Meshing Force Research Articles

Time-varying mesh stiffness calculation for gear pairs with misalignment errors in multiple degrees of freedom based on an analytical method

Misalignment errors (MEs) in multiple degrees of freedom (multi-DOFs) at the mesh position will lead to a change in the time-varying mesh stiffness (TVMS) and then affect the dynamic behaviour of gear pairs. Therefore, a new, more rapid, three-dimensional analytical model for TVMS calculation for gear pairs with three rotational and three translational MEs is established in this paper, and a new solution method based on potential energy theory is presented. In addition, the correctness of the new model is verified by the finite element method (FEM). Moreover, the effective contact line, uneven distribution of mesh force on the contact line, and mesh position change are taken into account. Finally, the TVMS under different ME conditions is calculated with the new analytical model. The results showed that the different MEs have dissimilar effects on the TVMS, and the relationship between the ME and TVMS is nonlinear. In addition, the region of single-pair and double-pair teeth in contact would also change with ME.

Study on vibration characteristics of hydraulic planetary transmission considering engine and torque converter excitation

Large-size vehicle loaders are mainly used in various construction projects with complex working environment, which has a serious impact on the internal vibration of the whole system. In this study, the vibration characteristics of hydraulic planetary transmission of vehicle loader considering the influence of the external excitation are evaluated theoretically and experimentally. The output torque of the four-stroke six-cylinder diesel engine is obtained by force calculation of crank and connecting rod mechanism. The equivalent stiffness and damping of the torque converter are solved according to the torsional dynamic equation of the torque converter. The trapped oil area, oil pressure, and flow rate of gear meshing part are extracted to obtain the trapped oil pressure of variable-speed pump. Then, the multi-degree-of-freedom gear system dynamic model is established to calculate the meshing force, which is applied to vibration response analysis model by using mode superposition method. Lastly, the vibration response test is performed on the experimental prototype to verify the calculation method. The conclusion shows that the generation principle of external excitation and its calculation method in this paper are feasible in the analysis of dynamic characteristics of hydraulic planetary torque converter.

Simulation based Approach to Study the Effect of Hypoid Gear Manufacturing Variability on In-Cabin Noise

<div class="section abstract"><div class="htmlview paragraph">With increase in demand for quieter product and reduction in masking noise, axle whine management plays a crucial role in the early product development process. Whine is tonal in nature and humans are more sensitive to tonal memory, hence this makes user to experience a very unpleasant ride which in turn results in bad product credibility. Dynamic mesh force excitation is the cause of the axle whine noise. Critical factors in consideration are gear micro geometry variability, misalignments, temperature of operation and resulting bearing pre-load, operating loads, and structural resonances that carry the excitation to the occupant’s ear. The variability associated with gear micro-geometry plays crucial role during optimization in the quest for robust gear design. This work attempts to study the influence of manufacturing variability on transmission error and dynamic mesh force with the aim of helping design engineers understand and consider manufacturing variability during target setting and gear design.</div></div>

Generation mechanism and evolution of five-state meshing behavior of a spur gear system considering gear-tooth time-varying contact characteristics

Teeth disengagement or back-side teeth engagement induced by backlash reduces the transmission quality and dynamic performance of gear systems, and the accurate interpretation of multi-state meshing behavior can provide guidance for structural optimization and performance evaluation. Therefore, the multi-state meshing behavior of the gear system is elaborated. A new nonlinear dynamic model of a spur gear system with five-state meshing behavior is established based on the time-varying backlash and contact ratio. The time-varying meshing stiffness and time-varying backlash considering the elastic contact of gear teeth, gear temperature rise and lubrication are included in the model. The five-state meshing behavior is clearly characterized by constructing five Poincaré maps, and its generation mechanism is revealed using dynamic meshing force time history, teeth relative displacement time history and phase portrait. The bifurcation and evolution of five-state meshing behavior are analyzed under the effects of load factor, meshing frequency and error coefficient. The results show that the mutation in the direction of dynamic meshing force leads to teeth disengaging and back-side single- or double-tooth contact, forming multi-state meshing behavior. Bifurcation caused by parameter changes greatly affects the evolution of five-state meshing behavior, particularly grazing bifurcation can decrease the number of teeth disengagement. Chaotic behavior or trajectory expansion inspires multi-state meshing vibration of the system. Previous gear system models could not reveal these phenomena due to ignoring the multi-state meshing behavior.

Dynamic Influence of Wheel Flat on Fatigue Life of the Traction Motor Bearing in Vibration Environment of a Locomotive

Wheel flat can cause a large impact between the wheel and rail and excites a forced vibration in the locomotive and track structure systems. The working conditions and fatigue life of the motor bearings are significantly affected by the intensified wheel–rail interaction via the transmission path of the gear mesh. In this study, a fatigue life prediction method of the traction motor bearings in a locomotive is proposed. Based on the L−P theory or ISO 281 combined with the Miner linear damage theory and vehicle–track coupled dynamics, the irregular loads induced by the track random irregularity and gear mesh are considered in this proposed method. It can greatly increase the accuracy of predictions compared with the traditional prediction models of a rolling bearing life whose bearing loads are assumed to be constant. The results indicate that the periodic impact forces and larger mesh forces caused by the wheel flat will reduce the fatigue life of the motor bearings, especially when the flat length is larger than 30 mm. Using this method, the effects of the flat length and relatively constant velocity of the locomotive are analyzed. The proposed method can provide a theoretical basis to guarantee safe and reliable working for motor bearings.

Mesh Force Modelling and Parametric Studies for Compound Oscillatory Roller Reducer

A compound oscillatory roller reducer (CORR) with a first-stage gear transmission and a second-stage oscillatory roller transmission is presented. The transmission principle of oscillatory roller transmission is introduced, and the tooth profile equation of the inner gear is derived. The analytical model of mesh force considering the installation errors and manufacturing errors is proposed. Then, parametric studies considering different errors on the mesh force are conducted. Results show that the design parameters are significant factors for mesh force. The mesh force is reduced by 17% as the eccentricity of disk cam increases from 2.5 mm to 4 mm. When the radius of the movable roller increases from 7 mm to 20 mm, the mesh force decreases by 8%. As the radius of disk cam increases from 125 mm to 170 mm, the mesh force is decreased by 26.5%. For the impacts of errors, the mesh force has a noticeable fluctuation when these errors exist including the manufacturing error of disk cam, the installation error of disk cam and the manufacturing error of movable roller change. The prototype of the reducer is manufactured and preliminary run-in test proved the feasibility of the transmission principle.

Vibration feature of spur gear transmission with non-uniform depth distribution of tooth root crack along tooth width

Gear tooth root crack is a typical fault of gear transmissions and it threatens the operation safety of gear systems. Therefore, precise detection of the tooth root crack is of a great significance for the prevention of serious safety accidents. As one of the commonly used methods, vibration-based detection has drawn great attentions. However, the mapping relationship between the vibrations and the severity of tooth root crack has not been established well, especially for the crack with non-uniform distribution due to the complexity of its dynamic modelling. To address this problem, an improved gear dynamics model considering non-uniform distribution of the gear tooth root crack fault is established in this paper by using the ‘slice method’. Compared with the traditional methods, the advantage of the proposed model is that it enables the accurate modelling of the non-uniform dynamic mesh forces along tooth width that will further excite the support bearings. Then, the dynamic simulation of the gear transmission with non-uniform tooth root crack is performed. The results indicate that the improved model can reveal the vibration characteristics of the gear transmission due to the extra roll and yaw motions caused by the non-uniform tooth root crack. In addition, the correlation between the vibration and the crack size and location is quantified. This study could provide theoretical reference for quantitative detection and diagnosis of gear tooth root crack.

Dynamics modeling and analyzing of spur gear pair with pitch deviation considering time-varying contact ratio under multi-state meshing

The pitch deviation caused by machining error affects the transmission stationarity of the spur gear pair. It results in time-variation of the contact ratio for the system by corner contact. Meshing characteristics of the spur gear pair with the pitch deviation are studied in order to analyze the time-varying contact ratio and motion characteristics of the system. The mathematical model of time-varying contact ratio is established to explore its time-varying characteristics. Dynamics model of the spur gear pair with the pitch deviation considering time-varying contact ratio under multi-state meshing is proposed. The transmission stationarity of the spur gear pair are explored by dynamic meshing forces. The influence of system parameters and operating parameters on dynamics characteristics of the spur gear pair with the pitch deviation considering time-varying contact ratio is studied by bifurcation diagrams, phase portraits and Poincaré maps. Results show that the pitch deviation as a short-period error in the spur gear pair leads to complex periodic motions. The possibility of the tooth disengagement, back-side teeth contacting and chaotic motion for the spur gear pair increases with the increase of complexity of pitch deviation values. The transmission accuracy and stationarity for the spur gear pair decrease when pitch deviation values are complex. This study provides a theoretical basis for dynamics modeling and stationarity analysis of the spur gear transmission system and other gear transmission systems.

Vibration characteristics and condition monitoring of internal radial clearance within a ball bearing in a gear-shaft-bearing system

Internal radial clearance is a key factor influencing bearing fatigue life. Moreover, bearings inevitably suffer from various wears and tears, which result in gradual increase of clearance and shorten bearing life. Monitoring bearing clearance changes using vibration can effectively indicate the bearing wear and provide good leading time to perform maintenances. Previous studies show that vibration at ball pass frequency on outer race (BPFO) can be based for clearance monitoring. However, such clearance induced vibration has not been well understood, especially under complicated dynamic interactions such as in a gearbox system. To fill this gap, this paper presents a nonlinear gear-shaft-bearing-housing vibration model with fourteen degree of freedom (DOF) to investigate the vibration responses under the dynamic gear meshing force and progressively changed radial clearances at first. Then, the model was verified through a two-stage spur gearbox. Furthermore, bearing characteristics with different radial clearances under the influence of gear are revealed and indicator based on modulation signal bispectrum-sideband estimator (MSB-SE) was proposed. Finally, vibration data from a run-to-failure gearbox test rig was utilized to verify the effectiveness of the MSB-SE indicator for bearing clearances monitoring. Simulation results show that BPFO is modulated on gear meshing frequency (GMF) and BPFO amplitude from envelope spectrum increases with bearing clearances under the influence of gear meshing. Indicator based on MSB-SE, possessing the capability of purifying the interferences of gear meshing and strong noises, is effective to capture the variance of bearing clearances. The experiment based on a run-to-failure gearbox test rig provided evidence for the effectiveness of the proposed indicator, which is more accurate than BPFO amplitude from conventional envelope analysis and time-domain indicators, such as RMS and kurtosis. These findings are of significance for bearing fault diagnosis and maintenance.

An Improved Dynamic Transmission Error Model Applied on Coupling Analysis of Gear Dynamics and Elastohydrodynamic Lubrication

Abstract In order to study the coupling effect of cylindrical gear dynamics and elastohydrodynamic lubrication (EHL), based on dynamic theory, a dynamic model of a six degrees-of-freedom (6DOF) coupled system is established. Based on the numerical solution of EHL, an improved dynamic transmission error (DTE) model was established. The oil film stiffness, oil film damping, and center oil film thickness obtained by the numerical solution were used in the gear dynamics analysis. The dynamics and EHL characteristics of the gear system are analyzed and compared with the general model. The results show that by using the improved DTE model, the comprehensive stiffness and comprehensive damping of the gear increase and fluctuate slightly. The amplitude of the tooth surface meshing force and dynamic response decreases. Under the action of tooth surface meshing force, the oil film pressure and thickness show oscillation characteristics.

An Analysis of the Gear Meshing Characteristics of the Main Planetary Gear Trains of Helicopters Undergoing Shafting Position Changes

The complex three-shaft three-reducer structural designs of helicopter transmission systems are prone to changes in the relative positions of shafting under the conditions of main rotor and tail rotor loads. These changes will affect the transmission characteristics of the entire transmission system. In this study, the planetary gear trains of helicopters were examined. Due to the fact that these structures are considered to be the most representative structures of the main reducers of helicopters, they were selected as the study objects for the purpose of examining the meshing characteristics of planetary gear trains when the relative positions of the shafting changed due to the position changes of the main rotor shafts under variable load conditions. It was found that by embedding the comprehensive time-varying meshing stiffness values of the main rotor shafts at different positions, a dynamic model of the relative position changes of the planetary gear trains could be established. Then, combined with the multibody dynamics software, the meshing characteristics of the sun gears, and the planetary gears, the planetary gears and the inner ring gears were simulated and analyzed under different inclinations and offsets of the shafting. The results obtained in this study revealed the following: (1) the average meshing force of the gears increased with the increases in the angle inclinations, and the meshing force between the sun gears and the planetary gears increased faster than the meshing force between the planetary gears and the inner ring gears. It was observed that during the changes in the shafting tilt positions, obvious side frequency signals had appeared around the peak of the meshing frequency in the spectrum. Then, with the continuous increases in the tilt position, the peak was gradually submerged; (2) the average meshing force of the gears increased with the increases in the offset, and the increasing trend of the meshing force between the sun gears and the planetary gears was similar to that observed between the planetary gears and the inner ring gears. It was found that when the shafting offset position changed, there were obvious first and second frequency doubling in the spectrum; (3) the mass center orbit radii of the sun gears increased with the increases in the shafting position changes, and the changes in the angular tilt position were found to have greater influencing effects on the mass center orbit radii of the sun gears than the changes in the offset positions. This study’s research findings will provide a theoretical basis for future operational status monitoring of the main transmission systems of helicopters and are of major significance for improvements in the operational stability of helicopter transmission systems, which will potentially ensure safe and efficient operations.

Input Torque Measurements for Wind Turbine Gearboxes Using Fiber Optical 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 6MW, 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 is 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.

Time-Domain Dynamic Modeling and Analysis of Complex Heavy-Duty Gearbox Considering Floating Effect

Expert insights into the time-domain dynamic behavior of heavy-duty gearboxes form the foundations of design evaluation and improvement. However, in the existing lateral–torsional coupling (LTC) modeling method for gearboxes that is normally used for frequency-domain dynamic behavior, the meshing forces are modeled as spring dampers with fixed acting points on the meshing gears to simulate only the transient LTC effect, and thus the steady state characteristic in the time domain cannot be obtained due to the unrealistic distortion of positions and orientations as the gear angles increase. In this paper, a novel and generally applicable LTC modeling method for heavy-duty gearboxes, mainly planetary gear sets with floating components, is proposed by using space-fixed spring dampers with floating acting points on the meshing gears to study the time-domain dynamic response and to support the dynamic design of heavy-duty gearboxes. Based on the proposed method, a LTC model of a 2 megawatt (MW) wind turbine gearbox with floating components considering the time-varying meshing stiffness, bearing stiffness, torsional stiffness, and floating effect was established. The simulated results of representative components were in accordance with experimental results on a test rig, and dynamic behavior was calculated.

Fault simulation and forecast of helical cylindrical gear of reducer based on ADAMS

Aiming at the problems that the dynamic speed and meshing force of the helical cylindrical gear of the reducer are difficult to actually measure and the test cost is high, the helical cylindrical gear transmission system of a crane reducer is used as the research object, and the 3D modeling software Solid Works is used to establish the assembly model of the gear transmission system, import the dynamic simulation software ADAMS to establish simulation model, and perform simulation analysis on it. The simulation results show that the gear speed is stable after acceleration, which is basically close to the theoretical calculation value; meshing at different multiples the gear meshing force amplitude is different at different frequencies. As the frequency multiplier increases, the amplitude shows a downward trend. Finally, Ensemble Empirical Mode Decomposition (EEMD) sample entropy and Extreme Learning Machine (ELM) are used for fault prediction to verify the effectiveness of this method. The above can provide a certain reference basis for the condition monitoring and fault prediction of the gearbox of metallurgical cranes, so as to ensure the safe operation of the crane.

Hybrid Dynamic Modeling and Analysis of High-Speed Thin-Rimmed Gears

Abstract In this study, a hybrid dynamic model of high-speed thin-rimmed gears is developed. In this model, the translational and angular displacements (including the rigid and vibration displacements) with a total of six degrees-of-freedom (DOFs) are selected as the generalized coordinates for each gear, and the meshing force distributions along the contact line and between the teeth are considered. Thus, the model can be implemented under stationary and nonstationary conditions. The condensed finite element models (FEMs) are developed with the centrifugal and inertia forces for gear bodies. This paper proposes a novel method to couple the lumped-parameter model and condensed finite element model for the hybrid dynamic model system, which considers the variation of the meshing tooth during the gear operation, namely, the variations of the acting point of meshing force. Based on the model, the dynamic analysis of high-speed thin-rimmed gears is conducted under stationary speed and acceleration processes. The effects of the flexible gear body, high speed, and tooth errors on the system dynamics and tooth load distribution are investigated. The analysis results are also compared with the current reference and pure finite element method to validate the proposed model.

Investigation of Load Sharing and Dynamic Load Characteristics of a Split Torque Transmission System with Double-Helical Gear Modification

A new dynamic model for a two-input two-path split torque transmission system which considers meshing error, time-varying meshing stiffness, and meshing-in impact is proposed. Time-varying meshing stiffness and meshing-in impact of each gear pair are accurately calculated based on tooth contact analysis and loaded tooth contact analysis. Equivalent displacements of eccentricity error and installation error along the meshing line of second- and third-stages gears are derived. The modified tooth surface of a third-stage double-helical gear is obtained by optimizing the amplitude of static loaded transmission error and meshing-in impact via nondominated sorting genetic algorithm-II (NSGA-II). Influence of modification on load sharing and dynamic load characteristics of split torque transmission system is investigated. The results indicate that the system’s dynamic meshing force increases when meshing-in impact is accounted for, which is unfavorable for the transmission. Following the modification of a double-helical gear, the dynamic load characteristics of the split torque transmission system are significantly improved, while its load sharing characteristics are improved to a certain extent.

Numerical calculation and experimental measurement for gear mesh force of planetary gear transmissions

In this paper, we propose a numerical method to calculate the dynamic mesh forces of planetary gear transmissions (PGTs). In this model, the effects of addendum modification are included in the time-varying mesh stiffness (TVMS) and then the influence of TVMS and relative mesh phase on the dynamic mesh forces are considered in the proposed model. To verify the proposed model, an experimental method is proposed to measure the dynamic ring-planet mesh forces by using gear tooth root-fillet strain signals. Static mesh experiments and dynamic mesh experiments are carried out to obtain the strain signals. To measure the dynamic mesh force, the loaded tooth contact analysis (LTCA) method and quasi-static experiments are carried to determine the coefficient matrix which is used to descript the relationship between the mesh force and tooth root-fillet strain. Finally, calculated mesh forces and measured forces are compared and discussed. Results indicate that the proposed experimental method can effectively measure the dynamic ring-planet mesh forces, and the numerical mesh forces agree well with the measured mesh forces.

Effect of silane coupling agents on flowability and compressibility of compound for bonded NdFeB magnet

Based on the difference of the Y-terminal functional group of the silane coupling agent (Y–Si(X)3), four different silane coupling agents were employed to pretreat the surface of the NdFeB powders. The effects of silane coupling agents on the flowability and compressibility of compounds for preparing bonded NdFeB magnets were studied. It is indicated that compounds pretreated by silane coupling agents have weaker friction and meshing force. The apparent density is increased by 0.3 g/cm3 compared with the compound without silane coupling agent, and the radial crushing strength is significantly increased by about 3–4 times. In addition, the epoxy resin is more evenly distributed on the surface of the compounds treated by silane coupling agents observed by scanning electron microscopy, and some agglomerated particles are produced. Also, the compressibility of compounds with silane coupling agents is significantly improved due to the fact that hardening exponents are reduced. However, the addition of silane coupling agents has almost no effect on the magnetic properties of bonded magnets. The special energy was used to manifest the flowability of magnetic powder particles representing the macroscopic performance of the force between powder particles, providing a new direction for the study of the interface compatibility of two-phase or multiphase composite materials.

Modeling and analyzing of nonlinear dynamics for spur gear pair with pitch deviation under multi-state meshing

The pitch deviation of gears caused by machining leads to corner contact in actual meshing process. In order to study the influence of the pitch deviation on motion characteristics of the spur gear pair, meshing state functions are constructed to reflect multi-state meshing including mesh-in at corner contact, mesh-out at corner contact, single and double pair drive-side teeth mesh, single and double pair back-side teeth contact and tooth disengagement. The nonlinear dynamics model of spur gear pair with pitch deviations under multi-state meshing is established. Variation laws of dynamic meshing forces and meshing states for the spur gear pair are analyzed. The influence of main parameters on nonlinear dynamics of the spur gear pair with pitch deviations is investigated. Results indicate that the pitch deviation results in complex periodic motion as a short-period error in spur gear pair. It mainly affects transmission stability of the spur gear pair by changing meshing states. This investigation provides a theoretical basis for the parameter design of the spur gear transmission system and the influence of pitch deviation in other gear transmission systems.

Lubrication Performance of Helical Gear Pair Under Dynamic Load Based on Local Involute Profile

Abstract The tribological properties of gear interface have been widely concerned in the past decades. In this study, based on the local involute profile and crown modification, a thermal elastohydrodynamic lubrication model is proposed for a helical gear pair. To discuss the influences of dynamic load on the tribological properties of helical gear pair, the dynamic mesh force of tooth surface is obtained by torsional vibration model. The influences of working conditions and surface roughness on the tribological properties of helical gear pair are investigated. The tribological properties are evaluated in terms of the average film thickness, friction coefficient, mechanical power loss etc. Results show that the dynamic effect of gears has a significant effect on the tribological properties, especially at a specific speed, such as resonance speed. In order to simulate gear lubrication accurately, it is recommended to adopt local involute tooth profile and consider tooth profile modification to calculate geometric clearance. The influence of input rotation speed on the dynamic characteristics and tribological properties of gear is more significant than that of input torque. The surface roughness significantly changes the distribution of interface pressure and film thickness. With the increase of roughness amplitude, the local fluctuation amplitude of pressure and film thickness increase and the dry contact occurs at the end of contact line. Meanwhile, the maximum subsurface stress moves toward the tooth surface, especial for the high frequency engineering roughness. This local stress concentration is harmful to the fatigue life of gear meshing process.