Time-varying Mesh Stiffness Research Articles

Nonlinear Dynamic Analysis of a Spur Gear Pair System with Wear Considering the Meshing Position

In this paper, a nonlinear dynamic model of a parallel shaft gear system consisting of two involute spur gears is developed to investigate the coupling effect between the gradual surface wear of gear teeth over time and nonlinear dynamic characteristics. A uniform wear model that accounts for how the volumetric wear of the gear teeth affect their meshing position, backlash, and stiffness is proposed. Additionally, a nonlinear dynamic model with six degrees of freedom is described that considers friction, time-varying gear backlash, and time-varying meshing stiffness. The proposed model significantly changes the mesh stiffness, not only in terms of value but also in terms of contact ratio. Furthermore, the nonlinear dynamic characteristics of the gear system vary significantly. It is found that the gradual wear of gear teeth affects the meshing position and further has a significant impact on the nonlinear dynamic characteristics of the spur gear system. This paper provides a basis for studying the nonlinear dynamic characteristics of the spur gear system as it experiences the gradual wear of teeth over time.

Optimization design of load-sharing and light weight of the twin rotors concentric face gear power-split transmission system based on gear tooth matching condition

To improve the dynamic load-sharing performance and achieve lightweight, the twin rotors concentric face gear power-split transmission system (TRFGPSTS) was optimized. The gear tooth matching conditions of the system based on the design characteristics of power flow closed-loop and virtual constraint configuration of the system are established, and the basic parameters of each gear meet the requirements of the system load-sharing layout and synchronous meshing of each branch movement are calculated. Based on the lumped mass parameter method, combined with the calculation results of gear tooth matching, the 45-DOF bending torsional coupling dynamic model of the system was established considering time-varying meshing stiffness, support stiffness, manufacturing and installation errors, and other factors, and the influence of each factor on the dynamic load-sharing performance of the system was analyzed. Combined with the influence law of various factors, a multi-objective dynamic load-sharing optimization design model was established with the best dynamic load-sharing performance and the minimum system quality as the objective function. Through the weighted combination method, the global optimal solution of the system optimization design was determined, and the influence of each parameter on the optimization objective function was analyzed. The correctness of the results was verified by using the genetic algorithm based on Pareto frontier distribution. The results show that the dynamic load-sharing characteristics of the system become worse with the increase of error and torsional stiffness, and are more sensitive to the influence of various error parameters. The optimization process focuses on the geometric parameters such as the Module, the teeth Number, the Pressure angle, and the Tooth width of each gear, which can reflect the degree of influence on the dynamic load-sharing and lightweight of the system. After optimization, the dynamic load sharing coefficient (DLSC) was reduced to 1.017, and the overall geometric volume was reduced to 1.473 × 108 mm3. It provides a theoretical basis for the design and research of a new type of helicopter power transmission system with a compact structure, good maneuverability, and high reliability.

Study on the vibration characteristics of toroidal involute worm meshing with different plastic gears

Toroidal involute worm (TI worm) meshing with plastic gear can serve as an efficient reducer for applications in smart home and automatic driving, owing to its stable transmission and high carrying capacity. This necessitates an in-depth analysis of the vibration and noise associated with TI worm drive to enhance the comfort in these scenarios. Utilizing the meshing characteristics of TI worm drive, a dynamic model including nonlinear factors such as time-varying meshing stiffness and meshing damping is established. The dynamic characteristics of TI worm meshing with different plastic helical gears are compared and analyzed. The TI worm drive experiments are carried out, and vibration signals are collected though accelerometers. The numerical analysis results align with the experimental findings. The results reveal that TI worm meshing with different plastics exhibits the maximum vibration amplitude in the axial direction of the worm. Specifically, PEEK demonstrates the highest amplitude, followed by POM, while Nylon has the smallest vibration amplitude. These findings serve as a valuable reference for mitigating vibration and noise of TI worm drives.

A time‐varying meshing stiffness model for gears with mixed elastohydrodynamic lubrication based on load‐sharing

AbstractIn mixed elastohydrodynamic lubrication (EHL), the load distribution between the gear meshing surfaces is shared by the oil film and the asperities of the gear's rough surface. Based on the load‐sharing concept, this paper proposes a time‐varying meshing stiffness (TVMS) model for gears with mixed EHL. The initial step involves the utilization of the Greenwood‐Williamson model to calculate the contact stiffness of surface asperities, while the lubricating film is assessed using a curve‐fitting formula to investigate the influence of gear surface morphology on TVMS. Subsequently, the incorporation of gear fillet foundation deformation and friction enables accurate TVMS determination. The proposed method is employed to examine the meshing stiffness of the gear pair under both dry lubrication and EHL conditions. Comparative analysis reveals favorable agreement between the proposed model and experimental results obtained under dry lubrication, thereby highlighting the superior performance of the proposed approach. Moreover, the time‐varying friction coefficient under EHL is computed, and the impacts of gear surface morphology parameters, temperature, speed, and load on lubrication conditions and TVMS are investigated. The findings presented in this paper contribute significantly to advancements in gear design and performance.

Analysis of electromechanical coupling vibration characteristics of motor-gear system based on bond graph theory

In recent years, there has been a growing demand in the field of engineering systems for research on the vibration response of motor-gear systems under non-steady state conditions. However, conventional modeling and simulation techniques are primarily restricted to systems within a single energy domain. When dealing with intricate electromechanical coupling systems, the modeling process becomes increasingly intricate. Consequently, this study adopts bond graph theory as the foundation for analyzing the vibration characteristics of motor-gear coupling systems. In addressing the vibration issue of the motor-gear transmission system, a nonlinear bond graph model is established that takes into account the time-varying mesh stiffness, static errors, and tooth surface friction, among other nonlinear factors. Then, according to the coupling relationship between motor mechanical systems and electrical systems, the electromechanical coupling dynamic model of the motor-gear system is established using the method of bond graph. In addition, the state equation of the system is derived based on the bond graph model. Finally, the system is simulated using 20-sim software, and the electromechanical coupled vibration characteristics of the simulation results are analyzed. This article lays the foundation for the design and analysis of motor-gear coupling systems and multi-energy domain complex systems.

An Analysis of Time-Varying Meshing Stiffness of High Contact Ratio Gears Considering a High-Order Modification Curve

An Analysis of Time-Varying Meshing Stiffness of High Contact Ratio Gears Considering a High-Order Modification Curve

Optimum design of planetary gear set using multi-objective optimization considering mass, power loss and transmission error

In this study, a multi-objective optimal design of a planetary gear set was performed considering mass, power loss, and transmission error. In particular, the results were analyzed from a novel perspective. The implemented multi-objective optimal design methodology minimizes the mass, power loss, and vibration with respect to seven macro-geometry variables. The peak-to-peak static transmission error (PPSTE) of each gear pair corresponds to the excitation source of the planetary gear set, and reducing the PPSTE helps minimize vibration. Thus, the PPSTE was calculated using the time-varying mesh stiffness obtained through an analytical model. The non-dominated sorting genetic algorithm-III (NSGA-III) was implemented as a multi-objective optimization algorithm, and the criteria importance through the inter-criteria correlation method was implemented for quantitative comparison purposes regarding the combination of objective functions. The results showed that obtaining an optimal macro-geometry design for a planetary gear set exhibiting good performance required simultaneous consideration of the mass, power loss, and vibration. Furthermore, the results confirmed that a versatile optimization algorithm, such as NSGA-III, should be used to capture the discontinuous solution distribution owing to various constraints of the planetary gear set.

Dynamic modeling and vibration analysis of an RV reducer with defective needle roller bearings

The rotate vector (RV) reducer's needle roller bearings (NRBs) are vulnerable to wear and flaws such as fatigue spalling of the raceway material. Studying the vibration mechanism and the excitation laws of NRB failures is essential to enhancing the RV reducer's performance and comprehending its operating state. This paper presents a dynamic model of the RV reducer based on the theory of contact multibody dynamics. It integrates internal and external raceway defects and takes into account the synchronous contact interaction between NRB groups and the crankshaft and cycloidal gears, as well as the time-varying mesh stiffness in the planetary gear transmission and cycloidal pin transmission. The study demonstrates that the rolling of bearings and the meshing of internal transmission components are the primary causes of vibration in a healthy RV reducer through experimental verification and dynamic simulation analysis of the system with and without NRB failures. Both the crankshaft rotation frequency and the cycloidal gear rotation frequency affect the vibration excitation source's characteristic frequency. The vibration amplitude of the RV reducer is influenced by bearing flaws on a periodic basis. The impact of internal raceway defects is greater than that of external raceway faults, resulting in a notable rise in vibration amplitude.

Effects of Macro-Pitting Fault on Dynamic Characteristics of Planetary Gear Train Considering Surface Roughness

The planetary gearbox plays a vital role in a wide range of mechanical power transmission systems, including high-speed trains, wind turbines, vehicles, and aircraft. At the same time, the planetary gear train inside the gearbox is regarded as the most susceptible to failure in the entire transmission system. To analyze the influence of surface roughness on the dynamic characteristics of the planetary gear train, a dynamic modeling method based on fractal theory is proposed. Firstly, the tooth surface contact model was established based on the W-M fractal function, and the time-varying mesh stiffness (TVMS) of the planetary gear train was calculated under healthy and tooth macro-pitting. Then, the lumped-parameter method is introduced to construct a planetary gear train translation-torsion dynamic model that comprehensively considers TVMS and tooth backlash. The vibration acceleration signals of the planetary gear train under different macro-pitting states and surface roughness are simulated and calculated, allowing a quantificative analysis of the influence of surface roughness on system vibration response. Finally, the correctness of the model for the planetary gear train is verified by experiments. The results show that compared with the planetary gear train modeling method based on Hertz contact theory, the root mean squared error of the vibration signal of this work under a macro-pitting fault state is reduced by 8.7%, further improving the reliability of the model.

Analysis of vibration and load-sharing characteristics of two-stage planetary gear transmission system with cracks

To reveal the influence of cracks on the dynamic characteristics of a two-stage planetary gear transmission system (TSPGTS), a nonlinear dynamic model with 30-DOF in bending and torsion was constructed, taking into account parameters of different crack degrees (depth, angle, length), as well as factors such as error, time-varying meshing stiffness (TVMS), and damping. In the model, the TVMS model of components with cracks was established using the potential energy method. By incorporating the crack factor into the model, time domain, frequency domain, phase diagram, and Poincare diagram characterizing the dynamic load characteristics were obtained, as well as the curve of dynamic load-sharing characteristics changing with cracks. The influence of different crack degrees on the displacement vibration and dynamic load characteristics of the system was analyzed, and vibration testing experiments were conducted on the system with planetary gear cracks. The results indicate that when the gear contains cracks, it will lead to a decrease in the stiffness of the system, when cracks appear on the II-stage planetary gear, the system will experience impact components with intervals of the rotation period of the II-stage planetary gear, and obvious sidebands will appear near the meshing frequency doubling. These situations will become more pronounced as the degree of cracks intensifies. As the degree of cracks in the II-stage planetary gear increases, the dynamic load-sharing characteristics of the TSPGTS will first improve and then deteriorate, and as the degree of cracks increases, the external and internal dynamic load coefficients continue to increase. Through experiments, the influence of normal gears and cracked gears on system vibration changes was compared and analyzed, and the theoretical results were in good agreement with experimental results, verifying the correctness of the theoretical model. This provides a theoretical basis for fault diagnosis and reliability research of the system.

Analysis of vibration characteristics of face gear powering-split transmission system

A nonlinear dynamic model of the face gear powering-split transmission system was established, and the nonlinear dynamic characteristics of the face gear powering-split transmission system were studied by considering various factors such as support stiffness, time-varying meshing stiffness, lateral clearance of teeth, comprehensive transmission error and bearing stiffness. The Runge–Kutta integral method is used to solve the differential dynamics of the system. The nonlinear behavior of the system is described by the phase plane diagram, time history diagram, frequency domain diagram, and Poincare section diagram. The influence of external excitation frequency on the nonlinear behavior of the system is described by the bifurcation diagram. In addition, the multi-scale method is used to analyze the superharmonic resonance of the system and determine the stability conditions of the superharmonic resonance of the system. The influence of meshing damping, time-varying meshing stiffness, and time-delay control parameters on the amplitude-frequency characteristics of the system is studied by numerical analysis of the superharmonic resonance. The results show that the system will show periodic chaotic alternating motion characteristics under external excitation. Selecting meshing damping, meshing stiffness, and time delay control parameters in a reasonable range can effectively reduce the superharmonic co-amplitude and keep the system in a stable state.

Analysis of the dynamic characteristics of the gear-shifting process under external excitation

This study focuses on a novel clutchless and synchronizer-free drive system designed explicitly for pure electric vehicles. In contrast to the majority of previous research that solely considers internal excitations, this study not only incorporates various non-linear factors such as time-varying friction forces, time-varying mesh stiffness, and damping associated with internal excitations but also places significant emphasis on analyzing the impact of non-linear external excitations, including road surface unevenness and motor torque fluctuations, on the shifting process of the drive system. Subsequently, a non-linear dynamic model of the shifting process is established. In the modeling process, departing from the traditional two-degree-of-freedom quarter vehicle dynamics model, this study adopts an elastic tire-body and elastic tire-tread tire model, enabling the establishment of a highly accurate multi-degree-of-freedom quarter vehicle dynamics model. The simulation results indicate that road surface roughness and motor torque fluctuation significantly impact the gear-shifting process and the impact of road surface roughness increases with the level of roughness. Moreover, the effect of motor torque fluctuation on the gear-shifting process varies depending on different factors and is not simply additive. Consequently, when formulating and optimizing shifting strategies, it becomes imperative to consider various external factors. The research methodology integrates these factors into the dynamic analysis of the shifting process, providing a theoretical basis for developing effective shifting control strategies while considering external excitations.

Research on Time-Varying Meshing Stiffness of Marine Beveloid Gear System

Beveloid gears have the advantages of compensating for axial error, providing smooth transmission, and eliminating turning error. Therefore, they are widely used in applications that require high transmission accuracy and stability. However, research on calculating the time-varying meshing stiffness of beveloid gears is still limited, and there is an urgent need to propose a method that can calculate the meshing stiffness of beveloid gears quickly and accurately. We first established the tooth profile expressions, assuming a pair of beveloid gears meshing with the same rack, and the contact line equations of parallel axis beveloid gear pairs were derived. Next, we analyzed the contact process of beveloid gears. We propose an analytical algorithm based on the slicing method to calculate the meshing stiffness of helical gears, straight beveloid gears, and helical beveloid gears. Then, the influence of different parameters on the meshing stiffness of helical beveloid gears was analyzed by changing the respective parameters. Finally, the finite element method (FEM) was used to verify the correctness of the analytical results, and then the errors were analyzed. The study demonstrates that the results obtained from the analytical algorithm we proposed have the same magnitude as those obtained by the FEM for the time-varying meshing stiffness calculation of beveloid gears.

Dynamic characteristics of spur gear system with tooth root crack considering gearbox flexibility

Considering the calculation efficiency and accuracy of meshing characteristics of gear pair with tooth root crack fault, the parametric model of cracked spur gear is established by simplifying the crack propagation path. LTCA method is used to calculate the time-varying meshing stiffness (TVMS) and transmission error (TE) of the cracked gear pair, and is verified by finite element (FE) method. The crack area share index is proposed to measure the degree of crack fault and the application range of simplified crack propagation path is obtained. The gearbox is introduced into the calculation as the condensation model, and the dynamic model of gear system with root crack fault considering gearbox flexibility is established. The effects of gearbox flexibility, rotational speed, crack degree and gear torque on the vibration response fault characteristics of gear system are analyzed. Compared with the simulation results without considering gearbox flexibility, the simulation results considering gearbox flexibility are smaller in amplitude and closer to the experimental results. The fault shock degree in acceleration waveform and fault frequency modulation in FFT spectrum are more obvious considering gearbox flexibility. The fault characteristics of vibration response are more obviously affected by the reduction of gearbox flexibility and the increase of fault degree than the change of rotational speed and gear torque. The research results provide theoretical and technical support for fault diagnosis and health monitoring of gear systems.

Quantitative assessment for tooth crack of the sun gear in a planetary gearbox

The sun gear in a planetary gearbox is prone to generate tooth crack due to its much more severe load conditions than other gears. The cracked sun gear will evolve into distinct tooth breakages and thus may deteriorate the operation status and even cause catastrophic loss of the whole transmission system. To avoid tooth crack induced malfunction of the planetary gearbox, the tooth crack of sun gear should be accurately identified during the early stage. Therefore, a novelty quantitative assessment methodology for tooth crack damage is proposed, which is a mathematical model for crack propagation of the sun gear in planetary gearboxes based on vibration acceleration signal. Firstly, the quantitative relationship between the crack propagation parameters along the gear thickness and width directions and the time-varying mesh stiffness is analytically formulated. The analytical time-varying mesh stiffness formulation is incorporated into a translational–torsional dynamic model to reveal the effects of crack propagation on the vibration signal of internal and external meshing pairs in the planetary gearbox. Secondly, a quantitative mapping function is established by taking the cracked section area as the variable and the residual of the summed envelope spectrum amplitude of the vibration acceleration signal as the assessment index. Finally, a numerical simulation as well as an experimental test is conducted and compared to verify the correctness of the proposed assessment methodology. We believe that the established model and the corresponding index provide a physically intuitive yet mathematically quantitative reference for diagnosing tooth crack in planetary gearboxes.

Dynamic modeling of spur gear transmission system with evolutive coupling fault of fatigue crack and wear

Parallel shaft spur gear system is widely used in various machineries because of its simple structure, high transmission efficiency and strong adaptability. The health status of spur gear greatly affects the safety and reliability of equipment in service, so it is quite necessary to reveal the evolution law of gear fault and clarify the fault dynamic response. Fatigue crack is a typical gear fault, and it is often accompanied by wear which is generally unavoidable in the life cycle of gear transmission system. The two typical gear faults reduce the meshing stiffness of the gear pair, whereupon a calculation approach of time-varying meshing stiffness for the coupling fault of fatigue crack and wear is proposed. First, the prediction models of fatigue crack and wear are respectively investigated, then a coupling fault evolution model is built by combining them. Using this model, a new approach for calculating the time-varying meshing stiffness is put forward. Via embedding the obtained meshing stiffness into the eight-degree-of-freedom dynamics model of spur gear transmission system as the fault excitation, an evolutive coupling fault dynamics model is constructed. Finally, the effectiveness and accuracy of the proposed evolutive fault dynamics model is verified by the experiment on a faulty single-stage gearbox. The proposed model has important guiding significance for the fault diagnosis and prognosis of spur gearbox.

Dynamic characteristics of rack vehicle-track coupled system on large slope line: theoretical modelling and experimental validation

Rack railway is a rail transportation that utilizes the gear-rack engagement to supply the traction forces on large-slope lines, primarily employed in mountainous regions where the conventional trains cannot operate due to the wheel-rail adhesion limitations. However, there is limited research on this system dynamics and further analysis is required to understand the interaction mechanism between the rack vehicle and track. Based on the typical vehicle-track coupled dynamics theory, the rack vehicle-track spatially coupled dynamics model is developed in this paper to evaluate the dynamic behaviours of the rack vehicle and the track system running on large-slope line with a detailed modeling of the longitudinal motion for the rack and sleeper, where the time-varying gear-rack mesh stiffness and the rail geometrical irregularity are incorporated. The excellent agreement observed between simulated and field-test results serves as strong validation for developed dynamics model. The results demonstrate that the vibration responses of the wheelset and the rack are dominated by the gear-rack mesh excitations. It should be noticed that the wheel loads are reduced by the presence of the gear-rack engagement. Furthermore, the vibration of the sleeper can explicitly reflect the gear-rack mesh characteristics through the transfer of the rack fastener.

Investigation of Effect of the Tooth Profile Modification on the Engagement Characteristics of the Gear Pair

As a vital component in the electric vehicle, the higher requirements for the gear precision are urgently needed in the service environment with high speed and heavy load. The tooth profile modification can greatly improve the accuracy and strength of the gear transmissions. Therefore, accurately obtaining the optimum modification parameters of the gear system under the action of specific loads is the premise of the research of gear vibration and noise reduction. The time-varying mesh stiffness calculation model is established in this paper considering the coupling effect of the tooth profile errors and structural coupling effect. The minimum variations of the loaded static transmission error are selected as the objective function to obtain the optimum modification parameters of gear pair. In addition, the effect of torque, modification amount and modification length on the engagement characteristics of the gear pair are studied. The research can provide some references for the design and optimization for the gear transmission system in the electric vehicle.

Dynamic performance analysis of a gear transmission system due to cavitation in gerotor pump

In this work, a dynamic model considering time-varying mesh stiffness and backlash was proposed and a test rig was built to investigate the vibration behavior of the idle gear set in a gerotor pump. The effects of input shaft speed, torque fluctuation, and gear tooth backlash on the dynamic performance of gerotor pump were analyzed according to the numerical and experimental results. Results demonstrate that when the torque fluctuation is neglected, the dynamic transmission error of drive gear is dominated by the first three mesh harmonic components; however, the RMS values of vibration acceleration have high frequency characteristics. When the torque fluctuation is considered, the dynamic responses of gerotor pump are modulated by the torque fluctuation. The effect of tooth backlash on the dynamic performance of gerotor pump is limited and mostly depended on the torque fluctuation condition. The gerotor pump noise is dominated by the low frequency component around the mesh frequency. The vibration energy can be divided into two parts in the frequency domain. In the low frequency band, the vibration energy is caused by the mechanical vibration. However, in the high frequency band, the cavitation phenomena may cause the concentration of the vibration energy and resonant of pump structure.

Dynamic modeling and analysis of planetary gear system for tooth fault diagnosis

This study concentrates on the dynamic modeling and analysis of a planetary gear system with tooth broken faults. Firstly, a systematic calculation method of time-varying mesh stiffness (TVMS) for the planetary gear system is developed. Analytical models are established for both external-external and external-internal gear pairs under both healthy and tooth broken conditions. Then, a translational-torsional nonlinear dynamic model is established for a planetary gear system and coupled with the TVMS calculation method, to achieve dynamic simulations with tooth broken faults on the sun, planet and ring gears. The influences of tooth broken faults at different gears and at different damage degrees on the vibration responses are revealed. In addition, 15 condition indicators are chosen to assess their sensitivity to the tooth broken faults in a wide range of operational speed rather than a specific operational speed, in order to provide comprehensive information for evaluating the vibration characteristics and health conditions. Finally, experimental vibration signals from a gear transmission test rig are applied to verify the proposed dynamic model. The modeling method and analysis results here could provide some suggestions for the vibration analysis and fault diagnosis of planetary gear systems, and further provide data basis for the development of fault diagnosis methods.