Mesh Stiffness Calculation Research Articles

On gear time-varying meshing stiffness calculation considering indexing feeds and processing characteristics of gear shaping processing

Gear shaping causes processing characteristics on gear tooth surfaces (PCGTS), e.g., gear tooth surface deviation (GTSD) and processing textures (PT), which can affect gear meshing performances. Gear shaping simulation is operated to obtain PCGTS with different indexing feeds. An improved contact stiffness method of gear tooth surfaces is developed based on a wedge-shaped contact form to obtain tooth contact stiffness with PT caused by gear shaping. An improved time-varying meshing stiffness (TVMS) method for gears manufactured by gear shaping is developed to calculate TVMS of gears with GTSD and PT under different indexing feed conditions. Effectiveness of the improved TVMS method is validated using the finite element method. Analysis results of the proposed contact stiffness method of gear tooth surfaces indicate that PT caused by gear shaping can aggravate fluctuations of gear tooth contact stress. Analysis results of the proposed TVMS method for gears also indicate that GTSD of gear tooth surfaces caused by gear shaping can reduce their TVMS. The proposed TVMS method of gears can effectively analyze TVMS of gear tooth surfaces via gear shaping and evaluate processing settings of gear shaping to improve gear meshing performances.

Time-varying mesh stiffness calculation and dynamic modeling of spiral bevel gear with spalling defects

Time-varying mesh stiffness calculation and dynamic modeling of spiral bevel gear with spalling defects

Loaded tooth contact analysis and meshing stiffness calculation for cracked spiral bevel gears

Tooth cracks may occur in spiral bevel gear transmission system of the aerospace equipment. In this study, an accurate and efficient loaded tooth contact analysis (LTCA) model is developed to predict the contact behavior and time-varying meshing stiffness (TVMS) of spiral bevel gear pair with cracked tooth. The tooth is sliced, and the contact points on slices are computed using roll angle surfaces. Considering the geometric complexity of crack surface, a set of procedures is formulated to generate spatial crack and determine crack parameters for contact points. According to the positional relationship between contact point and crack path, each sliced tooth is modeled as a non-uniform cantilever beam with varying reduced effective load-bearing tooth thickness. Then the compliance model of the cracked tooth is established to perform contact analysis, along with TVMS calculations utilizing three different models. By employing spiral bevel gear pairs with distinct types of cracks as examples, the accuracy and efficiency of the developed approach are validated via comparative analyses with finite element analysis (FEA) outcomes. Furthermore, the investigation on effects of cracks shows that tooth cracks can induce alterations in meshing performance of both entire gear pair and individual tooth pairs, including not only cracked tooth pair but also adjacent non-cracked tooth pairs. Hence, the proposed model can serve as a useful tool for analyzing the variations in contact behavior and meshing stiffness of spiral bevel gears due to different cracks.

A novel semi-analytical method for fillet foundation deflection calculation in presence of a tooth crack propagating into the rim of a spur gear

Existing analytical approaches for calculating fillet foundation deflections in spur gear struggle to adequately address the complexities introduced by tooth cracks, particularly as cracks propagate into the gear body. The finite element methods are accurate but computationally expensive. In response to these challenges, the paper introduces a novel semi-analytical method based on elastic circular ring theory that is capable of calculating the fillet foundation deflection for both healthy and cracked conditions. To establish the usefulness of the method, the fillet foundation deflections obtained are further used to obtain the time-varying mesh stiffness. A methodology centered on compatibility conditions is employed to achieve load distribution during double tooth engagement. The results are verified using two three-dimensional finite element models. In addition, the effect of different hub hole radii, crack lengths and orientations on the fillet foundation deflections and TVMS are investigated. The outcomes suggest that the proposed semi-analytical approach demonstrates superior accuracy in the calculation of spur gear mesh stiffness compared to alternative methods.

A stiffness model for EHL contact on smooth/rough surfaces and its application in mesh stiffness calculation of the planetary gear set

A method for calculating time-varying mesh stiffness (TVMS) is proposed by developing a stiffness model for elastohydrodynamic lubrication (EHL) contact and integrating it into three-dimensional loaded tooth contact analysis (3D-LTCA). To enhance the accuracy and efficiency of the calculation, three projects have been implemented: Firstly, the contact deformation is modified, and the average slope is employed for calculating contact stiffness. Secondly, the lubricant film and EHL contact deformation are introduced into the deformation compatibility equations. Finally, the polynomial chaos expansion (PCE) surrogate model is employed to enhance computational efficiency. The proposed method is used to calculate TVMS of the planetary gear set, and the influence of speed, torque, and rough surface are analyzed. The results show that the previous method underestimating the TVMS of planet-ring gear pair in double-teeth meshing area under fluid lubrication, while the proposed method effectively addresses the issue. The different effects of the torque and speed on TVMS under mixed and fluid lubrication also have been revealed, emphasizing the significance of surface topography in gear meshing characteristics.

Mesh stiffness calculation of defective gear system under lubrication with automated assessment of surface defects using convolutional neural networks

As typical failure mode of gear system, the tooth surface pitting, spalling can be detected in most long-running gear system, especially under heavy-load and high-speed condition, or under lubricant-starvation condition, the tooth pitting is characterized by irregular contour and random distribution. Most previous study on the defective gear system mainly based on manually detection of defective region, or just rely on geometric simplification of defects, leading to inaccurate results with low-efficient method, therefore, the machine-vision-based defect inspection method is proposed in the study of defective gear system. First, the pitting defects on gear tooth surface is detected and segmented based on involutional neural network U-net, then the tooth surface with segmented defective region is mapped to the elastohydrodynamic lubrication model of spur gear system, finally, the tribological behavior in addition to the mesh stiffness under lubrication condition of defective spur gear system are investigated and discussed. The results reveal that the machine-vision-based defects inspection could improve the accuracy and efficiency of the failure study for gear system.

Calculation of Time-Varying Mesh Stiffness of Internal Gears based on Precise Tooth Profile and Dynamic Analysis of Planetary Systems with Root Cracks

The internal meshing spur gear pair is the research object, and the potential energy method is applied to calculate the time-varying meshing stiffness of the internal gear. The internal gear tooth profile is divided into two parts: involute and transition curves, and the gear tooth stiffness is calculated by numerical integration based on accurate tooth profile, which improves the calculation accuracy. Analyzed the influence of different crack size parameters on the stiffness of internal meshing gears. A coupled dynamic model of a planetary system with internal gear crack faults was established, and the influence of cracks on the dynamic response of the planetary system was studied using Zoom-FFT spectrum and cepstrum analysis methods. The simulation results show that as the crack parameter size increases, the mesh stiffness of the internal gear pair gradually weakens, and the periodic vibration impact of the planetary-internal gear pair is also more severe. Cepstrum analysis can easily capture weak fault characteristic frequency information in the system and discover their variation patterns. The research results can provide a theoretical basis for the state monitoring and fault diagnosis of gear systems with crack defects.

A novel method for helical gear modeling with an experimental validation

Dynamic models are important for developing gear diagnostics methods since they allow physical phenomena occurring during operation to be studied in a relatively simple environment. The main challenge in gear modeling is the calculation of the time-variant gear mesh stiffness, and this challenge is even greater in helical gears. The mechanism of helical gears is more complex than in spur gears; the helix angle both adds an axial component to the contact force and also makes the contact line three-dimensional. This study suggests a novel dynamic model for helical gear vibrations that combines an existing validated dynamic model for spur gears with a unique extension for helical gears. The extension is based on a common method called “multi-slice”, according to which the helical tooth width is divided into infinitesimal slices, and each slice is treated as spur tooth. The suggested model introduces a novel implementation of the multi-slice method that overcomes the aforementioned challenges with only few parameters and calculations, depends on the tooth geometry. Furthermore, for the first time in helical gear modeling, the manufacturing profile errors are integrated to the model to generate scatter in the data that can better reflect the reality. The model is validated experimentally and for two different test-rigs by a qualitative comparison of the RMS of the vibration signal. The simulations and the measured data show similar behavior at different ranges of rotational speed and applied load, emphasizing the potential inherent in the model for future work on gear fault diagnosis.

Improved mesh stiffness model of spur gear considering spalling defect influenced by surface roughness under EHL condition

Gear tooth spalling is a typical tooth surface defects in gear transmission. The accurate calculation of the time-varying mesh stiffness of the gear pair with spalling defect under mixed elastohydrodynamic lubrication (EHL) plays an important role in tooth damage prediction and fault feature analysis. In this work, an improved time-varying mesh stiffness (TVMS) model of spalling faulty gear in mixed EHL line contact considering the coupled effect of rough surface topography is established. The proposed model considers the combined effect of spalling defect, surface roughness topography, and lubrication on the contact characteristics of the meshing interface to obtain a revised contact stiffness, which is further used to determine the comprehensive TVMS. The cylindrical contact coefficient for curved meshing interface with spalling fault is derived and incorporated into the statistical micro-contact Greenwood and Williamson model (GW model) to obtain the asperity contact stiffness. The revised contact stiffness at the rough faulty gear interface is obtained from the parallel action of the asperity contact stiffness and oil film stiffness under load-sharing concept. The dynamic responses of the gear system are analyzed using the established mesh stiffness model through the six degrees of freedom translational-torsional dynamic model, in which the dynamic meshing force is calculated iteratively using the transit surface topography and geometrical parameters along the line of contact instead of a quasi-static meshing force. Effects of multi-parameters as surface roughness, spalling fault size, and lubricant on the mesh characteristics and dynamic responses of the gear system are further investigated and discussed.

Time-varying mesh stiffness calculation of spiral bevel gear with spalling defect

Spiral bevel gears often operate at high speeds and under heavy loads, which can easily lead to the occurrence of spalling faults on the tooth surfaces, particularly in cases of inadequate lubrication. In this study, we introduce an efficient and accurate numerical calculation method to determine the time-varying meshing stiffness (TVMS) of a spiral bevel gear pair with actual spalling faults, taking into consideration the effects of multi-tooth meshing and the flexibility of the gear flank. In parallel, a finite element model is established to verify the accuracy of the proposed method across various spalling parameters. We delve into the impact of spalling length, width, location, and shape. The results demonstrate that the TVMS values obtained from our proposed method align closely with those obtained using the finite element method. The range of TVMS reduction caused by the spalling fault is primarily determined by the spalling location and width. Moreover, the magnitude of TVMS decline is strongly associated with the spalling length. Furthermore, the clearances between two conjugated tooth surfaces change due to the presence of the spalling fault, resulting in a redistribution of contact forces on the same contact surface and other meshing tooth surfaces.

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.

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.

Dynamics analysis of spur gears considering random surface roughness with improved gear body stiffness

In previous calculation of gear mesh stiffness, the gear body is typically treated as a deformable elastic ring, and the stiffness of the gear body is determined using Sainsot's empirical formula. In this paper, a new calculation method for determining the gear body stiffness based on the analytical finite element-potential energy method is proposed, and its feasibility is verified by the finite element method. A new nonlinear dynamic model for a spur gear system is established, taking into account the time-varying backlash and random surface roughness. This model aims to investigate the dynamical responses of the gear system under various parameters, including rotational speed, shaft hole radius and surface roughness. The results indicate that the proposed method for calculating the stiffness of the gear body is significantly more accurate than Sainsot's empirical formula, as it closely aligns with the finite element method. This leads to a more precise calculation of the time-varying mesh stiffness. The gear dynamics model incorporating random surface roughness exhibits a heightened degree of realism and complexity in its dynamic phenomena. This study provides a valuable reference for the future development of dynamic gear systems.

An improved time-varying mesh stiffness calculation method and dynamic characteristic analysis for helical gears under variable torque conditions

Based on the slice theory and numerical calculation methods, the time varying meshing stiffness (TVMS) calculation method of the gear pair was constructed for the helical gear pair of a electric continuously variable transmission (ECVT), and the TVMS of the helical gear were calculated under different constant and time-varying torque in this study. The obtained stiffness values were introduced into the established dynamics model of helical gear system, and the influence of changed TVMS, resulting from the variable speed and torque, on the nonlinear dynamic characteristics of gear pair was analyzed using the Runge-Kutta method. The results show that the proposed TVMS calculation method is very effective in computing the TVMS of helical gear pair under time-varying condition. The TVMS and dynamic transmission error (DTE) increase as the torque increase, and vice versa. Meanwhile, the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors at high speed and at low speed when torque is low, whereas not appear at steady high torque under low speed range. This study offers a method for gaining the TVMS of helical gear pair to analyze the dynamic characteristics of gear pair under actual working condition, and the influence of torque that change with the speed ratio on the dynamic performance of a gear system should be considered in the industrial applications.

Analytical calculation of mesh stiffness for spiral bevel gears with an improved global tooth deformation model

The paper proposes an efficient and accurate analytical calculation model of time-varying mesh stiffness (TVMS) for the spiral bevel gear. Firstly, an enhanced numerical loaded tooth contact analysis (NLTCA) model that considers both the global tooth and local contact deformations is developed. Based on Tredgold's approximation, spiral bevel gears are sliced into a series of spur gears. An improved global tooth deformation model is developed by accounting for axial tooth deformation and the effect of non-contacting tooth slices. The influence coefficient method is utilized to determine the local contact deformation. Subsequently, based on the results of the NLTCA model, a detailed calculation procedure for determining the TVMS of spiral bevel gears is described. Finally, some numerical examples are provided to illustrate the efficiency and accuracy of the proposed model through comparisons to finite element analysis (FEA) results. The proposed model can achieve satisfactory accuracy without FEA modification. Furthermore, ignoring the effects of the non-contacting tooth slices will lead to excessive elastic deformation when the length of the contact area across the tooth width is significantly shorter than the tooth width.

A novel analytical method for mesh stiffness calculation of helical gears with tooth profile modification

A novel analytical method for mesh stiffness calculation of helical gears with tooth profile modification

Mechanism analysis for GDTE-based fault diagnosis of planetary gears

Planetary gear transmission systems are widely utilized in production and daily life equipment, and their fault diagnosis is crucial. The most common tool in this field is the vibration monitoring. In this study, a different tool based on global dynamic transmission error (GDTE) is investigated from a mechanism analysis perspective. The aim is to explore the dynamic responses of planetary gears with different pitting degrees, establish the mapping relationships between GDTE and gear damages, and reveal the mechanism of GDTE-based fault diagnosis. To ensure the accuracy of dynamics model, a novel pitting modeling method based on matrix equations is proposed, which makes the growth process of pitting more physically grounded and the model for irregular pitting on tooth surfaces more practical by considering the termination during random expansion of pitting. Meanwhile, the matrix equation-based pitting model and potential energy method-based time-varying meshing stiffness calculation can be perfectly matched to ensure the precision of subsequent dynamic response analysis. More practical pitting model and accuracy stiffness ensure that the planetary gear dynamics model outputs more precise dynamic responses. Simple time and frequency domain analysis performed on GDTE show that GDTE contains rich information about gear health status. The mapping relationships between GDTE and gear pitting damages are revealed and validated by multi-body dynamics methods. GDTE provides a promising tool for gear condition monitoring and fault diagnosis.

A New Fast Calculating Method for Meshing Stiffness of Faulty Gears Based on Loaded Tooth Contact Analysis

Gear transmission systems are widely used in various fields. The occurrence of gear cracks, tooth pitting, and other faults will lead to the dynamic characteristics deterioration of the transmission system. In order to calculate the meshing stiffness of faulty gear pairs more effectively and precisely, this article improves the loaded tooth contact analysis (LTCA) method by analyzing the influence of different fault types on gear deformation, including bending-shearing deformation and contact deformation, which combines the accuracy of the finite element method (FEM) and the rapidity of the analytical method (AM). The improved LTCA method can model the fault areas accurately and optimize the deformation coordination equation under the actual meshing situation of the faulty gear tooth, making it suitable for calculating the meshing stiffness of faulty gears. Based on the calculation results of the finite element method, the accuracy of the improved meshing stiffness calculation method has been verified, and the sensitivity of different fault type parameters on meshing stiffness has been studied.

A new method for the single mesh stiffness calculation of helical gear considering slice coupling effect

In this paper, a new method is proposed, calculating the single mesh stiffness by cobining the silce principle and contact line analysis. Firstly, this method obtains the length of the contact line of the helical gear through contact line analysis. Secondly, along the tooth face width, the teeth of helical gear is separated into a series of equivalent thin teeth of spur gear. Thirdly, in order to consider the slice coupling effect, a weighting factor is introduced to modify the stiffness of the thin tooth itself according to the contribution of the thin tooth to the overall stiffness, which is distributed in parabola. Finally, by integrating the equivalent thin teeth of the spur gear along the contact line, the single mesh stiffness is calculated. The investigation shows that the influence of the slice coupling effect on single meshing stiffness related to the relationship between the value of transverse contact ratio and overlap ratio. When transverse contact ratio is greater, the influence of the slice coupling effect on the single mesh stiffness is very small an can be ignored, and when the overlap ratio is larger, which is significant.

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.