Mesh Stiffness Calculation Research Articles

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.

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.

Mesh stiffness calculation and vibration analysis of the spur gear pair with tooth crack, considering the misalignment between the base and root circles

AbstractAn improved variable cross‐section cantilever beam model for evaluating the time‐varying mesh stiffness (TVMS) of the perfect gear tooth is developed in which the tooth number of driving gear is less than 42 and that of driven is more than 42. The TVMS obtained by the proposed method is compared with the result without considering the misalignment between the base circle and gear root. Four types of root crack models and changes in TVMS of 13‐crack levels are presented. The fault vibration characteristic of a single‐stage spur gear reducer with root crack is analyzed and the correctness is qualitatively verified by the vibration signals of an experimental gearbox with crack or missing failure. The results presented in this paper are of great significance for a deep understanding of the possible causes of vibration and noise of gears and provide a theoretical foundation for the fault diagnosis of the gearbox.

Time-varying meshing stiffness calculation of an internal gear pair with small tooth number difference by considering the multi-tooth contact problem

Time-varying meshing stiffness calculation of an internal gear pair with small tooth number difference by considering the multi-tooth contact problem

Improved mesh stiffness calculation model of comprehensive modification gears considering actual manufacturing

Comprehensive modification refers to a modification method that simultaneously performs lead crown modification and tooth profile modification, which is widely used in precision gear manufacturing. However, there are few time-varying meshing stiffness (TVMS) models of spur gears with comprehensive modification. Based on the actual manufacturing methods and slice theory, an improved analytical model (IAM) for calculating the TVMS considering the coupling modification is established. The model does not require to use the finite element method (FEM) to determine the correlation coefficients. The IAM is compared with the published models and verified by the FEM. The results show that the IAM has higher accuracy and efficiency in the calculation of lead crown modification, tooth profile modification and comprehensive modification. The gear bearing capacity of machining method 2 is better than that of method 1. It is useful to improve the calculation accuracy to choose the number of slices reasonably according to the modification amount.

A refined analytical model for the mesh stiffness calculation of plastic gear pairs

The accurate calculation of the mesh stiffness for a plastic gear pair is significant for analyzing its transmission performance. During the meshing process, factors such as gear tooth meshing deformation and thermal effect will cause variations of the meshing position and tooth load particularly for plastic gears. These increase the difficulty for the accurate calculation of the gear mesh stiffness for plastic gear pair. To address this issue, the instantaneous gear tooth meshing deformation is obtained based on the potential energy method. Then, the effects of the above factors on the contact condition of a meshing tooth pair are analyzed, based on which the actual mesh position is accurately obtained via an iterative algorithm. Under the commonly-used elastic contact assumption, the contact stiffness at the new meshing position is calculated based on the Hertzian contact theory, and the algorithm of the mesh stiffness calculation for a plastic gear pair is derived by analyzing the influence of the elastic approach of meshing tooth on load distribution mechanism of gear pair. Compared with previous analytical methods, the proposed algorithm for mesh stiffness calculation shows higher accuracy, and it can also properly analyze the effect of load and other parameters on the mesh behavior of plastic gear pair, which can provide a theoretical basis for the performance analysis of plastic gear transmission system.

A critical review on FE-based methods for mesh stiffness estimation in spur gears

Several methodologies for the mesh stiffness calculation in spur gears have been found in the literature, in the last decades. Based on a bibliographical analysis and the research activities of the authors, attention has been devoted to three methodologies, which appeared as the most employed and cited. A critical review of such methodologies, FE-based method, hybrid analytical-FE method, and analytical approach, has been undertaken, highlighting some crucial aspects regarding the mesh stiffness estimation process. Within this framework, useful information related to the critical choice of parameters needed for the implementation of the methods are presented, in order to provide further guidelines for their application. Furthermore, the methods are compared in terms of results accuracy and computational time for full blank and lightweight gears. This comparison leads to useful guidelines regarding the definition of the most suitable approach depending on the specific requirements. Finally, two variants of an examined hybrid method are presented. They are based on linear, nonlinear FE analyses and analytical formulation. Their purpose is to compare the use of different analytical formulations against the FE formulation for the computation of the local deformation of gear teeth.

An improved analytical model for gear mesh stiffness calculation

Time-varying mesh stiffness is considered the primary excitation in gear vibrations. An accurate evaluation of mesh stiffness is of great importance when studying gear vibrations. Based on contact mechanics and gear engagement principle, we propose an improved analytical model to determine gear mesh stiffness. Gears with addendum modification and varying-load conditions are considered in the presented model. It is shown that the gear mesh stiffness is load-dependent, and the contribution of each component to the gear mesh stiffness is discussed. Experimental investigations based on strain gauges are presented to measure the mesh stiffness of an internal spur gear under different loading torques. In addition, existing methods are compared and discussed with the presented model. The results indicate that the proposed model can calculate time-varying mesh stiffness accurately under non-stationary conditions.

A Novel Loaded Tooth Contact Analysis Procedure With Application to Internal–External Straight Bevel Gear Mesh in a Pericyclic Drive

Abstract A comprehensive numerical loaded tooth contact analysis (LTCA) model is proposed for straight bevel gears that exhibit large number of teeth in contact, well beyond the involute line of action limits. This kind of contact is observed when the meshing gears have conformal surfaces, as in a pericyclic mechanical transmission, and is traditionally analyzed using finite element simulations. The pericyclic drive is kinematically similar to an epicyclic bevel gear train and is characterized by load sharing over large number of teeth in an internal–external bevel gear mesh, large shaft angles (175–178 deg), nutational gear motion, and high reduction ratio. The contact region spreads over a large area on the gear tooth flank due to high contacting surface conformity. Thus, a thick plate finite strip method (FSM) was utilized to accurately calculate the gear tooth bending deflection. Based on the tooth deformation calculation model and accounting for initial surface separation, a variational framework is developed to simultaneously solve for load distribution and gear tooth deformation. This is followed by calculation of contact stress, bending stress, mesh stiffness, and transmission error. The results demonstrate the high-power density capabilities of the pericyclic drive and potential for gear noise reduction. The model developed herein is applied with real gear tooth surfaces, as well.

Investigation of tooth crack opening state on time varying meshing stiffness and dynamic response of spur gear pair

In the traditional calculation model of mesh stiffness, the tooth root crack is commonly assumed to be in a closed state. However, the gear tooth crack will open under the action of mesh force in the meshing process, the mesh stiffness of the cracked tooth and the dynamic response of gear system is thus different from the closed crack case. There almost no studies have been done to investigate the influence of tooth crack open state on mesh stiffness and dynamic response. Thus, this paper presents an improved mesh stiffness calculation model of the cracked gear, in which the effective compression section and the neutral layer of the cracked tooth are re-evaluated considering the crack opening state. Then, the accuracy of the proposed model is verified by the finite element method (FEM), and the differences of mesh stiffness and statistical indicators of the dynamic response between the proposed and traditional models are further studied. The mesh stiffness calculation results show that the proposed model achieves a higher accuracy than the traditional model. And its precision gradually increases as the crack grows, which can be improved by 2.35% in the case of large crack (48.38% crack) when compared to the traditional model. The fault response signal obtained from the traditional model yields a larger amplitude than the actual situation, which will lead to errors of the statistical indicators used for fault diagnosis, in particular, the RMS value can be up to an average percentage difference of 11.98% under the large crack (48.38% crack). It is concluded that the proposed model has a higher accuracy for calculating the mesh stiffness of the cracked gear and can be further used to serve the fault diagnosis and detection of the gear system.

An improved mesh stiffness model for double-helical gear pair with spalling defects considering time-varying friction coefficient under mixed EHL

Based on “slice method”, the improved time-varying mesh stiffness (TVMS) calculation model of double-helical gear pair with actual involute, and transition curve is proposed. The nonlinear contact stiffness, improved fillet-foundation stiffness, time-varying friction coefficient with mixed elastohydrodynamic lubrication (EHL) are considered in the model. Based on the improved model, the mesh stiffness of double-helical gear pair under spalling defects are obtained, in which the depth of spalled region is varied along action line, then the influence of spalling length, axial position, and dislocation phase on TVMS are investigated. The results of the improved mesh stiffness model were compared with the finite element method (FEM) results, the resulting coincidence illustrated the superiority of the improved model.

Computational Studies on Mesh Stiffness of Paralleled Helical Beveloid Gear Pair

Considering the asymmetrical left and right tooth profiles including the transient curve at the tooth root region and the varying thickness along the axial direction due to the cone angle, we build the accurate profile curve model including the transient curve at the tooth root of helical beveloid gear with machining parameters to solve the problem of non-applicability of real digital model. According to the feature of gear shape varying along tooth width direction, we introduced the slicing method and derived its grouping formula. Finally, the efficient and accurate slicing-based mesh stiffness calculation model of paralleled helical beveloid gears was proposed using potential energy theory. Then, mesh stiffness was calculated using finite element contact model for comparison and verification. Finally, the impacts of macro geometry parameters including cone angle, normal pressure angle, helix angle, tooth width and addendum coefficient on single and total mesh stiffness were analyzed. The calculated mesh stiffness correlates well with the results from FEM with the maximum peak error is 4.8%. Results show that the tooth width shows an obvious incremental impact on average total mesh stiffness. When the pressure angle, helix angle, cone angle and addendum coefficient increase, the average total mesh stiffness increases first and then decreases. For the fluctuating value, it increases as the tooth width, helix angle and cone angle increase. However, the pressure angle and addendum coefficient show an opposite impact on the fluctuating value.

A new mesh stiffness model for modified spur gears with coupling tooth and body flexibility effects

Mesh stiffness calculation is critically important for gear dynamic analysis, and many mesh stiffness models have been developed for its convenience and efficiency. However, for a long time, how to properly consider the influence of gear body flexibility and tooth profile error during modeling process has been a challenging problem. The existing mesh stiffness models may either ignore the coupling flexibility of the adjacent teeth or the influence of tooth profile error, which may result in a large calculation error. In order to solve this problem, a new mesh stiffness model is proposed in this paper, in which both the gear body coupling flexibility and the influence of tooth profile error have been considered. The proposed model is compared with the existing mesh stiffness models and validated by the finite element method (FEM).

An improved mesh stiffness calculation model of spur gear pair under mixed EHL friction with spalling effect

Based on the “Potential energy method”, the improved mesh stiffness calculation model of spur gear pair with actual involute and transition curve is proposed. Furthermore, the spalling effect, nonlinear contact stiffness, improved fillet-foundation stiffness, time-varying friction coefficient under mixed elasto-hydrodynamic lubrication (EHL) condition are assimilated into the model. Then the results of the improved mesh stiffness model are obtained by numerical simulation, in which the mesh stiffness under mixed EHL friction and the effect of height for spalled area on the mesh stiffness are comprehensively investigated.

Influences of gear design parameters on dynamic tooth loads and time-varying mesh stiffness of involute spur gears

Gears are one of the most significant machine elements in power transmission due to the many advantages such as high load capacity, long life, and reliability. Due to the increasing power and speed values, the characteristics expected from the transmission elements are also increasing. Significant changes occur in the dynamic behavior of the gears at high speed due to the resonance. For this reason, determining the resonance frequencies is becoming an important issue for designers. This paper presents a method for determining the resonance regions of the gear system under different design parameters. The main purpose of this study is to examine the effects of different gear design parameters on spur gear dynamics. For this aim, the effects of these parameters on the mesh stiffness and contact ratio are examined, and the interaction of mesh stiffness, contact ratio, and dynamic response is presented. Different mesh stiffness calculation methods used to calculate time-varying mesh stiffness and a parametric gear dynamic model are proposed. To solve the equations of motions, a computer program is developed in MATLAB software. Five different design parameters, which are teeth number, pressure angle, reduction ratio, profile shifting factor, addendum factor, and damping ratio, are taken into consideration. The dynamic factor variation is calculated for 1600 rpm a constant pinion speed for each parameter for a single mesh period. Furthermore, the dynamic factor is calculated for the pinion speed between 400–30000 rpm and the frequency response and the resonance regions of the gear system are defined. As a result of the study, the profile shifting and the addendum factor are the most effective two parameters on the gear dynamics. Also, the contact ratio and mesh stiffness have a great effect on the dynamic response of the system. The methods decreasing dynamic load factors are also discussed at the end of the study.

Analytical and FE Determination of the Change of Single Stiffness for Cylindrical Gears with External Involute Teeth

The calculation of mesh stiffness with required accuracy is essential for determining the contact characteristics of gear pairs. The easiest approximation of the relative stiffness for the basic profile geometries is the so-called single stiffness. Standardized and analytical methods for the determination of the single and mesh stiffness of gears are used to achieve design goals considering the load capacity and the vibration excitation characteristics. Such methods involve the formulas of ISO 6336-1:2006 based on experimental relationships and the equations of Weber and Banaschek based on mechanical calculations. In this paper, guidelines are given to refine the analytical calculations. Our goal is to present the impact of the change of the applied pressure angle, module, load, rim thickness and tooth number on the maximal single stiffness. The profile geometry of the gears is generated with our program in MATLAB. The profile of gears is calculated by the tool geometry and the kinematics of production. The geometry is imported into Abaqus. The sensitivity of the models to different parameters is examined and compared to those obtained by analytical calculations. The benchmarks for the single stiffness are the two most widely used analytical calculation methods in Europe such as ISO 6336-1:2006 formulas and Weber and Banaschek equations.

A mesh stiffness method using slice coupling for spur gear pairs with misalignment and lead crown relief

The misalignment and lead crown relief make the tooth partial contact along the tooth width direction, which greatly affects the mesh characteristics. In this paper, a mesh stiffness model of spur gear pairs with misalignment and lead crown relief is proposed based on the slice theory, in which the slice coupling effect is presented. The slice coupling is simulated as a spring and the corresponding model is developed. Then, the solving procedure of the mesh stiffness model is studied. The deformation transfer model is derived between the springs in contact state and non-contact state. An iterative computation method is designed to calculate the mesh deformation of tooth pairs. Finally, the mesh stiffness model is validated by the finite element method for a spur gear pair with ideal tooth profile, misalignment and lead crown relief. Effects of the misalignment and lead crown relief on mesh stiffness of gear pairs are also studied. The results reveal that the proposed model is accurate, effective and efficient for spur gear pairs with ideal tooth profile, misalignment and lead crown relief. It is suggested that the slice coupling should be considered in mesh stiffness calculation of gear pairs with misalignment and lead crown relief.

Improved analytical calculation model of spur gear mesh excitations with tooth profile deviations

Time-varying mesh stiffness and gear transmission errors are the major excitations to gear dynamics system. How to calculate them more accurately and efficiently has attracted much attention worldwide for a long period. Especially consideration of the gear body structure coupling effect and the tooth profile error will make interactions between the tooth pairs in mesh more complicated so that traditional analytical methods could not applicable any more. In this paper, a comprehensive and general analytical gear mesh model is proposed by considering all the deformations including the teeth deformation, teeth contact deformation, fillet-foundation deformation, and gear body structure coupling effect, as well as tooth profile deviations. The calculation formulas for the mesh stiffness, load sharing ratio and static transmission error are derived, and the corresponding efficient solving method is also proposed. Based on the proposed model, the influencing mechanism of the gear body structure coupling effect on the total mesh stiffness is revealed. The proposed mesh stiffness calculation model is proven to be more general and comprehensive since the traditional analytical models are the special cases of the proposed model.

A semi-analytical method of time-varying mesh stiffness in concentric face gear split-torque transmission system

Concentric face gear split-torque transmission system (CFGSTTS) has great applied value in the field of aeronautical transmission due to the characteristic of high integration. Mesh stiffness, as one of the most primary sources of vibration, is vitally important for the dynamic performances of gear transmission system. The existing finite element method (FEM) and analytical method (AM) are not suitable for tackling the mesh stiffness calculation of closed-loop multi-branch system such as CFGSTTS. Thus, a semi-analytical method (SAM) is presented and verified, which combines the high precision of FEM with the high efficiency of AM. Additionally, the differences between the mesh stiffness of independent face gear drive and that of the same gear pair in CFGSTTS under accordant load is researched by applying SAM. The influence rules of distribution angle and load condition on the mesh stiffness of gear pairs considering system structure are also studied. Results demonstrate that the mesh stiffness of gear pairs in CFGSTTS is time-varying and tends to be consistent with each other by adjusting load parameters.

Time-varying mesh stiffness calculation of a planetary gear set with the spalling defect under sliding friction

Time-varying mesh stiffness (TVMS) is an important excitation source of a planetary gear set. Sliding friction and spalling defects have significant effect on the TVMS. Accurate evaluation of the TVMS can help obtain the vibration characteristics and further detect the spalling defects of the planetary gear set. In this paper, an improved analytical model is proposed to calculate the TVMS of the planetary gear set with sliding friction by considering the tooth profile beginning with the root circle. Then the impact of spalling defects on the TVMS is investigated. The results show that the improved analytical model can increase the precision of the TVMS. Besides, the effect of sliding friction and spalling defects on the TVMS is significant. This study offers a basis for dynamic performance analysis of a planetary gear set with spalling defects under sliding friction.