Meshing Force Research Articles

Mathematic modeling of vibration transmission path with discontinuous contact interfaces for gear dynamics in the gear-shaft-bearing-housing system

A twelve-DOF model of gear-shaft-bearing-housing system is developed to study the vibration transmission path characteristics under different internal excitations considering effect of discontinuous contact interfaces. Gear meshing generates vibrations to propagate through the transmission path consisting of gear, shaft, bearing, housing and multiple contact interfaces to reach the external transducer. Effect of contact interfaces between different transmitting components, including that between gear and shaft, inner race and outer race, outer race and housing are characterized using parallel contact stiffness and damping elements. Dynamic responses of transmitting components are analyzed for three internal excitations of single frequency sinusoidal force, dynamic meshing force of both normal and spalled gear tooth. An explicit mathematic model of transmission path function is constructed, which considers effect of transmission path on both amplitude attenuation and frequency structure shift for gear with/without fault. The transmission path function is verified with model predictions, finite element simulation and experiments.

A novel method for ring-planet gear mesh force identification via SVD-based Kalman filter

Dynamic mesh forces are the primary vibration excitations of the planetary gear transmission systems. However, they are difficult to measure directly, and conventional indirect measurement methods are typically only applicable to specific planetary gear sets with special measuring equipment. This study proposes a more widely applicable method for identifying the mesh forces of ring-planet gear pairs by utilizing vibration signals. To this end, a ring gear dynamic model is established based on the elastic theory of the thick ring in the form of a space state system. The state vector of the model is constructed from parameters of ring gear vibration responses, while the input state vector is derived from parameters of dynamic mesh forces. A singular value decomposition (SVD)-based Kalman filter is employed to provide a stable solution for the estimation of state vector and input vector simultaneously. A mesh force indirect measurement method based on tooth fillet strain signals collected by strain gauges is adopted to evaluate the accuracy of identification results. Results of dynamic experiments conducted in a planetary gear set test rig demonstrate that compared with measured results, the proposed method can obtain acceptable estimated mesh forces of ring-planet gear pairs by using vibration signals.

Study on the Dynamic Characteristics of Gears Considering Surface Topography in a Mixed Lubrication State

Based on mixed lubrication analysis, considering the influence of rough interface contact stiffness, contact damping, and interface friction on the gear transmission system, the relationship between interface contact and the overall performance of the gear transmission system has been established. First, the surface topography is characterized using statistical parameters of rough surfaces, and the contact stiffness and damping for tooth surfaces with different roughnesses are calculated. Subsequently, a six degree of freedom gear tribo-dynamics coupling model is developed. Finally, the established tribo-dynamics model is employed to investigate the relationship between surface roughness and the overall performance of the gear transmission system. This study provides a more intimate connection between the contact interface and the general behavior of the gear transmission system, enabling a better representation of real-world engineering problems. The research findings reveal that contact stiffness and damping decrease with increasing surface roughness. Higher roughness leads to greater gear vibration amplitude. Moreover, elevated surface roughness results in intensified meshing force and more significant energy loss. Surprisingly, when the roughness is appropriate, gears with rough surfaces lose less energy than those with smooth surfaces.

A model-based approach for gear train whine noise reduction by mesh phasing modification

Heavily loaded gear trains are known to produce a characteristic whine noise and vibration signature, which radiates through the bearings producing an undesired noise emission affecting the NVH behavior of the overall machine. This phenomenon is generated by the variable mesh stiffness and the static transmission error of the gear pairs in mesh within the gear train, but its magnitude strongly depends on how the meshing processes combine between each others. In this context, the present work proposes an analytical methodology to predict amplitude and direction of the variable loads applied on the bearings of idle gears, assuming quasi-static conditions. The dissertation demonstrates that the multiple mesh forces produce a variable load on idle gear bearings describing an ellipse whose shape depends on various geometrical parameters of the system. Numerical examples are shown to assess the potentials of the method and highlight how gear train spatial layout and phasing between different meshings may be used during gearbox design projects to limit the emitted noise of the gear train.

Tribological behavior of copper-iron particles as third body additives on copper matrix composites: Experiment link third body morphology to accommodation mechanism

The relationship between copper-iron third body composition, friction coefficient (COF), surface morphologies and accommodation mechanism were investigated by adding copper-iron powder to the friction surface. Results showed that the COF reduced with the increasing iron concentration in copper-iron third body. For adding Cu, Cu90%-Fe10%, Cu50%-Fe50%, Cu10%-Fe90% and Fe third body, the COF increased at speed of 100–600 rpm, while, it decreased when speeds ranged within 600–1500 rpm. At low speed, copper and iron acted as metal particles participated in friction, which increased the meshing force between the asperities, resulting in increasing the friction coefficient. At high speed, reduction of the COF could be attributed to the oxidative wear of iron reducing the adhesion of copper third body.

Research on load-sharing characteristics of six-branch herringbone gear transmission system

With the continuous development of high-speed and high-power gear transmission technology, as well as the new demand for vibration and noise reduction in high-speed and heavy-duty gear transmission systems in fields such as aviation and navigation, the herringbone gear branch transmission system is subject to higher requirements in terms of its dynamic characteristics. Therefore, it is necessary to conduct research on the dynamic characteristics of high-power and high-torque herringbone gear branch transmission systems. Based on the lumped parameter method, a bending-torsion coupling dynamic model of the transmission system is established, and a calculation method for load-sharing coefficients (LSCs) is proposed. The LSCs are obtained for each gear pair in the same meshing period of the transmission system that is under the interaction of time-varying internal excitation. The meshing force of the six-branch gearing system was calculated in ADAMS to verify the validity of the established dynamic model. The study results indicate that errors have a negative impact on the sharing performance. To maintain a sharing coefficient below 1.15, the manufacturing and installation errors of gears Z5 and Z6 should be restricted to 9 µm and 7 µm, respectively. Special attention should be given to gear Z5 in terms of support stiffness, with a value of 1 × 108 N/m being the most favorable. During the system design, the initial phase of the installation error should be considered and preferably selected near 180°. Additionally, the system exhibits easier load sharing with higher input power. The optimal torsional stiffness of the duplex shaft is 5 × 107 N/m, while the optimal bending stiffness is 5 × 108 N/m. This study aimed to provide a theoretical foundation for the design and optimization of marine high-power gear transmission systems.

Study on the Coupling Relationship between Wear and Dynamics in Planetary Gear Systems

The occurrence of wear is hard to avoid in gear systems because of their transmission principle. Wear will lead to a deviation of the system’s performance from the design objectives or even failure. In this paper, a dynamic wear prediction model considering the friction and wear of all meshing gears is proposed for planetary gear systems. The differences between different wear prediction methods are compared. The interactions among the wear, the dynamic response, and the uniform load performance of the planetary gears are investigated. The results show that considering friction and wear on all tooth surfaces can significantly reduce errors in the simulation. Wear mainly affects meshing stiffness in the double tooth contact region. The degree of fluctuation of stiffness and meshing force increases significantly with wear. The load-sharing factor in the dedendum and addendum regions decreases. Accordingly, the position of maximum wear on the tooth surface moves slowly towards the pitch line. Early wear improves the dynamic performance of the system. As the wear deteriorates, the higher harmonics of the meshing frequency increase significantly. The uniform load performance of planet gears exhibits the same trend of dynamic response as the others during the wear process.

An effective modelling approach for multi-body dynamic analysis of epicyclic gear train of GTF considering both structural flexibility and mechanical interactions

In this paper, a new modelling approach is proposed for the dynamic investigation of epicyclic gear train, and the novelty of this work lies in consideration of both structural flexibility and mechanical interactions during the analysis procedure. The method is of capacity to directly present the dynamic results of the supporting structure for convenient practical engineering evaluation and reduce the dimension of the system reasonably with appropriate assumptions for better computational efficiency. Firstly, the mechanical interactions among components are discussed in detail, and the principle that the structural flexibility works is also explained at length. Secondly, taking the epicyclic gear train of the geared turbofan (GTF) engine; for example, the dynamic model of the system is then established based on the developed hybrid user-defined elements. For model validation, the governing equations of the system are also derived by the lumped mass method. Thirdly, with the same values of the parameters, the results of normal dynamic meshing force obtained by the proposed model are compared with the ones by the lumped mass model. It can be stated from the data that (1) the maximum relative error between the theoretical value and the average value calculated by the two models is 8.28%, (2) the gear mesh frequency obtained by the two models are sufficiently close to the theoretical value, and (3) the fluctuation trends of the dynamic force keep basically consistent with each other. In summary, the comparison presented clearly indicates that the proposed model is indeed reasonable, which provides a new way for dynamic investigation and structural redesign of a large epicyclic gear train. Finally, as a practical engineering application, the vibration result of the deformable supporting structure of the GTF gear train is also presented, which directly provides valuable reference for vibration monitoring, fault diagnosis and other engineering problems in practice.

A novel design method of low noise spiral bevel gear tooth surface for polynomial ease-off topological modification

In order to reduce the vibration noise of the spiral bevel gear transmission system (SBGTS), a polynomial ease-off topological modification (PETM) method, which combines the load tooth contact analysis (LTCA) and vibration analysis of the spiral bevel gear, is proposed in this paper to minimize the normal dynamic meshing force and normal relative displacement to reduce the vibration noise. Firstly, a new PETM equation is established based on the complete conjugation principle, which is applied to the initial clearance calculation of LTCA and an explicit LTCA calculation method with PETM without requiring inverse processing parameters is constructed. The calculation accuracy is better than that of a general algorithm. Secondly, the dynamic response of an eight-degree-of-freedom dynamic system of spiral bevel gear is calculated according to the loaded transmission error and time-varying meshing stiffness calculated by the explicit LTCA of PETM. Finally, a PETM low-noise tooth surface design method is constructed with the goal of minimizing the normal meshing force and the normal relative displacement and the PETM equation coefficient as the optimization variable. The numerical results show that the new PETM low-noise tooth surface design method can improve the meshing performance and dynamic performance of spiral bevel gears to a greater extent than those before modification and after the second-order topological modification, thus greatly reducing the vibration noise of SBGTS.

Nonlinear Dynamic Modeling and Analysis for a Spur Gear System with Dynamic Meshing Parameters and Sliding Friction

The performance of gear systems is closely related to the meshing parameters and sliding friction. However, the time-varying characteristics of meshing parameters caused by transverse vibration are usually not regarded and the sliding friction has always been ignored in previous studies. Therefore, the influence of the transverse vibration on meshing parameters and sliding friction have not been considered. In view of this, a nonlinear dynamic model for a spur gear system is proposed. The dynamic meshing parameters (pressure angle, backlash, etc.) and the effects of the variations of these parameters on the dynamic mesh force (DMF) and sliding friction are emphasized. The differential equations of motion are derived by the Lagrange method and solved by the Runge–Kutta method. Then, the input speed and friction coefficient are used as control parameters to compare the dynamic responses of the new and previous models. The results show that the meshing parameters and sliding friction are affected by transverse vibration, leading to distinctive nonlinear dynamic responses. This paper can provide a basis for further research and give a better understanding of system vibration control.

The Non-Linear Excitation Load-Sharing Method of a High-Powered Nuclear Planetary Gear Train

The paper primarily employs the 3D calculation method of the helical gear-meshing line and meshing position, in addition to the traditional method of the gear-meshing stiffness calculation. This analysis and correction of load-sharing are beneficial for improving the assembly process of high-powered critical equipment. The dynamic models of rigid–flexible coupling, velocity–torque, and the meshing force of planetary gear trains in nuclear power plants are established based on the principles of gear dynamic characteristics. Based on an analysis of the vibration characteristics of a planetary gear train, a load-sharing method for the planetary gear train is proposed. This uniform load-sharing method is explored under different modification values to provide a reference for load-sharing research on high-powered key equipment. In this paper, a dynamic simulation analysis of the gearbox system is conducted, using virtual prototype software to study the load-sharing performance of the planetary gear system. Furthermore, via a vibration frequency analysis of the gear mesh force, the causes of planetary gear train vibration are discussed, particularly their impact on planetary load. This provides a basis for the assembly process of a nuclear power circulation pump gearbox, ensuring that the gearbox for the circulation pump has a longer life that meets the 40-year service life requirement, and provides a foundation for the study of planetary load characteristics.

Analysis of Dynamic Mesh Stiffness and Dynamic Response of Helical Gear Based on Sparse Polynomial Chaos Expansion

This paper presents an efficient method for obtaining the dynamic mesh stiffness and dynamic response of a helical gear pair. Unlike the traditional dynamic model that utilizes a time-dependent sequence, the mesh stiffness using the presented method is updated according to the gear displacement vector at each sub-step of the numerical calculation. Three-dimensional loaded tooth contact analysis (3D LTCA) is used to determine the mesh stiffness, and a surrogate model based on sparse polynomial chaos expansion (SPCE) is proposed to improve the computational efficiency, which is achieved by reducing the number of coefficients in the polynomial chaos expansion (PCE) model though a quantum genetic algorithm. During the calculation, the gear displacement vector at each sub-step is converted into the changes in center distance, misalignment angle, and mesh force, which are then introduced into the SPCE model to update the mesh stiffness for subsequent calculations. The results suggest that the SPCE model exhibits high accuracy and can significantly improve the computational efficiency of the PCE model, making it suitable for dynamic calculations. Upon updating the mesh stiffness during the dynamic calculation, the mesh stiffness declines, the dynamic transmission error (DTE) increases, and the frequency components of the responses change significantly.

Variable stiffness-based vibration prediction for full coupling model of gearbox

Vibration is a problem worthy of attention in gearbox design. In the case of high speed and heavy load, vibration may be more significant, so some methods are needed to predict the vibration of the gearbox. At present, most vibration prediction methods have the problems of few research parameters, low fidelity and a large amount of calculation. In this paper, a fully coupled finite method model of variable stiffness gearbox based on lumped parameter method (LPM) is proposed, and the vibration prediction of gearbox housing is carried out considering time-varying meshing stiffness (TVMS), transmission error and other factors. Firstly, a new stiffness calculation method was proposed based on finite element theory and potential energy method, which can accurately solve the TVMS of fault gear. At the same time, the comprehensive meshing error of gear was solved. At then, the TVMS and comprehensive meshing error of the healthy gear and fault gear were brought into the dynamic equation of two-stage spur gear to obtain meshing force. After that, the meshing force was used as an excitation to solve the vibration of the gearbox housing. Finally, the vibration regularity of the health gearbox and fault gearbox was compared. The sensitivity of each gear shaft to the vibration caused by fault and the vibration regularity of different bearing seats of the same gear shaft was explored. The effectiveness of the simulation results was verified by experiments.

Nonlinear Vibration Analysis of the Coupled Gear-Rotor-Bearing Transmission System for a New Energy Vehicle

Considering the effects of time-varying meshing stiffness, time-varying support stiffness, transmission errors, tooth side clearance and bearing clearance, a nonlinear dynamics model of the coupled gear-rotor-bearing transmission system of a new energy vehicle is constructed. Firstly, the fourth-order Runge–Kutta integral method is used to solve the differential equations of the system dynamics, and the time-varying meshing force diagram, time history diagram, phase diagram, FFT spectrum diagram, Poincaré map and bifurcation diagram of the system are obtained to study the influence of the external load excitation frequency on the dynamics characteristics of the system. In addition, the multiscale method is used to analyze the main resonance characteristics of the system and to determine the main resonance stability conditions of the system. The effect of time lag control parameters and external load excitation frequency on the main resonance of the system is analyzed by numerical methods. The results show that the gear-rotor-bearing coupled transmission system of the new energy vehicle has obviously nonlinear characteristics, avoiding the system instability interval reasonable selection of external load excitation frequency, meshing damping, time lag parameters and load fluctuations, which can be used to improve the stability of the transmission system of the new energy vehicle.

Dynamic simulation of a motor longitudinally mounted on a centralized electric drive systems for pure electric vehicles under torque ripple and transmission error excitations.

The dynamic characteristics of electric drive systems are crucial in electric vehicles. Based on the dynamic finite element method and previous studies, this study proposes and analyzes a new mathematical model for a motor longitudinally mounted on a centralized electric drive system of a pure electric vehicle. First, we analyze the largest torque ripple of a fractional slot concentrated winding inner-mounted permanent magnet synchronous motor designed for commercial electric vehicles. This torque ripple is identified as one of the excitations influencing the dynamic performance of the electric drive system. Second, a new dynamic mathematical model for the electric drive system is established. Third, we investigate the linear vibration responses of the system subject to torque ripple and transmission error. Finally, the relationships between critical motor parameters and dynamic mesh force are revealed. The results demonstrate that the proposed theoretical method can effectively determine the dynamic characteristics of the electric drive system, thereby providing valuable theoretical guidance for the design and optimization of the motor and electric drive system.

Dynamic characteristics of the rack and pinion lift mechanism of the combined command cabin

This paper presents a novel approach to accurately simulate the dynamic characteristics of a rack and pinion lifting mechanism while accounting for the coupling effect of meshing tooth pair stiffness and grease characteristics in real-world environments. A structure-grease coupling meshing stiffness model is established based on the deformation coordination conditions of the tooth pair meshing. The model incorporates the effect of the grease stiffness term on the time-varying meshing stiffness of the rack and pinion. By analyzing the dynamic characteristics of the combined command cabin lifting mechanism using this model, we find that the total meshing stiffness of the gear teeth is lower when the transient thermoelastic flow effect of the grease is considered. Furthermore, the total stiffness value decreases as the normal meshing force at the meshing point decreases. This research reveals that the worst lubrication conditions are typically found on the gear wheel teeth near the base circle, where the grease film temperature rise is highest, the grease film pressure is highest, and the grease film thickness is thinnest.

Dynamic Analysis of the Helical Gear Transmission System in Electric Vehicles with a Large Helix Angle

Sliding friction is an important excitation for the vibration and noise of a transmission system. However, in cases where the helix angle of the helical gear is large, the dynamic characteristics of the transmission system in an electric vehicle caused by friction excitation have not been explored explicitly. In this study, the helix angle of helical gears is 30.5 degrees. A mesh stiffness calculation algorithm is derived. A dynamic model of a Motor Two-Stage helical gear transmission system in an electric vehicle at a constant speed is established. A bench test is carried out to verify the model. A friction model is employed. The dynamic characteristics are analysed after considering the friction and the axial stiffness component. The friction makes the vibration in the x direction increase. The vibration in the y direction is suppressed by friction. The vibration in the z direction has a small change. The components in the frequency domain also change. The axial stiffness components only make the vibration and the meshing force increase in the time domain. This indicates that friction needs to be considered when dynamic characteristics are analysed. The study is good for the reduction of vibration and noise. This may provide a theoretical base for the condition monitoring of the gear system in electric vehicles.

Nonlinear excitation and mesh characteristics model for spiral bevel gears

Accurate and rapid calculation of nonlinear excitation (NE) and loaded mesh characteristics (LMC) of gear pairs is a key to achieving rapid iterative optimization of gear system designs. Therefore, an analytical model (AM) was proposed for the accurate and numerically efficient calculation of NE and LMC for application to spiral bevel gears (SBG). First, a tooth contact analysis of an SBG was completed using Coons surface technology and mesh theory to determine the contact path and unloaded transmission error (UTE). Thus, an AM for the principal and relative normal curvatures of the SBG was derived using the Euler's formula. Based on the traditional calculation method for a Hertz contact ellipse, the relation between the relative curvatures was introduced to derive the formulas for the major and minor axes using the Muller's method, which could improve the programmability and avoid numerical instability of the program. Then, analytical models for the contact ellipse and pressure distribution on the contact tooth surfaces were derived using the Hertz model and geometric theory. Subsequently, based on a single tooth LMC, infinitesimal method, elasticity, and series stiffness model, a single mesh stiffness (SMS) model for SBGs was developed, which incorporates transverse tooth stiffness, transverse gear foundation stiffness, axial tooth stiffness, axial gear foundation stiffness, and Hertz nonlinear contact stiffness. Based on the SMS and UTE, compliance and load matrices were established, and the time-varying mesh stiffness, loaded transmission error, mesh force, and multi-tooth LMC models of the SBG were determined. Finally, the proposed method was validated against nonlinear FE simulations using examples. Compared to FE simulations, the proposed method requires significantly less computational effort and can be further extended to optimization or system analysis problems.

Research on the Coupled Phase-Tuning Vibration Characteristics of a Two-Stage Planetary Transmission System

Mesh phasing has a dramatic impact on the static and dynamic behaviours of planetary gear systems. This research investigates the coupled phase-tuning mechanism of two-stage planetary gear systems and the corresponding relationship with the coupled vibration of the system. Due to the inherent meshing symmetry of the system, the phase-coupled tuning mechanism of the two-stage planetary system is derived based on meshing force relationships and coupling characteristics between different stages. The excitation and suppression relationships associated with the teeth number, harmonic order, and coupling vibration of the coupled system are clearly described. To study the effect of coupled phase tuning on the vibration response of a two-stage planetary gear system, a nonlinear dynamic model was established. The vibration responses under different tuning modes were calculated, and a coupled phase-tuning law for two-stage planetary systems was verified. Model 3 was used as a research tool to build a two-stage planetary transmission experimental platform, and the transverse vibration and torsional vibration of the first stage sun gear were analysed to further verify the correctness of the phase-coupling tuning law.

Three-dimensional fretting and sliding contact model in planetary gear system: Theoretical analysis and experiments

The failure of the interference fit between the planetary gear and the bearing outer rings in planetary gear systems has received increasing attention from researchers. In this study, a three-dimensional model for determining the fretting and sliding contacts between planetary gears and these outer rings is presented. A three-dimensional fretting numerical model under oblique forces was established to investigate the fretting contact between planetary gears and bearings, and an experiment was conducted to study the relative sliding between them. The results showed that if the torque generated by the difference between two meshing forces exceeded the limit torque of the interference fit, overall relative sliding would occur. Otherwise, fretting contact would occur in some regions of the interference fit surface under oblique forces. The relative sliding between the planetary gear and outer rings of the bearings can be reproduced experimentally; the relative sliding value decreased with an increase in the size of the interference fit.