1. Introduction
The performance of gear transmission systems directly influences the reliability and stability of the entire machine. Tooth meshing impacts represent one of the primary sources of dynamic excitation in gear systems, and their underlying mechanisms constitute a critical scientific problem in mechanical transmission research. Existing theoretical frameworks exhibit notable deficiencies in explaining the transient impact characteristics induced by abrupt changes in meshing forces, particularly regarding the force transmission mechanisms during multi-state meshing transitions. As the most representative form of gear transmission, the involute spur gear pair provides fundamental research outcomes concerning tooth meshing impact dynamics. These findings not only establish a theoretical foundation for optimizing the dynamic performance of spur gear systems but can also be extended to dynamic analyses of more complex gear systems, including helical gears and spiral bevel gears.
Throughout the tooth meshing process, tooth meshing impacts frequently occur, especially under high-speed and heavy-load conditions. These impacts affect the smoothness of tooth meshing, further inducing vibration, noise, wear, and fatigue damage in gear pairs. Therefore, investigating the generation mechanisms of tooth impacts and their influence on system dynamic characteristics holds substantial theoretical and engineering significance.
Previous research has categorized five meshing states in spur gear pairs: double-tooth drive-side meshing, single-tooth drive-side meshing, double-tooth back-side contact, single-tooth back-side contact, and tooth disengagement. When gear pairs transition from disengagement to drive-side meshing or back-side contact states, abrupt changes in meshing forces produce boundary impacts. During drive-side meshing and back-side contact states, frequent double/single-tooth alternation causes abrupt meshing force changes, generating switching impacts.
An existing impact model based on the impulse method demonstrated significant limitations: the computed impact force amplitudes were notably underestimated, and the predicted impact durations were excessively long, thus limiting precision. Consequently, I employ the energy method in this study to establish a tooth meshing impact model, systematically analyze the mechanisms and conditions of tooth impact occurrence, and construct a dynamic model of spur gear pairs incorporating tooth impacts.
2. Multi-State Meshing and Time-Varying Parameter Calculation for Spur Gear Pairs
To avoid seizure problems due to temperature rise and thermal expansion, a certain clearance must be reserved between tooth profiles, namely the backlash. Meanwhile, to ensure continuous transmission, the contact ratio must exceed unity. Due to backlash and contact ratio, spur gear pairs exhibit complex multi-state meshing characteristics during gear motion.
2.1 Multi-State Meshing Characteristics
I use the subscripts l = p and l = g to represent the pinion and gear, respectively. Parameters zl, ωl, Tl, θl, Rbl, Ral, and Il represent the tooth number, angular velocity, torque, dynamic angular displacement, base radius, addendum radius, and moment of inertia of the pinion (l = p) and gear (l = g), respectively. The parameters e(t), k(t), μ(t), Cg, and D represent the comprehensive transmission error, time-varying mesh stiffness, time-varying friction coefficient, damping coefficient, and half-backlash along the line of action, respectively. The contact ratio is denoted by εm. Table 1 presents the fundamental parameters of the spur gear pair I investigate.
| Parameter | Pinion | Gear |
|---|---|---|
| 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 ratio ν | 0.277 | 0.33 |
| Pressure angle α0 (°) | 20 | 20 |
| Contact ratio εm | 1.595 | |
According to the relative displacement x along the line of action and the half-backlash D, three meshing states exist in spur gear pairs: drive-side meshing when x > D, back-side contact when x < −D, and disengagement when −D < x < D. Due to the contact ratio εm = 1.595, both double-tooth and single-tooth states exist within drive-side meshing and back-side contact regions. Based on the relative displacement x and half-backlash D, the gap function for the no-impact model can be expressed as:
$$
f(x)=\begin{cases}
x-D, & x>D \\
0, & \left| x \right|\le D \\
x+D, & x<-D
\end{cases}
$$
2.2 Time-Varying Load Sharing Ratio
During double-tooth meshing, the load is not equally distributed between tooth pairs but follows a specific load sharing ratio. I compute the load sharing ratio using the following expression:
$$L_{jc}(t)=\begin{cases}
\frac{1}{\varepsilon_m}\left[1+\frac{\zeta(t)-\zeta_{Mi}}{\zeta_{Mod}-\zeta_{Mi}}\right], & \zeta_{Mi}\le \zeta(t)<\zeta_{Mod}-1 \\[6pt]
1, & \zeta_{Mod}-1\le \zeta(t)<\zeta_{Mi}+1 \\[6pt]
\frac{1}{\varepsilon_m}\left[1+\frac{\zeta(t)-\zeta_{Mi}}{\zeta_{Mod}-\zeta_{Mi}+1}\right], & \zeta_{Mi}+1\le \zeta(t)<\zeta_{Mod}
\end{cases}$$
where j = d represents drive-side meshing and j = k represents back-side contact; c = 1, 2 denotes the i-th tooth pair and the (i−1)-th tooth pair, respectively.
My analysis reveals that the load sharing ratio experiences abrupt changes at the moments tMi, tA, tC, and tMo, which correspondingly causes abrupt changes in the forces at these four moments. In the first double-tooth meshing region (tMi−tA), the i-th and (i−1)-th tooth pairs mesh simultaneously. In the single-tooth region (tA−tC), only the i-th tooth pair meshes, and the load sharing ratio equals unity. The second double-tooth region (tC−tMo) follows the same load sharing curve as the first, albeit with different tooth pairs involved.
2.3 Time-Varying Mesh Stiffness
The mesh stiffness of spur gear pairs is a key time-varying parameter characterizing the dynamic behavior of gear systems. Its value varies periodically with meshing time, primarily due to the changing number of meshing tooth pairs and the movement of contact points along the tooth profile. I calculate the Hertzian contact stiffness kh, bending stiffness kblc, axial compression stiffness kalc, shear stiffness kslc, and fillet-foundation stiffness kflc using the potential energy method, following established formulations:
$$k_h=\frac{\pi EB}{4(1-\nu^2)}$$
$$k_b=\int_{-\theta_b}^{\theta_b}\frac{3\{1+\cos\alpha_1[(\theta_b-\alpha_2)\sin\alpha_1-\cos\alpha_1]\}^2(\theta_b-\alpha_2)\cos\alpha_1}{2EB[\sin\alpha_1+(\theta_b-\alpha_2)\cos\alpha_1]^3}\,d\alpha_1$$
I further consider the coupling effect between adjacent teeth on the fillet-foundation stiffness. Figure 2 presents a comparison between the mesh stiffness with and without the coupling effect of adjacent teeth.
| Region | Without coupling | With coupling |
|---|---|---|
| Double-tooth region | Higher stiffness | Significantly reduced stiffness |
| Single-tooth region | Baseline stiffness | Unchanged stiffness |
After considering the adjacent tooth coupling effect, the stiffness values in the double-tooth meshing region decrease markedly, while those in the single-tooth region remain unchanged. The stiffness in the two double-tooth regions is identical, differing only in which tooth pairs are involved.
2.4 Time-Varying Friction Coefficient
In the dynamic analysis of spur gear transmission systems, the friction coefficient significantly affects simulation accuracy. I compute the time-varying friction coefficient using the following formula:
$$\mu_j(t)=\frac{1}{2}e^{b_1\left|SR_j(t)\right|P_{hj}^{b_2}(t)\left|\eta_M R_{avgi}\right|^{b_3}}\left[b_4\left|SR_j(t)\right|^{b_5}P_{hj}^{b_6}(t)\left|\log_{10}(\eta_M)\right|^{b_7}+e^{b_8\left|SR_j(t)\right|P_{hj}^{b_9}(t)}+b_{10}\right]$$
After incorporating the load sharing ratio, the time-varying friction coefficient exhibits abrupt changes at the tooth switching instants (tA and tC), no longer following a smooth curve. At the pitch point (tp), the friction coefficient becomes zero. The friction coefficient at the initial engagement point (tMi) exceeds that at the disengagement point (tMo).
3. Tooth Meshing Impact in Spur Gear Pairs
During the meshing process of spur gear pairs, transitions between different meshing states cause abrupt changes in the dynamic meshing force, leading to tooth meshing impacts. I classify tooth meshing impacts into two categories: tooth switching impacts and tooth boundary impacts.
3.1 Mechanism of Tooth Switching Impacts
Tooth switching impacts refer to impacts generated during drive-side meshing/back-side contact when the meshing state changes. At the moment tMi when the i-th tooth pair meshes in, the dynamic meshing force of this tooth pair rises rapidly from zero, producing the meshing-in impact. Simultaneously, this event corresponds to tC of the (i−1)-th tooth pair, where the meshing-in of the i-th tooth pair causes a sharp decrease in the dynamic meshing force of the (i−1)-th pair, producing the single-to-double tooth switching impact. At tA, coinciding with tMo of the (i−1)-th pair, the disengagement of the (i−1)-th pair produces the meshing-out impact, while the redistribution of load produces the double-to-single tooth switching impact for the i-th pair. Similar impact events occur at tC and tMo.
3.2 Mechanism of Tooth Boundary Impacts
When teeth transition from disengagement to the drive-side boundary (x = D), the dynamic meshing force jumps from zero to the drive-side boundary collision force, generating the drive-side boundary impact. Similarly, when teeth reach the back-side boundary (x = −D), the back-side boundary impact occurs. These boundary impacts may occur in either the double-tooth or single-tooth meshing regions. The boundary impact process continues until the maximum boundary impact force becomes insufficient to maintain disengagement (for drive-side boundary) or sufficient to enter back-side contact (for back-side boundary).
3.3 Energy-Based Modeling of Tooth Meshing Impacts
I assume the gears have rigid supports, are isolated from other inertial masses, and simplify the lubrication conditions. Therefore, at any instant during the meshing of spur gear pairs, the situation can be treated as contact between two cylinders. The radii of these equivalent cylinders are determined from the gear geometry. Based on the work-energy theorem and Hertz contact theory, the tooth meshing impact process can be divided into compression and recovery phases.
The impact force model I employ incorporates both elastic and damping components:
$$F_i(t)=k(t)\delta^n(t)+\lambda\delta^n(t)\dot{\delta}(t)$$
where k(t) is the time-varying mesh stiffness, n = 10/9 is the contact index for cylinder contact, δ(t) is the deformation, and λ is the hysteresis damping factor. The contact index n depends on the contact surface topology; for two cylinders, n = 10/9, whereas for spheres, n = 1.5.
For the compression phase, the work-energy equation takes the form:
$$\frac{1}{2}m_e[v_-^2-\dot{\delta}^2(t)]=\int_{\delta_-}^{\delta(t)}[k(t)\delta^n(t)+\lambda\dot{\delta}_c\delta^n(t)]\,d\delta(t)$$
where me = m1m2/(m1+m2) is the equivalent mass, v− is the initial relative velocity, and δc = 2v−/3 is the equivalent velocity in the compression phase. The maximum deformation can then be derived as:
$$\delta_{\max}=\left\{\frac{(n+1)m_e v_-^2}{2[k(t)\delta_c+\lambda\delta_c^2]}\right\}^{\frac{1}{n+1}}+\delta_-$$
The maximum impact force becomes:
$$F_{i\max}=k(t)\left[\frac{(n+1)m_e v_-^2}{2[k(t)\delta_c+\lambda\delta_c^2]}\right]^{\frac{n}{n+1}}$$
Similarly, for the recovery phase, the equivalent velocity is δr = −2v+/3, and the recovery phase duration can be obtained by integration. I calculate the hysteresis damping factor from the energy conservation principle:
$$\lambda=\frac{3k(t)(1-e^2)}{2v_-e}$$
where e is the coefficient of restitution. I solve the governing equations using a variable-step fourth-order Runge-Kutta method implemented in C language to obtain the deformation and impact force at any given instant.
3.4 Comparison Between Energy Method and Impulse Method
I compared my energy-based results with those obtained from the impulse method for a spur gear pair. Table 3 summarizes the results at different relative velocities.
| Relative velocity (m/s) | Energy method max force (N) | Impulse method max force (N) | Energy method duration (μs) | Impulse method duration (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 (up to 8.13 times higher at 1 m/s) and notably shorter impact durations (about 0.19 times at 1 m/s) compared to the impulse method. Both methods show that increasing velocity leads to larger maximum forces and shorter durations. The energy method aligns better with the transient characteristics of spur gear tooth impacts, where extremely short durations produce dramatic force changes. My approach offers several improvements: (1) it incorporates a damping force component; (2) it employs a cylinder-contact model with n = 10/9, more appropriate for gears with substantial face width; (3) it accounts for the coupling effect of adjacent teeth on the foundation stiffness; (4) it considers initial and final deformations in the compression and recovery phases, respectively; and (5) it enables calculation of impact force and deformation at any instant.
3.5 Dynamic Meshing Force Analysis
I computed the total dynamic meshing force for a spur gear pair operating at 800 W power with rotational speeds of 1000, 2000, 4000, and 6000 r/min. Table 4 lists the complete meshing cycle durations at these speeds.
| Speed (r/min) | Time (μs) |
|---|---|
| 1000 | 4557 |
| 2000 | 2278 |
| 4000 | 1139 |
| 6000 | 759 |
Due to the load sharing ratio, the dynamic meshing force undergoes significant changes at four special positions (tMi, tA, tC, and tMo). When the speed increases by a factor h, the dynamic meshing force becomes approximately 1/h of the original value, and the time becomes approximately 1/h of the original duration.
3.6 Coupled Analysis of Dynamic Meshing Force and Switching Impacts
I coupled the dynamic meshing force with the switching impact forces to analyze a complete meshing cycle. This analysis reveals that the meshing cycle can be divided into seven distinct phases: meshing-in impact (tMi−tMi1), first double-tooth meshing (tMi1−tA), double-to-single switching impact (tA−tA1), single-tooth meshing (tA1−tC), single-to-double switching impact (tC−tC1), second double-tooth meshing (tC1−tMo), and meshing-out impact (tMo−tMo1).
| Speed (r/min) | Meshing-in impact (μs) | First double meshing (μs) | Double-to-single impact (μs) | Single meshing (μs) | Single-to-double impact (μs) | Second double meshing (μs) | Meshing-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 |
As the speed increases, the overlapping regions between the dynamic meshing force and the impact forces become markedly larger. The meshing-in impact duration and meshing-out impact duration are relatively longer, causing significant overlap with the adjacent tooth pair events. When the speed increases by a factor h, the overlapping regions within a complete meshing cycle expand by approximately h times.
| Speed (r/min) | Meshing-in impact (%) | First double meshing (%) | Double-to-single impact (%) | Single meshing (%) | Single-to-double impact (%) | Second double meshing (%) | Meshing-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 |
Among the four types of switching impacts, the peak impact forces are ranked from highest to lowest as follows: double-to-single switching impact force, single-to-double switching impact force, meshing-in impact force, and meshing-out impact force. The meshing-in and meshing-out impact peak forces are approximately 1.1 times the meshing force mutation amplitude, while the double-to-single and single-to-double switching impact peak forces are approximately 4 times this amplitude.
| Speed (r/min) | Meshing-in peak force (N) | Double-to-single peak force (N) | Single-to-double peak force (N) | Meshing-out peak 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 |
In summary, as the rotational speed increases, the proportion of tooth meshing impact time within a complete meshing cycle increases significantly, while the stable meshing time decreases, leading to reduced system stability. The meshing-out impact exhibits a smaller maximum force than the meshing-in impact but has a longer duration; similarly, the single-to-double switching impact has a smaller maximum force but longer duration than the double-to-single switching impact.
4. Nonlinear Dynamic Modeling and Analysis of Spur Gear Pairs Considering Tooth Meshing Impacts
4.1 Dynamic Model Formulation
I construct a nonlinear dynamic model of spur gear pairs incorporating time-varying parameters, multi-state meshing characteristics, tooth switching impacts, and tooth boundary impacts. Using the Newton-Euler method, the governing equations for drive-side meshing and back-side contact are:
$$I_p\ddot{\theta}_p\pm [R_{bp}\hat{F}_{jp1}(t)+S_{jp1}(t)\hat{F}_{jfp1}(t)+R_{bp}\hat{F}_{jp2}(t)+S_{jp2}(t)\hat{F}_{jfp2}(t)]=T_p$$
$$I_g\ddot{\theta}_g\mp [R_{bg}\hat{F}_{jg1}(t)+S_{jg1}(t)\hat{F}_{jfg1}(t)+R_{bg}\hat{F}_{jg2}(t)+S_{jg2}(t)\hat{F}_{jfg2}(t)]=-T_g$$
For the disengagement state:
$$I_p\ddot{\theta}_p=T_p, \quad I_g\ddot{\theta}_g=-T_g$$
I introduce the following dimensionless parameters: ωn = √(kav/me) is the natural frequency, ω = ωh/ωn is the dimensionless meshing frequency, ξ = Cg/(2meωn) is the dimensionless damping coefficient, k(τ) = k(t)/(meωn²) is the dimensionless mesh stiffness, τ = ωnt is the dimensionless time, D = Dc/D is the dimensionless half-backlash, x = x/Dc is the dimensionless relative displacement, and F̃ = F/(meDcωn²) is the dimensionless force.
After dimensionless normalization, the equation of motion of the spur gear pair considering tooth meshing impacts becomes:
$$\ddot{x}+h_1(\tau,x)F_1(\tau,x)+h_2(\tau,x)F_2(\tau,x)=F_m+\varepsilon\omega^2\cos(\omega\tau)$$
where Fm is the dimensionless mean force (load coefficient), and the state-dependent functions hl(τ, x) (l = 1, 2) and the functions Fc(τ, x) (c = 1, 2) capture the impact contributions during different meshing phases. For the system without impacts, I formulate:
$$\ddot{x}+h(\tau,x)[k(\tau)f(x)+\xi\dot{x}]=F_m+\varepsilon\omega^2\cos(\omega\tau)$$
4.2 Boundary Impact Analysis
I investigate the boundary collision forces and maximum boundary impact forces for both the drive-side and back-side boundaries. Table 8 summarizes the key findings.
| Condition | Observation |
|---|---|
| Light load, high speed | Boundary impact forces significantly increase |
| Load coefficient exceeds critical value | Boundary impacts cease to occur |
| Small comprehensive transmission error | Small maximum boundary impact force |
| Large transmission error, small damping | Significantly increased boundary impact force |
| Damping and transmission error constant | Boundary maximum impact force positively correlated with collision force |
When the damping coefficient and comprehensive transmission error remain constant, the maximum boundary impact force exhibits a positive correlation with the boundary collision force. Under light-load and high-speed conditions, both quantities significantly increase. When the meshing frequency and load remain constant, the influence of comprehensive transmission error and damping coefficient on the boundary impact force becomes more complex without an evident simple pattern.
4.3 Typical Operating Conditions
I analyze four typical operating conditions using Poincaré maps, phase portraits, and force time histories. Table 9 summarizes the key findings for each condition.
| Parameters (F̄, ω, ξ, ε) | Without impact | With impact | Key observations |
|---|---|---|---|
| 0.5, 1.2, 0.05, 0.12 | Period-1 | Period-1 | Slight amplitude increase with impact |
| 0.08, 1.2, 0.5, 0.12 | Period-2 | Period-2 with three consecutive boundary impacts | Disengagement and boundary impacts occur |
| 0.005, 1.2, 0.3, 0.12 | Chaos with back-side contact | Chaos without back-side contact | Fikmax < Fk: no back-side contact |
| 0.005, 1.2, 0.9, 0.9 | Period-3 | Period-3 with reduced amplitude | Large damping and transmission error promote back-side contact |
My comparison with the impulse-method results reveals important differences. When the switching impact time becomes excessively long, the phase trajectory of the spur gear pair changes noticeably, potentially causing disengagement in systems that should not disengage. The impact force amplitude significantly influences system stability: larger boundary impact forces increase vibration intensity. With larger damping coefficients, the spur gear pair enters drive-side meshing more smoothly, and the number of consecutive drive-side boundary impacts decreases, improving system stability.
4.4 Transient and Steady-State Analysis
At parameters F̄ = 0.5, ω = 1.2, ξ = 0.05, ε = 0.12, D = 1, the system enters steady state after 221 meshing cycles (equivalent to 10.5 rotations of the driving gear). During startup, the displacement and velocity fluctuations are substantial, gradually damping until steady state is achieved. When the teeth reach the drive-side boundary, 21 consecutive boundary impacts occur. With smaller damping (ξ = 0.02), the number of consecutive boundary impacts increases markedly, indicating that damping significantly affects system stability.
The steady-state analysis reveals that the double-to-single switching point produces larger force amplitude changes (orange) compared to the single-to-double switching point (purple). The corresponding velocity and displacement changes exhibit the same hierarchy. Larger force mutation amplitudes produce more pronounced velocity and displacement variations in spur gear pairs.
4.5 Effect of Meshing Frequency
I investigate the effect of the dimensionless meshing frequency ω ∈ [0.5, 2.5] on the dynamic behavior of spur gear pairs with parameters F̄ = 0.1, ξ = 0.15, ε = 0.2, D = 1. The system without impact exhibits typical period-doubling bifurcation: period-1 → period-2 → period-4 → chaos → reverse period-doubling → period-1. Table 10 summarizes the critical frequencies and corresponding behavior transitions for the system with impacts.
| Critical frequency | Value | Transition |
|---|---|---|
| ω1 | 0.612 | Boundary impact region changes; no-impact system shows disengagement |
| ω2 | 0.703 | Boundary impact moves from double-tooth to single-tooth region |
| ω3 | 1.024 | Period-1 → Period-2 (period-doubling) |
| ω4 | 1.041 | Boundary impact region changes |
| ω5 | 1.21 | Boundary impact only in single-tooth region |
| ω6 | 1.242 | Period-2 → Period-4 (period-doubling with six consecutive boundary impacts) |
| ω7 | 1.281 | Period-4 → Period-8 |
| ω8 | 1.285 | Period-8 → Chaos (saddle-node) |
| ω9 | 1.313 | Chaos → Period-6 |
| ω10 | 1.33 | Period-6 → Period-12 |
| ω11 | 1.333 | Period-12 → Chaos |
| ω12 | 1.661 | Chaos → Period-2 (no-impact system still chaotic) |
| ω13 | 1.686 | Period-2 → Period-1 (no-impact system still chaotic) |
When boundary impacts occur in the single-tooth region rather than the double-tooth region, the displacement, velocity, and phase trajectory amplitudes become significantly larger. This indicates that boundary impacts in the single-tooth region have a more pronounced effect on spur gear dynamics. When six consecutive boundary impacts occur, the dynamics change significantly. With larger load coefficients (F̄ = 0.3), the bifurcation behavior becomes simpler, and chaos disappears entirely; the system exhibits only period-1, period-2, and period-4 motions.
4.6 Effect of Load
I analyze the effect of the load coefficient F̄ ∈ [0, 0.2] with parameters ε = 0.1, ξ = 0.5, ω = 1.2, D = 1. Table 11 summarizes the critical loads and corresponding transitions.
| Critical load | Value | Transition |
|---|---|---|
| F̄1 | 0.051 | Chaos → Period-4 (with impacts); Chaos → Period-8 at 0.0478 (without impacts) |
| F̄2 | 0.0538 | Period-4 → Period-2 |
| F̄3 | 0.0638 | Boundary impact moves from single-tooth to double-tooth region |
| F̄4 | 0.066 | Similar boundary impact region change (without impacts) |
| F̄5 | 0.0752 | Boundary impacts occur in both regions |
| F̄6 | 0.079 | Period-2 → Period-1 |
With the system incorporating impacts, since Fikmax < Fk, the spur gear pair does not enter back-side contact, whereas the no-impact system does. The system with impacts exhibits a noticeable delay in the transition from chaotic to periodic motion compared to the system without impacts. For the system with impacts, the phase trajectory at the back-side boundary immediately moves away, whereas the no-impact system smoothly enters back-side contact. At low frequency (ω = 0.5), the chaotic region shrinks, and the dynamics become simpler. At high frequency (ω = 2.1), the no-impact system maintains period-1 motion throughout, while the impact system transitions directly from chaos to period-1 via saddle-node bifurcation, indicating that impacts introduce more complex behavior even at higher meshing frequencies.
4.7 Effect of Damping
I investigate the effect of the damping coefficient ξ ∈ [0, 1] with parameters F̄ = 0.08, ε = 0.2, ω = 1.2, D = 1. Table 12 summarizes the critical damping values and corresponding transitions.
| Critical damping | Value | Transition |
|---|---|---|
| ξ1 | 0.094 | Period-1 → Chaos (19 consecutive boundary impacts before transition) |
| ξ2 | 0.677 | Chaos → Period-4 |
| ξ3 | 0.71 | Period-4 → Period-2 |
| ξ4 | 0.926 | Period-2 → Period-1 (no-impact system still period-2) |
With decreasing damping, the number of consecutive drive-side boundary impacts increases substantially. At ξ = 0.055, the system experiences 19 consecutive boundary impacts; at ξ = 0.08, 13 impacts; at ξ = 0.11, 10 impacts; at ξ = 0.37, 3 impacts; and at ξ = 0.7, only 2 impacts. This clearly demonstrates that damping effectively reduces the number of consecutive boundary impacts. When ε decreases from 0.2 to 0.12, the chaotic region narrows significantly, and the transition from period-2 to period-1 occurs earlier (ξ = 0.623 with impacts vs. ξ = 0.673 without impacts).
4.8 Effect of Comprehensive Transmission Error
I examine the effect of the comprehensive transmission error ε ∈ [0, 0.4] with parameters F̄ = 0.1, ξ = 0.5, ω = 1.2, D = 1. Table 13 summarizes the critical transmission error values and corresponding transitions.
| Critical ε | Value | Transition |
|---|---|---|
| ε1 | 0.1264 | Period-1 maintained while no-impact system becomes period-2 |
| ε2 | 0.1284 | Boundary impact frequency changes from two to one per cycle |
| ε3 | 0.1584 | Boundary impact moves from double-tooth to single-tooth region |
| ε4 | 0.1848 | Period-2 → Period-4 |
| ε5 | 0.1964 | Period-4 → Chaos |
Both systems exhibit typical period-doubling bifurcation sequences, but the impact system does not enter back-side contact since Fikmax < Fk. The impact system transitions from period-1 to period-2 via an abrupt jump, whereas the no-impact system does so through smooth period-doubling. When the phase trajectory changes significantly, the bifurcation and TLE curves exhibit corresponding jumps.
For smaller damping (ξ = 0.1), the system with impacts exhibits significantly more complex dynamics compared to the no-impact system, which shows only period-1 → period-2 → chaos. The impact system displays multiple bifurcation sequences: period-1 → period-2 (ε = 0.026) → period-4 (ε = 0.0895) → period-8 (ε = 0.09575) → chaos (ε = 0.097) → period-8 (ε = 0.12525) → period-4 (ε = 0.12625) → chaos (ε = 0.13125) → period-3 (ε = 0.188) → chaos (ε = 0.19375).
5. Concluding Remarks
This study systematically investigates the dynamic characteristics of spur gear pairs considering tooth impacts. Based on my comprehensive analysis, I draw the following main conclusions:
(1) The energy method exhibits significant differences from the impulse method in computing impact characteristic parameters. The energy method yields impact force values substantially higher than the impulse method while predicting considerably shorter impact durations. These differences confirm that the energy method more accurately captures the transient nature of tooth meshing impacts during spur gear meshing processes. Moreover, as velocity increases, the results from both methods tend to converge.
(2) Considering tooth switching impacts, a meshing cycle of the spur gear pair is divided into seven distinct motion phases. With increasing rotational speed, the overlapping regions between the dynamic meshing force and impact forces expand significantly. Abrupt changes in the dynamic meshing force constitute the fundamental cause of tooth meshing impacts, which induce sudden changes in velocity and displacement. The force mutation amplitude exhibits a positive correlation with the velocity and displacement changes.
(3) When only switching impacts occur without boundary impacts, the displacement and velocity responses exhibit minor changes. The double-to-single switching impact has a notably greater influence than the single-to-double switching impact. However, when boundary impacts occur with multiple consecutive drive-side boundary impacts, the dynamic characteristics of spur gear pairs change significantly. When the damping coefficient and comprehensive transmission error remain constant, the maximum boundary impact force positively correlates with the boundary collision force. When the meshing frequency and load remain constant, the effects of comprehensive transmission error and damping on the boundary impact force exhibit complex nonlinear relationships.
(4) System state parameters significantly alter the critical parameter influence domains and dynamic behavior transition patterns of spur gears. At higher or lower meshing frequencies, the system dynamics tend to simplify. Changes in the region where drive-side boundary impacts occur cause noticeable jumps in the bifurcation curves. With larger load coefficients, the dynamic transitions with increasing meshing frequency become simpler, and system stability improves markedly. When the comprehensive transmission error decreases, the critical point at which the system transitions from chaos to periodic motion occurs earlier with increasing damping. Larger comprehensive transmission errors make the spur gear pair more susceptible to chaotic behavior. With decreasing damping, the nonlinear dynamic transitions of the system with impacts become considerably more complex than the system without impacts. Additionally, smaller damping coefficients lead to significantly more consecutive drive-side boundary impacts.
This research provides a theoretical foundation for optimizing the dynamic performance of spur gears and offers valuable reference for studying tooth impact phenomena in other types of gear transmission systems.