesh-out ($t_{M_o}$).
3. Tooth Meshing Impact Analysis
3.1 Impact Classification and Mechanisms
Based on the multi-state meshing characteristics of spur gears, I classify tooth meshing impacts into two primary categories: tooth switching impacts and tooth boundary impacts.
Tooth switching impacts occur during the transition between different meshing states within the tooth surface meshing or tooth back contact process. These include:
- Mesh-in impact at time $t_{Mi}$
- Double-to-single tooth switching impact at time $t_A$
- Single-to-double tooth switching impact at time $t_C$
- Mesh-out impact at time $t_{M_o}$
Tooth boundary impacts occur when teeth reach the tooth surface boundary ($x = D$) or tooth back boundary ($x = -D$) from the disengagement state, resulting in tooth surface boundary impacts and tooth back boundary impacts, respectively.
For the coupling effect analysis, when spur gear teeth reach the tooth surface boundary with $F_{idmax} > F_d$, the system experiences boundary impact and remains in the disengagement state with reduced velocity: $\dot{x}_+ = e\dot{x}_-$ and $F_{i+} = F_{i-} – F_d$. When $F_{idmax} \le F_d$, the system enters the tooth surface meshing state. Similarly, at the tooth back boundary, if $F_{ikmax} < F_k$, the system maintains disengagement with $\dot{x}_+ = -e\dot{x}_-$; if $F_{ikmax} \ge F_k$, the system enters the tooth back contact state with $\dot{x}_+ = e\dot{x}_-$.
3.2 Energy-Based Impact Model
I assume that the spur gear has rigid supports, is isolated from other inertial bodies, and the lubrication conditions are simplified. At any instant of tooth meshing, the spur gear pair can be treated as two contacting cylinders. The equivalent contact radii of the driving and driven gears are:
$$R_{cp}(t) = \sqrt{R_{bp}^2 + R_{bp} \tan\alpha + \sqrt{R_{bg}^2 – R_{bp}^2}\sqrt{R_{ag}^2 – R_{bp}^2} – R_{bp}^2 + R_{bp}\omega_p t}$$
$$R_{cg}(t) = \sqrt{R_{bg}^2 + R_{ag}^2 – R_{bp}^2} – R_{bg}\omega_p t$$
The impact process is divided into a compression phase and a restitution phase. Based on the kinetic energy theorem, the compression phase is described by:
$$\frac{1}{2}m_e[v_-^2 – \dot{\delta}^2(t)] = \int_{\delta_-}^{\delta(t)} F_i(t)d\delta(t)$$
where $m_e = m_1m_2/(m_1 + m_2)$ is the equivalent mass, $v_-$ is the initial relative velocity, and $\delta_-$ is the initial deformation. The impact force is modeled using the spring-damper model and Hertz contact theory:
$$F_i(t) = k(t)\delta^n(t) + \lambda\delta^n(t)\dot{\delta}(t)$$
where $n = 10/9$ is the contact exponent for cylindrical contact based on the Palmgren model, and $\lambda$ is the hysteresis damping coefficient.
The equivalent velocity during the compression phase is determined as $\dot{\delta}_c = 2v_-/3$. The maximum deformation is:
$$\delta_{max} = \left[\frac{(n+1)m_e v_-^2}{2k(t)}\cdot\frac{2}{2\lambda v_- + 3}\right]^{\frac{1}{n+1}}$$
The maximum impact force is:
$$F_{imax} = k(t)\left[\frac{(n+1)m_e v_-^2}{2k(t)}\cdot\frac{2}{2\lambda v_- + 3}\right]^{\frac{n}{n+1}}$$
Through the energy conservation law during the entire impact process, the hysteresis damping coefficient is:
$$\lambda = \frac{3(1 – e^2)k(t)}{2ev_-}$$
The compression phase duration is approximated by:
$$t_s = \frac{\delta_{max} – \delta_-}{\dot{\delta}_c} = \frac{3(\delta_{max} – \delta_-)}{2v_-}$$
Similarly, the restitution phase duration is:
$$t_r \approx \frac{\delta_{max} – \delta_+}{(2v_+/3)}$$
where $v_+ = ev_-$. The total impact time is $t_{total} = t_s + t_r$.
3.3 Comparison between Energy Method and Impulse Method
Table 2 presents the quantitative comparison between the energy method and the impulse method for different relative velocities in spur gear impact analysis.
| Relative velocity (m/s) | Max impact force – Energy (N) | Max impact force – Impulse (N) | Impact duration – Energy (μs) | Impact duration – Impulse (μs) |
|---|---|---|---|---|
| 1 | 54.39 | 6.69 | 2.2 | 11.84 |
| 2 | 112.81 | 15.37 | 2.18 | 10.3 |
| 3 | 172.87 | 25.01 | 2.07 | 9.5 |
| 4 | 234.01 | 35.32 | 2.04 | 8.97 |
The energy method yields significantly higher maximum impact forces compared to the impulse method: 8.13 times higher at 1 m/s, 7.34 times at 2 m/s, 6.91 times at 3 m/s, and 6.63 times at 4 m/s. Conversely, the impact duration calculated by the energy method is substantially shorter, being 0.19, 0.21, 0.22, and 0.23 times the impulse method values at the corresponding velocities. Both methods confirm that as the relative velocity increases, the maximum impact force increases while the impact duration decreases. The energy method results better reflect the transient impact characteristics of spur gear meshing, where extremely short-duration force mutations occur. The superiority of the energy method lies in: the inclusion of the damping force component, the use of the more appropriate cylindrical contact model with $n = 10/9$, the consideration of adjacent tooth coupling effects on foundation stiffness, the incorporation of initial deformation in compression and final deformation in restitution, and the capability to calculate impact force and deformation at any given instant rather than only maximum values.
3.4 Dynamic Meshing Force Analysis
Based on the Newton-Euler method, the dynamic meshing force of spur gear pairs in different meshing states is derived as:
$$F_{jm} = \frac{I_g R_{bp} T_p + I_p R_{bg} T_g}{I_g R_{bp}^2 + I_p R_{bg}^2 + A_{j1} + B_{j2}}$$
where the friction-related terms are:
$$A_{jc} = \lambda_{jc}(t)\mu_{jc}(t)\gamma_{jc}(t)(I_g R_{bp} S_{pc}(t) + I_p R_{bg} S_{gc}(t))$$
Under constant power of 800 W, I analyzed the dynamic meshing forces at rotational speeds of 1000, 2000, 4000, and 6000 r/min. The dynamic meshing forces exhibit significant abrupt changes at the four critical positions ($t_{Mi}$, $t_A$, $t_C$, and $t_{M_o}$) due to the load distribution ratio. When the rotational speed increases by a factor of $h$, the dynamic meshing force decreases to approximately $1/h$ of its original value, and the time scale correspondingly reduces to approximately $1/h$.
3.5 Coupling Analysis of Dynamic Force and Impact Force
By coupling the dynamic meshing force with the tooth switching impact force, I divided a complete meshing cycle into seven distinct motion stages for spur gear pairs. Table 3 summarizes the time durations of each stage at different rotational speeds.
| Speed (r/min) | Mesh-in impact (μs) | Double mesh (μs) | Double-to-single impact (μs) | Single mesh (μs) | Single-to-double impact (μs) | Double mesh (μs) | Mesh-out impact (μs) |
|---|---|---|---|---|---|---|---|
| 1000 | 1.302 | 1700 | 0.907 | 1157 | 1.048 | 1700 | 1.33 |
| 2000 | 1.348 | 850 | 0.954 | 578 | 1.09 | 850 | 1.378 |
| 4000 | 1.395 | 425 | 0.991 | 289 | 1.134 | 425 | 1.426 |
| 6000 | 1.423 | 283 | 1.019 | 193 | 1.171 | 283 | 1.455 |
Table 4 presents the percentage proportions of each stage within a complete meshing cycle.
| Speed (r/min) | Mesh-in impact (%) | Double mesh (%) | Double-to-single impact (%) | Single mesh (%) | Single-to-double impact (%) | Double mesh (%) | Mesh-out impact (%) |
|---|---|---|---|---|---|---|---|
| 1000 | 0.0285 | 37.268 | 0.02 | 25.363 | 0.023 | 37.268 | 0.0292 |
| 2000 | 0.059 | 37.235 | 0.042 | 25.321 | 0.048 | 37.235 | 0.06 |
| 4000 | 0.122 | 37.152 | 0.087 | 25.263 | 0.099 | 37.152 | 0.125 |
| 6000 | 0.186 | 37.039 | 0.133 | 25.26 | 0.153 | 37.039 | 0.19 |
Table 5 lists the peak impact forces for different switching impact types.
| Speed (r/min) | Mesh-in impact force (N) | Double-to-single impact force (N) | Single-to-double impact force (N) | Mesh-out impact force (N) |
|---|---|---|---|---|
| 1000 | 54.603 | 213.882 | 201.41 | 53.48 |
| 2000 | 27.304 | 108.843 | 101.215 | 26.755 |
| 4000 | 13.653 | 54.66 | 50.889 | 13.377 |
| 6000 | 9.102 | 36.781 | 34.35 | 8.929 |
As the rotational speed increases, the proportion of time occupied by tooth meshing impacts within a complete meshing cycle increases significantly while the stable meshing time decreases, leading to reduced system stability. The overlapped regions between dynamic meshing force and switching impact show proportional expansion with increasing speed. Moreover, the peak forces of mesh-in and mesh-out impacts are approximately 1.1 times the amplitude of force mutation, while the peak forces of double-to-single and single-to-double switching impacts reach approximately 4 times the force mutation amplitude.
4. Nonlinear Dynamic Modeling and Analysis of Spur Gear Pairs
4.1 Dynamic Model Considering Tooth Meshing Impacts
Based on the Newton-Euler method, I constructed the nonlinear dynamic model of spur gear pairs incorporating time-varying parameters, multi-state meshing characteristics, tooth switching impacts, and tooth boundary impacts. The dimensionless governing equation is:
$$\ddot{x} + h_1(\tau, x)F_{c1}(\tau, x) + h_2(\tau, x)F_{c2}(\tau, x) = \bar{F} + \varepsilon \omega^2 \cos(\omega\tau)$$
where the state-dependent forcing terms are:
$$F_{c1}(\tau, x) = \begin{cases} \pm F_{jl}(\tau) & \tau_{Mi}^{(i)} \le \tau < \tau_{C1}^{(i-1)} \\ L_1(\tau)\bar{k}(\tau)f(x) + \xi\dot{x} & \tau_{C1}^{(i-1)} \le \tau < \tau_{Mi1}^{(i)} \\ L_1(\tau)\bar{k}(\tau)f(x) + \xi\dot{x} & \tau_{Mi1}^{(i)} \le \tau < \tau_{Mo}^{(i-1)} \\ L_1(\tau)\bar{k}(\tau)f(x) + \xi\dot{x} & \tau_{Mo}^{(i-1)} \le \tau < \tau_A^{(i)} \\ \pm F_{jl}(\tau) & \tau_A^{(i)} \le \tau < \tau_{A1}^{(i)} \\ \bar{k}(\tau)f(x) + \xi\dot{x} & \tau_{A1}^{(i)} \le \tau < \tau_{C}^{(i)} \\ \pm F_{jl}(\tau) & \tau_{C}^{(i)} \le \tau < \tau_{C1}^{(i)} \end{cases}$$
$$F_{c2}(\tau, x) = \begin{cases} \pm F_{3jl}(\tau) & \tau_{Mi}^{(i)} \le \tau < \tau_{C1}^{(i-1)} \\ L_2(\tau)\bar{k}(\tau)f(x) + \xi\dot{x} & \tau_{C1}^{(i-1)} \le \tau < \tau_{Mi1}^{(i)} \\ L_2(\tau)\bar{k}(\tau)f(x) + \xi\dot{x} & \tau_{Mi1}^{(i)} \le \tau < \tau_{Mo}^{(i-1)} \\ \pm F_{2jl}(\tau) & \tau_{Mo}^{(i-1)} \le \tau < \tau_A^{(i)} \\ \pm F_{4jl}(\tau) & \tau_A^{(i)} \le \tau < \tau_{A1}^{(i)} \\ \pm F_{4jl}(\tau) & \tau_{A1}^{(i)} \le \tau < \tau_{Mo1}^{(i)} \\ 0 & \tau_{Mo1}^{(i)} \le \tau < \tau_{C}^{(i)} \end{cases}$$
The gap function is defined as:
$$f(x) = \begin{cases} x – D & x > D \\ 0 & -D \le x \le D \\ x + D & x < -D \end{cases}$$
The dimensionless boundary maximum impact force is:
$$F_{ijmax} = \begin{cases} [1 + \lambda_{j1}\mu_{j1}g_1 + \lambda_{j2}\mu_{j2}g_2] \cdot k(\tau)\left[\bar{D}^{(x_+ – x_-)}\right] + \frac{e}{2}\bar{k}\left[\bar{D}(x_+ – x_-)\right]\left[\bar{D}(x_+ – x_-)^n – 1\right] & \tau_{Mi1}^{(i)} \le \tau < \tau_{Mo1}^{(i)} \\ [1 + \lambda_{j1}\mu_{j1}g_1]\bar{k}\left[\bar{D}(x_+ – x_-)\right] + \frac{e}{2}\bar{k}\left[\bar{D}(x_+ – x_-)\right]\left[\bar{D}(x_+ – x_-)^n – 1\right] & \tau_{Mo1}^{(i)} \le \tau < \tau_{C}^{(i)} \end{cases}$$
The complete dimensionless dynamic model of spur gear pairs considering tooth meshing impacts is:
$$\begin{cases} \ddot{x} + h_1(\tau,x)F_{c1}(\tau,x) + h_2(\tau,x)F_{c2}(\tau,x) = \bar{F} + \varepsilon\omega^2\cos(\omega\tau) & x \ne \pm \bar{D} \\ \dot{x}_+ = e\dot{x}_- & x = \bar{D} \wedge F_{idmax} > F_d \\ \dot{x}_+ = e\dot{x}_- & x = \bar{D} \wedge F_{idmax} \le F_d \\ \dot{x}_+ = -e\dot{x}_- & x = -\bar{D} \wedge F_{ikmax} < F_k \\ \dot{x}_+ = e\dot{x}_- & x = -\bar{D} \wedge F_{ikmax} \ge F_k \end{cases}$$
For comparison purposes, the impact-free dynamic model of spur gears is:
$$\ddot{x} + h(\tau, x)\bar{k}(\tau)f(x) + \xi\dot{x} = \bar{F} + \varepsilon\omega^2\cos(\omega\tau)$$
4.2 Boundary Impact Analysis
I investigated the boundary collision forces and boundary maximum impact forces at both tooth surface and tooth back boundaries. Table 6 summarizes the key findings.
| Condition | Tooth surface boundary | Tooth back boundary |
|---|---|---|
| Light load and high speed | Significant increase in maximum impact force | Significant increase in maximum impact force |
| Fixed meshing frequency, load above critical value | Boundary impact suppressed | Boundary impact suppressed |
| Small comprehensive transmission error | Small maximum impact force | Reduced risk of tooth back impact |
| Dashpot and transmission error constant | Positive correlation between collision and maximum impact force | Positive correlation between collision and maximum impact force |
When the damping coefficient and comprehensive transmission error remain constant, the boundary maximum impact force is positively correlated with the boundary collision force. Both forces increase significantly under light-load high-speed conditions. When the meshing frequency and load remain constant, the effects of comprehensive transmission error and damping on boundary impact forces exhibit complex nonlinear relationships.
4.3 Typical Operating Conditions
Using the variable-step fourth-order Runge-Kutta method, I obtained four typical motion regimes of spur gear pairs:
Case 1 ($\bar{F} = 0.5$, $\omega = 1.2$, $\xi = 0.05$, $\varepsilon = 0.12$, $D = 1$): Both impact and impact-free systems exhibit period-1 motion with stable operation. The phase trajectory amplitude slightly increases when tooth meshing impacts are considered. The force mutation at the double-to-single switching is significantly larger than at the single-to-double switching.
Case 2 ($\bar{F} = 0.08$, $\omega = 1.2$, $\xi = 0.5$, $\varepsilon = 0.12$, $D = 1$): Both systems exhibit period-2 motion with disengagement. For the impact system, three consecutive tooth surface boundary impacts occur, and the phase trajectory changes significantly near the boundary. The impact system shows more abrupt transitions when entering the tooth surface meshing state.
Case 3 ($\bar{F} = 0.005$, $\omega = 1.2$, $\xi = 0.3$, $\varepsilon = 0.12$, $D = 1$): The impact system remains in disengagement after tooth back boundary impacts ($F_{ikmax} < F_k$), while the impact-free system enters tooth back contact. Both systems exhibit chaotic motion, but the impact system shows enlarged phase trajectory amplitudes.
Case 4 ($\bar{F} = 0.005$, $\omega = 1.2$, $\xi = 0.9$, $\varepsilon = 0.9$, $D = 1$): Both systems exhibit period-3 motion. After two consecutive tooth surface boundary impacts, the impact system enters the double-tooth meshing region. Only when both damping coefficient and comprehensive transmission error are large does the system enter tooth back contact. With larger damping, the impact system transitions into meshing more smoothly, with fewer consecutive boundary impacts.
4.4 Transient and Steady-State Responses
For the impact system with parameters $\bar{F} = 0.5$, $\omega = 1.2$, $\xi = 0.05$, $\varepsilon = 0.12$, $D = 1$, the system reaches steady state after 221 meshing cycles, corresponding to approximately 10.5 rotations of the driving gear. During startup, the velocity and displacement fluctuations are substantial, gradually diminishing until stable operation is achieved. At the tooth surface boundary, I observed 21 consecutive boundary impacts during the transient period. When the damping coefficient is reduced to $\xi = 0.02$, the number of consecutive tooth surface boundary impacts significantly increases. These observations confirm that damping noticeably affects the stability of spur gear systems, and force mutation amplitudes directly correlate with velocity and displacement changes.
4.5 Effects of Meshing Frequency
I analyzed the bifurcation diagrams and TLE diagrams for both impact and impact-free systems with parameters $\omega \in [0.5, 2.5]$, $\bar{F} = 0.1$, $\xi = 0.15$, $\varepsilon = 0.2$, $D = 1$. Table 7 summarizes the critical meshing frequencies and corresponding dynamical transitions.
| Critical frequency | Value | Dynamic transition |
|---|---|---|
| $\omega_1$ | 0.612 | Impact-free system exhibits disengagement; impact system maintains stable meshing |
| $\omega_2$ | 0.703 | Boundary impact region changes from double-tooth to single-tooth region |
| $\omega_3$ | 1.024 | Impact system period-doubling: period-1 to period-2 |
| $\omega_4$ | 1.041 | Slight jump in bifurcation curve |
| $\omega_5$ | 1.21 | Jump in bifurcation curve and TLE |
| $\omega_6$ | 1.242 | Impact system period-doubling: period-2 to period-4 |
| $\omega_7$ | 1.281 | Impact system period-doubling: period-4 to period-8 |
| $\omega_8$ | 1.285 | Saddle-node bifurcation to chaos |
| $\omega_9$ | 1.313 | Saddle-node bifurcation to period-6 |
| $\omega_{10}$ | 1.33 | Period-doubling to period-12 |
| $\omega_{11}$ | 1.333 | Saddle-node bifurcation to chaos again |
| $\omega_{12}$ | 1.661 | Saddle-node bifurcation to period-2 |
| $\omega_{13}$ | 1.686 | Inverse period-doubling to period-1 |
When the load coefficient is increased to $\bar{F} = 0.3$, the bifurcation behavior becomes notably simpler, and chaotic motion disappears entirely. The critical transition points shift toward higher frequencies, indicating enhanced system stability under higher loads. At higher load levels, the impact system exhibits period-4 motion while the impact-free system maintains period-2 motion in certain frequency ranges.
For spur gear pairs, when only tooth switching impacts occur without boundary impacts, the displacement and velocity variations are relatively small since switching impacts have extremely short durations and occur only once. However, the double-to-single switching impact has a significantly greater influence than the single-to-double switching impact. When six consecutive tooth surface boundary impacts occur, the dynamic characteristics of the spur gear system change considerably. At specific parameter combinations, the impact system may undergo dynamic transitions earlier than the impact-free system, including earlier transition to chaos or to periodic motion.
4.6 Effects of Load
I analyzed parameters $\bar{F} \in [0, 0.2]$, $\varepsilon = 0.1$, $\xi = 0.5$, $\omega = 1.2$, $D = 1$. Table 8 summarizes the critical load values and transitions.
| Critical load | Value | Dynamic transition |
|---|---|---|
| $F_1$ | 0.051 | Impact system: chaos to period-4; Impact-free system: period-8 at $F=0.0478$ |
| $F_2$ | 0.0538 | Impact system: inverse period-doubling, period-4 to period-2 |
| $F_3$ | 0.0638 | Impact system: jump; boundary impact shifts from single-tooth to double-tooth region |
| $F_4$ | 0.066 | Impact-free system: jump; boundary impact shifts region |
| $F_5$ | 0.0752 | Impact system: slight jump; boundary impact in both regions |
| $F_6$ | 0.079 | Impact system: inverse period-doubling, period-2 to period-1 |
As the load coefficient increases, both systems transition from chaotic to periodic motion. The impact system exhibits delayed transitions compared to the impact-free system, indicating that ignoring impacts overestimates the stability margin of spur gears. The impact system does not enter tooth back contact state because $F_{ikmax} < F_k$; conversely, the impact-free system enters tooth back contact, leading to distinct mapping points in the region $x < -D$. Under low-speed ($\omega = 0.5$) and high-speed ($\omega = 2.1$) conditions, the dynamic behavior becomes simpler, with the high-speed case showing chaotic motion persisting even at high meshing frequencies for the impact system.
4.7 Effects of Damping
I analyzed parameters $\xi \in [0, 1]$, $\bar{F} = 0.08$, $\varepsilon = 0.2$, $\omega = 1.2$, $D = 1$. Table 9 summarizes the critical damping values and transitions.
| Critical damping | Value | Dynamic transition |
|---|---|---|
| $\xi_1$ | 0.094 | Saddle-node bifurcation: period-1 to chaos |
| $\xi_2$ | 0.677 | Chaos to period-4 |
| $\xi_3$ | 0.71 | Inverse period-doubling: period-4 to period-2 |
| $\xi_4$ | 0.926 | Inverse period-doubling: period-2 to period-1 |
At $\xi = 0.055$, the impact system experiences 19 consecutive tooth surface boundary impacts upon reaching the boundary. The number of consecutive boundary impacts decreases significantly as damping increases: from 19 at $\xi = 0.055$ to 13 at $\xi = 0.08$, 10 at $\xi = 0.11$, 3 at $\xi = 0.37$, and 2 at $\xi = 0.7$. The impact system transitions from period-2 to period-1 earlier than the impact-free system. When the comprehensive transmission error is reduced to $\varepsilon = 0.12$, the chaotic region width narrows for the impact system, and the transition node from chaos to periodic motion occurs earlier for both systems.
4.8 Effects of Comprehensive Transmission Error
I analyzed parameters $\varepsilon \in [0, 0.4]$, $\bar{F} = 0.1$, $\xi = 0.5$, $\omega = 1.2$, $D = 1$. Table 10 summarizes the critical transmission error values and transitions.
| Critical error | Value | Dynamic transition |
|---|---|---|
| $\varepsilon_1$ | 0.1264 | Impact system: jump; period-1 to period-2 |
| $\varepsilon_2$ | 0.1284 | Jump in bifurcation curve; two boundary impacts per cycle to one |
| $\varepsilon_3$ | 0.1584 | Jump; boundary impact region shifts from double-tooth to single-tooth |
| $\varepsilon_4$ | 0.1848 | Period-doubling: period-2 to period-4 |
| $\varepsilon_5$ | 0.1964 | Transition to chaos |
Both systems exhibit period-doubling bifurcation cascades, transitioning through period-1, period-2, period-4, and eventually into chaos. However, the impact system does not enter tooth back contact because $F_{ikmax} < F_k$. The impact system transitions to period-2 by a jump rather than smooth period-doubling. At smaller damping values ($\xi = 0.1$), the impact system exhibits notably more complex dynamic transitions compared to the impact-free system, including period-2, period-4, period-8, chaos, and intermittent periodic windows. The impact-free system shows simpler period-doubling behavior to chaos. Smaller damping leads to significantly more consecutive tooth surface boundary impacts in the impact system.
5. Conclusions
In this study, I systematically investigated the dynamic characterization of spur gears considering tooth meshing impacts. The main conclusions are summarized as follows:
(1) The energy method based on the kinetic energy theorem and Hertz contact theory more accurately characterizes the transient impact behavior of spur gears than the traditional impulse method. For spur gear pairs, the energy method calculates notably higher maximum impact forces and shorter impact durations, better reflecting the physical nature of instantaneous force mutations in gear meshing.
(2) With tooth switching impacts considered, a complete meshing cycle of spur gear pairs is divided into seven distinct motion stages. As the rotational speed increases, the overlapped regions between dynamic meshing force and impact force expand significantly. Among the four types of switching impacts, the peak forces rank from highest to lowest as: double-to-single switching impact, single-to-double switching impact, mesh-in impact, and mesh-out impact.
(3) When only tooth switching impacts occur, the displacement and velocity responses of spur gear pairs change slightly, with the double-to-single switching impact having a greater influence than the single-to-double switching impact. However, when boundary impacts with multiple consecutive occurrences are present, the dynamic characteristics of spur gears change considerably. The boundary maximum impact force is positively correlated with the boundary collision force when damping and transmission errors are constant.
(4) Key system parameters significantly affect the dynamic behavior of spur gears. Higher or lower meshing frequencies lead to simpler dynamic behavior. Increasing the load coefficient simplifies the dynamic transitions and enhances system stability. When the comprehensive transmission error decreases, both systems transition from chaos to periodic motion earlier with increasing damping. Larger comprehensive transmission errors more readily induce chaotic motion. When the damping coefficient decreases, the nonlinear dynamic transitions of the impact system become significantly more complex than those of the impact-free system. Smaller damping coefficients lead to a marked increase in the number of consecutive tooth surface boundary impacts.
6. Future Perspectives
This study is currently limited to theoretical modeling and numerical analysis without experimental validation. Future research should establish a gear transmission experimental platform to capture real-time meshing impact data using high-speed cameras and vibration sensors, thereby verifying the accuracy of the proposed model. Additionally, elastohydrodynamic lubrication theory should be incorporated to account for oil film thickness and lubricant viscosity effects on tooth meshing impacts in spur gears. Furthermore, tooth profile modifications, manufacturing errors including pitch and profile deviations, and tooth surface temperature fields should be integrated into the dynamic model to more comprehensively reflect the actual operating conditions of spur gear transmission systems.