During helicopter operations, continuous pilot control adjustments induce variations in aerodynamic parameters, which are subsequently transmitted to the transmission system through changes in engine output and main/tail rotor loads, thereby influencing the dynamic behavior of transmission system. To elucidate the influence patterns of pilot control parameters on helicopter transmission system dynamics, this study establishes a comprehensive rotor–engine–transmission coupled dynamic model. The helicopter flight dynamic model is developed based on a linear quasi-steady blade element aerodynamic model and uniform induced velocity theory. The engine thermodynamic model is formulated using the component-based method, while the transmission system is modeled via the lumped-parameter approach, incorporating internal and external excitation sources. A fully coupled dynamic framework is established and solved numerically using the Newton–Raphson algorithm, the finite difference method, and the Runge–Kutta integration scheme. Furthermore, parametric studies are conducted to systematically evaluate the effects of key control inputs—including main rotor and tail rotor collective pitches, as well as lateral and longitudinal cyclic pitches—on the dynamic characteristics of the transmission system. The results reveal that these control parameters significantly modulate the vibration amplitudes and dynamic load distribution, although they do not alter the quasi-periodic nature of the torsional meshing vibration. This work provides a robust theoretical foundation for the design and performance optimization of helicopter transmission systems under complex flight conditions.

Elliptical gears are widely used in the fields of aerospace and automation equipment due to their excellent variable speed characteristics and compact structure. To reveal the control mechanism of phase angle on the motion characteristics of two-stage elliptical gear transmission system (TSEGTS) and improve its stability under variable speed loading, this paper proposes a modeling method that integrates two-dimensional tooth profile envelope method and image processing technology to extract tooth profile and analyze system motion characteristics. By introducing the fluctuation coefficient as a stability indicator, the system studied the effects of phase angle, input speed, and load torque on angular velocity, angular acceleration, and torque. The results indicate that the phase angle has a significant periodic modulation effect on the motion characteristics of the system. The angular velocity and acceleration of intermediate shaft B and output shaft C, as well as the torque of output shaft C, exhibit a distribution with 90 degrees as the axis of symmetry within the range of 0 degrees-180 degrees. The average torque fluctuation coefficients of input shaft A and intermediate shaft B in the range of 0 degrees-90 degrees are 4.02 and 3.03, respectively, which are significantly higher than the 3.46 and 1.99 coefficients in the range of 90 degrees-180 degrees. As the input speed increases from 5 pi to 20 pi rad/s, the torque fluctuation coefficients of shafts A, B, and C increase from 2.87, 1.37, and 0.53 to 22.54, 14.19, and 8.43, respectively. When the load torque increases from 2 to 8 Nm, the fluctuation coefficients of the three axes decrease from 11.33, 7.12, and 4.22 to 3.51, 1.91, and 1.05, respectively. The increase in load torque effectively suppresses fluctuations and improves meshing stability. The experimental and simulation results are consistent, verifying the effectiveness of the model. The research results can provide theoretical basis for parameter optimization and dynamic performance control of TSEGTS.

The maneuvering flight actions of helicopters introduce additional excitation forces and stiffness, which subsequently affect the vibration characteristics of the power turbine rotor system. To analyze the dynamic behavior of the rotor system under various maneuvering flight conditions, a coupled bending-torsion finite element model was developed by integrating the Lagrange principle with the finite element method. This study investigated the dynamic characteristics associated with bending-torsion coupling of the rotor system during typical flight scenarios, including maneuvering turns, pitch, and synthetic actions. The results indicate that, although the bending-torsion coupling has a minimal impact on vibrations in the bending direction, it generates a torsional vibration frequency that is double the rotor’s rotational frequency in the torsional direction. Various maneuvering actions lead to static offsets in the bending direction, while the torsional direction exhibits vibrations at the same frequency as the rotor’s rotational frequency. Additionally, new subharmonic vibration frequency components are produced in both the bending and torsional directions.

Gas-bearing-supported gear–rotor systems are widely used in precision measurement equipment, where vibration performance is critically influenced by the bearing support characteristics. In this study, a nonlinear dynamic model of a gear–rotor system supported by gas bearings is established, in which the bending–torsional coupling effect and the bearing support stiffness are explicitly taken into account. Based on isothermal compressible gas lubrication theory, the Reynolds equation is solved using a finite difference scheme to obtain the gas film pressure distribution of the gas bearing. The influences of rotational speed from 500 to 1500 r·min−1 and gas film clearance from 7 to 11 μm on the gas film pressure characteristics and the equivalent support stiffness are investigated. The equivalent stiffness is subsequently incorporated into the dynamic equations of the gear–rotor system to analyze its vibration responses. The results show that increasing rotational speed significantly enhances the gas film pressure and enlarges the high-pressure region along the circumferential direction due to strengthened hydrodynamic effects. In contrast, an increase in gas film clearance leads to a pronounced reduction in pressure magnitude and load-carrying capacity, resulting in a nonlinear decrease in the equivalent support stiffness. Dynamic analysis indicates that rotational speed plays a dominant role in the vibration behavior of the gas-bearing-supported gear–rotor system, while variations in gas bearing stiffness substantially affect the vibration response characteristics.

The N-type double disc coupling has the characteristics of simplified structure and no need for lubrication. It is the main connecting component between the aircraft engine and the main reducer, mainly used to transmit torque and motion, and must withstand high speed, complex load conditions and comprehensive deviations between the transmission shaft systems. Under complex working conditions, the coupling can not only compensate the misalignment between the transmission shafting, but also produce various deformations, which are closely related to the stiffness of the coupling. Taking the N-type double disc coupling as the research object, strength calculation and analysis of the influencing factors of stiffness were carried out, and the stress distribution law of the profile under complex load conditions and the influence law of size parameters on stiffness were obtained. The effectiveness of the computational model was verified through experiments, providing theoretical reference for the design and optimization of double disc couplings.

To systematically clarify the migration and evolution mechanisms of metallic particles in the solid-liquid-gas three-phase flow field of a helicopter intermediate reducer under splash lubrication, this study innovatively develops a three-phase numerical model based on the CFD-DPM framework, and establishes a high-precision oil-churning visualization test rig for experimental validation via high-speed imaging technology. Under the baseline operating condition (gear rotational speed of 1500 r/min, dimensionless oil immersion depth lambda(h) = 6.78), the effects of multiple hydrodynamic forces (gravity, pressure gradient force, virtual mass force, Magnus lift, and Saffman lift) on particle trajectories and dynamic responses are comprehensively evaluated. Meanwhile, comparative analyses are conducted to quantify the regulatory effects of particle density (intrinsic property variation) and initial release location (meshing region, spallation region, sidewall/bottom) on migration behavior. Numerical results reveal that the pressure gradient force and virtual mass force dominate macroscopic particle transport: the former prolongs the quasi-steady stage by 40%-50%, while the latter accelerates particle velocity decay to near-zero within 0.3-0.5 s. In contrast, the Saffman lift and Magnus lift play secondary roles, mainly inducing high-frequency velocity fluctuations (velocity perturbation amplitude <= 15% for Saffman lift) and exhibiting strong medium dependence (Magnus lift doubles particle migration distance in the gas phase but is negligible in the oil phase). Increasing particle inertia significantly weakens flow-following capability, and the particle velocity response intensity follows the order: meshing region > spallation region > sidewall/bottom. Visualization experiments verify a complete particle transport cycle inside the gearbox (extrusion from the meshing region-airborne flight-wall collision-gravity-driven return), which is in good agreement with numerical predictions. Further experimental results show that increasing the rotational speed from 100 to 400 r/min significantly enhances oil splash height and particle adhesion on the top and side walls. At a high immersion depth (lambda(h) = 11.86), intense turbulent splashing leads to global dispersion of the oil-particle mixture; at a low immersion depth (lambda(h) = 1.70), particle motion is dominated by local entrainment with weak wall retention. Additionally, as the particle volume fraction increases from 0.1%, 0.5%, 1.0%, the oil-particle mixture exhibits a pronounced thickening effect, suppressing oil splashing and enhancing particle deposition. These findings provide robust theoretical support and technical guidance for the optimization of splash lubrication design and reliability assessment of helicopter transmission systems.

This paper proposes a novel dynamic modeling method for spiral bevel gear (SBG) systems that incorporates rotating mesh excitation. In this approach, shell elements are employed to model the flexible web and hollow shaft, while a semi-analytical continuum model is developed for the equivalent rim. A rotating mesh unit is introduced to achieve dynamic coupling of the gear pair. The accuracy of the proposed model is validated through comparison with finite element (FE) analysis results. Based on this model, the vibration characteristics under traveling wave resonance (TWR), the excitation conditions of TWR, and the influence of different excitation modes on the dynamic response are systematically investigated. The results demonstrate that the proposed method accurately predicts and characterizes the TWR phenomenon, addressing the limitations of existing modeling approaches. Under TWR conditions, the dominant vibration frequencies are shown to be linear combinations of the meshing frequency and the rotational frequency. Furthermore, incorporating rotating mesh excitation is identified as the critical factor for capturing TWR characteristics; in contrast, conventional stationary mesh excitation fails to reflect the spatial migration of traveling waves along the circumference and cannot accurately predict the resonant speeds.

[Objective]Splines are important connectors in aviation power transmission systems,and are affected by manufacturing errors,assembly errors or working loads,and often occur in parallel misalignment,angular misalignment and comprehensive misalignment,which affects the meshing length and bearing capacity of each tooth of the spline.In order to improve the stability of the transmission system and reduce the failure rate of aviation splines,the dynamic meshing stiffness characteristics of misaligned splines was analyzed.[Methods]Firstly,the concept of spline meshing length was introduced and equated to the waist length of the equivalent trapezoidal cross-section for sliced single spline teeth,whereby the meshing stiffness of a single spline tooth was acquired.Secondly,a dynamic model for the aviation spline-rotor system was established to solve the time-varying meshing length of splines.Finally,the slicing method was combined with the Ishikawa method to obtain the meshing stiffness of single spline teeth under various misalignment conditions,and the dynamic responses of the system were solved accordingly.[Results]The results show that the change amplitude of spline single-tooth meshing stiffness and the dynamic response of the rotor system increase when the static parallel dislocation and static angular dislocation amount are increased.When the static parallel dislocation amount remains unchanged,and the static angular dislocation amount changes,the change of spline meshing stiffness and the dynamic response of the rotor system is not obvious.When the static angular dislocation amount remains unchanged,the static parallel dislocation amount increases,the spline meshing stiffness and the dynamic response of the rotor system change significantly.

This article investigates the multiphysics coupling mechanism among Dynamic Thermal Elastohydrodynamic Lubrication (DTEHL), Load Tooth Contact Analysis (LTCA), and system dynamics in a Closed Differential Herringbone Gear Transmission System (CDHGTS). Based on gear meshing theory, ring gear flexibility, and rough surface contact, a dynamic contact lubrication model is developed using thermoelastic fluid dynamic lubrication theory. A gear heat transfer model integrating heat transfer and flexible deformation is established to capture dynamic thermal variations at each ring gear node. By coupling the DTEHL model, heat transfer model, and LTCA model, a nonlinear dynamic model incorporating flexible-tribo coupling is constructed. The nonlinear dynamic response of the CDHGTS under DTEHL conditions is analyzed, focusing on the influence of ring gear flexibility on system vibration. Furthermore, nonlinear dynamic behavior induced by ring gear flexibility, friction coefficient, and damping coefficient is examined. The proposed model is validated through bench vibration experiments. The study refines dynamic modeling and analysis methods for herringbone gears, offering theoretical guidance for evaluating the impact of coupled engineering factors on gear dynamics.

In order to study the comprehensive lubrication effect of the oil injection lubrication system for a tilt-rotor helicopter’s nacelle under different inclination angles, a calculation model of the nacelle was established using the CFD method. Considering the effects of oil reflux and cooling, the nacelle lubrication was analyzed. Through heat flow coupling simulation, a series of heat flow field distributions within the nacelle during the tipping process were obtained. The results indicate that the tilt angle has a significant effect on the performance of the oil injection lubrication system.

In spiral bevel gear transmission systems, spiral bevel gears are prone to fatigue cracks under heavy and alternating loads, endangering operational safety. The occurrence of root cracks leads to variations in time-varying meshing stiffness, which in turn affects the dynamic response of the transmission system, resulting in changes in tooth surface loading and further influencing crack propagation. Therefore, investigating the influence of the dynamic behavior of spiral bevel gear transmission systems on crack propagation has great significance for fault prevention and control. This study addresses the two-way coupling between crack propagation and dynamic response in spiral bevel gear transmission systems. A dynamic model of the spiral bevel gear system and a fatigue crack propagation model were established, and a coupled analysis method for crack propagation and dynamic behavior was proposed to investigate the effect of this coupling on crack growth. A series of experimental tests under various operating conditions were conducted on a spiral bevel gear crossed-axis transmission test rig to validate the effectiveness of the dynamic model.

Non-circular planetary gear systems exhibit significant time-varying transmission characteristics, which can easily trigger complex nonlinear dynamic responses during operation, adversely affecting vibration stability. This paper develops a 36-degree-of-freedom bending-torsional coupling nonlinear dynamic model for 4-6 order non-circular planetary gear systems, incorporating multiple excitation factors, including time-varying mesh stiffness (TVMS), meshing damping, backlash, friction, and comprehensive transmission error. A TVMS calculation method applicable to both internal and external meshing pairs is proposed based on curvature radius theory. The system's nonlinear dynamic equations are solved numerically using the Runge-Kutta method. Nonlinear dynamic analyses, including bifurcation diagrams, time histories, phase diagrams, and Poincare sections, were employed to systematically investigate the influence of key parameters-rotational speed, eccentricity, damping, and backlash on the system response. Multiscale methods were further used to examine their effects on main resonance characteristics. Quantitative results show that the system transitions from periodic motion to 2-periodic and chaotic states as rotational speed increases, with peak vibration amplitudes rising from 1.12 m/s 2 to over 5.6 m/s 2 across tested speeds. Eccentricity critically affects dynamic behavior: under identical speed conditions, the external meshing pair enters a quasi-periodic state at an eccentricity of 0.22, earlier than the internal pair at 0.32. Increasing tooth flank clearance can trigger chaotic responses, whereas enhancing system damping reduces peak amplitudes by up to 32%45%, significantly improving operational stability at medium to high speeds. Finally, a prototype test platform was established to validate the model. Experimental results and theoretical predictions show good agreement, peak to peak comparison has good consistency, confirming that the model effectively captures the system's main dynamic behavior. These findings elucidate the unique nonlinear dynamics of non-circular planetary gear systems, providing a quantifiable theoretical basis for structural optimization and engineering applications.

To elucidate the energy dissipation mechanism of a Standard straight bevel gear transmission system under splash lubrication, this article proposes a churning power loss prediction model tailored to such systems. This model decomposes the total power loss into three parts, namely circumferential resistance, end face stirring, and extrusion loss in the meshing area, and combines the control volume method with the bevel gear slicing method to achieve an engineering approximate quasi analytical model of stirring power loss. A transparent visualization test rig for churning power loss was developed, and systematic experiments were conducted over a range of rotational speeds (1,000-4,000 r/min), the dimensionless immersion depth (defined as the ratio between oil level and gear pitch radius) (0.5-1.5), and lubricant dynamic viscosities (0.0085-0.21228 Pa & centerdot;s). The results show that, as the speed increases from 1,000 r/min to 4,000 r/min, the total churning power loss rises from 135 W to 5524 W; meanwhile, the proportion of end-face drag loss decreases from 51.6% to 38.7%, whereas the contribution of squeeze loss in the meshing region increases from 48.2% to 61.3%. At 2,000 r/min, when the dimensionless immersion depth increases from 0.5 to 1.5, the total power loss increases from 283 W to 1559 W, and the share of mesh-region squeeze loss rises from 44.5% to 58%, becoming the dominant dissipation component. Under high-speed conditions of 4,000 r/min and dimensionless immersion depth of 1.0, increasing the lubricant viscosity from 0.0085 to 0.21228 Pa & centerdot;s causes the total power loss to increase from 774 W to 8176 W, with the dominant loss mechanism gradually transitioning from viscous shear to hydrodynamic squeeze. The theoretical predictions and experimental data trends are roughly consistent, confirming the accuracy and applicability of the proposed model under a wide range of operating conditions, and providing a quantitative basis for energy optimization of high-performance bevel gear transmission systems.

Abstract To calculate the spur gear single-tooth mesh stiffness (STMS), this study proposes two simulation-based methods: the action-line strain (ALS) and Mises strain (MS) methods. A 3D gear model was built in SolidWorks with GearTrax and simulated dynamically in ANSYS Workbench. The normal mesh force and tooth surface elastic deformation were extracted to determine STMS curves, and the time-varying mesh stiffness (TVMS) was derived based on the contact ratio. Results from both proposed methods align well with the classical torque-torsional angle (TTA) method in trend and magnitude, validating their effectiveness for spur gear stiffness analysis.

To meet the high maneuverability and rapid response demands of modern helicopters, it is crucial to examine the dynamic characteristics of key subsystems as an integrated system. This study develops a coupled dynamic model of the rotor-engine-transmission system, taking into account the influence of the rotor and engine on the transmission system. The effects of forward flight speed and rotor rotational speed on the vibration and load-sharing characteristics of the transmission system are analyzed. Results show that with increasing forward flight speed, the torsional vibration amplitude value of all gear pairs first decreases and then increases, with vibration magnitude of all gear pairs exhibiting a similar trend with vibration amplitude value. Correspondingly, load-sharing performance initially improves and then deteriorates. In contrast, increasing rotor speed continuously amplifies both the torsional vibration amplitude value and vibration magnitude, while progressively worsening load-sharing performance. However, variations in forward flight speed and rotor speed do not alter the distribution of vibration frequency components or the dynamic behavior patterns of the gear pairs. This research offers a new approach and methodological framework for investigating main reducer dynamics at the full-aircraft level, contributing valuable insights to helicopter system design and performance analysis. (c) 2025 The Author(s). Published by Elsevier Ltd on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

During helicopter maneuvering flight, the main rotor and tail rotor systems produce complex alternating dynamic loads. Under the combined excitation of external dynamic loads and internal structural excitations, the main gearbox presents obvious nonlinear dynamic responses. To explore the influence of maneuver-induced loads on the dynamic characteristics of the main gearbox, this paper proposes a modeling framework for analyzing the dynamic responses of the transmission system under diverse maneuver aerodynamic loads. The framework is established by integrating a coupled rotor-engine model with a lumped-parameter model of the transmission system. Based on the proposed framework, the vibration characteristics, load sharing performance, and dynamic load evolution of the transmission system under barrel roll and loop maneuvers are analyzed, where maneuver loads, time-varying mesh stiffness, gear backlash, as well as manufacturing and assembly errors are fully considered. The results show that load fluctuations and abrupt load variations occurring during barrel roll and loop maneuvers induce the torsional vibration responses of gear pairs to transition between quasi-periodic and chaotic states. Furthermore, the split-torque gear pairs show satisfactory load-sharing performance in the entry and recovery phases but degraded performance in the inverted flight phase, whereas an opposite trend is observed for the power-combining stage gear pairs. Nevertheless, the maximum dynamic load peaks appear in the entry and recovery phases, which degrades the operational stability of the transmission system. This study possesses important engineering application value, and the research findings provide a theoretical basis for the dynamic performance optimization of helicopter transmission systems.

The tilt-rotor synchronous transmission shaft is crucial in tilt-rotor aircraft, connecting the engine and rotor. The shaft on the flexible wing endures foundation motion and random loads during flight, and different loads impact it variably. This paper models the shaft dynamically, accounting for foundation excitation’s effect on vibration response and analyzing stress and deformation under random loads in helicopter mode. Results indicate that foundation excitation only impacts the action direction, not other directions’ responses. In helicopter mode’s random vibration, the maximum equivalent stress is found on the pinion output shaft.

Study on the effects of backlash, friction, and rotational speed on the nonlinear bifurcation characteristics of Non-Orthogonal Misaligned Face Gear Power Split-Flow Transmission System (NOMFGPSFTS), aiming to provide theoretical basis and technical guidance for improving power to weight ratio, prolonging service life, and ensuring the reliability of the power transmission mechanism of innovative helicopters, ships and automobiles. An 11 degree of freedom (DOF), bending, torsion and axis coupling nonlinear dynamic model is established by using the lumped mass method. The model uses Load Tooth Contact Analysis(LTCA) method to calculate the meshing stiffness of Non-Orthogonal Misaligned Face Gear (NOMFG), and combines the key nonlinear elements, including gear tooth surface friction, backlash, support stiffness and transmission error, to facilitate an accurate representation of the system dynamics. The system's dynamic differential equations are solved using the RungeKutta algorithm, and the system's nonlinear characteristics are demonstrated through time domain diagrams, Fast Fourier Transform (FFT) spectrograms, phase plane diagrams, Poincare maps and Maximum Lyapunov Exponent diagrams. Bifurcation diagrams are used to further reveal the effects of backlash, rotational speed, and friction coefficient on the system's nonlinear behavior. The study finds that with the increase of dimensionless backlash, the system transits from periodic-1 motion to chaotic motion, and may evolve into periodic-2 motion, showing obvious nonlinear vibrations. With the increase of rotating speed, the system transits from periodic-2 motion to chaotic motion, and finally realizes periodic-1 motion. With the increase of friction coefficient, the chaotic region of the system decreases. Finally, the correctness of the theoretical model is verified by experiments, which provides a theoretical basis for the stability study of the system.

The time and computational challenges posed by nonlinear contact elements complicate the modal analysis of complex gear transmission systems. To address these issues, this study proposes a finite element modeling approach for gear transmission systems, where the gear pair is represented as a spring element. An equivalent criterion for the gears is derived, and a method for improved meshing stiffness calculation is introduced. The proposed approach is experimentally validated. The results demonstrate that the equivalent modeling technique effectively simplifies the representation of gear pairs in 3D finite element analysis. Furthermore, the discrepancies between the finite element analysis and experimental data are all less than 10

Due to factors such as eccentric load and error, the gear will produce cracks and pitting corrosion and other failure modes, resulting in the change of the dynamic characteristics of the two-stage herringbone gear transmission system (TSHBGTS), which will have a serious impact on the stability of the operation of the high-torque engine. To investigate the impact of crack-pitting coupling on the vibration characteristics of the TVMS, a dynamic model with 48 degrees of freedom was developed. This model considers various factors such as errors, time-varying meshing stiffness (TVMS), torsional stiffness, support stiffness, tooth friction, and repulsion slot variations. The potential energy method is employed to calculate the TVMS for each herringbone gear pair in the system, accounting for the effects of crack-pitting coupling and changes in the retractor slot parameter. The dynamic model is solved using the Runge–Kutta numerical integration method, enabling the analysis of the system’s time-domain and frequency-domain responses under varying degrees of crack-pitting coupling. The vibration testing explores how different degrees of crack-pitting coupling affect the system’s time-frequency response characteristics. The results reveal that the system’s stiffness decreases with an increasing degree of crack-pitting coupling, initially dominated by pitting and eventually by crack influence. The time-domain response characteristics exhibit shock behavior that varies with the meshing cycle, becoming more pronounced with increased crack-pitting coupling. Additionally, the frequency-domain response is characterized by side frequency signals near the octave frequency, indicative of crack-pitting coupling effects. This behavior exacerbates as the crack-pitting coupling intensifies, leading to deteriorating vibration stability. Comparative analysis of test and theoretical data demonstrates consistency in trends, affirming the model’s accuracy. The research results are of great significance for the dynamic stability and quality evaluation of TVMS.