A novel method for vibration reduction in double-helical planetary gear trains via staggered phase tuning

Abstract

Planetary gear trains offer high power density and compact structure in advanced equipment but suffer from vibration challenges. A method based on staggered phase tuning is proposed to suppress vibrations in double-helical planetary gear trains. Using a helicopter main reducer as a case study, a multi-flexible-body dynamic model of the double-helical planetary gear train is developed by integrating substructure condensation with multi-body dynamics theory. The model considers the flexible effects of low-stiffness components, including thin-walled ring gears and complex casings. The effects of gear type, staggered phase, and its coupling with helix angle on system dynamics are examined. Double-helical configuration significantly reduces vibration compared to spur gears while providing self-balanced axial forces and a stable load-sharing coefficient (K_γ ≈ 1.01–1.03). The staggered phase p is a key parameter. A phase separation is observed: sun gear vibration is minimized at p = 0.5, whereas load sharing is optimized at p = 0.43, consistent with the theoretical inflection points of the internal and external gear contact ratios. The tuning effect shows strong coupling with operating conditions and system parameters, and the optimal phase varies with helix angle, highlighting the need for parameter co-design. The experimental results of staggered-phase tuning of the external helical gear pair agree well with the theoretical model in vibration reduction trends, confirming the effectiveness of this method. This study establishes a theoretical basis and optimization framework for high-performance dynamic design of double-helical planetary gear trains.

Keywords

staggered phase tuning vibration reduction method multi-flexible-body dynamic modeling double-helical planetary gear trains

1. Introduction

Planetary gear transmission systems have become a core mechanism for power transmission in high-end equipment owing to their advantages of high power density and compact structure, with helicopter main reducers being a typical example. The dynamic characteristics of the transmission system directly determine the overall reliability as well as the vibration and noise levels of the equipment. Common vibration reduction methods for gear transmission systems, such as gear type modification, gear shifting, and the addition of damping rings, are often limited by machining accuracy and operational adaptability, thereby imposing constraints on design and optimization. To expand the dynamic tuning dimensions of gear transmissions from a systemic perspective, a novel vibration reduction method based on staggered phase tuning is proposed for double-helical planetary gear trains. For a double-helical planetary gear train, this method suppresses system vibration through staggered phase tuning, a technique that adjusts the staggered amount without altering the macroscopic structure or tooth geometry. The mapping relationship between the staggered phase tuning angle and the system dynamic response is also investigated.

To predict the dynamic behavior of such systems, researchers have developed various modeling approaches. Among these approaches, the lumped-mass model^1–11 is widely used in the dynamic analysis of planetary gear trains, with a focus on gear-pair response analysis. Such models typically treat gear shafts and bearings as lumped components, using simplified stiffness to represent complex coupled interactions. While these models are simple and efficient (e.g., the dynamic load model by Zhou et al.,¹ the pure torsional model by Zhu et al.,³ the combined transmission model by Hu et al.,⁵ the multi-stage model by Wang et al.,⁷ studies on phase influence by Chen et al.,⁸ and nonlinear analysis by Mo et al.⁹), their inherent limitations stem from the lack of separate modeling for shaft systems, neglecting factors such as ring gear flexibility and housing support. This limitation prevents the precise calculation of shaft system responses, bearing reactions, and dynamic mesh loads. Finite element models^12,13 can more accurately reflect the elastic deformations and loading conditions of components. However, they are complex to model, involve a large number of degrees of freedom, require extensive computation time, and pose challenges in post-processing, making them unsuitable for dynamic design applications. Chen et al.¹² simultaneously considered spatial crack faults and complex foundation structures, establishing both a finite element model and a multi-body dynamic model of a gear-rotor system to analyze its dynamic characteristics. Qiao et al.¹⁴ obtained mesh characteristics (such as time-varying mesh stiffness and contact stress) using finite element methods. The mesh stiffness under single- and multi-stage cracks was then imported into a dynamic model of a two-stage cracked gear transmission for dynamic response analysis. The distributed mass–lumped parameter method aims to strike a better balance between efficiency and accuracy. For instance, Zhang et al.^15,16 took the main reducer of a helicopter as an example and proposed a dynamic modeling method based on experimental modal analysis, finite element methods, and substructure condensation. Ren et al.¹⁷ employed a hybrid finite element and lumped mass modeling technique, discretizing the system into shaft segments, gear elements, and bearing–housing elements, to establish a 162-degree-of-freedom dynamic model. This approach provides a valuable method for planetary gear train analysis.

The meshing phase is a key parameter in studying the vibration characteristics of gear transmission systems. Phase tuning in gear transmissions is primarily classified into two categories. The first involves adjusting the meshing frequency excitation phase by modifying the number of teeth on gears and the number of planetary gears in the transmission system. This aims to improve load distribution among transmission components and enhance system dynamic performance.^18,19 The second category involves achieving phase tuning through intentional tooth indexing. Recently, several researchers have conducted extensive studies on the meshing phase. Yan et al.¹⁸ investigated the coupling phase-tuning mechanism in two-stage planetary gear systems and its correlation with coupled system vibration. They described the relationship among system tooth counts, harmonic orders, and excitation/suppression effects of coupled vibrations. Zhang et al.²⁰ studied the influence of meshing phase on load sharing in herringbone planetary gears. Their results showed that fine-tuning the geometric phase of planetary gears, without altering the system transmission ratio or dimensions, can improve load sharing. Adjusting the tooth number relationships among planets, although slightly modifying the transmission ratio and dimensions, further reduces the system load-sharing coefficient while suppressing translational vibrations. Wang et al.²¹ derived the relationship between planetary meshing phase and planet/sun gear vibrations, defining a key parameter that depends on the mesh frequency harmonic order, ring gear tooth count, and total number of planets. Pizzolante et al.²² conducted a parametric study by evaluating the effects of gear layout and meshing phase differences. Dai et al.²³ examined the changes in the meshing phase modulation behavior of planetary gear trains (PGTs) due to various gear pin position errors. Guo et al.²⁴ investigated the effectiveness of meshing phase techniques in suppressing specific modal responses of multiple orders in typical split-path transmission systems. Their dynamic model accounts for three planar degrees of freedom per gear and incorporates time-varying mesh stiffness, thereby capturing meshing phase effects. In addition, other researchers have studied the offset angles between the left and right sides of herringbone teeth. Hou et al.²⁵ proposed a gear vibration reduction model with a gear-staggered tooth phase structure (GSTPS) to address high vibration and noise in gear transmission systems. Zhang et al.²⁶ introduced the principle of higher-order tuned gear transmission, defined the staggered time-phase angle for tuned gears, and investigated the influence of gear transmission parameters on dynamic mesh forces and vibration responses. Kang et al.²⁷ examined the quasi-static characteristics of double-helical gear pairs, focusing on the effects of key design and manufacturing parameters, including the nominal left-right offset angle, deviations (errors) from the nominal offset angle, and axial support conditions of the gears. Sun et al.²⁸ studied herringbone star gear trains and found that, although offset angles generate axial forces, selecting an appropriate offset angle reduces fluctuations in comprehensive mesh stiffness, thereby stabilizing dynamic mesh forces in herringbone star gear systems. Mo et al.²⁹ established a dynamic model of herringbone planetary gear transmission systems using a lumped-parameter method and observed that the offset angle significantly affects the load-sharing coefficient but has little influence on the maximum mesh force. Li et al.^30,31 developed a dynamic model for two-stage parallel-axis gear transmission systems considering time-varying mesh stiffness. They examined the impact of the meshing phase on nonlinear system vibration response and identified its role in suppressing vibration.

Despite significant progress in research, current studies still encounter the following critical issues: lumped-mass models lack the accuracy required to capture complex coupling effects, whereas high-precision finite element models are limited in dynamic design and analysis of complex systems owing to time-consuming computations and low efficiency.³² Existing research has predominantly focused on gear meshing analysis under rigid assumptions, often neglecting or oversimplifying the flexibility of key structural components, particularly complex, irregular, thin-walled housings, and their influence on vibration responses. This leads to notable discrepancies between theoretical predictions and actual operational responses. The meshing phase significantly influences the dynamic characteristics of planetary gear transmissions. Current phase-tuning methods for planetary gear trains primarily revolve around circumferential tuning relative to the sun gear, aiming to equalize the meshing stiffness among all planet gears at any given moment to reduce system vibration. However, under strict constraints on parameters, such as transmission ratio and load, such tuning approaches exhibit certain limitations. Although preliminary investigations have been conducted on tuning methods for fixed-shaft double-helical gears, systematic research on the application of double-helical gear tuning to planetary gear trains remains limited.

The remainder of this paper is organized as follows. Section 2 establishes a multibody dynamic model of the double-helical planetary gear transmission system by integrating substructure condensation techniques with multibody dynamics theory, with explicit consideration of flexible, low-stiffness components. Section 3 compares and provides a mechanistic explanation of the integrated dynamic responses, including vibration, meshing force, and load-sharing characteristics, under identical operating conditions for three different gear types. Sections 4 and 5 investigate the vibration suppression mechanism of staggered phase tuning in a double-helical planetary gear transmission system. Section 6 verifies the effectiveness of staggered-phase tuning for vibration reduction through experiments on staggered-phase tuning of an external helical gear pair.They identify the optimal phase corresponding to different optimization objectives, such as vibration reduction and load equalization, and clarify the system dynamic behavior under the coupling effects of operating conditions, helix angle, and other parameters. The findings provide a theoretical foundation for the dynamic design of high-reliability helicopter transmission systems and for vibration reduction optimization of double-helical planetary gear trains based on staggered phase tuning.

2. Multibody dynamic modeling of a double-helical planetary gear train considering flexible low-stiffness components

In the planetary gear transmission system of the helicopter main reducer studied here, key components such as the ring gear and housing exhibit thin-walled and complex, irregular structural characteristics, leading to significant deformation under actual loads. If modeled under traditional rigid assumptions, their dynamic behavior cannot be accurately characterized, resulting in considerable deviation between simulation results and physical reality. Therefore, to balance model accuracy and computational cost, a system-level dynamic model was constructed using the substructure condensation method. Each substructure was condensed into a super element,²⁶ and a system-level multi-flexible-body coupled dynamic model was formed using multipoint constraint (MPC) methods and inter-substructure coupling relationships.

2.1. Modeling of component flexibility based on substructure condensation

The nodal coupling relationships for gears primarily involve two types of constraint paths: the shaft–hole connection zone and the tooth meshing zone, including couplings at gear shaft holes and among individual gear teeth. In shaft–hole coupling modeling, a master node coordinate system is established at the geometric center of the shaft hole. The shaft–hole–bearing contact interface is discretized into sets of slave nodes using a radial layered discretization method. For spur gear tooth coupling modeling, a symmetric mapping strategy based on the pitch circle cross-section is adopted. Specifically, a set of master nodes is arranged along the intersection line of the tooth width centerline and the pitch circle. For each gear tooth, three uniformly distributed MPC master nodes are created along the tooth width direction, and a topologically structured network of slave nodes with a circumferential gradient distribution is established in the tooth face meshing zone. The selection of condensation points for helical gears differs from that for spur gears. For helical gears, the condensation points are distributed along the helix line, and three uniformly spaced MPC master nodes are created along the tooth width direction at the pitch circle position of each tooth. During geometric cleanup of the helical gear model in 3D software, the Cartesian coordinate system must be adjusted. Taking a helical planetary gear as an example (Figure 1), a Cartesian coordinate system is established on one end face of the helical gear: the symmetry axis of one tooth is taken as the y-axis, the shaft hole axis as the z-axis, and the x-axis points to the right. The angular difference between adjacent master nodes within the same tooth is expressed as follows:

Δ p_{t} = \frac{b}{3} \cdot \frac{360}{p_{z}}

(1)

where b represents the tooth width, and p_z denotes the lead. For a right-hand helical gear, the angular offset of any master node relative to the y-axis within a given tooth is expressed as follows:

φ_{i, j} = \frac{360}{z} \cdot (i - 1) + Δ p_{t} \cdot (j - \frac{1}{2})

(2)

where z represents the number of teeth, i denotes the tooth number (i = 1,2,…,z), and j indicates the master node number within the same tooth (j = 1,2,3). The coordinates of any master node can be calculated based on the angular offset, as follows:

{\begin{cases} x_{i, j} = r_{p} \cdot \cos φ_{i, j} \\ y_{i, j} = r_{p} \cdot \sin φ_{i, j} \\ z_{i, j} = - \frac{b}{3} (j - 1) + \frac{b}{6} \end{cases}

(3)

where r_p represents the pitch radius. Once the coordinates of the gear tooth master nodes are obtained, the master nodes for the teeth can be created, enabling fully flexible modeling of the gear.

Figure 1.

Diagram of master node locations for helical gears.

The condensation method for the remaining gears is similar. The ring gear is connected to the upper and lower casings via bolts, requiring the creation of master nodes on the bolted interfaces with the casings. The condensation process is illustrated in Figure 2. The shaft system involved in the assembly mainly consists of the sun gear shaft and the planet carrier shaft. As shown in Figure 3, master nodes for the sun gear shaft are placed at the power input and bearing connection points, whereas master nodes for the planet carrier shaft are located at the planet gear connection points on the carrier, bearing connection points, and the power output point. To meet multifunctional design requirements, such as structural load-bearing, component mounting, vibration damping, and light weighting, the casing structure typically integrates various functional geometric features, including bearing holes, mounting bolt surfaces, and lugs, resulting in significant structural complexity. The distribution of key master nodes on the casing is shown in Figure 4.

Figure 2.

Condensed finite element model of gears.

Figure 3.

Condensed finite element model of the shaft system.

Figure 4.

Condensed finite element model of the casing.

To verify the dynamic characterization accuracy of the condensed superelements, a comparative study is conducted on the errors and node counts of each substructure condensation model.³³ The specific results are shown in Figure 5. The results indicate that within the frequency range of ≤ 3f_m (≤ 2152 Hz), the frequency errors are all ≤ 1.88%, and the mode shape errors are ≤ 3.21%. This demonstrates that the condensed model can accurately retain the key dynamic characteristics of the original components.

Figure 5.

Summary of the condensation model of each substructure.

2.2. Integration of the multi-flexible-body coupled dynamic system model

In multibody systems, interactions between bodies are modeled as force elements. In practical engineering applications, connections between components are typically realized through such force elements, most of which act between marker points on two adjacent bodies. When the relative position, velocity, or other kinematic quantities between components change, these force elements apply corresponding forces or moments to the connected bodies, with magnitudes determined by their characteristics.³⁴ Therefore, during the construction of the system dynamic model, based on flexible multibody dynamics theory, the topological structure of the double-helical planetary gear transmission system was analyzed (Figure 6), and the specific modeling workflow is illustrated in Figure 7. By applying multipoint constraint (MPC) methods and considering inter-substructure coupling relationships, a system-level multi-flexible-body coupled dynamic model was formed (Figure 8). The fundamental parameters of the overall system and each substructure are detailed in Tables 1–3.

Figure 6.

System topology diagram.

Figure 7.

System modeling workflow.

Figure 8.

System coupled dynamic model.

Table 1.

Gear parameters.

	Sun gear	Planet gear	Ring gear
Number of Teeth	34	36	106
Normal Module (mm)	2.5	2.5	2.5
Pressure Angle (deg)	25	25	25
Helix Angle (deg)	15	15	15
Effective Face Width (mm)	52.5	52.5	52.5

Table 2.

Bearing stiffness parameters.

Bearing stiffness	kxx (N/m)	kyy (N/m)	kzz (N/m)
ID	kxx (N/m)	kyy (N/m)	kzz (N/m)
Sun Gear Shaft Bearing	2×10⁸	6×10⁸	4×10⁸
Planet Carrier Bearing1	2×10⁷	2×10⁷	1×10⁵
Planet Carrier Bearing2	4×10⁶	4×10⁶	2×10⁶
Planet Gear Bearing	5×10⁸	5×10⁸	5×10⁸

Table 3.

Material properties.

Component	Material	Density (kg/m³)	Young’s modulus (GPa)	Poisson’s ratio
Upper Casing	Cast Aluminum	2.70×10³	70	0.340
Lower Casing	Cast Magnesium	1.77×10³	42.5	0.275
Sun Gear Shaft	Alloy Steel	7.78×10³	195	0.250
Planet Carrier	Alloy Steel	7.78×10³	195	0.250
Gears	Gear Steel	7.85×10³	206	0.275

For a system with multiple meshing frequencies and potential high-frequency content, the choice of time step critically affects the accuracy of the computed vibration responses.³⁵ In conducting multi-flexible-body dynamic simulations of the double-helical planetary gear transmission system, a fixed-step integration method was adopted to balance the accuracy of time-domain responses with the feasibility of frequency-domain analysis. The specific settings are as follows:

The SODASRT 2 integrator, based on a variable-order numerical differentiation algorithm, was employed to solve the system’s dynamic equations. This method ensures stability in solving stiff problems while effectively handling the strongly nonlinear dynamic behavior induced by time-varying mesh stiffness in gear transmission systems. The total simulation time was set to 20 s, with a sampling frequency of 20,000 Hz, corresponding to an integration step size of 5×10^-5s. The selection of this step size was based on the following considerations: The highest frequency of interest is the meshing frequency and its 6th harmonic component (approximately 4,304 Hz). According to the Nyquist sampling theorem, the sampling frequency must be at least twice the highest analysis frequency. The adopted sampling frequency of 20,000 Hz is substantially higher than the theoretical lower limit of 8.6 kHz, ensuring adequate retention of high-frequency vibration information and avoiding frequency aliasing. Regarding the temporal resolution required for gear contact force calculations, this step size guarantees a minimum of 27 computation points per meshing cycle (meshing cycle ≈ 1.394ms), satisfying the requirements for precise integration of time-varying mesh stiffness excitations.

3. Comparative analysis of dynamic behavior for planetary gear trains with different gear types

Planetary gear transmission systems are widely used in critical power transmission applications, such as helicopter main reducers, because of their compact structure, high transmission ratio, and strong load-bearing capacity. However, their complex multi-path coupled transmission characteristics and internal excitations, such as time-varying mesh stiffness, make vibration and noise issues particularly prominent. These problems directly affect the reliability, comfort, and stealth performance of aircraft. Gear type design, a fundamental factor influencing gear system dynamics, serves as both the origin and a key aspect of vibration control. Spur, helical, and double-helical (herringbone gears), as the three basic gear-type configurations, exhibit distinct dynamic performance characteristics. Spur gears are simple to manufacture but suffer from severe meshing impacts, which readily induce vibration. Helical gears improve meshing smoothness and significantly enhance transmission stability through helix angle design; however, they generate notable axial forces, posing challenges to bearing life and system layout. Double-helical gears theoretically combine the advantages of a high contact ratio and self-balanced axial forces, making them an ideal choice for high-performance transmission systems. However, existing research has predominantly focused on comparing gear types for single gear pairs, lacking a comprehensive comparison and mechanistic explanation of the integrated dynamic responses, such as vibration, meshing forces, and load-sharing characteristics, of the three gear types under identical operating conditions within a complete planetary gear system. Therefore, this chapter aims to establish a unified dynamic analysis framework. Using the same multi-flexible-body coupled dynamic modeling approach, while keeping fundamental gear parameters, material properties, and bearing stiffness unchanged, only the gear-type geometry is modified to helical or double-helical (Figure 9). The dynamic responses—including meshing forces, vibration accelerations, and load-sharing characteristics—of the three gear types under rated operating conditions are compared. This provides an essential benchmark and theoretical foundation for subsequent chapters, which focus on the in-depth vibration reduction optimization of staggered phase tuning in double-helical gears.

Figure 9.

Diagram of different gear type structures.

3.1. Comparison of meshing force excitation and vibration responses for different gear types

To reveal the intrinsic impact of different gear types on the dynamic response of the system, the meshing forces, axial forces, and load-sharing characteristics among planet gears in external gear pairs were compared. Figure 10(a) shows the time-domain curves of the meshing force for the external gear pairs of the three planetary gear train configurations. Their dynamic characteristics differ significantly: the spur gear exhibits severe fluctuations in meshing force, with noticeable dynamic shocks (reaching 19.2 kN), which originate from abrupt changes in mesh stiffness owing to its low contact ratio and constitute the primary excitation source for vibration and noise. The helical gear exhibits a relatively smooth meshing force curve, benefiting from the higher contact ratio provided by the helix angle. Its peak value (approximately 6.1 kN) is substantially lower than that of the spur gear (16.2 kN), confirming the effectiveness of helical gears in improving transmission smoothness and suppressing dynamic impact forces. In contrast, the double-helical gear demonstrates the best meshing smoothness, with its meshing force amplitude stably suppressed at a low level (around 0.4 kN) and exhibiting minimal fluctuation. This indicates that the double-helical structure, through load distribution between the left- and right-hand helical flanks, maximally mitigates meshing impacts at the excitation source, laying the foundation for achieving lower system vibration. Furthermore, the comparison of axial forces (Figure 10(b)) clearly illustrates the differences in force flow among the gear types. Theoretically, the spur gear generates no axial force, and the simulation results align with this expectation. The helical gear produces a significant and steady axial force (approximately 2.2 kN), which is an inevitable result of the helix angle effect during torque transmission. For the double-helical gear, its axial force fluctuates within a very narrow range around zero (−79.5 to 82.1 N, with a mean close to zero). This result directly verifies the core advantage of the double-helical design: the axial forces generated by the left- and right-hand helical flanks achieve self-balance, thereby eliminating the net axial load at the system level and significantly alleviating axial bearing pressure.

Figure 10.

Meshing forces for different gear types.

The fluctuation characteristics of meshing force directly determine the level of system vibration response. Therefore, after clarifying the excitation characteristics, the mesh stiffness and vibration acceleration responses of different gear types are compared below.

Mesh stiffness, as a core internal excitation in gear systems, directly determines dynamic load and vibration levels through its fluctuation characteristics. Figure 11 compares the time-varying mesh stiffness curves for different gear types. The results show that the mesh stiffness of the spur gear fluctuates drastically within the range of 1.81–3.63 × 10⁹ N/m, with an average value of approximately 2.80 × 10⁹ N/m. This wide variation contributes significantly to system vibration. In contrast, both helical and double-helical gears demonstrate superior meshing smoothness. The mesh stiffness of the helical gear remains stable within a very narrow range (5.06–5.15 × 10⁹ N/m), with a nearly constant smooth curve. This is attributed to its high contact ratio, which ensures continuous multi-tooth contact and suppresses dynamic fluctuations at the excitation source. The mesh stiffness of the double-helical gear lies between the two (2.40–2.69 × 10⁹ N/m, with an average of about 2.53 × 10⁹ N/m), exhibiting periodic fluctuations and an overall stiffness level lower than that of the helical gear. This result indicates that the double-helical structure successfully achieves self-balancing of axial forces while inheriting the smooth transmission characteristics, thereby balancing dynamic performance and reliability.

Figure 11.

Time-varying mesh stiffness of different gear types.

The root mean square (RMS) value of vibration acceleration serves as an effective metric for assessing system vibration levels. A comparison of the vibration acceleration RMS values for the three gear types under rated operating conditions is shown in Figure 12. The results indicate that the double-helical configuration demonstrates a clear advantage in vibration suppression. At the measurement point on the sun gear shaft, the vibration acceleration RMS value of the double-helical gear is reduced by 87.7% compared to the spur gear (188.1 m·s^-2), dropping to 23.2 m·s^-2. At the measurement point on the planet carrier, a reduction of 85.0% is observed, reaching 10.4 m·s^-2. In contrast, although the helical configuration also shows improvement, the reduction at the sun gear shaft is only 28.2%, whereas at the planet carrier, it remains comparable to that of the spur gear. In the frequency domain, the meshing frequency f_m and its harmonics remain the dominant excitation frequencies for vibration acceleration at all measurement points. Similarly, the amplitudes of the meshing frequency and its harmonics for the double-helical gear are significantly lower than those for the spur and helical gears. Therefore, the double-helical configuration is advantageous in reducing vibration levels at key measurement points, demonstrating its significant engineering application value in transmission systems requiring high operational smoothness.

Figure 12.

Time/frequencydomain diagrams of shaft system vibration acceleration for different gear types.

In summary, the double-helical configuration exhibits the lowest vibration acceleration among the three gear types, which is directly attributed to its smoother meshing force and self-balanced axial force. This establishes a clear excitation-response relationship.

3.2. Comparison of load-sharing and bearing force characteristics for different gear types

Beyond vibration suppression, another critical performance indicator for planetary gear trains is the load-sharing characteristic among planet gears, which directly affects system reliability and fatigue life. This section analyzes the load-sharing coefficient and bearing force distribution for different gear types.

Load-sharing performance is a key indicator for evaluating the reliability and load-carrying efficiency of planetary gear transmissions. By analyzing the distribution of meshing forces among the planet gears, the time-domain variation of the load-sharing coefficient K_γ was calculated (Figure 13). The spur gear exhibits severe fluctuations in its load-sharing coefficient (1.01–1.80), which is directly related to the intense impact of its meshing force and represents a major limitation on its load-carrying capacity and reliability. The helical gear shows improved load-sharing performance, with a reduced fluctuation range (1.01–1.18), indicating that the higher contact ratio helps suppress dynamic variations in load distribution. The double-helical gear demonstrates outstanding and stable load-sharing performance. Its load-sharing coefficient not only approaches the ideal value of 1 most closely but also exhibits minimal time-domain fluctuation (stably maintained between 1.01 and 1.03). This confirms that the double-helical structure effectively optimizes torque distribution among multiple planet gears, thereby achieving near-ideal static and dynamic load sharing. This constitutes a prominent advantage, as it enhances overall system reliability and fully exploits the load-carrying potential of multiple planet gears operating in parallel.

Figure 13.

Real-time load-sharing coefficient for different gear types.

The load-sharing performance is further reflected in the bearing force distribution across the shaft system and planet gear bearings, as shown in Figures 14 and 15. In the helical gear configuration, owing to the superposition of axial forces from the planet gears on the sun gear, the sun gear shaft bearings experience significant axial loading. This substantial axial force poses a severe challenge to the selection, service life, and system reliability of the sun gear shaft. Therefore, the double-helical configuration, which features self-balanced axial forces, offers a significant advantage in this regard.

Figure 14.

Bearing support forces for different gear types in the shaft system.

Figure 15.

Comparison of planet gear bearing support forces for different gear types.

In summary, the comparison of three gear types reveals a consistent trend: the double-helical configuration not only achieves the lowest vibration response but also exhibits the most stable meshing force and the best load-sharing performance. This establishes the double-helical gear as the optimal choice for high-performance planetary gear trains, providing a solid foundation for the staggered-phase tuning study.

4. Optimal value of the staggered phase for vibration reduction in double-helical planetary gear trains

The superior performance of the double-helical planetary gear train in balancing transmission smoothness, load sharing, and axial forces was clearly established in the previous section. However, the full vibration-reduction potential of this configuration has not yet been exploited. The left- and right-hand helical flanks are not required to mesh simultaneously, and a key relative positioning parameter, the staggered phase, exists between them. Although the importance of this parameter has been recognized in existing studies, most analyses have been limited to qualitative discussions or single operating conditions. Systematic answers to several critical engineering questions remain unclear. These include how the staggered phase quantitatively affects complex multi-source coupled vibrations in planetary gear trains, whether the optimal value of the staggered phase varies with operating conditions (speed and load), and how it influences load-sharing characteristics while reducing vibration. The tuning mechanism is particularly complex in planetary gear trains because the meshing phase relationships of multiple planets are intertwined with the staggered phase. This complexity is greater than that in fixed-axis gear pairs. Therefore, based on the developed dynamic model, an in-depth study on staggered phase tuning is conducted in this chapter. First, a systematic investigation is conducted under rated operating conditions. The effects of the staggered phase on system vibration, mesh forces, and load-sharing characteristics are analyzed. The optimal values of the staggered phase for different objectives, such as vibration reduction and load equalization, are identified. Potential trade-offs between these objectives are revealed. Subsequently, the evolution of tuning behavior is examined under varying input speeds and load torques.

4.1. Analysis of staggered phase tuning mechanism and optimal phase

In traditional herringbone gear meshing, a meshing phase difference does not exist. To introduce a phase difference in a single gear pair, the staggered phase-tuning method²⁶ can be applied: for a herringbone gear with a tooth width of b/2, the two corresponding helical gear halves are offset circumferentially by a specific angle, referred to as the staggered phase angle (Figure 16). This method effectively introduces a meshing phase difference while ensuring that the overall transmission ratio remains unchanged and that the meshing of each layer complies with gear meshing principles. The complementary effect on meshing stiffness achieved through the staggered phase angle can significantly improve gear transmission performance. Gears designed using this method are defined as staggered phase-tuned gears.

Figure 16.

Schematic diagram of staggered phase tuning structure.

For a single double-helical gear, it is assumed that one helical flank remains fixed, whereas the other is rotated by a certain angle in the circumferential direction. When the rotation corresponds to a standard tooth-space width of πm_t/2 (corresponding to an angular pitch of π/z), the tooth space of one flank aligns with the tooth of the opposing flank. Consequently, the resulting mesh-stiffness function curve is shifted by half a meshing period (T_m/2) relative to the standard mesh-stiffness curve. When the rotation corresponds to a standard tooth pitch of πm_t (corresponding to an angular pitch of 2π/z), the resulting mesh-stiffness function curve is shifted by one full meshing period (T_m) relative to the standard curve. Owing to the periodicity of the mesh-stiffness function, the two curves coincide completely.

The relationship between the staggered phase p and the staggered phase angle φ is defined as follows:

p = \frac{φ}{2 π / z}

(4)

In the equation, z represents the number of teeth. Owing to the periodicity of gear meshing, the staggered phase angle cannot exceed one standard tooth pitch angle (2π/z). For a standard herringbone gear, both the staggered phase angle and the staggered phase are zero.

The vibration responses of key components in the double-helical planetary gear train under different values of the staggered phases were studied. The relationship between the root mean square (RMS) value of vibration acceleration and the staggered phase for the central components—the sun gear shaft and the planet gears—is shown in Figure 17. Evidently, the RMS curves of vibration acceleration for the two components are essentially symmetric about p = 0.5. Therefore, the range from p = 0 to p = 0.5 is considered in the following discussion. The vibration acceleration of the sun gear shaft exhibits the largest variation (approximately 26 m·s^-2), whereas that of the planet gears shows a relatively smaller variation (approximately 14 m·s^-2). As the staggered phase increases, the vibration acceleration of the sun gear shaft reaches its minimum at p = 0.5. For the planet gears, the vibration acceleration first decreases and then increases with increasing staggered phase, with the minimum value obtained at p = 0.43.

Figure 17.

RMS values of vibration acceleration for different staggered phases.

The staggered-phase tuned gear pair can be considered as the superposition of two single-side gear pairs with different initial meshing phases. Let the initial meshing phases of the two gear pairs be p⁽¹⁾ and p⁽²⁾,respectively, with p⁽¹⁾ ≠ p⁽²⁾。Setting p⁽¹⁾ = 0 and p⁽²⁾=p, the comprehensive mesh stiffness of the staggered-phase tuned gear pair can be expressed as:

k_{d} (t) = \bar{k} + \frac{1}{2} [Δ k^{p^{(1)}} (t) + Δ k^{p^{(2)}} (t)]

(5)

where

\bar{k}

is the average stiffness of a single-side meshing pair, and Δk^p ⁽¹⁾(t) and Δk^p ⁽²⁾(t) are the time-varying stiffness variations of the two meshing pairs, respectively.

For a helical gear pair, the mesh stiffness curve can be approximated as a trapezoidal wave. Within one meshing cycle, the time-varying mesh stiffness can be expressed in piecewise form as:

k (t) = {\begin{cases} \frac{k_{\max} - k_{\min}}{α_{1} T} [t - (n - p) T] + k_{\min}, & n T < t < (n + α_{1} - p) T \\ k_{\max}, & (n + α_{1} - p) T < t < (n + α_{2} - p) T \\ \frac{k_{\min} - k_{\max}}{(α_{3} - α_{2}) T} [t - (n + α_{2} - p) T] + k_{\max}, & (n + α_{2} - p) T < t < (n + α_{3} - p) T \\ k_{\min}, & (n + α_{3} - p) T < t < (n + 1 - p) T \end{cases}

(6)

Where α₁,α₂ and α₃ are coefficients related to the contact ratio, defined as:

{\begin{cases} α_{1} = \min (ε_{α}^{'}, 1 - ε_{α}^{'}, ε_{β}^{'}, 1 - ε_{β}^{'}) \\ α_{3} = {\begin{cases} 1 - ε_{γ}^{'} & if ε_{β}^{'} > 1 - ε_{α}^{'} \\ ε_{γ}^{'} & else \end{cases} \\ α_{2} = α_{3} - α_{1} \end{cases}

(7)

Where ε_α, ε_β and ε_γ are the transverse contact ratio, axial contact ratio, and total contact ratio of the helical gear pair, respectively; ε′_α=mod(ε_α,1),ε′_β=mod(ε_β,1) and ε′_γ=mod(ε_γ,1) are their respective fractional parts.

After expressing the mesh stiffness as the sum of its average value and the variation, the time-varying stiffness variation can be expanded into a Fourier series:

Δ k (t) = \sum_{n = 1}^{\infty} (a_{n} \cos \frac{2 n π t}{T} + b_{n} \sin \frac{2 n π t}{T})

(8)

For a helical gear pair, the Fourier coefficients a_n and b_n are expressed as:

a_{n} = \frac{k_{\max} - k_{\min}}{2 {(n π)}^{2}} [\frac{1}{α_{1}} \cos (2 n π (α_{1} - p)) + \frac{1}{α_{3} - α_{2}} \cos (2 n π (α_{2} - p)) - \frac{1}{α_{3} - α_{2}} \cos (2 n π (α_{3} - p)) - \frac{1}{α_{1}} \cos (2 n π p)]

(9)

b_{n} = \frac{k_{\max} - k_{\min}}{2 {(n π)}^{2}} [\frac{1}{α_{1}} \sin (2 n π (α_{1} - p)) + \frac{1}{α_{3} - α_{2}} \sin (2 n π (α_{2} - p)) - \frac{1}{α_{3} - α_{2}} \sin (2 n π (α_{3} - p)) + \frac{1}{α_{1}} \sin (2 n π p)]

(10)

For a staggered-phase tuned gear, after superimposing the stiffness variations of the two meshing pairs, the Fourier coefficients of the n-th harmonic component are given by:

a_{n}^{d} = a_{n}^{p^{(1)}} + a_{n}^{p^{(2)}}, b_{n}^{d} = b_{n}^{p^{(1)}} + b_{n}^{p^{(2)}}

(11)

Setting p⁽¹⁾ = 0 and p⁽²⁾=p, the following expressions can be derived:

a_{n}^{d} = - Q_{x} {\frac{\sin [n π (α_{1} - p)] \sin (n π α_{1})}{α_{1}} + \frac{\sin [n π (α_{2} - α_{3})] \sin [n π (α_{2} + α_{3} - p)]}{α_{3} - α_{2}}}

(12)

b_{n}^{d} = Q_{x} {\frac{\cos [n π (α_{1} - p)] \sin (n π α_{1})}{α_{1}} + \frac{\sin [n π (α_{2} - α_{3})] \cos [n π (α_{2} + α_{3} - p)]}{α_{3} - α_{2}}}

(13)

Where,

Q_{x} = \frac{2 (k_{\max} - k_{\min}) \cos (n π p)}{{(n π)}^{2}}

(14)

The amplitude of the n-th harmonic component of the staggered-phase tuned gear is:

J = \sqrt{{(a_{n}^{d})}^{2} + {(b_{n}^{d})}^{2}} = | Q_{x} | \sqrt{\frac{\sin^{2} (n π α_{1})}{α_{1}^{2}} + \frac{\sin^{2} [n π (α_{2} - α_{3})]}{{(α_{3} - α_{2})}^{2}} + 2 \frac{\sin (n π α_{1}) \sin [n π (α_{2} - α_{3})]}{α_{1} (α_{3} - α_{2})} \cos [n π (α_{1} - α_{2} - α_{3})]}

(15)

From the above expressions, it can be observed that the amplitude J of the time-varying mesh stiffness variation of the staggered-phase tuned gear is related to the staggered phase p and the coefficients α₁,α₂,α₃. That is, the mesh stiffness amplitude of the staggered-phase tuned gear depends not only on the staggered phase but also on the contact ratio of the original gear.

Research findings on vibration reduction in herringbone planetary gear trains using staggered phase-tuning theory indicate that the optimal staggered phase for a planetary gear train is inherently related to the mesh stiffness curve coefficients of the internal and external gear pairs.²⁵ The contact ratio correlation coefficients for the internal and external gear pairs were calculated separately.

1. For the external gear pair, the total contact ratio is ε_γ = ε_α + ε_β = 1.4325 + 0.8650 = 2.2975. The corresponding mesh stiffness curve coefficients are α₁ = 1.1350, α₂ = 0.5675, and α₃ = 1.7025. The inflection point is determined as 1 − α₂ = 0.4325 ≈ 0.43.

2. For the internal gear pair, the total contact ratio is ε_γ = ε_α + ε_β = 1.4985 + 0.8650 = 2.3635. The mesh stiffness curve coefficients are α₁ = 1.1350, α₂ = 0.5015, and α₃ = 1.6365. The inflection point is determined as 1− α₂ = 0.4985 ≈ 0.50.

Thus, a preliminary inference is made that the vibration acceleration of the planetary gear train is minimized when the staggered phase p equals the smaller of the inflection points of the mesh stiffness curves for the internal and external gear pairs, that is, p = min(p_w, p_n). This corresponds to the optimal value of the staggered phase. Concurrently, the vibration response of the central component, the sun gear, is minimized at p = 0.5.

For the three main staggered phase parameters p = 0, p = 0.43, p = 0.5, the shaft center orbits of the central components—the sun gear shaft and the planet carrier—in the double-helical planetary gear transmission system were studied (Figures 18 and 19). The fluctuation of the sun gear shaft’s center orbit is relatively smaller at a staggered phase of p = 0.5. By contrast, for the planet carrier, the fluctuation of the center orbit is minimized at p = 0.43. This indicates that the staggered-phase method is effective in altering both the acceleration and vibration displacement of the planet carrier.

Figure 18.

Sun gear shaft center orbits under different staggered phases.

Figure 19.

Planet carrier center orbits under different staggered phases.

The influence of different staggered phase values on the dynamic meshing force and the load-sharing coefficients was investigated. The results are shown in Figures 20 and 21. Under rated operating conditions, a larger staggered phase corresponds to smaller fluctuations in the dynamic meshing force and a lower load-sharing coefficient. This trend is consistent with the variation in the vibration response of the sun gear shaft.

Figure 20.

Dynamic meshing force under different staggered phases.

Figure 21.

Load-sharing coefficient under different staggered phases.

A comparison of bearing forces for various components under different staggered phases is shown in Figure 22. Consistent with the earlier findings, the radial bearing force on the sun gear shaft reaches its minimum when the staggered phase is p = 0.5. For planet carrier bearing 2, the radial bearing force is minimized at p = 0.43. In contrast, the axial bearing forces on all components show no significant variation with changes in the staggered phase.

Figure 22.

RMS of bearing forces under different staggered phases.

4.2. Behavior of staggered phase tuning under different operating conditions

For several critical staggered phases of a double-helical planetary gear transmission system, the root mean square (RMS) values of vibration acceleration for various components were studied under different input speeds. The results are shown in Figure 23. At higher rotational speeds (1300–2000 r·min^-1), the vibration-reduction effect of the staggered phase method is pronounced, and the vibration level is consistently lowest at p = 0.43.

Figure 23.

Effect of rotational speed on vibration acceleration RMS under different staggered phases.

The effect of input speed on the RMS of the dynamic meshing force was investigated under different staggered phase conditions. The results are presented in Figure 24. A comparison of the curves for different staggered phases reveals that within the range close to the rated speed (1300–2000 r·min^-1), the dynamic meshing force RMS is also minimized at p = 0.43. This further confirms the effectiveness of this phase in suppressing system vibration under high-speed operating conditions.

Figure 24.

Effect of rotational speed on dynamic meshing force RMS under different staggered phases.

Figure 25 illustrates the effect of the staggered phase on the vibration response under different load torques. The regulatory effect of phase tuning on system vibration exhibits selectivity with varying loads: for the sun gear shaft, the vibration acceleration RMS remains at a relatively low level when p = 0.5, whereas for the planet gears, a lower vibration response is observed at p = 0.43. This phenomenon indicates that the vibration-reduction effect of the staggered phase differs among system components, allowing for targeted selection based on specific vibration-suppression objectives.

Figure 25.

Effect of load torque on vibration acceleration RMS under different staggered phases.

5. Coupling analysis between helix angle and staggered phase

Previous research has revealed the regulatory role of staggered phase tuning in the dynamic performance of double-helical planetary gear trains. However, the design of gear systems is a multi-parameter co-optimization process.^36,37 As another fundamental characteristic parameter of double-helical gears, the helix angle not only determines the axial dimensions, contact ratio, and load-carrying capacity of the gear, but also fundamentally influences the excitation characteristics of mesh stiffness and the balance of axial forces. A profound coupling relationship may exist between the helix angle and the staggered phase. Different helix angles can fundamentally alter the dynamic characteristics of the system, thereby causing a shift in the corresponding optimal staggered phase. Neglecting this coupling relationship may limit the applicability of conclusions regarding the optimal staggered phase derived from a single helix angle. Currently, in studies on double-helical planetary gear trains, the helix angle and staggered phase are often treated as isolated parameters, and an in-depth investigation of their synergistic influence mechanism at the dynamic level remains lacking. Therefore, building upon earlier work, this section introduces the helix angle as a key variable to conduct a coupling analysis of the helix angle and staggered phase.

5.1. Coupled effect of helix angle and staggered phase on system vibration

To investigate the coupling effect of the helix angle, a comparative analysis was conducted under rated operating conditions using different helix angles and staggered phases. The shaft center orbits of the sun gear shaft and planet carrier are shown in Figures 26 and 27. Evidently, for any given staggered phase, when the helix angle is increased from 0° to 15°, the range of the shaft center orbits for both the sun gear shaft and planet carrier converges significantly. This demonstrates that increasing the helix angle effectively enhances the meshing contact ratio and transmission smoothness of the system, reduces dynamic excitation, and thereby improves the operational stability of the rotors. Under the same helix angle (e.g., β = 15°), the staggered phase exhibits a clear modulating effect on the shaft center orbit. For the planet carrier, the orbit is most compact and exhibits the smallest fluctuation at p = 0.43, confirming the effectiveness of this phase in suppressing the vibration of the supporting structure.

Figure 26.

Sun gear shaft center orbit under different helix angles/staggered phases.

Figure 27.

Planet carrier center orbit under different helix angles/staggered phases.

The variation in the root mean square (RMS) of vibration acceleration at key measurement points with respect to the helix angle and staggered phase is shown in Figure 28. For any given staggered phase, the RMS values of vibration acceleration for both the sun gear shaft and planet gears decrease monotonically as the helix angle β increases, indicating that increasing the helix angle reduces the overall vibration level of the system. Under the same helix angle, the RMS values of vibration acceleration at all measurement points corresponding to the staggered phase p = 0.43 are lower than those at the other two phases. This demonstrates that this phase value exhibits consistent superiority for vibration suppression across the helix angle range of 0° to 15°.

Figure 28.

Variation of vibration acceleration RMS with helix angle under different staggered phases.

5.2. Coupled effect of helix angle and staggered phase on system load-sharing characteristic

The coupling relationship between the load-sharing coefficient K_γ and variations in helix angle and staggered phase is shown in Figure 29. The results indicate that when β = 0°, load-sharing performance is highly sensitive to the staggered phase. Without tuning (p = 0), the load-sharing coefficient is as high as 2.2, indicating severely uneven load distribution. As the phase increases, load-sharing performance improves rapidly, reaching approximately 1.3 at p = 0.5. This demonstrates that, for a spur gear system, staggered phase tuning effectively improves load sharing. When β = 8°, introducing a helix angle enhances overall load-sharing performance. When β = 15°, the load-sharing coefficients under all three phases are reduced to below 1.03 and further approach 1 as the phase increases. This indicates that a sufficiently large helix angle ensures outstanding load-sharing performance, at which point the role of staggered phase tuning shifts from correcting imbalances to fine-tuning.

Figure 29.

Variation of load-sharing coefficient with staggered phase under different helix angles.

6. Experimental study on vibration reduction via staggered-phase tuning

To verify the feasibility of vibration reduction using staggered-phase tuned gears, an experiment was conducted on the staggered-phase tuning of an external helical gear pair. Three groups of staggered-phase angles (p = 0, 0.2, 0.4) were selected as the experimental research contents. A helical gear pair with 36 teeth, a module of 4 mm, a pressure angle of 20deg, and a helix angle of 10deg was adopted to test the vibration response of the staggered-phase tuned gear transmission.³⁸ Meanwhile, a dynamic model consistent with the experimental setup was established using the same flexible multi-body dynamics modeling method for simulation calculations, and the computed vibration responses were compared with the experimental results.

The test rig and the locations of the measurement points are shown in Figure 30. The motor torque ranges from 0 to 140 N·m, with a power of 11 kW. The rotational speed of the input-end motor ranges from 0 to 8000 rpm, and that of the loading-end motor ranges from 0 to 2500 rpm. An NI 9234 signal acquisition card was used, with a sensitivity of 8 mV/μm and a measurement range of 2 mm. The arrangement of the measurement points for vibration displacement is shown at points a, b, c, and d in Figure 30. The vibration acceleration sensors are capable of measuring vibration responses in three directions; therefore, only one sensor was arranged at the input end and one at the output end, respectively. The arrangement of the measurement points for vibration acceleration is shown at points e and f in Figure 30.

Figure 30.

Experimental bench and measurement points.

The vibration acceleration and vibration displacement at the input end of the parallel-axis helical gear pair under different load torques were tested for staggered-phase angles of p = 0, p = 0.2, and p = 0.4, respectively. Figures 31 and 32 show the comparison between the experimental and simulation results of the RMS values of vibration acceleration and vibration displacement at the input end under different load torques, respectively. It can be observed from the figures that, for the helical gear pair, regardless of the staggered-phase angle, both the vibration acceleration and vibration displacement at the input end increase with increasing load torque. In both the experimental and simulation results, the vibration acceleration and vibration displacement at staggered-phase angles of p = 0.2 and p = 0.4 are lower than those at p = 0, which verifies that the theoretical model of vibration reduction via staggered-phase tuning and the experiment are in good agreement regarding the vibration reduction trend. The results at p = 0.4 are lower than those at p = 0.2, which can be attributed to the fact that the optimal staggered-phase angle for the helical gear pair is p = 0.35, and p = 0.4 is closer to this optimal value.

Figure 31.

The influence of torque on acceration in x-direction of input.

Figure 32.

The influence of torque on displacement in x-direction of input.

7. Conclusion

Using the established multi-flexible-body coupled dynamic model of the helicopter main reducer planetary gear transmission system, a systematic study was conducted on the effects of different gear types, staggered amounts, and the coupling relationship between the helix angle and staggered phase. The following conclusions were drawn:

(1) Through the mechanisms of self-balanced axial forces and frequency-domain cancellation of impact energy, the double-helical configuration exhibits excellent overall dynamic performance. Compared with spur and helical gears, the double-helical gear significantly reduces the RMS value of system vibration acceleration while maintaining superior load-sharing characteristics (K_γ ≈ 1.01–1.03).

(2) The staggered phase is a key design variable for performance tuning. Its optimal value is determined by the mesh stiffness characteristics of the system, and varies with the objective (vibration reduction or load sharing). Sun gear vibration is minimized at p = 0.5, whereas planet gear vibration and system load-sharing performance are optimized at p = 0.43. These values closely correspond to the theoretical inflection points of the contact ratios of the internal and external gear pairs.

(3) A significant synergistic coupling effect exists between the helix angle and staggered phase, necessitating integrated design. Increasing the helix angle generally improves system smoothness and load-sharing performance, while the staggered phase modulates vibration suppression under a given helix angle. Parameter coupling analysis shows that neglecting the interaction between these two factors may render conclusions based on a single parameter ineffective in practical applications.

This study demonstrated a method for vibration reduction in double-helical planetary gear trains via staggered phase tuning, overcoming the limitations of traditional phase-tuning approaches and providing a systematic dynamic design framework for such systems. The findings provide a theoretical basis for enhancing the dynamic performance of high-reliability helicopter transmission systems.

Footnotes

ORCID iD

Aiqiang Zhang

Author contributions

Conceptualization: X.H.W. and A.Q.Z.; Methodology: A.Q.Z. and X.H.W.; Software: P.S.; Validation: A.Q.Z. and P.S.; Formal analysis: A.Q.Z. and P.S.; Investigation: P.S. and X.T.; Resources: P.S.; Data curation: P.S.; Writing—original draft preparation: P.S.and Y.W.; Writing—review and editing: P.S., X.H.W. and A.Q.Z.; Visualization: P.S.; Supervision: A.Q.Z. All authors have read and agreed to the published version of the manuscript.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by the Hubei Provincial Natural Science Foundation of China (2024AFB469), the Open Fund of State Key Laboratory of Mechanical Transmission for Advanced Equipment (Grant Nos. SKLMT-MSKFKT-202503 and SKLMT-MSKFKT-202333), the Doctoral Scientific Research Foundation of Hubei University of Automotive Technology (BK202333), and the Excellent Young and Middle-aged Science and Technology Innovation Team Project of Hubei Provincial Colleges and Universities (T2022027).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.*

Appendix

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