Meshing Force Research Articles

Effect of rotor eccentricity on the dynamic performance of a traction motor and its support bearings in a locomotive

Rotor eccentricity is a common fault of traction motors during operation. To ensure the stability and safety of the traction motor, it is essential to analyze the effect of rotor eccentricity on the dynamic performance of the traction motor, particularly the motor bearings. This study focuses on the extraction of the fault vibration features of a locomotive in the presence of rotor eccentricity under complex internal and external excitations from the traction power transmissions and wheel-rail interactions. A spatial locomotive-track coupled dynamics model with an eccentric rotor motor is applied in this study, and the detailed mechanical structures of the traction motor and motor bearings are comprehensively considered. The results indicate that rotor eccentricity can deteriorate the vibration conditions of the traction motor and intensify the interactions between the components of the motor bearings. Due to the effect of the time-varying mesh forces, the variation in the loaded region of the non-driving end bearing is more significant than that of the driving end bearing.

Self-Excited Vibration Analysis of Gear-Bearing System with Multipoint Mesh and Variable Bearing Dynamic Coefficients

The gear-bearing system is the most important part of integrally centrifugal compressors. According to statistics, the majority of integrally geared compressor accidents are caused by excessive vibration of the geared rotor. However, its complicated dynamic characteristics and inevitable vibration faults in actual operation present significant challenges throughout the analysis and design stages. In this paper, the coupled self-excited vibration of the gear system characterized by multipoint meshing and oil film bearing supporting is investigated. Firstly, the structure of the gear system in an integrally geared compressor is used as a research object. The modeling approach of meshing excitation, including time-varying mesh stiffness, gear meshing error, and tooth backlash are introduced. However, the variable stiffness and damping coefficient equations of journal bearing and oil film thrust bearing are modeled and utilized to approximate the variable bearing force and simplify the vibration computation under the assumption of Newtonian fluid. Then, a dimensionless modeling method of the gear system considering gyroscopic moment of gear disk, variable meshing force, as well as variable stiffness and damping coefficient is proposed. Based on the dynamic model, the influence of the bearing dynamic coefficients and load on the vibration of the entire gear system is studied. Among which, the vibration displacement and meshing force are examined using frequency-domain and time-domain analysis methods. The results suggest that the flexible support can restrain the system’s nonlinear motion, whilst increasing load on the gear system can improve gear operation stability and reduce load fluctuation.

Input torque measurements for wind turbine gearboxes using fiber-optic strain sensors

Abstract. Accurate knowledge of the input torque in wind turbine gearboxes is key to improving their reliability. Traditionally, rotor torque is measured using strain gauges bonded to the shaft. Transferring the resulting signal from the rotating shaft to a stationary data acquisition system while powering the sensing devices is complex and costly. The magnitude of the torques involved in wind turbine gearboxes and the high stiffness of the input shaft pose additional difficulties. This paper presents a new alternative method to measure the input torque in wind turbine gearboxes based on deformation measurements of the static first-stage ring gear. We have measured deformation using fiber-optic strain sensors based on fiber Bragg gratings because of their advantages compared to conventional electrical strain gauges. The present study was conducted on a Siemens Gamesa Renewable Energy gearbox with a rated power of 6 MW, in which a total of 54 fiber-optic strain sensors were installed on the outer surface of the first-stage ring gear. The gear mesh forces between the planets and the ring gear cause measurable deformations on the outer surface of the stationary ring gear. The measured strains exhibit a dynamic behavior. The strain values change depending on the relative position of the strain sensors to the planet gears, the instantaneous variations of the input torque, and the way load is shared between planets. A satisfactory correlation has been found between the strain signals measured on the static ring gear and torque. Two signal processing strategies are presented in this paper. The first procedure is based on the peak-to-peak strain values computed for the gear mesh events, and therefore torque can only be estimated when a gear mesh event is detected. The second signal processing procedure combines the strain signals from different sensors using a Coleman coordinate transformation and tracks the magnitude of the fifth harmonic component. With this second procedure, it is possible to estimate torque whenever strain data of all sensors are available, leading to an improved frequency resolution up to the sampling frequency used to acquire strain data. The method presented in this paper could make measuring gearbox torque more cost-effective, which would facilitate its adoption in serial wind turbines and enable novel data-driven control strategies, as well as a more accurate assessment of the consumed fatigue life of the gearboxes throughout their operation.

Prediction of tribological and dynamical behaviors of spur gear pair considering tooth root crack

Variations in gear tooth meshing stiffness and tribological characteristics caused by tooth root cracks seriously affect the dynamic performance of gears, limiting the development of durable high strength gears. Therefore, the potential energy method was applied to obtain the time-varying meshing stiffness of the spur gear pair, and the finite element method was applied to verify the calculation results. Secondly, the load distribution of the gear with the cracked tooth root was derived using coordinated deformation conditions, and the effect of the tooth root crack on the tribological characteristics of the tooth surface was obtained based on the elastohydrodynamic lubrication (EHL) model. Finally, the tribo-dynamic behaviors of the gear under the action of the root cracks were investigated. The results indicated that root cracks change the load distribution ratio in the double-tooth meshing zone, which leads to corresponding changes in the tooth surface film thickness and friction coefficient. In addition, when the cracked teeth are engaged in meshing, the reduction in meshing stiffness causes vibration and shock, which causes the appearance of sidebands near the main meshing frequency. The amplitude of the dynamic meshing force increased with the increase in the degree of root cracks.

Nonlinear dynamic modeling and analysis of spur gear based on gear compatibility conditions

In the traditional gear dynamic model, the mesh stiffness is obtained under static load. When the mesh stiffness changes greatly with the change of load, the traditional dynamic model is not accurate enough to analyze the gear system. Therefore, this paper proposes a new dynamic model with a more realistic dynamic mesh force based on gear compatibility conditions. The proposed dynamic mesh force model is applicable to the gear system with and without tooth modification and verified by the finite element method. Afterward, the paper compares the differences in the dynamic responses between the proposed model and the traditional model. The results indicate the dynamic responses of the proposed model are generally consistent with those of the traditional model for the standard involute gear system. For the lead crown gear system, compared with the proposed model, the traditional model will lose important nonlinear characteristics in the high-speed area. At most speeds, the amplitudes of vibration and dynamic mesh force of the proposed model are greater than that of the traditional model, especially in the bifurcation, quasi-periodic and chaotic regions.

A Calculation Method for Tooth Wear Depth Based on the Finite Element Method That Considers the Dynamic Mesh Force

Gear wear is a progressive material removal process that gradually changes the tooth profile shape and dynamic mesh force, where the dynamic mesh force affects the tooth surface wear. To describe this process, a spur gear dynamic model that includes the mesh stiffness and unloaded static transmission error (STE) of the worn tooth profile is proposed for calculating the dynamic mesh force. Then, based on the finite element method (FEM), a dynamic contact analysis model that considers the dynamic mesh force is proposed for calculating the time-varying contact stress and relative sliding distance of the tooth surface mesh point. Finally, combined with the Archard wear model, a tooth wear depth calculation method that considers the worn tooth profile and the dynamic mesh force is proposed. In addition, the wear depth and dynamic characteristics under different wear times are studied.

Dynamic modeling of central bevel gear transmission system in aero-engine

Aero-engine is the most critical part of an aircraft. As its key component, the central bevel gear transmission system is always designed only considering the meshing vibration and ignoring structural characteristics of shafts and bearings, and comprehensive vibration characteristics is not well addressed. The transmission system may suffer severe vibration. To accurately study the dynamic characteristics of the system, a model of a central bevel gear transmission system is developed considering the effects of bevel gear mesh stiffness, shaft stiffness, bearing stiffness, and unbalance excitation in this paper. It has been validated by finite element software ANSYS. On this basis, the influence of variation of bearing stiffness and mesh stiffness on the natural characteristics of the central bevel gear transmission system is investigated. The influence of unbalance on the radial vibration and gear meshing force is studied by the analytical calculation method. A case study is carried out. The results show that variation of stiffness within a certain range will cause dramatic changes in the natural characteristics of the system. The unbalance of the driven bevel gear shaft has an effect on the radial vibration and meshing force of the system. The proposed model can reflect radial vibration characteristics, axial vibration characteristics, torsional vibration characteristics, and some coupled natural characteristics of the system. This research provides a theoretical basis to optimize dynamic performance and reduce the vibration of the system and will be helpful in designing the central bevel gear transmission system.

A revised method to calculate time-varying mesh stiffness of helical gear

Time-varying mesh stiffness (TVMS) of gear plays vital role in analysing dynamic characteristic of gear transmission. So accurately evaluating the TVMS is important and essential. In this paper, a revised method to calculate the TVMS of helical gear is proposed. Based on slice method, the helical gear is sliced into pieces along the tooth width direction. The proposed method corrects the fillet foundation stiffness within multi-tooth in contact and considers the non-linearity and load-dependence of the Hertzian contact stiffness. The effect of the axial mesh force is considered. Finally, an equivalent helical gear model is established in ANSYS to study the mesh stiffness. The results show the proposed method has high effectiveness compared with FEM (finite element method).

Electromechanical Coupling Dynamic Characteristics of Differential Speed Regulation System Considering Inverter Harmonics Under Variable Operating Conditions

The differential speed regulation system driven by inverters, motors and differential gear trains is a complex electromechanical system that achieves speed regulation and energy savings efficiently and economically. In this paper, the complete drive trains dynamic model of the differential speed regulation system with the inverter power supply-motor-differential gearbox-load is established. The inverter power supply considers constant voltage to frequency ratio control, sinusoidal pulse width modulation and inverter, the motors are modeled based on the equivalent circuit method, and the differential gear trains consider a lateral-torsional model with time-varying meshing stiffness and damping. The dynamic properties of the system are analyzed and compared with the variable frequency drive model considering inverter power supply and the direct drive model using ideal power supply under variable operation conditions. The results show that the variable frequency drive model varies more smoothly in the time domain and contains additional harmonic components in the frequency domain resulting from the coupling of the inverter triangle carrier frequency to the fundamental frequency of the power supply than the direct drive model. It also shows that the variable frequency drive model is a more realistic and rational model than the direct drive model. The complex coupling relationship can be seen in the inverter power supply-motor-differential gearbox-load system, where both inverter harmonics and mechanical meshing force harmonics are found in the dynamic response of the motor’s electromagnetic torque, rotor speed; the inverter harmonics are also reflected in the mechanical meshing force harmonics.

Rotor Profile Design and Motion Simulation Analysis of Hydrogen Circulating Pump

With the development of automotive fuel cells towards high power and high efficiency, higher requirements are put forward for the speed and boost ratio of hydrogen circulation pump, while high speed and boost ratio increase the pulsation characteristics and accelerate the aging of hydrogen fuel cells. Cam type hydrogen cycle pump has the advantages of compact structure, high efficiency under low boost ratio, and good adaptability to low temperature, so it has become an important research direction. The efficient and reasonable rotor profile design can improve the volumetric efficiency of the cam rotor pump. It can reduce the pressure pulsation and flow pulsation in the pump. The equation of multi-stage circular arc rotor profile is established according to the meshing principle of the rotor. Motion Simulation module in UG NX is employed to conduct dynamic interference detection of rotor mechanism and simulation analysis of rotor mechanism meshing motion on the rotating parts of the hydrogen circulating pump. So the angular velocity curve and meshing force change curve of the driven gear is obtained. In order to verify the rationality of rotor profile design, the meshing force is compared with the standard spur gear.

Research on dynamic load spectrum of electric vehicle transmission system based on underdamping characteristic

Overall, the electric vehicle transmission system shows an underdamped characteristic. Under changeable road conditions and high-frequency response for the motor, the resulting dynamic load environment may cause multiple failure modes, such as contact fatigue failure and bending fatigue fracture, for the transmission component, which limits the electric vehicle transmission component lifespan and system reliability. To reflect the dynamic load characteristics of the electric vehicle transmission system using a permanent magnet synchronous motor as a power source and accurately calculate the dynamic load of the transmission system, a high-speed helical gear-rotor-bearing coupling mechanical model for the electric vehicle transmission system was built based on simulating actual operation working conditions of the electric vehicle and considering the external load excitation caused by the Electromagnetic torque of the permanent magnet synchronous motor and vehicle driving resistant change as well as internal excitation caused by gear time-varying meshing rigidness and meshing error. Through simulation calculation of the mechanical model, the dynamic meshing force of the gear pair and dynamic contact force of the support bearing was obtained. Based on the Hertz contact theory, the stress-time history was obtained for the key parts, the rain flow counting method was adopted for the statistics collection and analysis of the stress-time history, and the fatigue load spectrum for various key parts of the transmission system was obtained. The result lays a foundation for the fatigue life prediction and reliability analysis for the pure electric vehicle transmission system.

Influence of Different Modified Gears on the Vibration and Noise of Electric Vehicle Reducer

The tooth profile crowning modification was applied onto paired gears for reducing the noise of an electric vehicle reducer. The simulated gear contact spots are compared before and after modification, and are validated by a contact spot experiment. Based on the rigid-flexible coupling model of the gear transmission system, the time-varying meshing stiffness, time-varying meshing force, and the vibration and noise of the gear pairs with different modified gears obtained by simulating calculation are analyzed. The results showed that the selection of modified gear has a great influence on the modification effect. In the way of tooth profile crowning, it is not advisable to modify the pinion independently, as it may increase the frequency and degree of meshing impact between the helical gear pair, making the transmission become less smooth; while modifying the wheel and pinion at the same time can effectively reduce the time-varying meshing stiffness and force, and the vibration and noise. Also, the optimized gear modification scheme is verified by the noise test.

Effects of spalling fault on dynamic responses of gear system considering three-dimensional line contact elasto-hydrodynamic lubrication

Spalling is a common fault in gear system, which directly degrades the performance of the gear system and increases the risk of downtime of the entire system. Until now, lots of efforts have been done to study the underlying mechanism of the spalling fault and its effects on dynamic characteristics for the purpose of accurate diagnosis. However, the lubrication characteristics affected by the spalling fault are usually neglected, which degrades the modelling accuracy of gear system with spalling fault. Therefore, a new dynamic modelling method with spalling fault considering three-dimensional (3D) line contact elasto-hydrodynamic lubrication (EHL) is proposed in this paper. With the established model, the contact characteristics of the lubrication film between gear teeth affected by the spalling fault and the dynamic responses of the gear system can be studied. The main differences of the proposed model are that the meshing force over the lubrication film is calculated iteratively instead of a given quasi-static meshing force and the comprehensive meshing stiffness is obtained considering the effects of both spalling fault and lubrication film. The effects of spalling fault on film pressure distribution, film thickness distribution, film stiffness, comprehensive meshing stiffness and dynamic responses are studied based on the proposed method. Finally, the proposed 3D line contact EHL model is verified by comparing to a traditional EHL model. Effects of the spalling fault size on the comprehensive meshing stiffness and the dynamic responses of the gear system are investigated and discussed. The results reveal that this work can provide a more reasonable analytical method for understanding the contact characteristics and dynamic responses of gear system with spalling fault.

Nonlinear dynamics of a spur gear pair with tooth root crack based on an amplitude modulation function

PurposeGear transmissions are widely utilized in practice. This paper aims to uncouple the crack feature from the cracked time-varying mesh stiffness (TVMS) and investigate the effects of the crack on the nonlinear dynamics of a spur gear pair.Design/methodology/approachAn approximate method to simulate the cracked TVMS is proposed by using an amplitude modulation function. The ratio of mesh stiffness loss is introduced to estimate the TVMS with different crack depths and angles. The dynamic responses are obtained by solving a torsional model which takes the non-loaded static transmission error, the backlash and the cracked TVMS into account. By using the bifurcation diagram, the largest Lyapunov exponent (LLE) and dynamic mesh force, the influences of crack on nonlinear behaviors are examined. The dynamic characteristics are identified from the phase diagram, Poincaré map, dynamic mesh force, time series and FFT spectra.FindingsThe comparison between the healthy and cracked gear pairs indicates that the crack affects the system motions, such as the obvious changes of impact force and unpredictable instability. Besides, the additive and difference combination frequencies can be found in periodic-1 and -2 motions, but they are covered in periodic-3 and chaotic motions. Deeper crack is an important determinant of the nonlinear behaviors at a higher speed.Originality/valueThe research provides an interesting perspective on cracked TVMS and reveals the connection between crack and nonlinear behaviors of the gear pairs.

A new analytical model for evaluating the time-varying mesh stiffness of helical gears in healthy and spalling cases

Accurately evaluating the time-varying mesh stiffness (TVMS) of a mating gear pair is one of the most important aspects in gear vibration dynamic simulation. Existing work on estimating TVMS of helical gears with tooth spalls generally assumes the defects are in flat bottom shape which deviates from the spalling features as observed in curved bottom shape in practice. The cliff like spalling feature results in inaccurate calculation of TVMS of helical gears with such spalls. This paper proposes an advanced method to evaluate the TVMS of helical gears with tooth spalls with curved-bottom features. In the proposed method, the gear tooth is sliced along the center line of tooth spall in order to easily evaluate the equivalent cross-section area and area moments of inertia of the gear tooth which enables the proposed method adapts to tooth spall(s) with curved-bottom features in arbitrary position, angle and size. The proposed method corrects the foundation stiffness errors of existing methods and takes the influence of axial mesh forces into consideration. Its effectiveness on modelling tooth spalls with curved-bottom features for helical gears is validated by FEM (finite element method).

Vibration signal analysis of planetary gear with amplitude, frequency and phase modulation

In this paper, the vibration signal of planetary gear with amplitude, frequency and phase modulation is studied. The proposed mathematical model is employed to in- vestigate the modulation behavior of planetary gear. Based on this model, the ampli- tude modulation (AM) sidebands are analyzed to verify the correctness of theoretical calculation by Inalpolat and Kahraman. Then, the frequency modulation (FM) side- bands and phase modulation (PM) sidebands are also illustrated through an exam- ple analysis. The effects of parameters of planetary gear such as number of planets, teeth of sun and planet phasing relationships on the AM, FM and PM sidebands are analyzed. Finally, the specific expression of transmission error, time-varying mesh stiffness and dynamic mesh force including gear manufacturing error is developed. Time history signal and acceleration spectra of gear mesh interface excitations including AM, FM and PM are investigated for the meshes of sun-planet and ring- planet. The results show that gear parameters have important influence on the mod- ulation behavior. Additionally, manufacturing errors can be introduced to predict the sidebands of planetary gear. The amplitude, frequency and phase modulation study are extremely significant for the noise and vibration reduction, especially the fault diagnosis of planetary gear

Nonlinear dynamics of hypoid gear with coupled dynamic mesh stiffness

This study proposes a dynamics model for hypoid gears, considering mesh stiffness that is dependent on dynamic mesh force. Owing to the fundamental role of mesh stiffness in gear dynamics, its proper and accurate description is critical in any modeling effort. Unlike many works in the literature, which adopted mesh stiffness arrived under quasi-static condition, this study establishes the interaction between mesh stiffness and dynamic response to allow the real time variation of mesh stiffness under dynamic condition. To overcome the unaffordable computational time involved in the determination of dynamic mesh stiffness, a parametric surface approach is employed. A set of parametric simulations is carried out to show evident impact of dynamic mesh stiffness on hypoid gear dynamic response, which is more significant when load becomes lower. The introduced model provides a physics based tool to quantify the interaction between mesh stiffness and dynamic mesh force, capable of capturing more complicated dynamic response of hyoid gear mesh.

Dynamic Characteristics Analysis of Gear-Bearing System Considering Dynamic Wear with Flash Temperature

The influence of the dynamic wear model considering the tooth contact flash temperature on the dynamic characteristics of a gear-bearing system is studied. Firstly, the meshing stiffness model, based on flash temperature theory, is established. Then, the changing of tooth surface temperature and meshing stiffness in the process of gear meshing is analyzed. Next, the initial tooth surface wear is calculated based on the Archard theory, and the dynamic wear model of the system is established. Finally, the effects of initial wear, friction factors, and damping ratio on the system response are studied. The results show that with the increase of fractal dimension D, the uncertainty and the fluctuation amplitude of backlash decrease, and the meshing force decreases. Therefore, the initial tooth surface wear is reduced, and the stability of the system response with a dynamic wear model is improved; with the increase of the friction coefficient, the tooth surface flash temperature rises, and the root mean square value of the vibration displacement of the system amplifies, which indicates that the system tends to be unstable; with the increase of damping ratio, the system changes from unstable quasi-periodic and chaotic motion to the stable periodic motion. The increase of damping accelerates the energy loss of the system and makes the system prone to be stable.

Research on dynamic load sharing characteristics of double input face gear split-parallel transmission system

The face gear power-split system has huge superiorities over the traditional transmission form in the application of modern rotorcraft, and it has become the research trend of the industry in recent years. Thus this paper took the double input face gear split-parallel transmission system used in the rotorcraft as the research target, and established its dynamics model through the lumped parameter theory. Based on the Newtonian second law, the dynamics equations were built and solved to gain the meshing forces and load sharing coefficients of the transmission system. Simultaneously, the impacts of the eccentric errors, support stiffness, and torsional stiffness on the load sharing characteristics were studied. The results show that the meshing forces and load sharing coefficients of each gear pair have periodic changes; the eccentric errors of each drive stage gear have only a significant effect on the corresponding drive stage. Moreover, the changes in the support stiffness of the split-torque shafts and double gear shafts mainly affect the load distribution of the parallel stage, and the shaft torsional stiffness is less sensitively to maintain load balance. In addition, the increment of the shaft stiffness increases the load sharing coefficients of the corresponding gear pairs.

An improved mesh stiffness model of helical gear pair considering axial mesh force and friction force influenced by surface roughness under EHL condition

An improved mesh stiffness calculation model of helical gear pair is proposed, in which the transverse gear tooth stiffness, transverse gear foundation stiffness, in addition to axial gear tooth stiffness, axial gear foundation stiffness, that influenced by surface roughness under elastohydrodynamic lubrication (EHL) condition are fully considered. The time varying friction coefficients are obtained by elastoplastic model under varying surface roughness, and then the influence of friction force on mesh stiffness calculation are investigated. The results show that the improved mesh stiffness calculation model could obtain the mesh stiffness under actual lubrication condition compared with conventional method, in which the axial mesh forces and friction forces acting on the gear tooth and gear foundation are always ignored in the mesh stiffness calculation for helical gear pair.