Introduction

The two-speed transmission system can achieve two-speed output and is used in helicopters, tanks, loaders, automobiles, and other fields. The two-speed transmission system is adopted in the main reducer of the helicopter, thereby achieving cruise and hovering of the helicopter (starting and stopping). Consequently, fuel consumption can be lowered. A typical structure of a two-speed transmission system consists of a friction clutch, an overrunning clutch, and a planetary gear transmission system [1–3]. The overrunning clutch is in the overrun state and the system output is a high-speed gear output when the control friction clutch is combined; the opposite is true when the control friction clutch is released and the overrunning clutch is in the combined state and the system output is a low-speed output. The dynamic characteristics analysis of a two-speed transmission system is separated into three categories: shift process analysis, low-speed gear analysis, and high-speed gear analysis. This study primarily examines the low-speed transmission characteristics of a two-speed transmission system. The study of the two-speed transmission system’s nonlinear dynamic characteristics can serve as a foundation for the best possible design and manufacturing of the transmission system. The two-speed transmission system’s operational performance has a direct impact on the functionality and lifespan of helicopters and other equipment.

During low-speed transmission, the overrunning clutch and the planetary gear transmission system form a coupling system. Nonlinear characteristics of the planetary gear transmission system are mainly investigated in three aspects: modeling method, solution method, and stability evaluation. The models mainly include pure torsional and bending-torsion coupling models. The time-varying meshing stiffness, lateral clearance, and comprehensive meshing error of the gear teeth are usually considered during modeling. Wei et al. [4] proposed a method to establish the dynamic model of the planetary gear system by using shafting elements such as shaft segments, flexible ring gear, and flexible planetary carriers. This method can guide the design of the planetary gear system with high reliability and low vibration. Huang et al. [5] proposed a nonlinear pure rotational dynamics model for a multistage closed planetary gear set consisting of two simple planetary stars. Gui et al. [6] developed a centralized parameter model for a typical planetary gear system with different error types.

Analytical and numerical methods are typically employed for solving the nonlinear dynamics of a planetary gear transmission system. The analytical methods typically include multi-scale, harmonic balance, modal superposition, and Fourier series. Numerical methods include the iterative, Euler, Runge-Kutta, and Gill methods. Lin et al. [7] applied the multi-scale method to investigate the parameter conditions of an unstable planetary gear system. Chaari et al. [8] used an iterative method to calculate the nonlinear dynamic response of the planetary gear system. Zhang et al. [9] solved the vibration response of the 2K-H planetary gear reducer with planetary suspension by employing the Fourier series method. Moreover, the authors analyzed the influence of gear error, planetary suspension, and the initial meshing phase on the dynamic characteristics of the system. Sun et al. [10, 11] analyzed the dynamic characteristics of a closed planetary gear system under the dynamic coupling of the star gear train and planetary gear train by utilizing the Gill integral method with variable step length. Wu et al. [12] studied the nonlinear dynamic characteristics of the compound planetary wheel drive system based on the harmonic balance method. Xu et al. [13] established a new gear tooth modification model according to tooth top modification characteristics and tooth profile modification of the planetary gear train.

Dynamic stability and motion state research on a planetary gear transmission system is mainly based on the nonlinear dynamic model of the system. The numerical algorithm is used to study the system’s stability and possible motion state under different parameters. Xiang et al. [14] analyzed the planetary gear system’s motion and various nonlinear dynamics via the global bifurcation diagram, FFT spectrum, Poincare diagram, phase diagram, and maximum Lyapunov exponent. Zhou et al. [15] explored the reliability of the shearer planetary gear system and the sensitivity analysis based on reliability. The results show that the structural parameters of the solar wheel significantly impact the system’s reliability compared to other mechanisms. Li et al. [16] established the reliability prediction model of a helicopter planetary gear train when subjected to partial loading. Xiang et al. [17] identified the influence of system motion on backlash variation by using a global bifurcation diagram, maximum Lyapunov exponent (LLE), FFT spectrum, Poincare diagram, phase diagram, and time series. Zhou et al. [18] analyzed the effects of meshing frequency, meshing damping, and tooth clearance on the bifurcation and chaos characteristics of the system via phase orbits, Poincare plots, and time history curves.

Kumar [19] proposed a deep learning model and tested the proposed deep learning model on both gear and rotor datasets. The result is more efficient than the current method. Vashishtha [20] proposed an optimal selection of hyperparameters (HPs) based on a deep learning model to study worm gearboxes and found that this method is more efficient than conventional methods. Zhou [21] proposed a health index prediction method based on discrete probabilistic entropy and a bearing health prediction method based on long and short term memory, and the comparison shows that the method is superior to other time series prediction models. Vashishtha [22] proposed an intelligent defect identification scheme for tapered roller bearings based on the ELM model. The original vibration signal of bearing test bench is decomposed into different modes to remove noise. Permutation entropy (PE) is used as a measure to select the prominent pattern. Good experimental results are obtained. Vashishtha [23] proposed a deep learning-based bearing fault identification method. The results show that the language fuzzy classifier achieves the maximum performance with the least computation time. Vashishtha [24] proposes a new method for identifying defects in centrifugal pump impellers, and the end result is improved system reliability ACMD performance.

From the above analysis, it can be seen that there have been many studies on the nonlinear characteristics of a pure planetary gear transmission system but less research on two-speed transmission, especially the dynamic characteristics of the coupling system composed of an overrunning clutch and gear transmission system. The overrunning clutch and the gear drive system inevitably influence each other. Therefore, in order to explore the vibration mechanism of low-speed gear in a two-speed transmission system, this paper intends to use the centralized parameter method to establish a system of pure torsional nonlinear dynamic differential equations considering the number of planetary wheels, tooth side clearance, and gear transmission error for the low-speed gear in the two-speed transmission system. The variable-step fourth-order Runge-Kutta method is used to solve the dimensionless differential equation system, and the influence of gear modulus and other parameters on the nonlinear dynamic characteristics of the two-speed transmission system is analyzed by taking the excitation frequency as the control parameter. The research results can provide a reference for the dynamic design and optimization of such two-speed transmission systems in the future and provide theoretical guidance for the mobility and economy of two-speed transmission systems.

Structure composition and working principle of a two-speed transmission system

A two-speed transmission system shown in Fig 1 comprises the input shaft, double-row planetary gear of the planetary gear train, friction clutch, overrunning clutch, and output shaft structure. The overrunning clutch is set between the planet carrier and the rack; the friction clutch is set between the planet carrier with the output shaft, power input by the input shaft, and output by the output shaft. When the friction clutch is released, the overrunning clutch is in the engaged state, and the planetary rack is in the fixed state. The power is transmitted through the sun wheel, a double row of planetary wheels, and finally, the inner gear ring output, which corresponds to the low gear state. When the friction clutch is engaged, the overrunning clutch is overrunning, and the planetary rack is connected with the inner gear ring. The power is also successively passed through the sun wheel, a double row of planetary wheels, and the inner gear ring output. The output shaft is at a high speed, corresponding to the high-speed stop state.

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Fig 1.

A two-speed drivetrain: (a) Schematic with (1) the sun wheel, (2) the first stage planetary wheel, (3) the planetary shelf, (4) the input axis, (5) the overrunning clutch, (6) the second stage planetary wheel, (7) the inner gear ring, (8) the friction clutch, and (9) the output shaft; (b) 3D model.

https://doi.org/10.1371/journal.pone.0298395.g001

A certain type of two-speed transmission system is applied to the helicopter. During 70%–80% of the working time, the transmission system is in the low-speed gear state, while the double-row planetary wheels are stationary in the high-speed gear state. Therefore, the focus is placed on the dynamic characteristics of the low-speed gear state in this paper.

Low-speed transmission dynamics model of a two-speed transmission system

The centralized mass method is adopted to establish the dynamic model of a two-speed transmission system running in low gear, as shown in Fig 2. The model takes the planetary frame as the reference coordinate system and the locked state. The fixed shaft transmission is the planetary gear system with double rows of planetary wheels. The three degrees of freedom of the solar wheel were fixed on the planetary shelf, and the origin was placed in the center of the planetary shelf. The coordinate system origin of the double row of planetary wheels is taken as the center of each planetary wheel, which is also fixed on the planetary shelf.

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Fig 2. Dynamics model of a two-speed transmission system in low gear.

https://doi.org/10.1371/journal.pone.0298395.g002

In Fig 2, R_bs represents the radius of the base circle of the solar wheel; R_bp1 is the base circle radius of the first stage planetary gear; R_bp2 is the base circle radius of the second stage planetary gear; R_br is the radius of the base circle of the inner gear ring; T_D is the driving torque; T_L is the load torque. K_sp1 is the time-varying meshing stiffness of the solar wheel and the first planetary wheel; C_sp1 is the meshing damping of the solar wheel and the first stage planetary wheel; K_p1p2 is the time-varying meshing stiffness of the second stage planetary wheel and the first stage planetary wheel; C_p1p2 is the meshing damping of the second stage planetary wheel and the first stage planetary wheel; K_p2r is the time-varying meshing stiffness of the inner gear ring and the second planetary wheel; C_p2r is the meshing damping between the inner ring gear and the second stage planetary wheel. Parameters , and are angular accelerations of the sun wheel, the first planetary wheel, the second planetary wheel, and the inner gear ring, respectively. Parameters J_s, J_p1, J_p2, and J_r are the inertia of the sun wheel, the first planetary wheel, the second planetary wheel, and the inner gear ring, respectively.

Low gear dynamics equation of two-speed transmission system

As shown in Fig 2, K_sp1(t) is the time-varying meshing stiffness of the solar wheel and the first planetary gear pair. Its value can be regarded as a rectangular wave, as shown in Eqs 1−4: (1) (2) (3) (4)

Where k_sp1 is the average value of time-varying meshing stiffness; k_sp1r is the RTH harmonic amplitude; φ_sp1r is the phase angle of the RTH harmonic; ε_sp1 is the coincidence degree of the solar wheel and the first stage planetary gear. The R-value is taken as five since the first five harmonics have relatively accurate accuracy.

It is assumed that all gears are unmodified involute spur gears, the bending deformation of input and output shafts is neglected, and the centralized parameter method and Newton’s law are employed. Then, the pure torsional nonlinear mathematical equation of the two-speed transmission system can be expressed as follows: (5)

Where θ_s, θ_p1i, θ_p2i, θ_r1 are torsional vibration displacements of the sun wheel, the ith planetary wheel of the first stage, the ith planetary wheel of the second stage, and the inner gear ring (i = 1, 2, N), respectively; R_bs, R_bp1i, R_bp2i, and R_br are the radii of the base circle of the sun wheel, the ith planet wheel of the first stage, the ith planet wheel of the second stage, and the inner gear ring (i = 1, 2, N), respectively; is the derivative with respect to time; the input torque T_D(t) is the fluctuation value that can be expressed as T_D(t) = T_Dm + T_DaT(t), where T_Dm is the average torque value, and T_DaT(t) is the instantaneous fluctuation value that can be expressed as T_DaT(t) = T_DaTsin(ω_aTt+ϕ_aT). Parameter e_sp1i(t) is the static transfer error between the solar wheel and the ith planetary wheel of the first stage. Fabrication and assembly of the gear can be regarded as e_sp1i(t) = êsin(ω_et+ϕ_e). Lastly, T_L is the load torque applied to the output shaft of the transmission system.

Let , where x_s, x_p1i, x_p2i and x_r1 are the equivalent line displacements of the sun wheel, the first-stage planetary gear, the second-stage planetary gear, and the inner gear ring, respectively. The dimensional motion equation of the system can be obtained via the linear variation of Eq (5): (6) m_s, m_r, m_p1i, m_p2i are the equivalent masses of the solar wheel, the inner ring gear, the first stage planetary wheel, and the second stage planetary wheel on the meshing line m = J/R².

The transmission error of the gear pair X is introduced via Eq (7), where X_sp1i, X_rp2i, and X_p1ip2i, represent relative displacements between the solar wheel and the first planetary wheel, between the inner gear ring and the second planetary wheel, and between the first planetary wheel and the second planetary wheel, respectively.

(7)

A dimensionless time scale τ = tω_n and displacement scale b_c are defined as follows: , , ; , ω_T = ω_aT/ω_n. Eq (6) can lead to a dimensionless equation, as shown in Eq (8): (8)

Where

; ; ; ; ; ; ; ; ; ; ; ; ; .

Where the dimensionless integrated meshing error can be expressed as: (9)

The non-dimensional gap nonlinear function can be expressed as: (10)

Low gear dynamic response of a two-speed transmission system

The low-speed gear model parameters of a two-speed planetary gear drive are shown in Table 1. The fourth-order variable step size Runge-Kutta method is used to solve Eq (8), and the initial value is q1(0) = 0, q2(0) = 0, q3(0) = 0; q1’(0) = 0, q2’(0) = 0, q3’(0) = 0. The solution interval is [0, 6000]. The excitation frequency is taken as the control parameter. Then, the transfer error bifurcation characteristics between the sun wheel and the first-stage planetary wheel, the first-stage planetary wheel and the second-stage planetary wheel, and the second-stage planetary wheel and the inner gear ring are investigated. Simultaneously, the transfer error bifurcation characteristics of the solar wheel and the first planetary wheel under different gear moduli are explored.

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Table 1. Low-speed gear model parameters of a two-speed planetary gear transmission.

https://doi.org/10.1371/journal.pone.0298395.t001

A change in transmission error between the sun and the first planetary gear with excitation frequency ω_h

The excitation frequency ω_h is taken as the control parameter, and the dimensionless excitation frequency is increased from 0.000 to 3.000. Then, a dimensionless dynamic transfer error bifurcation diagram of a pair of solar wheels and the first-stage planetary gear train is obtained, as shown in Fig 3.

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Fig 3. The sun and the first planetary gear with respect to the excitation frequency ω_h global bifurcation diagram.

https://doi.org/10.1371/journal.pone.0298395.g003

As shown in Fig 3, the motion state of the system exhibits rich bifurcation characteristics with frequency changes: when ω_h≤1.19, 2.78≤ω_h≤3, the system is in a period-1 motion; when 1.19≤ω_h≤1.41, 2.48≤ω_h≤2.78, the system is in a multiperiodic motion; at 1.41≤ω_h≤2.48, the system is in a chaotic motion.

It can be clearly seen in spinning Figs 3 and 4 that there is a significant jump at ω_h = 0.62 and ω_h = 0.78, and at 0.62≤ω_h≤ 0.78, the gear pair is in a quasi-periodic motion. Take ω_h = 0.63, the quasi-periodic motion is formed by the combination of two or more non-reducible frequencies, the phase diagram is a closed curve band with a certain width, the corresponding Poincare section is a closed curve, and its time domain diagram, phase diagram, and Poincare cross-section are shown in Fig 4(A); When the ω_h≤ 0.62, the gear pair will increase slightly with the increase of the excitation frequency ω_h; when 0.78≤ω_h≤1.19, the gear pair increases significantly with the increase of the excitation frequency ω_h, and there is a tendency to enter chaotic motion, but it is still in the period-1 motion. Take ω_h = 1.03, the phase plane diagram of the period-1 motion is a closed circle, the Poincare section is a single point, and its time domain diagram, phase diagram, and Poincare section are shown in Fig 4(B); When 1.19≤ω_h≤1.41, the motion of the gear pair is period-2 motion; when the excitation frequency is located at 1.41≤ω_h≤2.48, the gear pair enters chaotic motion; at this time, take ω_h = 1.92, as shown in Fig 4(C), the time domain response diagram shows aperiodic motion, the phase plane diagram is disordered, and the Poincare diagram shows many discrete points; When the excitation frequency is located at 2.48≤ω_h≤2.78, the approximate motion of the system changes from period-2 motion to period-1 motion; at this time, ω_h = 2.71, as shown in Fig 4(D), the phase plane is two closed circles, and the Poincare section is approximately two single points. Finally, when the excitation frequency is 2.78≤ω_h, the gear pair moves in a period-1 motion.

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Fig 4.

Time domain diagram, phase diagram, and Poincare cross-section diagram corresponding to different excitation frequency points of the solar wheel and the first stage planetary gears: (a) 0.63, (b) 1.03, (c) 1.92, (d) 2.71.

https://doi.org/10.1371/journal.pone.0298395.g004

The first and second planetary gears’ transmission error with respect to the excitation frequency ω_h

Fig 5 shows the dimensionless transmission error bifurcation diagram of the first stage planetary gear and the second planetary gear pair drawn with the excitation frequency ω_h as the control variable in the low-speed mode of two-speed planetary gear transmission. From Fig 5, it can be seen that there are obvious jumping phenomena at ω_h = 0.28, ω_h = 0.49, ω_h = 0.61, and ω_h = 0.72, respectively. When the two-speed planetary gear transmission is located at 0.10≤ω_h≤0.18, the motion does not change much; in the low frequency band 0.58≤ω_h≤0.61 and 0.75≤ω_h≤0.77 and high frequency band 2.88≤ω_h, the system has a period-1 motion; the gear pair has a period-2 motion in the system in the low frequency band 1.18≤ω_h≤1.22 and the high frequency band 2.68≤ω_h≤2.79; the gear pair has a chaotic motion in the low frequency band ω_h≤0.75, except for four jump points At 0.77≤ω_h≤1.18 and 1.22≤ω_h in the middle and high frequency bands, ω_h≤2.68 exhibit chaotic motion; at 2.79≤ω_h≤3.0, the system turns to period-1 motion.

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Fig 5. A change in the amplitude of first and second planetary gears with respect to the excitation frequency ω_h.

https://doi.org/10.1371/journal.pone.0298395.g005

It is clear from Figs 5 and 6 that when the frequency is small, ω_h< 0.75, the system is mostly in a chaotic motion, but there are several jumps in this chaotic motion, such as at ω_h = 0.28, ω_h = 0.49, ω_h = 0.61, and ω_h = 0.72, indicating that the system vibrates during the frequency increase. With the slow increase of frequency ω_h, at 0.75≤ω_h≤0.77, it enters a period-1 motion; when ω_h = 0.78, the system suddenly jumps from a quasi-periodic motion state to a chaotic motion, which presents a "cataclysm" feature, indicating that the system enters the chaotic motion through the "cataclysm" pathway, as shown in Fig 5(A), because the quasi-periodic motion is formed by the combination of two or more non-reducible frequencies, and its phase diagram is a closed curve band with a certain width. The corresponding Poincare section is a closed curve. The system was in a chaotic state at 0.77≤ω_h≤1.18 and 1.22≤ω_h≤2.68, and ω_h = 1.55 was selected, and its time domain diagram, phase diagram, and Poincare cross-sectional diagram are shown in Fig 5(B). At 1.18≤ω_h≤1.22 and 2.68≤ω_h≤2.79 are in a period-2 motion, ω_h = 2.70, and the time domain diagram, phase diagram, and Poincare cross-sectional diagram are shown in Fig 5(C); the system returns to period-1 motion at 2.88≤ω_h, and ω_h = 2.92 is selected, and its time domain diagram, phase diagram, and Poincare cross-sectional diagram are shown in Fig 5(D).

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Fig 6.

Time domain phase diagram and Poincare cross section diagram corresponding to different excitation frequency points of the first stage and second stage planetary gears: (a) 0.78, (b) 1.55, (c) 2.70, (d) 2.92.

https://doi.org/10.1371/journal.pone.0298395.g006

A change in the transmission error for a secondary planet gear and internal gear ring with excitation frequency ω_h

Fig 7 shows the dimensionless transmission error bifurcation diagram of the second stage planetary gear and the inner ring gear pair drawn in the low-speed gear mode of two-speed planetary gear transmission with the excitation frequency ω_h as the control variable. It can be seen that there is an obvious jump phenomenon at ω_h = 0.61 of the gear pair. When the two-speed planetary gear transmission is ω_h≤0.29, the motion response of the two-speed planetary gear transmission does not change much; in the low frequency band 0.29≤ω_h≤0.61, the middle and low frequency band 0.79≤ω_h≤0.81 and the high frequency band 2.91≤ω_h range, the system has a period-1 motion; in the middle frequency band 1.14≤ω_h≤1.21 and the high frequency band 2.71≤ω_h≤2.91, there is a period-2 motion; in the 0.61≤ω_h≤0.79, 0.81≤ω_h≤1.14 and 1.21≤ω_h≤2.71, there is chaotic motion.

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Fig 7. A change in the transmission error for a secondary planet gear and internal gear ring with excitation frequency ω_h.

https://doi.org/10.1371/journal.pone.0298395.g007

It can be clearly seen from Figs 7 and 8 that when the frequency is small, ω_h≤ 0.61, the system is in an unstable period-1 motion. Take ω_h = 0.5, and its time domain diagram, phase diagram, and Poincare cross-sectional diagram are shown in Fig 8(A); With the slow increase in frequency ω_h, a jump occurs at ω_h = 0.61, and the system suddenly jumps from a period-1 motion to a chaotic motion, which shows the characteristics of "cataclysm", indicating that the system enters the chaotic state through the "cataclysm" pathway. The system is in period-1 motion at 0.79≤ω_h≤0.81, 0.81≤ω_h≤1.14 in chaotic motion, period-2 motion at 1.14≤ω_h≤1.21, and chaotic motion at 1.21≤ω_h≤2.71, taking ω_h = 1.80, and its time domain diagram, phase diagram, and Poincare cross-sectional diagram are shown in Fig 8(B). At 2.71≤ω_h≤2.91, the system enters a period-2 motion, taking ω_h = 2.64, and its time domain diagram, phase diagram, and Poincare cross-sectional diagram are shown in Fig 8(C); at 2.91≤ω_h, it returns to a stable period-1 motion, taking ω_h = 2.95, and its time domain diagram, phase diagram, and Poincare cross-sectional diagram are shown in Fig 8(D).

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Fig 8.

Time domain diagram, phase diagram, and Poincare cross-section diagram corresponding to different excitation frequency points of transmission errors of the second stage planetary gear and inner ring gears: (a) 0.51, (b) 1.80, (c) 2.64, (d) 2.95.

https://doi.org/10.1371/journal.pone.0298395.g008

Influence of gear modulus on the transfer error bifurcation between the solar gear and the first stage planetary gear

Fig 9 shows the bifurcation characteristics of the gear pair formed by the sun gear and the first-stage planetary gear under the condition that the gear module m is different. With the gradual increase of modulus, the system vibration amplitude difference will first briefly rise, as shown in Fig 9(A)–9(C), and the subsequent overall trend of vibration amplitude difference is gradually contracted, as shown in Fig 9(D)–9(H); From Fig 9, it can be seen that the system has experienced quasi-periodic, period-1, period-2, chaotic, and multi-periodic motions over the entire excitation frequency range.

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Fig 9.

Influence of tooth modulus on transmission error bifurcation between the solar gear and the first planetary gear pair: (a) m = 1.5, (b) m = 2, (c) m = 2.5, (d) m = 3, (e) m = 4, (f) m = 5, (g) m = 6, (h) m = 8.

https://doi.org/10.1371/journal.pone.0298395.g009

It can be seen from Fig 9(A) that when the system is in the low excitation frequency range, the motion amplitude of the system is not large, and then there will be an obvious jumping phenomenon, gradually entering the chaotic motion; Fig 9(B)–9(H) shows that the system will gradually enter the chaotic motion prematurely with the increase of gear modulus and then end the chaotic motion prematurely. The chaotic motion is not rich enough to respond to each other and other types of motion characteristics, but with the increase in gear modulus, the fluctuation amplitude of the system motion can be suppressed to a certain extent.

Therefore, from the overall perspective of Fig 9, with the increase in modulus, the vibration amplitude difference of the system can be suppressed to a certain extent. At the same time, the excitation frequency required for the system response to enter the chaotic motion will gradually decrease with the increase in gear modulus, and the excitation frequency range span of the chaotic motion of the system will gradually decrease, and finally it will transition into a period-1 motion through a period-2 motion in the chaotic motion.

Conclusions

Using the centralized parameter method, a six-degree-of-freedom pure writhing mechanical model of a two-speed transmission system with low-speed gear was created in this research. The model takes into account mesh stiffness, damping, tooth clearance, and overall gear error. The Runge-Kutta method was used to numerically solve the nonlinear system, with the excitation frequency serving as the control parameter. To investigate the transfer error variation between the solar wheel and the first planetary wheel, the first planetary wheel and the second planetary wheel, and the second planetary wheel and the inner ring, the global bifurcation, time domain diagram, phase diagram, and Poincare cross-section were utilized. Furthermore, the transfer error bifurcation features of the solar wheel and the first planetary wheel were thoroughly explored under different gear moduli, and the following findings were reached:

1. The two-speed transmission system has a rich cycle conversion phenomenon in the process of increasing the excitation frequency in the low-speed gear mode, which is accompanied by quasi-periodic motion, period-1 motion, period-2 motion, multi-periodic motion, and chaotic motion, but eventually converges to period-1 motion.
2. The increase in gear modulus will advance the chaotic motion, and the transformation phenomenon of motion characteristics will gradually become less obvious. In the low frequency band (ω_h≤0.5), when the gear modulus is relatively small, it is generally accompanied by the jumping phenomenon, and the conversion frequency between various motion responses is higher, so it is relatively better to use high module gears in the low frequency band. When the excitation frequency is in the medium frequency band (0.5<ω_h≤2), the chaotic motion of high modulus gear is too concentrated, and most of them are maintained in the high amplitude difference range, which is not conducive to the service life of the gear, and the long working time will also increase the relative error between parts, which is not conducive to the transmission of torque. There are hidden dangers to safety performance, so in the medium frequency band, it is more reasonable to use relatively low module gears when meeting the transmission torque requirements; When the excitation frequency is a high frequency band (2<ω_h≤3), the amplitude difference of the three pairs of gear pairs tends to be stable, and the gear modulus has little effect on its change, so in the high frequency segment, the modulus size can be reasonably selected according to the use requirements.

Limitations and deficiencies

1. This paper adopts the lumped parameter method in modeling and the Runge-Kutta method in solving differential equations, which is a commonly used modeling and solving method for system dynamics. However, the lumped parameter method is a simplified modeling method, which usually regards the shaft as a point mass or a rigid connection, and has certain limitations in capturing the deflection and deformation of the shaft.
2. In future studies, under the condition that the amount of calculation is appropriate, if the accuracy is required to be high, the potential energy method or Timoshenko theory can be considered for modeling, which will make the results more accurate.

Supporting information