Review
Mesh stiffness models for cylindrical gears: A detailed review

https://doi.org/10.1016/j.mechmachtheory.2021.104472Get rights and content

Highlights

  • Review on gear mesh stiffness models.

  • Guidelines and implementation procedure for gear mesh stiffness models.

  • Gear transmission trends on gear mesh stiffness.

Abstract

Gear transmissions have always been a subject of study for many different reasons which change due to the constant evolution of technology. Consequently, new topics must be addressed and different ways to solve the upcoming problems must be discovered. Gear mesh stiffness is a central topic for both gear design and gear dynamic modeling due to its ability to represent the gears’ behavior, making it a subject of high interest in the field of gear transmissions. Different types of models are used to establish the mesh stiffness of parallel axis cylindrical gears, namely, analytical, finite element, hybrid and approximated analytical models. Also, a dedicated section to mesh stiffness models for polymer gears is included. In this work, implementation guidelines for each class of model are presented along with relevant literature, providing a broad range of information in great detail. Lastly, the main conclusions for each type of model are discussed and an overview of the future evolution of gear mesh stiffness is given.

Introduction

Gears, the most common machine element used to transmit motion and power, are constantly adapting to today’s requirements. Whatever the shifts in demands that are awaiting, gears will prevail and find their way. The mesh stiffness of gears characterizes their behavior, which is crucial for their development. Whether the topic is design, optimization, dynamics or noise, the gear mesh stiffness plays one of the leading roles. So, in order for gears to keep adjusting throughout time, a proper definition of the gear mesh stiffness must exist.

Gear mesh stiffness modeling is a central topic of research since it is the core element of a gear pair. In dynamics, it is the main responsible for the noise, vibration and dynamic loads, thus, the usage of an appropriate gear mesh stiffness representation largely affects the results of dynamic models [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In the design stage or for optimization purposes, gear mesh stiffness can be utilized as a tool to establish both the macro- and micro-geometry while considering the dynamic performance of gears due to its clear dynamic influence [13], [14], [15], [16].

Stiffness is the resistance of a body to the deflection induced by an applied load. In its simplest form (linear single degree of freedom system such as a spring), stiffness (k) is defined as the ratio of an applied load (L) over the corresponding generated displacement (δ), Eq. (1). k=Lδ

For a linear system with n degrees of freedom, the stiffness is defined by a n×n matrix. Each element of the matrix (kij) is obtained according to Eq. (2), this is, the coefficient between the load applied in the degree of freedom i and the displacement produced by that load in the degree of freedom j. kij=Liδj

When the issue is the gear mesh stiffness, a similar description is usually applied, so the gear mesh stiffness represents the load on the gear mesh required for a given displacement. The deformations occurring during the loading process of a gear tooth are described by Attia [17], enlightening the complexity and multiple phenomena affecting the gear mesh stiffness. According to Attia [17], first the Hertzian deformation takes place at the contact point on the profile. Then, the load is transmitted to the tooth’s body which causes bending, shear and compression deformations. The combination of these deformations at every cross-section defines their magnitude and direction. The load then reaches the tooth root and is also transferred to its adjacent parts. Finally, the load gets to the gear body and, if it is strained, there can be angular tooth deformation with respect to the gear center [17].

When discussing the stiffness of gears, it is important to leave a note on the transmission error (TE). Transmission error represents the difference between the perfect position (unmodified, geometrically perfect and infinitely rigid gears) and the actual position of a gear, in other words, it is the relative displacement of the output gear with respect to the input gear — mathematically expressed in its angular form by Eq. (3). TE=θ2rb1rb2θ1where θi is the rotational angle of gear i and rbi is the base radius of gear i. Notice that the transmission error is negative when the output gear (gear 2) lags behind its conjugate position. The transmission error in Eq. (3) can be modified to its linear form by multiplying it by rb2, becoming a displacement along the line of action.

Depending on the working conditions of the gear pair, two types of transmission error are commonly established, the static transmission error and the dynamic transmission error. The transmission error can be related to the gear mesh stiffness since it can be viewed as the displacement caused by an applied load, which, in this scenario, is referred to as static or loaded-static transmission error.

Gear mesh stiffness can be evaluated by means of torsional stiffness or linear stiffness, which can be related to each other. The torsional mesh stiffness, defined in Eq. (4), is given by the ratio of the applied torque (T) and the transmission error. Notice that the concept of stiffness is related to the elastic deflections and, therefore, the no-load transmission error (associated to the manufacturing errors) should, by definition, be removed from the transmission error when calculating the gear mesh stiffness since it does not translate as elastic deflections but as rigid-body displacement. Although, contributions of the no-load transmission error should not be disregarded when modeling a gear pair as they can remarkably modify its behavior. kt=TTE

The linear mesh stiffness, established in Eq. (5), is the ratio of the applied load (L) and the displacement (δ) in the load’s direction, usually along the line of action. In this particular situation δ is the linear transmission error. kl=Lδ

Either way, the relation between torsional and linear gear mesh stiffness is established according to Eq. (6). kt=T2TE=Lrb2TE=Lrb22TErb2=Lrb22δ=klrb22

The determination of stiffness requires an exact description of the load and displacement [18], which makes the computation of gear mesh stiffness a complex task. The complexity of the gear mesh stiffness arises from the many variables that affect the gear geometry and the contact conditions that change the description of the load and displacement.

The gear mesh stiffness can be obtained numerically or experimentally, although, experimental methods are mostly used for empirical studies, validation of numerical models or as monitoring techniques. So, experimental methods are not addressed in this work since the main concern is the gear mesh stiffness modeling. Regardless, some investigations on the measurement of stiffness and transmission error in gears are presented. For instance, [19], [20], [21], [22], [23], [24], [25], [26], [27] perform experimental measurements of the transmission error and [28], [29], [30], [31], [32], [33], [34] measure the stiffness of a single tooth, teeth pairs and of damaged teeth. Encoder error on the measurement of gear transmission error is studied in [35] and different methods for the measurement of angular speed and gear transmission error are reviewed and discussed in [36] and [37], respectively.

The literature research conducted on numerical methods for gear mesh stiffness modeling divided the different models into four classes: analytical, finite element, hybrid and approximated analytical. In short, every model resorts to different techniques to describe the gear mesh stiffness: analytical models use analytical expressions; finite element models employ the finite element method; hybrid models apply both analytical expressions and the finite element method and the approximated analytical models make use of simple and computational inexpensive approximate analytical expressions. All of the mentioned models have their advantages/disadvantages and are usually developed with a specific purpose, which may vary, for example, from the study of gear dynamics, profile modifications, teeth deflections, cracked teeth and the effects of gear geometrical parameters/errors on the mesh stiffness.

Early gear investigations were not directly concerned with gear mesh stiffness but rather with the strength of gear teeth. The question to answer was: “What is the breaking load of gear teeth?”. Lewis’ bending strength equations [38] answered that question and unified the around forty-eight “rules” for calculating the bending strength in existence at that time. None of those rules took into account the actual tooth form (unlike Lewis’ equations) and they could present differences up to 500%. The Lewis’ formula is based on the fact that a parabola enclosed on the gear tooth defines a beam of uniform strength. Later, it was found by empirical evidence (photoelastic technique) that Lewis’ equations were wrong, as expected since it violated the Saint Venant’s principle, as stated by Wellauer and Seireg [39]. Many authors then improved Lewis’ equations to be in agreement with the results from the photoelastic technique while others developed completely new approaches [39]. The works of Baud and Peterson [40] and Walker [41] both presented equations for the gear teeth deflections. However, to deduce those formulae, Baud and Peterson [40] considered the tooth as a non-uniform cantilever beam while Walker [41] resorted to experimental measurements taken on a fixed single tooth.

Constantin Weber and Kurt Banaschek [42], [43] presented a series of studies on the deflection of gears where it is established that the total deformation of meshing gear pairs emerges from three sources: tooth bending, gear body deformation and Hertzian contact. The developed expressions became a reference for the calculation of gear deflections and the foundation for future investigations which are currently employed in several different types of models [42], [43]. The works of Timoshenko and Goodier [44] and Timoshenko and Woinowsky-Krieger [45] give essential information of the theory of elasticity and the theory of plates and shells. These books [44], [45] present solutions to engineering problems of practical importance which are the basis for many engineering investigation, where gears are no exception.

The modeling of a gear tooth as a cantilever plate was first accomplished, according to [39], by C. W. MacGregor in 1935 [46] where the deflections and moments on a thin plate with infinite length under a concentrated load on the free edge were calculated. The investigations of Holl [47] and Jaramillo [48] on cantilever plates were fundamental for the development of many other studies including the semiempirical solution presented by Wellauer and Seireg [39]. This semiempirical solution relies on the superposition principle and the moment-image method (developed procedure) to obtain the bending moment distribution on a finite cantilever-plate under any transverse loads. The results were in agreement with experimental tests conducted on thin plates and tooth-shaped thick plates [39]. Attia [17] investigated the deflections of gear teeth with thin rims resorting to strain-energy theories. The gear teeth deflection accounted for the bending, shear and compression deformations of the gear tooth and adjacent part of the gear body, circumferential deformations of the gear body as well as the impact of deflections in the neighboring teeth [17]. Umezawa et al. [49] obtained a numerical solution by the finite difference method for the deflections due to a concentrated load of a finite length cantilever thick plate. New boundary conditions were applied which produced results in consonance with experimental deflections on a cantilever thick plate [49]. Then, Umezawa [50] expanded the previous work [49] for a rack-shaped cantilever plate with finite width, meaning that the deflections and moments were determined for a variable thickness cantilever plate under transverse loads applied at any location on the surface [50]. Seager [51] developed a set of equations describing the loading and deflections of a pair of involute helical gear teeth with the purpose of analyzing the static/quasi-static behavior of gear pairs and selecting the most adequate profile modifications. This work [51], based on Seager’s dissertation, discusses several topics such as convective effects, separation distance, contact stiffness, profile modification and load distribution [51].

Terauchi and Nagamura [52] used two dimensional elastic theory and conformal mapping functions to determine the tooth deflections. The normal tooth load is approximated by a set of concentrated loads and the mapping functions are used to establish the tooth profile to finally reach the tooth deflections. The method was compared with well-known tooth deflection formulas to prove its validity. A discussion on the Hertzian contact deformation is also presented [52]. In a subsequent work Terauchi and Nagamura [53] employed the previously developed calculation method [52] to compute the deflections of several spur gear teeth. From these results simple and approximate expressions for the tooth deflections and contact Hertzian deformations were proposed [53]. Cardou and Tordion [54] seek the solution to the two difficulties found when calculating the gear tooth flexibility by the complex potential method: (i) the indeterminacy of the displacements and (ii) the singularity at the point of interest (teeth contact point). While the first problem was solved by selecting a proper reference point for the calculations, the second was dealt with by calculating the displacements at a certain depth under the surface. Flexibility curves were compared with other works and found to be in agreement [54]. Steward [55], [56] presents a 3D elastic model for the meshing of spur gears. A 3D finite element analysis was applied to establish the influence coefficients that allow to determine the tooth centerline deflections curves for different tooth geometries. Concerning the contact compliance, it is included by 2D Hertzian contact theory with semiempirical correction coefficients for the regions close to the tooth tip. Deflections results agreed with experimental measurements [55], [56]. Yau et al. [57] use the Rayleigh–Ritz energy method to compute the deflections of tapered plates under concentrated loads, which are implemented to simulate gear teeth. When comparing the determined deflections with theoretical and experimental results, the differences found were assigned to the neglected shear deformations in the theoretical models and base distortion in the experimental tooth models [57].

Stegemiller and Houser [58] developed a model to determine the base deflections of wide face width gear teeth. This model is based on several finite element analysis and applies the moment image method presented in [39]. The base rotations and translations obtained with the model are in agreement with the acquired finite element results [58]. Kim et al. [59] adopted the finite prism method to calculate the load sharing, pressure distribution, mesh stiffness and tooth fillet stresses on webbed spur gears. The conducted analysis allowed the development of simple formula that can estimate with reliability the tooth root stress accounting for the effects of thin rims and webs [59]. Litvin et al. [60] resorted to the finite element method and tooth contact analysis to consider a distributed contact force in the calculation of transmission error, loaded tooth deflections, load sharing ratio, real contact ratio and tooth bending stress [60]. Guilbault et al. [61] integrate the finite strip method with a pseudo 3D model of the tooth base to compute the tooth bending stiffness and fillet stresses. The procedure is intended to be a fast and precise gear design tool. These facts were proved by the acceptable precision of the results and the reduced time required in processing the model when compared to 3D finite element analysis [61].

Blankenship and Singh [62] estimated the forces and moments produced and transferred through the gear mesh interface in a dynamic model. The developed formulation allows the comparison, in a mathematical level, of several simplifying assumptions commonly applied in gear dynamic systems [62]. Velex and Maatar [63] treated the gear meshing from a dynamic perspective. The transmission error and dynamic mesh stiffness are obtained as a result of the proposed model which separates the rigid body displacements (flank deviations, misalignments and eccentricities) from the elastic displacements (contact deflections) in its formulation. Both quasi-static and dynamic solutions show agreement with experimental and analytical results from the literature. The impact of geometrical errors and profile modifications on the gear dynamics is evaluated resorting to the developed methodology [63]. Ajmi and Velex [64] developed a model to simulate the quasi-static and dynamic behavior of solid wide-faced gears. Gear bodies were modeled resorting to two node shaft finite elements in bending, torsion and traction; tooth bending and shear deflections including coupling effects were modeled with the Pasternak’s foundation model; contact deflections were introduced with the classical approximation based on semi-infinite elastic spaces; tooth shape deviations and alignment errors were included as normal deviations. It was found that gear body flexibility is relevant for the quasi-static and dynamic conditions of wide-faced gears and that the tooth coupling effects are less significant in dynamic conditions [64].

Smith [65] presented a comprehensive explanation and analysis to the design, development, metrology and troubleshooting of noise and vibration of gear. Among the several elucidated topics it stands out the transmission error measurement and modeling of spur and helical gears, including the thin slice assumption, as well as practical guidelines for a complete setup of experimental procedures [65]. Linke et al. [66] conduct a complete analysis to cylindrical gears covering every gear topic from the fundamental principles of gearing to the manufacturing. The load capacity and running performance of gears is thoroughly presented, incorporating the meshing characteristics and stiffness description/modeling [66].

The aforementioned investigations shed light on the developments of gear stiffness related studies throughout time. Some non-gear researches are mentioned due to their contributions in the development of gear investigations. The contents presented in those works contain useful information for any gear study as they explain gear phenomena in a direct and clear way. Besides that, a lot of experimental results are also shown. All the classes of gear mesh stiffness models currently employed can be found is these investigations.

This review is divided into seven sections, beginning with an introduction that defines the gear mesh stiffness, highlights its importance and presents some of the first developments on the deflections, stress and load sharing of gears. Then, a specific section for each class of gear mesh stiffness model is presented where not only the existing works are analyzed but also a description of the models’ implementation procedures is performed. That being said, analytical models are the first ones to be studied (Section 2) with emphasis to the potential energy method (complete description on how to obtain the gear mesh stiffness for spur gears) followed by the literature review. Section 3 is dedicated to the finite element models where guidelines for the key aspects are given together with the presentation of some works. The next section explores the hybrid models (Section 4) and the approximated analytical models are detailed in Section 5. Section 6 analysis polymer gears and describes works related to their mesh stiffness. Throughout these sections there are discussions focusing critical aspects of the models. This study ends with an analysis of the different models studied, stating their advantages, disadvantages and purposes, as well as an overview of the gear mesh stiffness trends.

Within the field of gear transmissions, the mesh stiffness is a topic of high interest and therefore a review with great insight involving all types of gear mesh stiffness models is on demand. The aim is to provide a literature review along with implementation procedures/guidelines suited for any researcher and practitioner whose work relates to gears, giving a detailed source of information on the mesh stiffness models of parallel axis cylindrical gears.

Section snippets

Analytical models

Analytical models express the gear mesh stiffness through the usage of analytical expressions acquired from mechanics of materials. The most common analytical method found in the literature is the potential energy method (PEM). There are also some references to the Ishikawa method, which is very similar to the potential energy method but the gear teeth are simplified as the combination of a rectangle and a trapezoid [67]. For these reasons, the potential energy method is going to be described.

Finite element models

The finite element method (FEM) is a procedure used to find numerical solutions of equations that define the behavior of any system. These problems are usually defined resorting to the laws of physics, algebraic equations, differential equations or integrals. Concerning structural analysis, FEM is a powerful tool to calculate displacements, stresses and deformations of loaded structures [109], [110], [111].

This numerical method is characterized for dividing a continuous domain of a problem into

Hybrid models

Hybrid models are characterized by having two components, one of them is the finite element method and the other usually is an analytical method — that is why these models can also be named analytical-FE models. While the finite element method is used to compute the global deformations, the analytical method is applied to determine the local (Hertzian) deformations. By doing this, the mesh does not need to be as refined as when only using the finite element method. That being said, the major

Approximated analytical models

Approximated analytical models have the lowest computational cost off all models as they only require a few simple numerical steps to obtain the gear mesh stiffness. In this group of models, complex phenomena that involve many steps and high numerical computation are represented by a direct and effortless expression.

The simplicity of these models is specially highlighted while using iterative processes. In these cases, as the same calculations have to be constantly repeated, with a simple

Polymer gears

A dedicated section to polymer gears is presented since the majority of gear mesh stiffness investigations are devoted to steel gears and there are significant differences on the gear mesh stiffness modeling for each type of material. Polymer gears have a lightweight nature, self-lubricating ability, are quieter, more resistance to corrosion and have lower mass production costs than steel gears [141], [142], [143], [144], [145], [146], [147]. On the other hand, this type of gears have, when

Remarks on gear mesh stiffness models

As it was demonstrated along the review, gear mesh stiffness has always been a topic of great concern among gears’ researchers and the fact that it is still being studied proves its importance and complexity.

In the analytical models, the most commonly used method is the potential energy method. This method is extensively used due to its good results and high flexibility in incorporating geometrical modifications to the tooth, for example, tip relief, tooth cracks and spalling. The potential

Acknowledgments

The authors gratefully acknowledge the funding through several projects and grants whom without this work would not have been possible:

  • National Funds through Fundação para a Ciência e a Tecnologia (FCT), Portugal under the PhD grant SFRH/BD/147889/2019;

  • LAETA, Portugal under project UID/50022/2020.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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