2.1 Dynamics model

A nonlinear dynamic model of the curve-face gear system is established as shown in Fig 1. The curve-face gear transmission system consists of a curve-face gear and a non-circular gear, where the non-circular gear serves as the driving gear and the curve-face gear acts as the driven gear. define i=p,g. p as the driving gear and g as the driven gear; j=x,y,z, denote the vibration displacement of gear i along each coordinate axis direction; denotes the rotational angular displacement of gear i around its respective axis; M_i and I_i respectively denote the concentrated mass and concentrated moment of inertia of gear i; T_i denotes the torque applied to gear i; The rolling bearings supporting the gears are equivalent to linear springs and linear dampers in the Y and Z coordinate directions. The support stiffness coefficients are denoted as K_ij, and the support damping coefficients are denoted as C_ij; denotes the meshing frequency of the gear pair.

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Fig 1. Dynamic model of curve-face gear transmission system.

https://doi.org/10.1371/journal.pone.0335920.g001

The curve-face gear transmission system has 6 degrees of freedom, and its generalized coordinates can be expressed as .

The radial clearance of the rolling bearing is analyzed in the Y and Z coordinate directions, which can be represented using the nonlinear clearance function from equation:

(1)

where B_i represents half of the bearing radial clearance.

The relative micro-displacement X_n along the meshing line due to transmission error and vibration is expressed as:

(2)

where is the pitch curve pressure angle of the curve-face gear. R_p is the base circle radius of the non-circular gear, R_g is the base circle radius of the curve-face gear. e(t) is the comprehensive transfer error of the gear pair, and can be expressed by the equation . In the equation, E_m represents the amplitude of transfer error, represents the initial phase, and t represents the time.

The dynamic meshing force F_n of the curve-face gear transmission system and its component forces F_i along each coordinate axis direction are expressed as:

(3)

Where K(t) is the time-varying meshing stiffness of the gear pair. It can be expressed as the equation , where K_m is the average value of meshing stiffness, K_a is the fluctuation amplitude of the time-varying meshing stiffness, C_n is the meshing damping of the gear pair, and its expression is , where is the relative damping ratio of the meshing gear pair, and M_e is the equivalent mass of the gear pair.

In Eq (3), the function of gear backlash is represented by f(X_n,B) as the following equation:

(4)

where B is the half of the total backlash.

The time-varying meshing stiffness is a critical parameter in the dynamic equations of gear systems. Its magnitude is related to the contact ratio, and when the contact ratio is non-integer, the mesh stiffness becomes a periodic function of the rotation angle. In this paper, the ISO draft formula for tooth stiffness calculation is adopted to approximate the tooth stiffness of the curve-face gear [19]. The approximate value of the tooth stiffness per unit width is:

(5)

(6)

Where is the helix angle of helical gears, taken as 0, and are the number of teeth in the noncircular gear and curve-face gear, and x₁ and x₁ are the modification coefficients of the noncircular gear and curve-face gear, taken as 0. After considering the tooth width factor, the tooth stiffness of the orthogonal curve-face gear is:

(7)

The meshing stiffness can be approximated as the stiffness in the double-tooth meshing zone. The meshing stiffness is expressed as:

(8)

where k₁ is the tooth stiffness of the noncircular gear, k₂ is the tooth stiffness of the curve-face gear, and , is the meshing stiffness of the single-tooth meshing zone. Taking into account the problem of unequal load distribution, a common practice is to utilize 1.5 times the stiffness of single-tooth meshing as the meshing stiffness for the zone with double-tooth meshing.

Based on Newton’s second law, the 6-degree-of-freedom motion differential equations for the curve-face gear transmission system are established as follows [20]:

(9)

where is the coefficient of sliding friction; λ is the friction force direction function, and its expression is , L_p is the time-varying friction arm direction and its expression is , where R_ap is the addendum circle radius of the non-circular gear, ε is the contact ratio of the curve-face gear pair, P_bp is the base circle pitch, and n is the rotational speed. is the angular acceleration produced by the curve-face gear, whose expression is , where i₁₂ is the transmission ratio of the curve-face gear transmission system.

Since the curve-face gear transmission system is a variable transmission ratio system, the expression for the transmission ratio i₁₂ is:

(10)

where R is the radius of the curve-face gear pitch curve, a represents the semi-major axis, k is the eccentricity ratio, and n₁ and are the order and the rotation angle of the noncircular gear.

The nominal dimension is defined as b_c, the inherent frequency as , dimensionless parameters, , , , , b = B/b_c, , , , , , , where τ is the dimensionless time, and is the dimensionless meshing frequency.

The above equations of dimensionless parameters are plugged into Eq (9), and the following differential Eq (11) of dimensionless motion of the system is obtained.

(11)

where , , , , , , , , tooth meshing damping ratio in gear pair.

The functions of non-dimensional backlash and non-dimensional bearing radial clearance are represented by Eq (12) and Eq (13), respectively.

(12)

(13)

The primary design parameters of the curve-face gear transmission system under study are as follows: modulus m = 4mm, number of teeth for non-cylindrical gears z_p = 18, number of teeth for curve-face gear z_g = 36, order of non-cylindrical gears n₁ = 2, order of curve-face gear n₂ = 4, and normal pressure angle = 20^°. The selection of standard dynamical parameters is provided in Table 1.

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Table 1. Standard parameters of the curve-face gear transmission system.

https://doi.org/10.1371/journal.pone.0335920.t001

2.2 Methods

2.2.1 Impact state.

As illustrated in Fig 2, the backlash delineates the phase plane into three distinct regions. During the meshing process of the system, the motion trajectory may manifest five impact states, referred to as I. Specifically:

1. I = 0: Non-impact state.
2. I = 1: Tooth surface impact state.
3. I = −1: Tooth back impact state.
4. I = 2: Double-sided impact state.
5. I = −2: Complete disengagement state.

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Fig 2. Distribution of the system’s phase trajectories on the plane.

https://doi.org/10.1371/journal.pone.0335920.g002

By combining the above 5 types of impact states, the motion pattern I/nT of the gear pair is obtained, where I represents the meshing impact state during gear meshing, n denotes the number of periods of the system’s meshing motion, and T = 2π/ represents the meshing period time of the system.

To characterize the disengagement severity of the gear pair during meshing impact processes, the disengagement duty cycle (DC) is utilized. DC is defined as the ratio of the duration of disengagement phenomena to the total operational time during gear rotation. In the non-impact state, DC = 0.0. In single-sided impact (tooth surface impact, tooth back impact) and double-sided impact states, 0 < DC < 1. A higher DC value indicates a longer duration of disengagement and a more severe disengagement condition in the gear pair.

Due to manufacturing and assembly errors, elastic deformation of gear teeth, and the time-varying characteristics of meshing stiffness, coupled with the alternating occurrence of single-tooth and double-tooth meshing in gear pairs, additional dynamic loads or impacts are generated during the meshing process [27,28]. The dynamic load coefficient () is commonly used to evaluate the deviation between the actual dynamic load and the theoretical load in gear dynamics research. Since the dynamic load during the meshing process varies over time, continuous sampling of the dynamic load coefficient for each meshing cycle (T = 2nπ/) is conducted. The maximum value in the obtained dataset is the maximum dynamic load coefficient () under operating conditions. The larger the () value, the greater the alternating impact stress and the more severe the impact noise.

2.2.2 Co-simulation method.

In the schematic diagram of two-parameter co-simulation presented in Fig 3, the two-parameter co-simulation is developed based on cell mapping. This method involves selecting two critical parameters from the system parameters to construct a two-parameter plane. The system parameters, denoted as u and v, form a (u, v) parameter plane that is divided into a finite number of grids. By calculating the periodicity and impact counts of stable phase trajectories at the center points a(i, j) of each grid, the motion state of the system within that grid can be determined. The motion state of phase trajectories at each point on the parameter plane is computed, and grids exhibiting the same motion state are marked with identical colors and symbols. This process elucidates the pattern types and distribution rules of periodic motion across the entire two-parameter plane. In this study, a 400  400 grid system is employed for calculations.

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Fig 3. Schematic diagram of two-parameter co-simulation.

https://doi.org/10.1371/journal.pone.0335920.g003

3.1 Nonlinear dynamics under standard parameters

The fourth-order variable-step Runge-Kutta method is utilized to solve the motion differential equations of the curve-face gear transmission system. Various research tools, including bifurcation diagrams, impact state diagrams, disengagement duty cycle diagrams, phase diagrams, time history diagrams, dynamic load coefficient diagram, and Poincaré maps, are employed to analyze the nonlinear dynamic characteristics of the system under varying meshing excitation frequencies, thereby revealing the transition laws of periodic motion. With the standard parameters outlined in Table 1, the dimensionless meshing frequency is selected as the control parameter, varying within the range [0.1, 2.5]. Through numerical simulations, the meshing displacement bifurcation diagram, impact states: disengagement duty cycle (I: DC) diagram, and maximum dynamic load coefficient () diagram of the system as illustrated in Fig 4.

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Fig 4. (a) Bifurcation diagram of mesh displacement; (b) I: DC diagram; (c) diagram.

https://doi.org/10.1371/journal.pone.0335920.g004

Fig 4(a) presents the bifurcation diagram of the curve-face gear transmission system. In this diagram, red indicates the non-impact state (I = 0), while black represents the tooth surface impact state (I = 1). It is evident that when < 0.569, the system exhibits 1-0-0 periodic motion with impact state I = 0. In this regime, the gear pair operates in a state without impact, characterized by a disengagement duty cycle (DC) of 0, which implies that no tooth surface separation occurs during the meshing period, allowing for smooth system operation. The displays a gradual upward trend. As illustrated in Fig 5(a) for = 0.48, the superposition of the phase diagram and Poincaré map for the 1-0-0 periodic motion is shown, where • denotes points formed by the trajectory crossing the Poincaré section. A single periodic mapping point is generated when the trajectory intersects the Poincaré section. As shown in Fig 6(a), the FFT spectrum appears only at the meshing frequency . As the system crosses = 0.569, a grazing bifurcation (which is marked as “G Bif” in Fig 4) occurs, leading to a transition from 1-0-0 periodic motion to 1-1-0 periodic motion via this bifurcation, with the impact state changing from I = 0 to I = 1 (from non-impact to tooth surface impact). This transition corresponds to the phenomenon of "tooth surface separation-contact-tooth surface separation" within the meshing period. Simultaneously, experiences a sudden change, accompanied by intensified impact vibration noise and instability in system operation. Fig 5(b) illustrates the phase diagram at the onset of grazing bifurcation, where the Poincaré section exhibits one stable periodic mapping point, indicating that the gear pair experiences one tooth surface impact within a meshing period. At the grazing bifurcation point and . Grazing causes the system to jump around the bifurcation critical value of the control parameter = 0.66, resulting in a sharp increase in both the disengagement duty cycle (DC = 0.66) and the maximum dynamic load coefficient ( = 1.311). The I remains unchanged.

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Fig 5. Superpositions of the phase diagram and Poincaré mapping of the gear pair: (a) = 0.48 1-0-0, (b) = 0.569 grazing, (c) = 0.86 1-1-0, (d) = 1.02 2-2-0, (e) = 1.072 grazing, (f) = 1.25 2-1-0, (g) = 1.37 4-3-0, (h) = 1.384 8-6-0, (i) = 1.53 chaotic, (j) = 1.7 4-3-0, (k) = 1.81 2-2-0, (l) = 2.24 1-1-0.

https://doi.org/10.1371/journal.pone.0335920.g005

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Fig 6. FFT spectrum: (a) = 0.48 1-0-0, (b) = 1.02 2-2-0, (c) = 1.37 4-3-0, (d) = 1.53 chaotic.

https://doi.org/10.1371/journal.pone.0335920.g006

After surpassing the jump threshold, the system stabilizes into a 1-1-0 periodic motion, with both DC and rapidly decreasing. Fig 5(c) illustrates the phase diagram of the 1-1-0 periodic motion. As increases further, the system crosses the period-doubling bifurcation (which is marked as “PD Bif” in Fig 4) threshold at = 0.988, transitioning from 1-1-0 to 2-2-0 periodic motion (as shown in Fig 5(d) for = 1.02). In this state, two tooth surface impacts occur, with the trajectory crossing the Poincaré section twice, resulting in two stable discrete points. As shown in Fig 6(b), the FFT spectrum exhibits subharmonic /2 and its harmonics n/2 (where n is an integer).

At = 1.072, a grazing bifurcation transitions the system from 2-2-0 to 2-1-0 motion. The DC and increase linearly, which reduces the load capacity of the gear pair and amplifies vibration noise. Fig 5(e) illustrates the phase diagram corresponding to this grazing bifurcation. The transition process from 2-2-0 to 2-1-0 is depicted in Fig 5(d), 5(e), and 5(f). At = 1.33, period-doubling bifurcation drives the system from 2-1-0 to 4-2-0 motion. During this bifurcation, the period number (n) and tooth surface impact count (p) double, while continues to rise, exacerbating vibration noise. At = 1.354, the system undergoes a grazing bifurcation, transitioning from 4-2-0 motion to 4-3-0 motion. Fig 5(g) presents the phase diagram for 4-3-0 motion at = 1.37, featuring four periodic mapping points on the Poincaré section. As shown in Fig 6(c), the spectral lines of the FFT spectrum are distributed at discrete points of n/4. At = 1.37, the system enters 8-6-0 periodic motions, respectively, through a continuous series of period-doubling bifurcations. Fig 5(h) displays the superposition of the phase diagram and Poincaré map for 8-6-0 periodic motion, revealing eight stable periodic mapping points on the Poincaré section. As continues to increase, the system enters chaotic motion at = 1.396. Throughout this transition, the impact state of the system remains unchanged. Fig 5(i) shows the superposition of the phase diagram and Poincaré map for chaotic motion, where the phase diagram features numerous intersecting closed or open curves, and the Poincaré section exhibits many scattered discrete points. As shown in Fig 6(d), continuous frequency components appear in the FFT spectrum. As continues to rise, the system experiences continuous inverse period-doubling bifurcations, gradually transitioning from chaotic motion into 8-6-0, 4-3-0, and 2-2-0 subharmonic motions at = 1.642, = 1.672, and = 1.714, respectively. At = 1.864, the system stabilizes into a 1-1-0 periodic motion. The DC, , and the number of motion periods all exhibit a rapid decreasing trend. Fig 4(j), 4(k), and 4(l) illustrate the superposition of phase diagrams and Poincaré maps for the 4-3-0, 2-2-0, and 1-1-0 motions, respectively.

3.2 Pattern types and transition laws of periodic motion in (ωomega, b) parameter plane

This section employs a two-parameter co-simulation to investigate the nonlinear dynamic characteristics of curve-face gear transmission systems. By taking dimensionless meshing frequency (ω) and dimensionless backlash (b) as key parameters, where ω [0.1, 2.5] and b [0.0, 1.0] while keeping other parameters constant, the pattern types, distribution regions, and transition laws of periodic motions in the (ω, b) parameter plane were identified through two-parameter co-simulation, as illustrated in Fig 7. Fig 7(a) presents a three-dimensional bifurcation diagram of the system’s meshing displacement as parameters vary. The blue color indicates that the system’s impact state is I = 2, characterized by double-sided impact. It is observed that the non-impact state occurs only in the low-frequency region, while the tooth surface impact state exists in the high-frequency region with larger b values. Conversely, double-sided impact primarily occurs in the high-frequency region with smaller backlash values. The system exhibits various bifurcation behaviors during meshing motion, including grazing bifurcation, period-doubling bifurcation, and inverse period-doubling bifurcation. Furthermore, the displacement of the meshing pair increases with the increasing b value.

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Fig 7. Meshing motion law diagram of the system in the (, b) parameter plane: (a) relative micro-displacement bifurcation diagram of gear pair, (b) pattern types and distribution regions of periodic motions.

https://doi.org/10.1371/journal.pone.0335920.g007

Fig 7(b) illustrates the distribution of periodic motions within the (ω, b) parameter plane, providing an intuitive representation of the types, distribution areas, and variation laws of these motions. In this parameter plane, I/nT is utilized to identify the impact state at the backlash, with distinct colors representing various types of motion patterns. The primary periodic motions present in the (ω, b) parameter plane include: 0/1T periodic motion, 1/1T periodic motion, tooth surface impacts indicated by different colors, double-sided impact subharmonic periodic motion, double-sided impact quasi-chaotic motion denoted by 2/nT in dark gray regions, and tooth surface impact quasi-chaotic motion represented by 1/nT in light gray regions. When b < 0.4125, a significant number of double-sided impact periodic motions and chaotic behaviors emerge in the (ω, b) parameter plane, leading to extreme instability in system operation characterized by pronounced vibrations and noise, alongside complex dynamic behaviors. As b increases, particularly within the range of [0.4125, 1.0], the system ceases to exhibit double-sided impact periodic motions in the (ω, b) parameter plane; rather, only tooth surface impact periodic motions occur, resulting in a more stable operation with reduced vibrations and noise, alongside improved performance metrics. At lower frequencies, the system demonstrates 0/1T non-impact periodic motion. As ω increases, a grazing bifurcation is observed, with the periodic motion shifting from non-impact 0/1T motion to tooth surface impact 1/1T motion. Subsequently, a period-doubling bifurcation occurs, resulting in a transition of the periodic motion from 1/1T state to 2/2T subharmonic periodic motion. Subsequently, a series of continuous period-doubling bifurcations takes place, leading to higher-periodic motions such as 1/4T, 1/8T, and so forth, ultimately culminating in chaotic motion. As ω further increases, this chaotic motion degenerates into lower-periodic motions, including 1/8T, 1/4T, 1/2T, and returning to 1/1T through a series of continuous inverse period-doubling bifurcations. Moreover, when the parameter b is within the range [0.4125, 1.0], variations in b do not affect the periodic motion characteristics of the system; rather, variations in b only influence the displacement of the meshing pair. The grazing bifurcation value, period-doubling bifurcation value, and inverse period-doubling bifurcation value remain unchanged with respect to b, indicating that the system behaves similarly to a linear system.

To investigate the global bifurcation characteristics of the system more clearly, single-parameter bifurcation diagrams were obtained by traversing the two-parameter plane transversely at b = 0.15, 0.25, and 0.35, as illustrated in Fig 8.

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Fig 8. Single-parameter bifurcation diagrams of mesh displacement under different dimensionless backlash b: (a) b = 0.15, (b) b = 0.25, (c) b = 0.35.

https://doi.org/10.1371/journal.pone.0335920.g008

Fig 8(a) presents the single-parameter bifurcation diagram of the system at b = 0.15. In the low-frequency region, the system exhibits non-impact 0/1T motion. As ω increases, the system transitions from a non-impact state to a double-sided impact state, eventually entering a stable tooth surface impact state, represented as 0/1T 1/1T 2/1T 2/2T 1/2T 1/1T. Fig 8(b) presents the global single-parameter bifurcation diagram of the system at b = 0.25. Compared with the bifurcation diagram at b = 0.15, the non-impact state interval in the low-frequency region remains unchanged, but the double-sided impact state interval contracts and gradually transforms into a tooth surface impact state, with the tooth surface impact state interval expanding. However, high-period double-sided impact motions such as period 4 and 8 appear in the double-sided impact state interval. Fig 8(c) depicts the global single-parameter bifurcation diagram at b = 0.35. As b increases, the non-impact state region at low frequencies remains unchanged, but the double-sided impact state interval significantly contracts and gradually transforms into a tooth surface impact state, resulting in a noticeable expansion of the tooth surface impact state interval. However, the motions in the double-sided impact state interval degenerate into low-period double-sided impact motions such as 2/4T and 2/2T.

Figs 9, 10, and 11 depict the phase diagram and Poincaré map superposition, along with the time history and dynamic load coefficient diagram for the double-sided impact motion under different dimensionless backlash b and dimensionless meshing frequency. By integrating the phase diagram with the time history diagram, one can observe the number of motion cycles (n), the number of tooth face impacts (p), and the number of tooth back impacts (q) of the gear pair, thereby determining the impact state (I) of the gear. The irregularity of the dynamic load coefficient () is notable, as it can exhibit positive, zero, and negative values. When reaches zero, the gear teeth engaged in meshing experience alternating dynamic loads. Prolonged operation of the system within this parameter range can result in fatigue fractures at the tooth root or damage to the tooth surface. Therefore, it is evident that gear backlash significantly influences both the characteristics of meshing impact and separation.

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Fig 9. 1-1-1 double-sided impact motion, when b = 0.15, = 0.82: (a) superposition of phase diagram and Poincaré map, (b) time history diagram, (c) dynamic load coefficient diagram .

https://doi.org/10.1371/journal.pone.0335920.g009

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Fig 10. 2-2-1 double-sided impact motion, when b = 0.15, = 0.82: (a) superposition of phase diagram and Poincaré map, (b) time history diagram, (c) dynamic load coefficient diagram .

https://doi.org/10.1371/journal.pone.0335920.g010

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Fig 11. 4-4-2 double-sided impact motion, when b = 0.15, = 0.82: (a) superposition of phase diagram and Poincaré map, (b) time history diagram, (c) dynamic load coefficient diagram .

https://doi.org/10.1371/journal.pone.0335920.g011

The DC cloud diagram of the curve-face gear transmission system in the (ω, b) parameter plane is depicted in Fig 12. Fig 12(a) illustrates the three-dimensional DC cloud diagram within the (ω, b) parameter plane. It is observed that when b > 0.4125, the system exhibits a non-impact meshed 0/1T periodic motion region with DC = 0.0. As ω increases, the system traverses the grazing bifurcation, where the impact state transitions from I = 0 to I = 1, resulting in an instantaneous increase in the DC value with a relatively small amplitude. Subsequently, the system undergoes a jump, which causes a drastic increase in DC to 0.6–0.7, indicating severe disengagement and a reduced load-bearing capacity. As ω continues to increase, the system enters quasi-periodic motion, during which DC gradually increases, resulting in worsening disengagement characteristics and load-bearing capacity. A further increase in ω drives the system into chaotic motion, where DC reaches its peak, corresponding to the most severe disengagement and weakest load-bearing capacity. As ω increases further, the system experiences an inverse period-doubling bifurcation, causing chaotic motion to degenerate into low-periodic motion. During this process, DC gradually decreases. Ultimately, a smooth DC surface is formed in the 1/1T periodic motion region. For b[0.4125, 1.0], in the bilateral impact state region, as b increases, the system transitions from single-periodic motion to high-periodic motion, accompanied by an increase in DC. As shown in Fig 12 (b), the (ω, b) parameter plane diagram of impact states (I: DC) illustrates the distribution range and variation pattern of disengagement duty cycles across different impact states. When b is large, DC exhibits a linear strip-like distribution. With increasing ω, the variation follows the sequence: 0:0.0 1:0.0–0.1 1:0.1–0.2 1:0.6–0.7 1:0.4–0.6 1:0.7–0.8 1:0.6–0.7 1:0.4–0.6. When b is small, the DC exhibits an increasing trend along both the ω and b axes.

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Fig 12. Impact states: disengagement duty cycle (I: DC) diagram in the (, b) parameter plane: (a) DC cloud diagram, (b) I: DC plane distribution diagram.

https://doi.org/10.1371/journal.pone.0335920.g012

Fig 13 illustrates the maximum dynamic load of the curve-face gear transmission system within the (ω, b) parameter plane. Fig 13(a) presents a three-dimensional cloud diagram of , where forms a dynamic load surface that gradually increases along the ω-axis in the 0/1T periodic motion region. It reaches its peak in the double-sided impact 2/1T motion region and subsequently decreases with increasing b when b is small. Fig 13(b) depicts the plane distribution diagram, revealing the variation pattern of . As the system traverses the jump, experiences an instantaneous linear increase due to severe hard impacts on the gear pair. When b is large, increasing ω drives the system into quasi-periodic motion, where intensified vibrations and noise cause to rise again. Subsequent continuous period-doubling bifurcations lead to chaotic motion, during which rapidly peaks in the chaotic region. As ω continues to increase, chaotic motion degenerates into low-periodic motion, resulting in a decrease in , forming a smooth dynamic load surface in the 1/1T periodic motion region.

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Fig 13. Maximum dynamic load coefficient in the (, b) parameter plane: (a) cloud diagram, (b) plane distribution diagram.

https://doi.org/10.1371/journal.pone.0335920.g013

3.3 Pattern types and transition laws of periodic motion in (ωomega, xi) parameter plane

With dimensionless meshing frequency (ω) and meshing damping ratio () serving as control parameters, where ω [0.1, 2.5] and [0.04, 0.16], the nonlinear dynamic behavior of the curve-face gear transmission system is investigated within the (ω, ) parameter plane, while other parameters remain fixed. Fig 14 illustrates the distribution of meshing motion in this parameter space. Fig 14(a) presents a three-dimensional bifurcation diagram of meshing displacement against parameters (ω, ), indicating that non-impact motion predominates in the low-frequency region. At lower values of , a jump occurs in the low-frequency region, transitioning the system from non-impact motion to tooth surface impact motion. In the high-frequency region, continuous period-doubling bifurcations propel the system into high-period motion, ultimately culminating in chaotic motion. Conversely, at larger values of , the gear pair demonstrates single-period motion across all parameter ranges. Fig 14(b) depicts the meshing motion distribution map in the (ω, ) parameter plane, highlighting red regions of 0/1T non-impact motion, yellow regions of 1/1T tooth-surface impact motion, color-coded regions representing subharmonic motions, and gray regions indicating chaotic motion. The transition laws are as follows: in the low , low-frequency region, the system exhibits 0/1T non-impact motion. As ω increases, grazing bifurcation occurs, transitioning from 0/1T non-impact motion to 1/1T tooth-surface impact motion. This is followed by a jump, which maintains the motion type. Further increases in ω trigger period-doubling bifurcations, transitioning from 1/1T motion to subharmonic motions (1/2T, 1/4T, 1/8T), ultimately leading to chaotic motion. Inverse period-doubling bifurcations gradually reduce chaotic motion into low-period motions (1/8T 1/4T 1/2T), eventually stabilizing into 1/1T periodic motion. Increasing leads to an expansion of the 0/1T and 1/1T motion domains while shrinking the subharmonic motion domains. When > 0.1436, only 0/1T and 1/1T meshing motions exist in the (ω, ) parameter plane. In conclusion, for systems with small , a judicious increase in effectively reduces chaotic motion.

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Fig 14. Meshing motion law diagram of the system in the (, ) parameter plane: (a) relative micro-displacement bifurcation diagram of gear pair, (b) pattern types and distribution regions of periodic motions.

https://doi.org/10.1371/journal.pone.0335920.g014

Fig 15 illustrates the single-parameter bifurcation diagrams of the system for various values of , clearly depicting the transition process among different types of periodic motion. It is evident that as increases, the period-doubling bifurcation points progressively lag, while the inverse period-doubling bifurcation points advance. With the increase of both ω and , and in conjunction with Fig 14, the fundamental laws governing the impact motion of the curve-face gear system within the (ω, ) parameter plane can be summarized as follows:

1. When = 0.05, 0/1T (G Bif) 1/1T (PD Bif) 1/2T (PD Bif) 1/4T (PD Bif) ... chaos... (PD Bif) 1/8T (PD Bif) 1/4T (PD Bif) 1/2T (PD Bif) 1/1T, Fig 15(a) shows.
2. When = 0.09, 0/1T (G Bif)1/1T (PD Bif) 1/2T (PD Bif) 1/4T (PD Bif) 1/2T (PD Bif) 1/1T, Fig 15(b) shows.
3. When = 0.15, 0/1T (G Bif) 1/1T, Fig 15(c) shows.

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Fig 15. Single-parameter bifurcation diagrams of mesh displacement under different meshing damping ratios : (a) = 0.05, (b) = 0.09, (c) = 0.15.

https://doi.org/10.1371/journal.pone.0335920.g015

Fig 16 illustrates the DC cloud diagram and the distribution of impact states within the (ω, ) parameter plane. Fig 16(a) demonstrates the variation of DC with respect to . For < 0.1436, the system stabilizes into single-period motion with negligible variation in DC. Fig 16(b) emphasizes the evolution of impact states. An increase in results in a delay of the jump points, thereby enhancing the critical speed threshold for tooth separation. Within the range [0.04, 0.1436], the DC follows a complex trajectory along ω, marked by an increase, decrease, increase, and decrease. For > 0.1436, DC evolves smoothly along the ω-axis, transitioning from 0:0.0 to 1:0.0–0.2, then to 1:0.2–0.3, followed by 1:0.3–0.4, and finally to 1:0.4–0.5, indicating a gradual rise.

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Fig 16. Impact states: disengagement duty cycle (I: DC) diagram in the (ω, ) parameter plane: (a) DC cloud diagram, (b) I: DC plane distribution diagram.

https://doi.org/10.1371/journal.pone.0335920.g016

Fig 17 illustrates the of the curve-face gear transmission system within the (ω, ) parameter plane. In Fig 17(a), the relationship between and control parameters is presented. In the low-frequency 0/1T motion region, forms a continuously rising surface along the ω-axis. Upon crossing a grazing bifurcation or jump, experiences abrupt changes; notably, it increases linearly for small values of . In the subharmonic motion region, manifests as a tongue-shaped plateau, with the plateau height decreasing as increases. Fig 17(b) displays the contour map of . For small , during the jump, spikes to between 1.8 and 2.1, indicating a significant impact of loads on the gear pair. As increases, the impact magnitude during the jump gradually diminishes. In the subharmonic region, demonstrates a gradient decrease along the -axis.

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Fig 17. Maximum dynamic load coefficient in the (, ) parameter plane:(a) cloud diagram, (b) plane distribution diagram.

https://doi.org/10.1371/journal.pone.0335920.g017