Categorization of gear stiffness calculation methodologies, workflows, and software

The desire for more compact, efficient and quieter powertrains in the Electrification Age has once again brought gears into the spotlight. In the field of modern gear design and analysis, loaded tooth contact analysis is one of the most important simulations performed. Gear stiffness plays a fundamental role in this simulation and is calculated slightly differently in about every software on the market. The need to accurately resolve gear stiffness to capture pitting, bending, scuffing, efficiency, and noise characteristics of a gear pair makes the topic very relevant to the entire gear industry. Due to the gear stiffness being calculated in the background of most software and the fact the calculation is nontrivial in nature, not all gear engineers are familiar with the gear mesh stiffness methods being used in their software of choice and what limitations come along with them. The goal of this work is to categorize all the key gear mesh stiffness methodologies and workflows used in both industry and academia, and provide the key references where the methodologies originated from. The advantages and disadvantages of each gear mesh stiffness methodology are covered in detail, along with what methodologies are used in the most common gear software on the market. Finally, recommendations for what gear stiffness methodologies to use in different scenarios are provided.

1 Introduction

The Electrification Age is upon us. The dominance of the internal combustion engine (ICE) is dwindling, particularly in the automotive industry. Internal combustion engines are being replaced or supplemented by more efficient and versatile electric motors in the form of electric drive modules (EDMs). The increased focus on global warming, the uncertainty of oil prices, and society’s obsession with innovation have fueled, and continue to fuel, the Electrification Age. There is a global race to see who will emerge from the Electrification Age on top, and gears certainly play an essential role in this contest.

While the internal combustion engine may slowly lose dominance, gears are as crucial as ever. The need for five to 10-speed transmissions has been removed due to the use of a more efficient electric motor. Most EDMs use either a single-speed or a two-speed reduction drive, with much debate over which to choose [326, 457, 387, 438]. Most auto-motive applications use a single-speed gearbox, while some luxury brands use two-speed gearboxes. Commercial applications, on the other hand, often employ two-speed or even three-speed gearboxes due to the significantly larger torque operating range. Although the need for a greater number of gears has decreased due to electric motors running more efficiently over a wider range of speeds, their complexity has increased; the American Gear Manufacturers As-sociation (AGMA) recently estimated that electric gearboxes contain a third fewer gears compared to their internal combustion counterparts [436]. The complexity comes from pushing for higher contact ratios to minimize the noise, smaller package spaces coming from the desire to include all the components of the powertrain into a single assembly (i.e., motor, gearbox, differential, inverter, control module, oil pump, oil cooler), and lower viscosity oils to increase efficiency. The race of companies to get their products to market has only made the design of gears for electrification more difficult, as the development time has substantially decreased.

By losing the noise masking that internal combustion engines can provide, gears for EDMs need to be even quieter. The increased focus on noise has resulted in the tolerances of not only the gears but also all other structural components of the gearbox being lowered to help decrease the system’s noise, vibration, and harshness (NVH). While electrification gears are fewer in quantity than their internal combustion counterparts, they are typically more expensive due to the tighter tolerances, lower surface finishes, and greater complexity. The fear of gear noise can sometimes lead to noise anxiety among both designers and managers, resulting in costly design modifications for minimal gains in NVH performance.

The total contact ratio of a gear set represents the average number of teeth in contact for a full mesh cycle. NVH is widely known to improve with increased contact ratios, but the improvement is in step-like increments (not linear). A theoretical contact ratio can be calculated using pure geometry, but the actual contact ratio depends on load, microgeometry, stiffness, and misalignment; for this reason, it must be calculated numerically. The total contact ratio is the sum of the profile and helix contact ratios. The Electrification Age has seen a push from theoretical total contact ratios in the 1.3-3.0 range for conventional ICEs to total contact ratios in the 3.0-5.0 range for EDMs, mainly due to noise concerns; the contact ratio in the past has gone down as far as 1.1 [91]. While ICEs typically have a profile contact ratio between 1.3 and 1.8, they are now closer to 2.0 for EDMs. Gears with a profile contact ratio of 2.0 or greater are referred to as high contact ratio (HCR) gears, as opposed to low contact ratio (LCR) gears; HCR gears started gaining popularity in aerospace beginning in the late ’60s as a way to increase load carrying capacity and decrease noise [74]. While ICEs typically have a helix contact ratio between zero and 2.0, they are now closer to 2.0-4.0 for EDMs. It is still possible to get away with 3.0 total contact ratio gear designs in the Electrification Age and still meet NVH targets by going to tighter tolerances and clearances (partially to reduce sidebands [137, 303, 308, 287]), adding bias control [364], minimizing misalignment, ensuring proper load sharing, and performing system-level microgeometry optimizations. (Figure 1)

Microgeometry is added to high-performance gears to make the gear set as a system perform better under loaded conditions where elasticity starts to cause errors in the transfer of uniform motion, and natural tip engagement and backlash cause shocks as the teeth come into contact [28, 67, 181]; microgeometry is one of the strongest levers available to tackle gear whine [271]. Several options for microgeometry are available, categorized into either profile or helix forms. Common profile microgeometry includes tip and root relief, and common helix microgeometry includes crowning and slope relief. Figure 29 in the Appendix illustrates the microgeometry of an example e-mobility gear set, which is commonly categorized into profile errors, helix errors, and pitch errors. Figure 32 in the Appendix  illustrates the impact of different microgeometry forms on the contact patch of the gear set.

To make gears quieter, the helix angles are being increased to drive up the helical contact ratio. With higher helix angles come multiple issues, including increased bending stress (especially for narrow-width gears [319]), increased axial loads (which decrease efficiency), increased overturning moments (which increase misalignment), and increased effects of unintentional bias in the grinding process. Planetary gear sets, in particular, are very sensitive to overturning moments because the bearings are directly underneath the planet gears. There is an increase in the use of tapered roller bearings (TRBs) in EDMs, which are widely known to be less efficient and more expensive, to handle the increased axial loads and overturning moments introduced by high-helix gear designs. TRBs are often preloaded to remove clearance and reduce misalignment, but in planetary gear sets, they are sometimes not preloaded to increase efficiency. Along the same lines, gear-tip diameters are being increased to drive up the profile contact ratio. With increased tip diameters come more issues stemming from operating closer to the base circle at very low roll angles. This increases the need for well-designed flank-to-fillet transitions to avoid whine issues caused by flanks with steps or by flanks that don’t have an involute far enough down for the mating gear to land properly. It also increases the need for well-designed microgeometry to avoid scuffing, which is already exacerbated by the lower viscosity of the oils being used. In EDMs, oil viscosity is being reduced to nearly half its current level in conventional ICE gearboxes to meet increased demand for improved efficiency. Operating closer to the base circle means dealing with lower radii of curvature at higher velocities, further increasing scuffing concerns. One way to address the need for higher helix angles and tip diameters is to make better use of the existing space.

While a gear pair has a certain effective face width and a certain effective working depth, it is not guaranteed that the entire profile and helix will come into contact with each other. Inefficient use of both the profile and helix space is more likely to occur if a microgeometry optimization is replaced with best practices and/or rules-of-thumb, or if the microgeometry analysis does not adequately capture the effects of misalignment (possibly due to not including the deformations of the housing and any asymmetric shafts). If the misalignment is not predicted correctly in the analysis stage, a portion of the flank, predominantly in the helix direction, can come out of contact. When a microgeometry optimization is not performed, designers tend to err on the side of caution and overcompensate for the amount of relief added to the gears to protect them against durability issues, particularly scuffing. With this approach, light loads can lead to less efficient use of the flank surface. Even at high loads, when more of the flank surface comes into contact from elastic deformations, this can unintentionally lead to part of the flank never coming into contact because of too much relief. (Figure 2)

The importance of gear mesh stiffness cannot be understated. To accurately design the microgeometry for a gear set, the deflection and twist of the contact zone, teeth, and bodies must be adequately accounted for. The gear mesh stiffness partially determines the deflections. For clarity, gear stiffness is a property of a single gear (typically pre-processed), and gear mesh stiffness is a property of a gear pair (typically post-processed). When the gear stiffness is combined with the contact stiffness, which depends on the microgeometry of the gears and the relative misalignment of the gear set, it creates a gear mesh stiffness. The relative misalignment of the gear set is dependent on the gear support stiffness and any clearances in the support system, and is highly nonlinear, changing with load and temperature. The contact stiffness predominantly comes from the compression of the two mating flanks. The tooth stiffness primarily arises from bending and shearing of the tooth. The rim and body stiffness can be the most difficult to capture, especially for thin-rimmed gears or gears with large webs and asymmetric lightening features. The gear-support stiffness comprises the stiffnesses of any shafts, bearings, housings, or other elastic components in the gear system.

The gear stiffness and its support stiffness are required for a gear system analysis (GSA). GSA is a highly nonlinear analysis performed on a gear system to resolve the misalignments and loading on components with relative movement between them (i.e., gears, bearings, splines, radial/axial clearance fits). Gears are often the most challenging compo-nents in the gearbox to design due to the sliding nature of their contact; hence, the analysis is named after them. In a GSA, any components contributing to the relative misalignment between moving parts are modeled, including gears, bearings, shafts, radial/axial fits, spline connections, snap rings, couplings, clutches, housings, and mounts. GSA is typically performed with specialized software, as opposed to a general finite element (FE) or multibody dynamic (MBD) software. The specialized software is specifically designed to allow gear systems to be pre-processed, solved, and post-processed very efficiently. The unique features that allow this efficiency include automated gear and bearing contact pairing and discretization, the inclusion of all common gear rating standards (which is not an easy task to implement), automated multi-gear layout creation for assemblies such as planetaries and differentials, links to bearing catalogs from popular industry brands, the inclusion of manufacturing process in geometry creation, both coefficient-based (common in Europe) and direct (common in the United States of America) macrogeometry input options, quick model creation from sketched two-dimensional CAD layouts, advanced post-processing of all common gear results, and customer support engineers with background in gear design and analysis. (Figure 3)

The gear mesh stiffness is calculated as part of a loaded tooth contact analysis (LTCA), which can be integrated, coupled, or decoupled with the gear-system analysis. The actual total contact ratio is also output from an LTCA. LTCA is the analysis of a loaded gear pair over the range of the mesh cycle through the discretization of the geometry of the mating gears for both stiffness and contact to determine outputs such as contact pressure, contact pattern, contact temperature, contact efficiency, transmission error (TE), mesh stiffness, and directly or indirectly bending stress. LTCA is a more complex version of the much simpler tooth contact analysis (TCA). TCA is an analysis that models the meshing of a gear pair over the range of the mesh cycle through the discretization of the geometry of the mating gears, only for contact and not for stiffness, and models a predetermined interference between the mating gears; the contact pattern is one of the few results obtained from the analysis. TCA can be used as a preprocessing analysis in an LTCA workflow to predict where the anticipated lines of contact will be, thereby improving solution time.

The transmission error (TE) of a gear set is defined as the positional error between the driving and driven gears when compared to a theoretically perfect transfer of motion. The positional error is commonly quantified as a linear displacement in the tangential direction along the transverse plane of the tooth at the base circle diameter. TE is caused by multiple factors, including the inherent nature of changing numbers of teeth in contact, profile/helix/pitch/radial errors from manufacturing/damage/wear, flank microgeometry, elastic deformations from contact/thermal/modal/centrifugal forces, and system-level misalignment. As shown in Figure 4, the transmission error has both a high-frequency component (mesh error at mesh frequency) and two low-frequency components (pitch errors at the shaft frequency of each gear in the mesh). Typically, only the mesh error is in focus in an LTCA; the gear runout, an effect of the errors in the manufacturing process, is often ignored.

Both TE and mesh stiffness are outputs of an LTCA (mesh stiffness can be used as an input for further analyses). The mesh stiffness is composed of the same harmonics as the transmission error and can be used as an input to a torsional vibration analysis, where dynamic transmission error would be a calculated output. Ultimately, it’s the peak-to-peak transmission error (PPTE) that is of interest to a gear designer, due to its direct correlation with gear noise. The PPTE is commonly defined as the peak-to-peak value of the TE over a full mesh cycle of the gear, excluding the effects of shaft harmonics and lower-frequency components.

There are multiple workflows for determining gear mesh stiffness and transmission error. They all involve three different sub-analyses: a sub-analysis to determine the relative misalignment between the two gears (system deflection analysis), a sub-analysis to determine where the gears are contacting (tooth contact analysis), and a sub-analysis to determine the deflection at the gear mesh (loaded tooth contact analysis); they are all part of the overall gear system analysis. Ideally, these analyses are integrated together in a single solution. These analyses can be decoupled and calculated separately, but fidelity is lost in the process. An entire section of this work is dedicated to this topic, along with an overview of commercial and academic software available on the market.

Gear mesh stiffness (broken down into gear contact, tooth, rim, and body stiffness) is the primary focus of this work, along with quasi-static loaded tooth contact analysis and transmission error. Fatigued or damaged gear geometry (both macro and micro), contact search analysis, elasto-hydrodynamic analysis, and gear-support stiffness (bearings, shafts, housings) are not the focus of this work. The history of gear stiffness and related topics in the field is covered first. This is followed by a section categorizing the different gear stiffness methodologies and workflows. Then, a section covers the gear stiffness calculation methodologies used in academia and commercial software. Finally, the work concludes with a discussion on when to use the different methods and workflows, highlighting subtleties in the calculation process and identifying general areas for improvement in the field of loaded tooth contact analysis. The intended readers are engineering students, subject matter experts, and technical team managers and directors. This work directs the reader to references that can be explored for more detailed formulations, unlike a highly mathematical work that focuses on specific equations and derivations.

The fidelity of gear noise, vibration, and harshness (NVH) calculations relies heavily on gear mesh stiffness and the fidelity of the system deflection analysis (SDA) and loaded tooth contact analysis (LTCA) modeling methodologies to calculate NVH characteristics such as sound power and sound pressure resulting from transmission error (see Figure 51 in the Appendix for an illustration). The two main gear NVH failure modes extensively studied for gears are whine [399, 346, 334, 430, 270, 273, 435, 224, 315, 380, 290, 286, 437, 298, 227] and rattle [363, 398, 228, 123, 229, 202, 228, 234, 296, 302, 450, 428, 451, 376, 420, 405, 390, 404, 370, 361, 415]. Whine is a tonal noise at the meshing frequency of the gear set; it is dominated by gear mesh stiffness effects, making LTCA a fundamental analysis to assess its severity (often performed quasi-statically in the frequency domain). Rattle is a broadband noise generated by the gear set due to the constant hammering of the gears as they traverse the backlash of the mesh and affect one another (often performed dynamically in the time domain). Figure 50 of the Appendix illustrates the difference between whine and rattle in the frequency domain. To get all the way from transmission error at the gear mesh to sound pressure at some distance from the gearbox requires a number of sub-analyses to be performed, as illustrated in Figure 48 of the Appendix and covered by multiple authors [331, 366, 375, 170]; Figure 49 shows illustrations fundamental to understanding gearbox acoustics, including how loud typical enclosed gearboxes are. Gear NVH is a vast topic that has been covered extensively [51, 98, 115, 215, 305], and this work will not focus on the subject specifically; OSU’s gear noise short course covers this subject in depth [379].

While this work mainly focuses on the design and analysis of cylindrical involute gear geometry [367, 55, 443, 32, 136, 142, 257], helical gears in particular, none of the gear stiffness methods to follow are constrained to any one type of gear geometry [263] (see Figure 23 of the Appendix for various gear geometry types). Cylindrical gears are by far the most common geometry to study due to both their popularity and the simplicity of their geometry, followed by bevel and hypoid gears. Research on the stiffness of bevel and hypoid gears is less common, partly because their geometry is heavily dependent on the manufacturing methods used, which creates numerous complications. These complications arise because the specifics of the manufacturing methods are closely guarded by the two leading global bevel gear manufacturers, Gleason and Klingelnberg, each with their own unique and complex machine settings, which makes achieving exact geometry difficult [373]. For the best understanding of the subject matter in this work, the following literature can be referenced: general elasticity [80], history of elasticity [39, 12], contact mechanics [126], finite element method [339], general dynamics [372], nonlinear dynamics and chaos [186], gear dynamics [379, 258], and fundamental acoustics [235]. (Figure 5)

This work focuses on applying the methods to the field of e-mobility, which has become a significant presence in the automotive and commercial industries; this material is also relevant for defense and aerospace audiences. With the advent of the Electrification Age, the hypoid gear, once almost always found in the final drive, has been largely removed, partly due to efficiency concerns and partly because the power source is now integrated into the axles and wheels. The three most common EDM architectures in the e-mobility industry are the e-axle (mounted to the axle, common in heavy-duty trucks and tractors, and acting as un-sprung weight), the e-drive (mounted to the sub-frame, common in light-duty automotive vehicles, and acting as sprung weight), and the e-hub (integrated into the wheel, and acting as un-sprung weight). Helical parallel-axis and planetary gear sets are the dominant architectures in EDMs. Bevel gears, in the form of differentials, are still present in e-axles and e-drives, but are eliminated in the case of e-hubs. Smaller e-hub applications can eliminate gears altogether by implementing direct drive. For a review of the latest electrification architectures and layouts, see the following [324, 460, 424, 417, 434].

One of the main drivers of gear research has been national defense interests, particularly in rotorcraft technology (mainly due to more stringent weight requirements in aerospace applications). A notable example is the infamous Bell-Boeing V-22 Osprey tiltrotor military aircraft, which features vertical takeoff and landing (VTOL) capabilities; it is the first of its kind [163, 393] (See Figure 6). The push for legislation worldwide to lower emissions of both greenhouse gases and noise has been a secondary driver; a third, less discussed, driver is pride. In the U.S., the National Aeronautics and Space Administration (NASA) leads the way in gear technology (researching alongside the U.S. Army Research Laboratory), relying on academic partners such as The Ohio State University (OSU) Gear Lab, University of Illinois (when Litvin was present), and ANSOL to further assist in the effort.

A little history: the National Advisory Committee for Aeronautics (NACA) was established in 1915, primarily for research related to World War I. In 1958, after the launch of Sputnik, NACA was converted to the NASA Lewis Research Center (named after George W. Lewis, former director of NACA). In 1999, the NASA Lewis Research Center was renamed the NASA Glenn Research Center (in honor of John Glenn, the first American to orbit the Earth) [230]. The U.S. Army Research Laboratory operated a lab at NASA Glenn Research Center from 1970-2020 (in Cleveland, Ohio), collaborating with NASA for 50 years.

Following the technical reports of NASA [190], it is evident how gear technology has evolved over time and which aspects were critical to national defense during different decades. The main contributors out of NASA for gear research include individuals such as Anderson for gear and bearing contact [107, 108, 94, 85, 61, 69], Coy for general gear design [88, 102, 141, 125, 112], Handschuh for thermal behavior of spiral gears [333, 352, 249, 218, 241, 199], Krantz for gear contact fatigue and wear [188, 236, 272, 292, 454, 343], Lewicki for gear tooth root cracks [138, 200, 189, 242, 177], Litvin for gear geometry [110, 155, 219, 253, 111], Oswald for gear NVH [179, 191, 169, 201, 178, 174, 180, 170], Townsend for gear lubrication [222, 97, 132, 119, 127], and Zarkrajesek for conditional monitoring [166, 214, 185, 204, 161, 162]. All of these topics have gear mesh stiffness at the core. The references are listed in ascending order of year, and it is interesting to scroll through the list and see how the development of gear mesh stiffness went alongside research performed at NASA. The most recent papers out of NASA include research on magnetic gears [389, 413], hybrid composite gear bodies [407], and asymmetric tooth profiles [459]. Figure 22 in the Appendix illustrates the gear-design process, along with an outline of the focus of this article, which encompasses not only gear-mesh stiffness but also its impact on all aspects of durability and NVH.

Now to the heart of the work, gear stiffness and its role in the broader gear system analysis (GSA), an analysis not taught in either university or industry to the depth that is about to be introduced. This work referenced a library of more than 3,000 gear articles, seven AGMA gear-training courses, OSU’s gear noise short course, and more than 75 gear books (along with 150 more books on fundamental and classic theory), all since Timoshenko first published on gear stiffness in 1926 [18] (some literature dates back prior to 1900). The author has been building this library for nearly a decade. Still, there are inevitably missing pieces of literature due to the vastness of gear research published worldwide in many different languages. The large size of this work is partially because of the need to create terminology to categorize all of the gear software used around the world, but it is also required to explain gear system analysis (GSA), which is fundamental to the understanding of gear mesh stiffness and has never been critically reviewed in detail before.

2 History

Gear design and analysis have evolved significantly over the years. What started as an art has almost entirely transitioned to a science; microgeometry is one of the last pieces that could still be considered an art. The field of gear design started with best guesses, moved to rules of thumb, transitioned to simplified analytical/empirical equations, and has made it all the way to computer-aided engineering (CAE). This section provides a brief history of the most important papers on gear stiffness, transmission error, noise, and loaded/unloaded tooth contact analysis (LTCA/TCA). While an effort was made not to omit any critical pieces of work, there are likely relevant papers that have been missed due to the large number of papers on the topic, written worldwide over the years, in multiple languages. (Figure 7)

As the introduction mentions, gear mesh stiffness is heavily tied to gear durability analysis. While this section will not review durability in great detail, it is vital to know some history on the subject. The four main failure modes of gears are bending, pitting, scuffing, and wear (reference Figure 43 of the Appendix); while these four failures can lead to catastrophic gearbox failures, other failure modes such as efficiency and noise, vibration, and harshness (NVH) can destroy an entire gearbox platform as a whole, if severe enough. One of the first topics studied in gear durability was speed factors (aka dynamic factors) around 1868 by E. Walker [120, 15], who examined how dynamic effects increased gear loads as a function of speed. In 1922, stress distribution in gears was first studied with the photo-elastic method [17]. Starting in 1925 and published in 1931, Buckingham further investigated the dynamic loads on gear teeth [21]. Bending was one of the earliest modes of failure investigated, dating back to Lewis’s 1892 work on determining the bending strength of gear teeth [11]; Wellauer and Seireg later pointed out flaws in this approach in 1960 [46]. Gear efficiency was studied by Hyde, Tomlinson, and Allan in 1932, and in the discussion, Merritt addressed wear and pitting [22]. Three years later, in 1935, Way [49] studied the pitting of pure rolling contacts. Finally, general scuffing was first studied in 1936 by Bowden and Ridler [25] and applied a year later to gears by Blok [26]; a number of review papers have been written on the scuffing phenomenon over the years [56, 89, 122, 207, 464]. Gear stiffness began to come into play as researchers investigated analytical models to predict these failures, for which determining the load distribution was fundamental. Lastly, while this work focuses on gear mesh stiffness, the topic is often included in some fashion in the literature on gear dynamic models and NVH, for which many reviews have been written over the years [145, 98, 261, 445, 75, 340, 291].

The earliest paper found on the topic of gear stiffness goes back to 1926 (eight years after the end of World War I) when Timoshenko and Baud came up with analytical formulas to predict the contact stiffness, tooth bending stiffness, and tooth shear stiffness by treating a single gear tooth as a cantilever beam and the contact as rollers [18]. In 1927, Lewis, leading a special research committee on the strength of gears, took Timoshenko’s work and extended it to multiple teeth in contact [19]. Lewis’s research aimed to investigate the effects of elasticity on gear loads. Lewis indirectly mentioned transmission error when discussing how errors in the gear profile cause accelerations and decelerations in the gear velocity, thereby increasing loading. In 1929, Baud and Peterson, using the same fundamental equations, went on to determine the effect of gear deflections on contact ratio, in the process creating one of the earliest gear teeth load sharing curves [20]; the load share curve was a different take on Lewis’ deformation plot as a function of the number of teeth in contact [19]. They concluded that increasing the loads on gear teeth can cause an increase in contact ratio due to the deflections of the gear teeth, and they had experimental results to back it up. (Figure 8)

In the 1930s, experiments for gear-tooth deflection started to be performed, in addition to gears first being modeled as plates by MacGregor in 1935 [24]. In 1932, Hyde et al. created a machine to measure tooth deflection, as well as one to measure one of the earliest forms of static transmission error [22]. Later, in 1938, H. Walker did more extensive work in tooth deflections and started the discussion around profile modifications to improve gear design (aka microgeometry) [28]. Over the next decade, there was little activity on the subject. World War II, which lasted from September 1939 to September 1945, is likely the reason for the lack of publications during the early 1940s.

Not all progress stopped during the war. Though the origins of the finite element method are somewhat contested [452], it is generally agreed that the method was introduced in 1941 by Hrennikoff, far from the war at the University of British Columbia [29]. The method was originally called the “framework method,” then the “matrix stiffness method,” and finally settled on the “finite element method” in 1960, coined by Clough [43]. Wire-resistant strain gauges also were invented during the war, which would later be used to help determine load distributions in gear teeth for correlation back to LTCA [323]. Lastly, the first primitive form of the slice method was introduced by Merritt in his book, first published in 1942, to estimate the load distribution in helical gears [30].

After the war, the need for gear fatigue testing was highlighted, and research began in 1946, later published in 1964 [63]. Four years after the conclusion of the war, in 1949, Weber used the work of H. Walker to validate his new approach for determining analytical gear tooth stiffness [35]; this was again published a year later in 1950 by Weber and Banaschek [35] (which is also a common citation). Weber not only included the three stiffness components introduced by Timoshenko (contact stiffness, tooth bending stiffness, and tooth shear stiffness) but also included tooth normal stiffness and an approximation of the gear body stiffness. The tooth stiffness was derived using Castigliano’s second theorem, published in 1879, which uses a partial derivative of strain energy to formulate the deflection relative to a load [6]; this is still the approach most commonly used today for analytical tooth stiffness. Weber used Hertz’s work in 1882 [8] to derive a new formula for contact stiffness with plane strain assumptions. An assumption of a small contact patch compared to the depth of integration was also used, which makes the equation deviate at higher loads. In 1958, Richardson, out of the Massachusetts Institute of Technology (MIT), independently derived the contact stiffness using Hertz theory, similar to Weber’s work, and went on to extend the derivation to include plane stress, the change in radius of curvature at the point of contact, and not include the assumption of a small contact patch that Weber did. In addition, Richardson first introduced the concept of extended tip contact (ETC) in his dissertation [42]. (Figure 9)

Rewinding a few years, in 1953, Tuplin would create one of the first gear mass-elastic models that would begin the study of gear dynamics [40]. Four years later, in 1957, Harris, working out of Cambridge University, introduced the modern form of loaded transmission error, along with the infamous “Harris Map,” and noted the important constraint that gears can only be optimized for a single load level [41]; the term “transmission error” appeared six years later in 1963 from a paper by Harris and his colleagues Gregory and Munro on a method for measuring the parameter [57]. In 1967, Seager, doing his dissertation at Cambridge University [71], introduced one of the first versions of the coupled-slice method, which he later wrote about in 1970 [79].

Rewinding a few years, in 1961 at Gleason, Baxter introduced 3D tooth contact analysis (TCA) to determine the effects of misalignment on bevel and hypoid gear contact patterns [47]. More than10 years later, at Gleason in 1973 [87], Wilcox and Coleman published one of the first papers on using the finite element method to determine gear stress since the finite element method was coined in 1960. Wilcox, later in 1977, would combine his recent work and the work of Baxter to formulate one of the first known combinations of the FEM and TCA, but without using an analytical method for contact [93]. Vedmar, out of Lund University in 1981, would become the first to publish a version of the modern hybrid finite element method, including analytical contact in addition to TCA [106]. A year later, Krenzer would also extend Baxter’s work to LTCA [103], but instead used a simplified version of the multi-slice method introduced by Weber and Banaschek about 1950, one with only a single slice.

A decade before, in 1972, Conry and Seireg [154], out of the University of Illinois and Wisconsin, respectively, wrote a paper that would later be the foundation for the load distribution program (LDP) out of The Ohio State University (OSU); this paper was based on a more general formulation published a year earlier [81]. A flat plate method was used to calculate the gear tooth stiffness, based on papers published between 1935 and 1960 [24, 34, 46]. This method would soon be replaced by a tapered plate method developed in the late ’80s and early ’90s, incorporating a Rayleigh-Ritz method for load distribution, which was compared to a finite element methodology for validation [129, 140, 194]. Also in 1972, Schmidt, from the Technical University of Munich, introduced the influence coefficient method (ICM) to model coupling (aka convective) effects in the multi-slice method (MSM)[83], based on previous work from Kagawa in 1961 for a plate [50]. (Figure 10)

In the 1980s, the idea that transmission error was directly linked to gear noise was widely accepted, but not completely. When the renowned OSU Gear Consortium was created in 1981 by Houser, one of its main goals was to develop analytical models to predict transmission error [109]. One of the first analytical methods researched at OSU before LDP came from a model initially developed between 1968 and 1970 by the Army to predict noise and dynamic loads in helicopters by modeling the compliance of the gear mesh teeth as beams and the contact as rollers [73, 76]. This work originated from investigations in the early 1960s by the Army into helicopter noise [52, 54]. The work concluded that no analytical models were being used to predict gear NVH (identified as a gap). It led to the creation of a helicopter vibration and noise reduction program at Boeing in the 1970s [92, 96]. In 1983 [116], OSU would publish the first AGMA FTM paper on gear mesh stiffness and LTCA, where they would introduce LDP using the 3D plate method of Conry and Seireg [154]. In the same year, Winter and Podlesnik published a set of experimental tooth stiffness curves [120, 121, 124], which other software have used for validation [335]. Still again, in 1983, Neupert created a version of the multi-slice method (MSM) that used the finite element method (FEM) to determine the stiffness, which allowed the coupling of neighboring teeth and improved calculation of rim stiffness; this would later be turned into the software STIRAK out of RWTH Aachen University [349]. Three years later, in 1986, OSU published the first paper to optimize microgeometry by minimizing transmission error in a spur gear [131]; this paper would point out that beams, which are 2D formulations that assume a uniform load across the face width, need to be replaced with plates, which are 3D formulations, to capture a non-uniform load across the face width of the gear (required for helical gears).

In the late ’80s, Vijayakar, inspired by Wilcox’s work at Gleason, developed another version of the hybrid finite element method (HFEM), replacing Hertz’s approach with Boussinesq’s for the contact stiffness. Vijayakar first co-published an article with Wilcox and Busby at Gleason in 1988 discussing the difficulties of modeling contact with the finite element method [152]. Three years later in 1991 [172, 171], Vijayakar would use the work of J.M. de Mul out of SKF [130] (work which used the theory of Boussinesq [9] to model the contact of arbitrarily curved bodies) to create another version of the HFEM using the Boussinesq method for contact stiffness; this work would eventually be turned into the commercial software Transmission3D (T3D), shown in Figure 26 of the Appendix. OSU’s LDP introduced the HFEM in 1996, which was necessary to model gears with thin rims [210]. Additionally, in the late ’80s, optimizations started to be performed, for example, the work out of OSU on minimizing static gear transmission error [131], and the work out of the California University of Santa Barbara by Vanderplaats et al. on multi-objective gear optimization [151]. In 1988, KISSsoft published a paper on their renowned component-level gear design software [143], which would implement a gear LTCA algorithm in 2005. Finally, in 1989, Petersen would extend the work of Weber to include a cutter simulation for more refined fillet geometry [158] (see Figure 30 of the Appendix for an example, and Figure 47 for examples of the effect of cutter geometry on various cylindrical gear tooth macrogeometry features).

From the ’90s onward, the development of gear loaded tooth contact analysis (LTCA) software would boom, particularly in universities. For instance, GATES [175] out of the University of New Castle was created from the dissertations of Steward [160] and Haddad [167] in the early ’90s, improving upon the work Vedmar performed on the hybrid finite element method (HFEM) [106] by extending it to multiple teeth along with other minor improvements. In 1995, the HFEM was implemented by Gosselin et al. out of Laval University to study bevel and hypoid gears [198]. In 1996, the stiffness coupling method (SCM) implementation of the multi-slice method (MSM) was introduced into software in the form of LVR out of the Technical University of Dresden, in collaboration with OSU [205]. BECAL, a specialized bevel gear LTCA software, has been developed at the Technical University of Dresden on behalf of the Drive Technology Research Association since the early 1990s and has been continuously adapted to the growing requirements of the industry ever since [474, 386]. (Figure 11)

This movement in software was partially sparked by the conclusion that transmission error was directly correlated to noise (aka sound power). Opitz loosely made the link in 1968 when he noticed both sound power and transmission error increased with load, just on separate plots [75]. One of the first hard links of transmission error to noise was by Niemann and Baethge out of FZG in 1970 [78]. Smith out of Cambridge would again make the link in 1978 [98]. Even with this established evidence, there was doubt on the strength of the link for years. Finally, in 1994, a paper published by Houser at OSU, Oswald and Valco at the Lewis Research Center, and Drago and Lenski at Boeing again linked transmission error to noise measurements [187]. A year later, Palmer and Munro would again make the link [203] and also show the impact of long [78], intermediate [156, 165], and short reliefs [41]. These papers were the catalyst needed for gear loaded tooth contact analysis software to take off in the automotive industry and continue branching out further from there. Software would start to be migrated over from older platforms like DOS (est. 1981) to newer platforms like Windows (est. 1985), which was in 2004 for Windows LDP.

Lastly, while not directly related to stiffness, in 1996 Paul reminded gear designers that, no matter how good their macrogeometry and microgeometry are, any unwanted plus material can destroy the NVH of their gear set [211]. Two years later, in 1998, the Ferrium family of gear steels was introduced [223], which was designed using a computer and is the most robust gear material tested by NASA [292, 325].

The 2000s and onward was really when commercial LTCA software started to boom (with gear software high-lighted as early as 1980 by Drago [99]), beginning with the bombshell paper by Romax [497] introducing system deflection analysis to commercial software in 2000 [232]; system deflection analysis is described in detail later, but centers around modeling the gearbox as a system (often including housings and asymmetric shafts) to determine the relative misalignments at the gear set so they can be counteracted with gear microgeometry. SDA was implemented in academic software as early as 1983, as reported by Houser at OSU [116]. In 2000 and 2002, Houser introduced the modern form of gear microgeometry design, which involves employing large design of experiments (DoEs), “running many” cases to map out the design space for the gear designer [233, 250]; reference Figure 25 of the Appendix. In 2003, commercial acoustic software was integrated with a gearbox NVH analysis to predict sound pressure levels [259]; this had been studied in academia in the ’90s out of the University of Kentucky [170] in 1991 and the University of New Castle in 1994 [192, 195, 196], with simpler methodologies discusses as early as the ’80s [113]. In 2004, Sainsot and Velex would improve upon the work of Weber for gear tooth foundation stiffness, which is the method that is now commonly used for the multi-slice method (MSM) when the FEM is not used [265]. Generalized multibody dynamic (MBD) software such as SIMPACK [499] began introducing specialized gear stiffness models, such as the dynamic floating method implementation of the finite slice method (FSM) introduced in 2006 [280]. Investigations in the late 2000s then focused on comparing MBD and FEM versions of models for gear rattle, including methods based on modal reduction [284, 300]. In 2007, Hemmelmann’s dissertation [289], out of RWTH Aachen University, created the fundamental work that would later be turned into ZaKo3D in 2010 [306], which has been developed ever since [381]; ZaKo3D has a state-of-the-art implementation of manufacturing simulations combined with the hybrid finite element method, which allows high-fidelity assessments for effects such as waviness [475] and topological modifications [418]. (Figure 12)

2010 onward saw the creation of a new role in the gear industry specifically for gear system analysis engineers, and the emergence of commercial gear software as mainstream, replacing older internally developed company programs [377]. Modern gear design groups at larger companies in the industry often split gear design work between component-level gear designers specializing in software such as KISSsoft [493] and internally developed programs (often a role for more experienced gear engineers), and system-level gear designers specializing in software such as MASTA [494], Romax [497], and Transmission3D [501] (often a role for less experienced gear engineers, supervised by more experienced engineers). System-level gear design roles require extensive software knowledge in FE packages such as Abaqus [476] and ANSYS [481] for stiffness reductions and surface vibration analysis (software such as ANSA [480] and HyperMesh [489] are often used for preprocessing efforts) and vibroacoustic packages such as ACTRAN [477] and COUSTYX [484]. Gear manufacturing engineers often use specialized gear manufacturing software, such as the Gear Production Suite (GPS) offered by Dontyne [486], which is also integrated into Romax (manufacturing modules are also available in KISSsoft and MASTA).

In 2011, at the Technical University of Munich at FZG, LTCA and contact durability analysis were first combined to predict the location on the surface where the pitting was going to start [318] (this has yet to make its way into most mainstream commercial gear software). On the topic of commercial gear software, between 2012 and 2023, SMT wrote more than 20 articles on the topics related to its gear system analysis software MASTA (quasi-static) and DRIVA (dynamic). Topics included both quasi-static semi-coupled and decoupled system deflection analysis [365], dynamic analysis with their basic LTCA solver [348], their hybrid finite element method (HFEM) implementation [382, 369, 354], a new homegrown dynamic solver with advanced methods to reduce run-time called ATSAM [421], their automation capabilities via scripting [412], and tooth interior fracture analysis with their HFEM solver [360].

While experimental mesh stiffness analysis is not of intense focus in this work, between 2015 and 2019, a number of papers out of the Indian Institute of Technology Indore explored experimental techniques using photoelasticity [356], strain gauges [371], digital image correlation [384], laser displacement [396], and modal analysis [411]. In 2020, single-tooth stiffness was explored experimentally and numerically using the FEM [423]. These works are relevant, as LTCA methodologies rely on experimental validation to become proven.

In the late 2010s, instead of focusing on minimizing the peak-to-peak transmission error (PPTE) a number of papers began to focus on reducing the tonality of the gear whine through techniques such as topography scattering out of RWTH Aachen University at WZL [368, 406] and in-equidistantly spaced teeth in gears out of the Technical University of Darmstadt [409, 427]. In 2018, research was conducted on various techniques to extract gear mesh stiffness from an LTCA run using the FEM [395].

Since 2020, research related to gear mesh stiffness includes topics on machine learning [461, 475], digital twins [469], the hybrid finite element method [469, 473, 459], the multi-slice method [446, 465], the FEM [470], and EHD effects on stiffness [472]. One of the latest review papers on gear stiffness was written in 2021 by Marafona out of the University of Porto [439], and categorized the techniques into four categories: analytical, finite element, hybrid, and approximated analytical models (which have gained momentum recently [471]).

The following sections will categorize all the various gear stiffness methodologies, workflows, and software used in NATO countries and Switzerland (both academic and commercial). They will be followed by a discussion on subtleties in modeling techniques and areas identified for future research.

3 Categorization of Methodologies

This section will cover the different methodologies for calculating gear mesh stiffness for a loaded tooth contact analysis (LTCA). As mentioned before, this work does not try to focus on a particular geometry type. Still, most published research is performed with cylindrical gears due to their simple geometry and wide use in industry. This work will focus on gears with no defects of any kind resulting from fatigue and wear; these types of effects are studied as part of the gearbox conditional monitoring field (focused mainly in aerospace, where failure detection is absolutely critical), and several review papers have been written over the past five years [394, 455, 426].

The primary objective will be to capture all methodologies developed for both academic and commercial software worldwide. The goals of this section are to explain the fundamental concepts of LTCA, come up with a single figure that can visualize all the different gear stiffness methodologies, create a set of terminology that can be used to categorize the different approaches, create a figure to visualize the most common discretization approaches, and explain the concept of coupled vs. decoupled workflows. This section focuses on the component-level, while the next section will focus on the system-level workflows (it may be necessary to jump between sections at times). Another section will categorize all the software found in NATO countries and Switzerland into a series of tables using the terminology in this section. Due to the large amount of explanation required to distinguish among the various software on the market and to explain the difficult subject of gear system analysis, an Appendix of many figures will be provided.

The reader should understand that the reason there are so many methods for gear stiffness covered in this section is that gear designers require different balances of speed and fidelity at various stages in the gear-design process (which is covered in the software section). Approximations and reductions in gear stiffness are a necessary evil used in gear software to allow much faster design iterations. As Dr. Box famously said in 1976 [439], “Since all models are wrong, the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad.”

Many review papers over the years have either directly covered the topic of gear stiffness or indirectly included it as part of the review due to its strong link to the main topic of review, such as gear dynamics. Most dissertations on the subject of LTCA have some of the strongest reviews on gear mesh stiffness. Marafona et al. published one of the most recent reviews in 2021, which divided the review into four sections: fast analytical methods (FAMs), finite slice methods (FSMs), finite element methods (FEMs), and hybrid finite element methods (HFEMS) [439]. All four methods were covered in detail, and more theory and equations were provided than in this work. A final comparison of the methods with respect to accuracy, computation time, implementation difficulty, and required resources was presented. Marafona focused heavily on recent Chinese research on gear LTCA, which emphasized the multi-slice method but also included many other methods, such as those used in Windows LDP. From 2013 to 2021, Marafona covered more than 30 articles originating in China, with a significant spike in written papers about 2018. In the same year, Natali et al. wrote a shorter review with a greater focus on comparing various models to the FEM [442]. A year prior, Sainsot and Velex, out of the University of Lyon, published a review on contact stiffness [429].

Figure 13 shows an illustration of the different components of gear stiffness. A gear is commonly broken down into four components: local contact stiffness, global tooth stiffness, rim foundation stiffness, and gear body stiffness. For methods like the FEM, all four components can be captured with ease for any geometry. For methods like the FSM, differentiating between global tooth stiffness, rim foundation stiffness, and gear body stiffness can become difficult, especially for thin-rimmed gears, which often have to resort to the FEM to capture correctly [252, 210, 309] (studied as early as 1964 by Attia, before the FEM [62]). Due to the large computation time associated with using the pure FEM for contact, it is much more popular to use a semi-analytical method to replace the local FEM contact; this creates issues of its own that will be discussed, which involve efforts to not double account for stiffness at the local contact to global tooth interface.

What makes gear software so unique is that the system deflection analysis (SDA), used to determine misalignment at the gear mesh interface, and the loaded tooth contact analysis (LTCA), used to determine load distribution on the gear teeth, are usually split into two separate analyses. This is done so that a very basic model of the gear mesh stiffness can be used in the SDA and not slow down a solution that can involve resolving over a dozen nonlinear clearances in the system. A much higher-fidelity model is then used in the LTCA to evaluate important parameters, such as contact and bending stresses. To split the SDA and LTCA from one another, some boundary must be chosen as the split line, and attempts must be made not to double account for stiffness at this interface or to violate St. Venant’s principle, whose premise is that the difference between the effects of two different but statically equivalent loads become very small at sufficiently large distances from the loads [5, 68, 31]. The split line is typically chosen between the rim and body interface (as shown in Figure 13), far enough away from the root of the teeth so that the boundary condition does not artificially increase the stiffness of the tooth, increasing the stress (typically a radial distance of 1.00-1.20 times the whole depth is the minimum distance chosen, which is not always possible for thin-rimmed gears).

Many gear body geometry types are shown in Figure 27 of the Appendix, and all need to be evaluated for compliance in the gear design process. Simple analytical methods focus only on an elastic ring of material below the tooth and cannot capture nonlinearity in the geometry, such as thinning of the webs or asymmetric lightening features. For this reason, the gear body’s stiffness is predominantly modeled using FEM. With the gear body being modeled as FEM, it makes modeling the teeth with FEM advantageous to more easily integrate the gear teeth and body together.

See the next section for a detailed definition of LTCA, but for now, focus on Figure 14 to illustrate the interaction of the different components of gear stiffness with one another when used in an LTCA. The setup of an LTCA involves grounding the wheel at the bore and applying a load on the pinion bore to determine how much the bore deflects torsionally, with a specific interest in the tangential deflection at the pitch circle, which corresponds to the transmission error of the gear mesh. The local contact, the global tooth and rim, and the gear body all act as springs in series. The discretized stiffness elements across the gear’s face width act as springs in parallel. The FSM has many techniques to reduce computation speed, with the hope of having little impact on fidelity. While it is common to discretize the local contact and global tooth and rim across the face width of the gear, the gear body is sometimes reduced to a single DoF in the FSM. Neighboring DoF can either be decoupled or coupled; it is common to couple the tooth DoFs and less critical to couple the body DoFs; the contact stiffness is rarely, if ever, coupled in the FSM. When the FEM is utilized, all of the couplings are inherently captured, giving it an inherent advantage at the cost of speed and difficulty of implementation. While not illustrated, a “gap function” determines what slices will come into contact as they approach one another, taking into account the microgeometry applied to each of the gear flanks.

Figure 15 illustrates the split of the SDA and the LTCA in a typical gearbox computational analysis and defines three methods for combining the two analyses. In the decoupled method, the SDA only provides the misalignment to the LTCA, and the LTCA returns no feedback to the SDA. In the semi-coupled method, the SDA provides mis-alignment to the LTCA, and the LTCA sends a more refined load distribution back to the SDA; this continues until convergence. In a gear system analysis, the coupling of the SDA and the LTCA is inherently captured. While the SDA and the LTCA each have their own gear stiffness models, this section only focuses on the higher-fidelity models used in the LTCA.

For the reader to understand all the software and methods that are about to be described on the topic of gear stiffness, a large amount of effort is required to define all terminology, attempt to cite the papers where they originated from, and create illustrations to make the explanations clearer. The author sought to create terminology that was common yet also allowed for unique three-letter acronyms to convey large amounts of information in the tables; for this reason, not all terminology is commonly used. More than 50 terms, listed in alphabetical order by acronym, describing the different methods discussed on the subject of gear stiffness. They include:

• Axial Interaction Method (AIM): A method used in the coupled-slice method (CSM) to allow interaction among neighboring axial slices of the same tooth, implemented by coupling the slices together in some manner (commonly referred to as including “convective” effects); the FEM inherently captures these effects. Three of the major axial interaction methods used in the FSM include the Rayleigh-Ritz method (RRM), the stiffness coupling method (SCM), and the influence coefficient method (ICM).
• Beam Method (BM): An analytical elasticity method typically based on either the work of Euler-Bernoulli (flexure-dominated long beams, 1744 [3]) or Timoshenko (shear-dominated short beams, 1921 [16]), using a single beam element for the entire profile. Timoshenko improved upon the Euler-Bernoulli model by accounting for shear distortion and rotational inertia. Several reviews comparing the two methods have been published [314, 304]. Beams were first investigated by Leonardo da Vinci about 1493 [1], and later by Galileo in 1638, Mariotte in 1686, Jacob Bernoulli in 1705 (father of Daniel Bernoulli, who had Leonhard Euler as a pupil), and Parent in 1713. Timoshenko applied it to gears first in 1926 [18], and Nagaya included dynamic effects in 1981 [104]. For more information, reference the following sources [336, 38].
• Boundary Element Method (BEM): An alternative numerical approach to the finite element method (FEM), only requiring a discretization of the surface (reducing the problem by one dimension), as opposed to the surface and volume when using the FEM. The BEM was developed over a number of years and was first introduced to the world by Brebbia in 1978 when he published the first book on the subject [135]; a full history is covered here [269]. The BEM is most commonly used to determine bending stress for FSM methods that do not capture the value in the LTCA [134], but has also been used for LTCA in limited cases [266, 276].
• Coarse Element Method (CEM): In this application of the FEM, the elements are only refined enough to allow the solution to actually be solved without any numerical issues, but not refined enough to capture effects such as microgeometry changing the radius of curvature, or small contact patch sizes seen at very low loads (which must be evaluated for NVH).
• Circumferential Interaction Method (CIM): A method used in the coupled-slice method (CSM) to allow interaction among neighboring teeth in terms of stiffness. The CIM, as related to the FSM, has had few publications in the literature and is most commonly incorporated using a method such as the FEM, which inherently links all the teeth into a single stiffness matrix.
• Conformal Mapping Method (CMM): A method that uses two-dimensional elastic theory combined with conformal mapping functions to determine tooth stiffness. The method originated in a 1962 paper [53] and was implemented in 1980 [100] and 1981 [105] by Terauchi et al., from Japan.
• Coupled-Slice Method (CSM): A multi-slice method (MSM) that has some type of coupling effects included between neighboring slices, most commonly an axial interaction method (AIM) paired with an end effect method (EEM), and in some cases a circumferential interaction method (CIM). The method’s origins lie mostly in German universities, such as the Technical University of Dresden’s implementation of LVR [205, 248] with the stiffness coupling method (SCM) and the Technical University of Munich’s (FZG) implementation of RIKOR [147] with the influence coefficient method (ICM). A more extensive review can be found in Figure 42 of the Appendix. Reference Figure 44 in the Appendix for the most recent comparison of the CSM and DSM.

Regarding which method is more computationally efficient, for the SCM, you would have to find the number of operations needed to solve the banded symmetric matrix using banded Gauss elimination, which depends on the bandwidth and the size of the matrix. In contrast, with the ICM’s symmetric matrix, the number of operations would depend on the matrix’s size. The method with the fewest operations would be more computationally efficient. However, you also have to consider the accuracy and ease of creating the system matrix.

• Dynamic Reduction Method (DRM): A method used to reduce the number of DoF in the model, to facilitate faster solutions. For dynamic problems, the most popular model reduction technique is component modal synthesis (CMS), introduced in 1960 by Hurt and later simplified by the Craig-Bampton technique in 1968 [72]. Dynamic reduction methods to include gear contact have been studied [337, 317, 284, 300].
• Dynamic Floating Method (DFM): A Lagrangian-type approach where the stiffness is defined at the base of the tooth of the gear, independent of the line-of-action (LOA); reference Figure 40 of the Appendix. This approach originates from multibody dynamic software and was first incorporated with a single tangential DoF [280], with later implementations using a single torsional DoF [350, 419].
• Decoupled-Slice Method (DSM): A multi-slice method (MSM) that has no form of coupling between any of the neighboring slices. This is an older method mentioned as early as 1943 by Merritt to determine load distribution in helical gears (only a brief description is provided on page 199). By the 1960s, research was underway to study the “convective” effects that were deemed crucial to include (see Figure 42 of the Appendix). Reference Figure 44 in the Appendix for the most recent comparison of the DSM and CSM, which shows large variation due to the lack of coupling effects.
• Elastic Base Method (EBM): A method used to determine the global tooth stiffness, whether for a single slice, single tooth, or incorporating the entire gear if the method allows. Examples of EMBs include the beam method (BM), the plate method (PM), the shell method (SM), the finite element method (FEM), and the potential energy method (PEM).
• End Effect Method (EEM): A method used in the coupled-slice method (CSM) to attempt to capture the buttressing effects that happen near the end-faces of the gears, as the tooth section transitions from plane strain to plane stress; reference Figure 24 of the Appendix. One of the earliest forms of the end effect method was the moment image method (MIM) [330], but more recent versions incorporate a general softening method (GSM), similar to that used in KISSsoft [493]. The finite element method inherently captures this effect.
• Elastohydrodynamic Analysis (EHDA): Lubrication analysis performed at the gear contact interface to capture effects such as friction and load distribution in either boundary, mixed, or full-film lubrication conditions, according to Stribeck. Stribeck created his renowned curve, which relates the fluid film thickness to speed and load under various conditions. Gumbel combined Stribeck’s curves into a single curve in 1914 by introducing a dimensionless parameter. Finally, Hersey introduced a more refined dimensionless parameter in the same year, the ratio of the product of viscosity (η) and rotational speed (N) to the average load (P), which is referred to as the Hersey number. Reference Figure 33 of the Appendix for an illustration of the Stribeck curve along with an experimentally derived version performed recently for various gearbox oils.

EHDA is required to capture the true load distribution at the contact patch for lubricated gears (as Figure 34 of the Appendix shows) by solving for the friction coefficient (if it’s not assumed constant) and resolving the corresponding shear force at the contact interface, all to evaluate the changes to the forces at the contact patch, which directly affect the gear stiffness; models also assume simple Coulomb friction models [255]. When temperature effects are included in the analysis, this becomes TEHDA [297]; friction and temperature effects cannot be ignored at high pitch line velocities [243], especially for plastic gears. Models can further be distinguished on whether they allowed mixed lubrication or remained in a single form of lubrication [231, 311]. Due to the complexity, this is sometimes performed in a post-evaluation analysis (PEA). A number of papers have either reviewed this subject [95, 254, 425], explored this subject analytically [295, 173, 297, 397, 231, 453, 86, 467, 311], or explored this subject experimentally [48, 431, 231, 328, 114, 221]. The flank must be modeled all the way down to the surface finish to capture true EHD effects, as the gear flanks are known to be dependent on the tooth flank finishing process [353].

• Elastic Ring Method (ERM): The elastic ring method (ERM) is a method used to account for a portion of the rim thickness at the foundation of the gear tooth. For gears with lightening modifications, the finite element method (FEM) is the most common approach for modeling the gear body, as analytical solutions are typically unavailable. In 2004 [265], Sainsot and Velex came up with an expression for the foundation stiffness of a gear tooth, using the work of Muskhelishvili [59]. In his book, first published in 1933, Muskhelishvili derived the displacement of a two-dimensional circular ring based on a complex power series representation of displacements, stresses, and external loads. Prior to the work of Sainsot and Velex in 2014, Weber, in 1949 [33], derived an expression for the displacement of the gear tooth foundation, modeling the gear as an elastic half-plane. Attia in 1964 [62], and later Cornell in 1981 [101], published alterations to Weber’s original work. The altercations led to small changes in the coefficients, L∗, M∗, P∗, Q∗, used to calculate the deflection of the tooth foundation. Sainsot and Velex compared their work to a two-dimensional finite element model and Weber’s original work. The method of Sainsot and Velex was shown to come far closer to the finite element method results than the work of Weber. It should be noted that the derivation relies on the assumption of linear stress distributions as the root of the gear tooth. Both Sainsot, Velex, and Weber assumed linear stress distributions at the root of the gear tooth. O’Donnell [45, 60] and Seager [71] concluded that cubic variations of the normal stress and parabolic distributions of the shear stress were more accurate than the linear distributions used by Weber. Sainsot and Velex found very little improvement when assuming the cubic and parabolic distributions, and they found no simple deflection equation when they were used. For these reasons, Sainsot and Velex assumed linear stress distributions in their derivation.
• Extended Tip Contact (ETC): Contact that occurs outside the line-of-action (LOA) due to tooth deflections, increasing the contact ratio (reference Figure 39 of the Appendix). This phenomenon is typically a software switch that is turned on and off. This subject is covered in great detail by Langlois in the following two references [369, 382], and was introduced by Richardson in 1958 [42].
• Fast Approximate Method (FAM): Analytical-based methods that rely heavily on either empirical or numerical correlation (or both), which allows very fast calculations. This method was used heavily in early gear dynamic models, which assumed a constant mesh stiffness and eventually moved to variant mesh stiffness [145]. Calculations can be performed for either a single tooth pair or for multiple teeth in contact. Early examples include Umezawa [133, 84] and Cai [184, 197], and a commonly used method in gear software is ISO 6336 [278, 447] (this method is based on Weber’s work [33] combined with a polynomial curve-fit from Schafer [82]). FAMs have gained popularity in research lately with cosine models [385], Fourier series models [351], Heaviside models [383, 440], using a method such as the FEM [176, 441, 320] and the PEM [458] for tuning.
• Flexible Body Method (FBM): A family of methods that assume some or all of the gear body is flexible and are part of the more general flexible multibody dynamics field of study [262].
• Finite Difference Method (FDM): The FDM is a 2D numerical technique used for solving differential equations by approximating the derivatives with finite differences. The method has been used for gear stiffness discretizing both along the helix [27] and along the profile [247].
• Finite Element Method (FEM): A popular method for numerically solving differential equations using elements such as brick, quads, tetrahedrons, or general p-type elements [146]. The method can be applied directly, or it can be reduced and curve-fit to analytical equations for quicker analysis [150, 159]. The method is applied to gears to determine stress and calculate stiffness; it can be used in either 2D or 3D. The FEM is commonly used for validating and tuning lower-fidelity models, but is not practical for general design due to its high computational cost [152]. For practical design purposes, the local contact in the FEM is replaced with a semi-analytical contact method, referred to as the hybrid finite element method (HFEM). The history of this method was reviewed in the previous section. This method has been used both quasi-statically [93, 294, 345, 225] (implemented most recently by RIT-IGD [490] through multiple published papers [316, 357, 378, 392]) and dynamically with a single tooth [148] and multiple teeth [422]. There are two methods of refinement (see Figure 28 of the Appendix for an illustration): h-type, where the number of elements is increased (most commonly with adaptive methods [277, 281, 267, 260, 408]), and p-type where the orders of the polynomials of the shape functions are increased [139]; used in Transmission3D [459]. The FEM has also been combined with the BEM into a single solution [283]. A number of papers have reviewed the FEM in gear LTCA [410, 442, 439].
• Finite Prism Method (FPM): A method where finite prisms are used in the place of finite elements, which was originally designed for prismatic structures; considered a simplification of the finite element method (FEM). This has only been seen in academia [144]. For further information, review the following references [336, 128].
• Finite Slice Method (FSM): A method where the gear tooth is “sliced” at the bottom of the tooth, and an elastic base method (EBM) is incorporated to determine the stiffness of the “sliced” portion. The method is split into single-slice methods (SSMs), such as the thin-slice method (TSM) and the long-slice method (LSM), and multi-slice methods (MSMs), such as the decoupled-slice method (DSM) and the coupled-slice method (CSM).
• Finite Strip Method (FsM): A method where finite strips are used in the place of finite elements, which was originally designed for prismatic structures; considered a simplification of the finite element method (FEM). This methodology is used in HyGEARS from Involute Simulation Software Inc [217]. For further information, review the following references [336, 216].
• General Interaction Method (GIM): A general method used in the multi-slice method to couple slices from neighboring teeth together (not slices on the same tooth as described by the AIM). Due to the lack of research on the topic, only a general category was created for the use in this work. Typically, only the FEM can provide this type of coupling, and a GIM can be created by tuning some analytical formula to numerical models such as the FEM.
• Hybrid Finite Element Method (HFEM): The HFEM models the tooth, rim, and body with the FEM using a coarse mesh and replaces a fine mesh of elements at the contact zone with a semi-analytical contact formulation. A technique, first introduced by Vedmar [106], must be used to remove the local contact from the global tooth stiffness analysis (see Figure 36 of the Appendix). This method is implemented in software such as MASTA from SMT [369], which uses a Hertz representation for the contact stiffness, and Transmission3D/Calyx from Ansol, which uses a Boussinesq representation for the contact stiffness [172]. This method is considered the state-of-the-art for gear LTCA.
• Half-Plane Method (HPM): A popular method used to determine contact stiffness based on the work of Hertz studying a semi-elastic half-plane [8]. This method was later applied to gears by Weber [33] and Richardson [42]. The HPM is considered a reduced version of the half-space method (HSM). Assumptions include: the surfaces are continuous and nonconforming, the contact area is small with respect to the bodies, the strains are small, the bodies are considered elastic, isotropic, and homogeneous, and the surfaces are frictionless [126].
• Half-Space Method (HSM): A method used to determine contact stiffness of a half-space based on the work of Boussinesq for the normal component [9] and Cerruti for the tangent component [7] (which allows the inclusion of friction). This method was first used by SKF for bearings [130] and later applied to gears by Vijayakar in Transmission3D/Calyx [172]. The history of the theory can be found here [126, 12]. Near the end faces with a free surface, a quarter-space solution can be employed for improved fidelity [44, 77, 117, 307].
• Influence Coefficient Method (ICM): An axial interaction method (AIM) that was first applied by Kagawa to plates in 1961 [50] and later extended to gears by Schmidt in 1972 [83]. Reference Figure 41 of the Appendix for an illustration and Figure 42 for more history. The Technical University of Munich’s (FZG) LTCA software RIKOR was one of the first to use this method in 1988 [147], as described in [329].
• Line-of-Action (LOA): The line-of-action is a theoretical line that is tangent to the base circle of both the pinion and wheel and is the path the gear follows if the gear were to be modeled rigidly (reference Figure 35 in the Appendix). Elastic effects can create extended tip contact (ETC) outside the LOA at high loads.
• Long-Slice Method (LSM): A method that models the tooth as a finite plate along the helix. The finite plate is modeled as a continuum and is not discretized along the face width for stiffness. The first implementation was created in 1972 by Conry and Seireg [154] using the flat plate method [24, 34, 46]. Fidelity was later increased by using the Raleigh-Ritz method (RRM) with tapered plate geometry [129, 140, 194]. Both of these methods are implemented in Windows LDP out of OSU [321].
• Misalignment Along the Line-of-Action (MLOA): The MLOA is the relative misalignment of a gear pair that acts along the LOA. Houser wrote on the subject in 2006, later corrected in 2008 [282], and stated that the MLOA was the only critical component when considering gear whine, and that the MOLOA can be ignored. Foltz later covered this subject again, including derivations for both the MLOA and the MOLOA. While Houser illustrated misalignment by showing changes at the contact surface, Foltz illustrated the misalignment by showing changes perpendicular to the contacting surface along the LOA [456]. Rouverol and Pearce in 1988 [149], Nakagawa et al. in 1989 [157], Smith in 1994 [193], and Haigh in 2003 [256] highlighted the connection between misalignment and transmission error. See Figure 35 in the Appendix for an illustration.

The range, not the magnitude, of misalignment is commonly limited to be no greater than 75 microns, with lower limits typically used for noise-sensitive applications; similar relative angular misalignment limits exist for bearings. While any magnitude of misalignment can be corrected for at a single load [41], it is the range of misalignment that creates the challenge. As the load strays away from a single load the microgeometry was optimized for (typically around 30% load for e-mobility applications), the NVH characteristics will increase; this is why transmission gearboxes that need to operate anywhere from zero to full load are so difficult to optimize microgeometry for, and the range of torque a single gear set sees should be increased with caution.

• Misalignment Along the Off-Line of Action (MOLOA): The MOLOA is the misalignment perpendicular to the MLOA. See Figure 35 of the Appendix for an illustration.
• Multi-Slice Method (MSM): An implementation of the finite slice method (FSM) where the tooth slice is further sliced along the face width of the tooth. When the neighboring slices are not connected together in any way, this is referred to as the decoupled-slice method (CSM). When neighboring slices are connected via an axial interaction method (AIM), the resulting method is called the coupled-slice method (CSM).
• Off Line-of-Action (OLOA): The OLOA is perpendicular to the LOA previously described and is the direction in which the friction forces act. Gear noise excitation outside the LOA was covered by Borner and Houser in 1996 [206]; these excitations become more significant as the peak-to-peak transmission error (PPTE) decreases.
• Potential Energy Method (PEM): An elastic base method (EBM) where elastic strain energy is paired with Castigliano’s second theorem to solve for displacements. In 1879, Castigliano published a book [70], later translated to English in [6], with the following theorem: for a linear structure, the partial derivative of the strain energy with respect to an applied force is equal to the component of displacement at the point of application of the force that is in the direction of the force. Weber first applied this method in 1949 [33], and it has since become the most popular method in modern software. While Weber applied this method to determine bending (in-plane), shear, and normal (in-plane) components, it has also been used to determine axial stiffness [400].
• Plate Method (PM): An elastic base method that either assumes a flat plate method (FPM) [24, 34, 46] or a tapered plate method (TPM) for determination of the tooth stiffness [129, 140, 194], as incorporated into Windows LDP. The flat plate method does not incorporate shear (thin plates), similar to Kirchhoff-Love plate theory [10], while the tapered plate method does incorporate shear (thick plates), similar to the Mindlin plate theory [36]. For more information, reference the following [336, 38].
• Quasi-Static Floating Method (QFM): A relative frame method (RFM) that uses the quasi-static domain with the frame in either the normal or transverse direction. This method is similar to the quasi-static grounded method (QGM), except that the small changes in the frame orientation as the gear mesh deflects are accounted for. This can loosely be considered a hybrid Eulerian and Lagrangian approach, or can be thought of as the QGM with corrections to account for deflections that change the normal orientation.
• Quasi-Static Grounded Method (QGM): A relative frame method (RFM) that uses the quasi-static domain with the frame in either the normal or transverse direction. This method assumes that the frame orientation is grounded based on the initial conditions and that no changes in the frame’s deflection are accounted for. This is an Eulerian approach, as opposed to the Lagrangian approach used in the dynamic floating method (DFM).
• Rigid Body Method (RBM): A method where the entire gear body is considered rigid, and either a small amount of penetration of mating surfaces is repelled with a penalty stiffness (whose stiffness can be representative of some portion of the gear tooth), or no separation is allowed by either imposing Lagrange multipliers as additional constraints or using a very high penalty stiffness. This method is commonly only used in multibody dynamics when the gears themselves are not of key interest, and they are just required to transmit forces to the components of interest; this method is also used for rattle, which requires many time-steps to be solved efficiently, and the gears are rarely even in contact due to excessive rattling [363]. A review of contact models in multibody dynamics can be found here [288, 332], and example use cases in the software IVRESS can be found here [268, 327].
• Rim Compensation Method (RCM): A method used to compensate for stiffness at the foundation of the gear tooth, composed of a select portion of the gear rim. The method is typically implemented either semi-analytically using an elastic ring method (ERM) or inherently in the finite element method (FEM).
• Refined Element Method (REM): A method of refining the contact zone elements in a pure FEM solution (FEM for global tooth and local contact). In this method, the elements are refined enough to capture effects such as microgeometry and very narrow contact patches observed at low loads.
• Relative Frame Method (RFM): A method used to determine the frame of reference for an LTCA. Three methods exist in the literature: the quasi-static grounded method (QGM), where the frame is completely fixed based on the initial conditions and placed in either the normal or tangential direction (Eulerian approach), the quasi-static floating method which is similar to the QGM except that the small changes in the frame orientation as the gear mesh deflects are accounted for (hybrid Eulerian and Lagrangian approach), and the dynamic floating method (DFM) where the frame rotates with the body of the gear (Lagrangian approach). Reference Figure 40 in the Appendix for an illustration of the approaches.
• Rayleigh-Ritz Method (RRM): An axial interaction method (AIM) utilizing a Rayleigh-Ritz approach to re-solve the nonlinear load distribution by minimizing potential energies. This method is used in Windows LDP, which utilizes a tapered plate for the elastic base method (EBM) [129, 194].
• Stacked Beam Method (SBM): An implementation of the beam method (BM), but instead of a single beam element for the entire profile, beams are stacked on top of one another up the profile based on the work of Laskin [73] and Cornell[101]. This method has been implemented in a separate program from OSU called the spur gear transmission error program (STEP) [246].
• Stiffness Coupling Method (SCM): A type of axial interaction method (AIM) where only the neighboring slices are coupled together, as opposed to the influence coefficient method (ICM), where all slices have some level of influence on one another. This method was developed primarily by the Technical University of Dresden from many dissertations over the years and incorporated into the software LVR [205, 248]. OSU has also used this method in a face gear LTCA program, referencing the work out of Dresden for the stiffness coupling term [246]. Reference Figure 41 of the online Appendix for an illustration.
• Slice Direction Method (SDM): A method used in the quasi-static domain to reduce the stiffness of the gear down to a single DoF, either in the normal or transverse plane (reference Figure 46 of the online Appendix). The direction is further defined to be either normal to the contact and tangent to the base circle or transverse to the contact and tangent to the operating pitch circle diameter; reference Figure 40 of the online Appendix for an illustration. This method can be applied with a quasi-static grounded method (QGM) or a quasi-static floating method (QFM).
• Shell Method (SM): A method similar to the plate method (PM), except using shells with the Rayleigh-Ritz method (RRM) and applying the technique to spiral bevel gears; this method originated at OSU, similar to the plate method (PM) [182, 183]. The main difference between a plate and a shell is that a plate is flat, and a shell is curved (making it ideal for spiral bevel gears). The most recently published literature can be found here [391, 449]. For more information on plates vs. shells, see the following references [336, 38].
• Single-Slice Method (SSM): An implementation of the finite slice method (FSM), which discretizes the tooth as a single slice, as opposed to the multiple slices that the multi-slice method (MSM) uses. This method is further categorized into a thin-slice method (TSM) and a long-slice method (LSM).
• Smooth Particle Method (SPM): A method where the smooth particle hydrodynamics (SPH) theory is applied to the study of solid mechanics. While SPH is more commonly associated with gearbox lubrication analysis [237], it has also been applied to solid mechanics [239, 274]. Recently, it has even been used, albeit in a limited manner, for loaded tooth contact analysis (LTCA) [341]. For a full history of how SPH was developed in the late 1970s for gas dynamic problems in astrophysics, see [239].
• Static Reduction Method (SRM): A method used to reduce the number of DoF of the model in order to facilitate faster solutions. For quasi-static problems, Guyan reduction, introduced independently in 1965 by Guyan and Irons, is the most popular model reduction technique [64, 65].
• Thin-Slice Method (TSM): A method used to either investigate a “thin” slice of the gear tooth along its face width or to analyze gears with very small face widths where load distribution is assumed uniform across the helix. This method is typically the starting point for investigating elastic base methods (EBMs) and is what Weber used when investigating the potential energy method (PEM) [33]. Once the EBM theory is developed, this method is typically transitioned to a multi-slice method (MSM), which commonly incorporates an axial interaction method (AIM).

With the terminology out of the way, the next step is to organize and visualize all the methods discussed into a single figure. Figure 52 of the online Appendix attempts to do this while also leading the reader to the state-of-the-art method, the hybrid finite element method (HFEM). Figure 16 shows illustrations of the rigid body method (RBM), the single-slice method (SSM), the multi-slice method (MSM), and the finite element method (FEM). The next section focuses on categorizing software following the terminology generated in this section.

4 Categorization of Workflows

The previous section focused on the different methodologies for calculating the various components of stiffness of a gear pair in mesh. This section focuses on how the mesh stiffness determined from the previous section is integrated into actual workflows. There are essentially six different types of analyses, as shown in Figure 17, that can be performed during the design of a gear pair, where the load distribution of the gear pair can either be evaluated or is an input into the analysis:

• Tooth Contact Analysis (TCA): TCA is an analysis where an interference (microns in magnitude) is imposed between two gears modeled rigidly to resolve the “unloaded” contact pattern and make corrections to the microgeometry, as required; a rigid body misalignment may or may not be included in the analysis. The analysis is performed over the full mesh cycle of the gear pair to determine all areas on each gear flank that penetrate one another; at the end, the resolved contacting areas are typically combined into a single contact pattern plot. A TCA mimics a contact pattern test commonly performed on gears, bevel gears in particular, where the gear flanks of several teeth are first coated with a thin layer of dyed gear marking compound (i.e., Steel Blue DYKEM [462]) and then rotated with minimal load to resolve the “unloaded” contact pattern, to determine if adjustments to the assembly are required.
• Loaded Tooth Contact Analysis (LTCA): LTCA involves loading an elastically modeled gear mesh to solve for the effects of load distribution on the deflection of the gear teeth to resolve its impact on durability results. Unlike a TCA, which imposes an interference, an LTCA solves for this value by modeling the stiffness of the contact, tooth, rim, and body; gear mesh stiffness is an analysis output. Other outputs of an LTCA can include contact stress (to evaluate pitting), bending stress (to evaluate bending fatigue), contact temperature (to evaluate scuffing), power loss (to evaluate efficiency), and transmission error (to evaluate whine). A TCA can be performed beforehand to rapidly identify anticipated points of contact before the analysis.
• Post Evaluation Analysis (PEA): PEA is an analysis performed after an LTCA to resolve outputs not included in the LTCA solution. This is required to evaluate bending stress for methodologies such as the plate and slice method, which do not discretize the fillets of the gear teeth in the solution. A PEA can also determine the stresses in the gear body, especially in larger, thinner bodies with lightening features. Lastly, the load distributions can be used in a decoupled TEHD analysis that would slow down the LTCA solver if directly incorporated.
• Sub-System Deflection Analysis (S-SDA): S-SDA involves solving for the relative misalignment of the gear pair bodies due to the deformation of the shaft and bearings, but excluding the effects of the housing due to additional complexity. Typically, there are limits on gear and bearing misalignments, which, if not met, may require the system to be stiffer or have smaller clearances. The gear stiffness is captured in the analysis, but typically using a very low-fidelity model to drastically reduce solution computation time.
• System Deflection Analysis (SDA): SDA involves solving for the relative misalignment of rotating components such as gears and bearings. This analysis models the gearbox as a system (typically including housings and asymmetric shafts) to determine the relative misalignments of these components so they can either be reduced to meet best practices or corrected with microgeometry [232]. The stiffness of the housing and any asymmetric shafts is commonly statically or dynamically condensed down to a set of nodes, whose stiffness matrices are then imported into the SDA software. For dynamic problems, the most popular model reduction technique is component modal synthesis (CMS), introduced in 1960 by Hurt and later simplified by the Craig-Bampton technique in 1968 [72]. For quasi-static problems, Guyan reduction (aka static condensation), introduced independently in 1965 by Guyan and Irons, is the most popular model reduction technique [64, 65, 232]. One major weakness of reduction techniques is that there is no guarantee that the eigenvalues and eigenvectors of the reduced problem will be good approximations of those of the original problem. For this reason, work has been done on estimating the error resulting from Guyan reduction [342]. An SDA is commonly solved in the frequency domain to evaluate NVH. Similar to the S-SDA, the relative misalignments at the gears and bearings can be assessed to determine if they pass best practices. It is common for the loads resolved from an SDA to be later used as deflection/force inputs for sub-component PSA, whether on the gear itself or any other structurally loaded component with hard-to-resolve loading conditions.

SDA is what some gear engineers focus on day in and day out, and is where the industry currently lacks trained talent. SDA is a large and complicated subject on its own. If five gear engineers were given a modern e-mobility system and asked to create an SDA model in popular software such as MASTA [365] or Romax [232] (example shown in Figure 5), five different models would be created, all very similar but slightly different based on what assumptions or simplifications were used.

• Gear System Analysis (GSA): GSA involves a comprehensive analysis that is commonly used to determine the durability, efficiency, and NVH of the gearbox system as a whole (coupled together). This system-level analysis can be performed as a single integrated solution or decoupled into multiple sub-analyses (much more commonly used to reduce computation time). GSA involves the deflection, stress, and life analysis of not just the gears [359, 402, 403, 358], but also their supporting shafts, bearings [285, 299], housings, and mounts; including any clearances that are designed into the system (e.g., splines, radial fits, axial fits). A GSA is most commonly solved in the time domain. Due to the inclusion of friction, the removal of the linearity condition that surfaces cannot come out of contact, and the inclusion of modal effects, the dynamics added to the solution inherently include the off-line of action (OLOA) NVH components that are often neglected in an SDA. In dynamic formulations, the controls can also be critical for capturing the coupling between mechanical and electromechanical components, especially when backlash becomes influential [322, 293, 213, 264]. The analysis has “gear” dominating the name, as it is the most difficult component in the rotating system to design due to the macro-sliding at the interface (something bearings only experience on the micro-level). Pitch line velocities in e-mobility gears are starting to exceed 50 m/s, far from the 15-25 m/s upper limit typically seen in ICEs but not near the 100 m/s seen in high-speed turbo gears.

Many other types of gear computational analyses are required during the design and analysis of gear sets and include 1D torsional vibration analysis to quickly model rattle and find resonant frequencies (see Figure 37 of the Appendix), hydraulic system analysis to design the hydraulic system including oil jet sprays (if required), component modal analysis to ensure gear body or gearbox housing mode shapes fall above or below specified targets, component stress and fatigue analysis for the gears with large bodies and their supporting structures, windage and churning analysis as a special sub-analysis in a much larger gearbox efficiency analysis, and finally vibroacoustic analysis to resolve the sound power generated by the gearbox system. The large depth of knowledge required to be a gear designer is one of the hurdles the industry faces in getting younger engineers into the field.

There are two main domains in which solutions are solved: the quasi-static domain (with dynamic effects option-ally captured by d’Alembert’s principle [2]) and the dynamic domain. Besides GSA, which is commonly solved in the dynamic domain, most other solutions will remain in the quasistatic domain. For the dynamic domain, solutions such as the SDA are commonly computed in the frequency domain, whereas a GSA is frequently computed in the time domain. A number of methods have been introduced to address the high computational cost of time-domain methods by combining them with hybrid methods. SMT introduced its new solver ATSAM just recently [421], as a way to more efficiently tackle gear blank tuning of asymmetric gear blanks with lightening features [388] (researched as early as 1999 [226, 244]). ATSAM is a dynamic model that utilizes multiple linearized, modal models of the system. It solves them sequentially in the time domain by passing the final conditions from one to the next as initial conditions. It was designed as an efficient way to use modal models to solve time-domain dynamics in systems where the modes change slowly relative to the excitations. Methodologies such as static-mode-switching [337, 317] out of the Catholic University of Leuven (KU Leuven) and the University of Stuttgart (building on previous work [284]) or the harmonic balance method out of OSU [153] are other innovative hybrid approaches used to try and bridge the cap between frequency domain and time domain solutions as it pertains to nonlinear problems.

There are many families of time integration methods an engineer can choose from when working in the dynamic domain; the two main methods are explicit and implicit (hybrid methods also exist). Explicit methods only use known information about the current state of the system and, in doing so, do not require solving a set of equations to implement; they require a small step size to converge (conditionally stable) and are easier to implement. On the other hand, implicit methods use information about the current and next state of the system, which requires solving a system of equations to evaluate; they do not need a small step size to converge (unconditionally stable), but are more challenging to implement. Explicit methods are suitable for highly dynamic systems (i.e., gear meshes with rattle [168]), while implicit methods are ideal for steady-state systems (i.e., gear meshes with whine) [240].

The earliest and simplest integration techniques are the explicit Euler method (forward Euler method) and the implicit Euler method (backward Euler method). The explicit Euler method was invented by Euler himself around 1768 [4], and the implicit Euler method was developed in the 1950s at the University of Wisconsin for the integration of stiff equations [37]. One of the most popular methods for families is the generalized Runge-Kutta method, a family of implicit and explicit iterative methods that includes the Euler methods [13, 14, 208, 209]; the explicit 4th- and 6th-order Runge-Kutta methods are particularly popular for rattle investigations. Over time, added features such as automatic time-stepping [347] and predictor-corrector methods [268] have improved the speed and fidelity of solutions.

Wrapping up this section, as mentioned in the previous section, coupling all analyses into a single integrated solution is ideal. Decoupling analyses inherently reduce the fidelity of the solution. When you decouple the tooth contact analysis from the loaded tooth contact analysis, the change in contact as the teeth deflect will not be captured. If you decouple the loaded tooth contact analysis from the system deflection analysis, the change in load distribution as the gear settles on a system-level misalignment will not be captured. In addition, you need to stitch together the loaded tooth contact analysis and the system deflection analysis at a boundary on the gear body; this can lead to double-counting some stiffness and loss of fidelity at the boundary. Decoupling gears with thin rims becomes increasingly difficult because the split line (the location of the imposed boundary conditions) must be very close to the teeth.

The coupling of solutions also extends to macrogeometry and microgeometry optimizations. Essentially, three Tiers of workflows are used in industry for gear macro- and microgeometry optimization (as shown in Figure 18). Tier 1 methods use a GSA approach, so they are inherently coupled; Type A methods are solved dynamically, while Type B methods are solved quasi-statically. Tier 2 methods are the semi-coupled methods previously discussed; Type A methods couple the SDA and LTCA together during the solution (two-way coupling), while Type B Methods use a one-way coupling of only the misalignment feeding into the LTCA. Tier 3 methods are of the decoupled type; Type A methods find their optimal solution, then plug the optimized microgeometry back into the SDA and tune it to the desired results. Type B methods do not add this correction step, and should be avoided if possible.

5 Categorization of Software

An attempt was made to categorize all major specialized gear software providers in NATO countries and Switzer-land that offer loaded tooth contact analysis (LTCA) capabilities. The following specialized gear software providers were contacted to properly categorize their software based on the terminology of the previous section, using the format (”Year Established,” ”Country of Origin,” ”Website”): ANSOL – Transmission3D (T3D) / Calyx (2007, USA, [501]), Dontyne – Gear Production Suite (GPS) (2006, England, [486]), FVA GmbH – FVA Workbench (2010, Germany, [485]), Gleason – GEMS (1865, USA, [487]), Gleason – KISSsoft (1998, Switzerland, [493]), Hexagon – Romax (1989, England, [497]), Involute Simulation Software – HyGEARS (1995, Canada,[494] ), MDESIGN – LVR (2005, Germany,[495]), and SMT – MASTA (2002, England, [494]). Very little information could be found publicly about specialized gear software providers outside of NATO countries and Switzerland, so they were not included in this work. It should also be noted that some software providers, such as GWJ Technology – eAssistant SystemManager (Germany,[#]) offer system-level analysis but no LTCA capabilities. [503]

The following specialized academic gear software providers were contacted, using the format (”Abbreviation,” ”Country of Origin,” ”Website”): The Ohio State University – Gear Lab – LDP, (OSU, USA, [502]), Rochester Institute of Technology – Gear Research Laboratory – IGD (RIT, USA, [490]), RWTH Aachen University – Laboratory for Machine Tools and Production Engineering – STIRAK / ZaKo3D (WZL, Germany, [500]), Technical University of Dresden – Institute of Machine Elements and Machine Design – BECAL (TUD, Germany, [483]), and the Technical University of Munich – Institute of Machine Elements – Gear Research Center – RIKOR (FZG, Germany, [496]).

Typically, these software providers supply software when a company becomes a member of their consortium, which comes with an annual fee based on the company’s size.

The two following 1D torsional vibration analysis (TVA) software providers are briefly mentioned, using the for-mat (“Year Established,” “Country of Origin,” “Website”): Gamma Technologies – GT-Suite (1994, USA, [488]) and SIEMENS – Amesim (1847, Germany, [479]). While these suppliers also have their own internal LTCA algorithms, they will not be of focus. Instead, the author highlights their capabilities for 1D torsional vibration analysis in the time and frequency domains. In the past ten years, both providers have recently added frequency domain solvers that can handle gears. Cummins Inc.’s longtime internal frequency TVA tool, TVSIM, partially inspired these solutions.

While specialized gear software providers are the focus of this work, it would not be complete without mentioning some of the many general multibody dynamic software providers out there; this work will give five examples, as there are quite a few options out there with their own specialized gear LTCA algorithms. Using the format (“Year Established,” “Country of Origin,” “Website”): Advanced Science and Automation Corporation – IVRESS (1998, USA, [491]), AVL – EXCITE (1948, Austria, [482]), CONTECS Engineering Services GmBH – SIMDRIVE 3D (2001, Germany, [498]), Dassault Syste´mes- SIMULIA – Simpack (1981, France, [499]), Hexagon – ADAMS (1992, Sweden, [478]).

In addition, the author will be creating a new LTCA software out of Purdue, titled the “Gear System Analysis Program (GSAP).” The software will be developed in MATLAB and implement a new version of the coupled-slice method (CSM) with several improvements.

Please note that software changes companies every so often, so these tables could become invalidated at any time. The intent of this review is not to reveal trade secrets but rather to inform gear engineers using the software what theory is being utilized in the background. The purpose of this review is not to pit software providers against one another, so no comparisons outside of the same gear software will be given. As the terminology is new and can be quite confusing, it is highly advised to discuss any questions with your software provider. It should also be noted that gear software is ever-evolving, and the values in this table a year from now could have been changed through regular software updates or major software releases. Lastly, note that many software providers integrate other LTCA software algorithms into their software, similar to Klingelnberg KIMOS [492] integrating BECAL for LTCA; making all these connections is not the focus of this work.

Table 1 lists, alphabetically, the different LTCA methodologies used in software with significant market share in the American e-mobility industry. Note that the ”variant” type is only valid within a single software and is not mutually exclusive to other software. The year in the table represents when the solver was first released, even if it was later released by a previous institution it had previously resided in. Geometry is either ”exact” and represents the fillet with a cutter simulation, or the geometry is ”simple” by either assuming simplified geometry like tapered plates or using a methodology that does not include the fillet explicitly in the formulation. The ”local” and ”global” acronyms are in the previous section. Table 2 lists additional software originating in Germany and Canada, as well as the newly created GSAP from Purdue. Table 3 was created to distinguish between different implementations of the finite slice method (FSM), and Table 4 was created to distinguish between different implementations of the hybrid finite element method (HFEM).

Figure 19 divides the methodologies previously described for SDA and LTCA into six tiers (in ascending order of increasing fidelity), along with descriptions, use cases, recommended methods, applicable software from Table 1, and explicit rationales for each. The table focuses on the American e-mobility industry, and the recommended software is based solely on theory. The decoupled-slice method (DSM) was not included in the table due to the large deviations it experiences outside of spur gear sets with no misalignment or helix modifications [446] (illustrated in Figure 44 of the Appendix). Figure 45 of the online Appendix illustrates two standard methodologies for verifying LTCA results: a contact patch comparison and a peak-to-peak transmission error (PPTE) comparison. The differences between Tier 1, Tier 3, and Tier 5 solvers (as implemented by SMT) are shown in the PPTE comparison.

6 Discussion

There are two main topics for discussion in this section: subtleties in calculation methodologies and areas for research in gear system analysis (GSA). Even between two different software programs using the same calculation methodology, there can still be significant differences in the results, which comes down to the various subtleties in the implementations of each method. Often, these subtleties are left out of articles because they are deemed less critical in explaining the methodology or as a way to maintain a competitive edge (proprietary information). While the field of gear LTCA is a very popular topic that has been well-researched worldwide, areas still exist for improvement; this will be covered at the end of the section.

Subtleties often stand out more in finite slice methods (FSMs), such as the coupled-slice method (CSM), because many different sub-methods must be combined to capture the 3D coupling (aka convective) effects. Methodologies such as the finite element method (FEM) are much more stable, as the field is well researched and has many applications beyond modeling gears. Subtleties also stand out more in decoupled workflows, which require stitching methods together at the boundary between the system deflection analysis (SDA) and the loaded tooth contact analysis (LTCA). The following are a few subtleties in the implementation of different LTCA software programs:

• Simplification of Tooth Geometry: Some models, such as the plate method (PM), assume flat or tapered geometry and do not accurately capture the true form of the fillet. The actual deformation at the point of contact between the gear teeth is rarely captured accurately; instead, a small degree of interference is allowed. Other simplifications in geometry are sometimes used to calculate the radius of curvature of a gear, sometimes exclud-ng the effects of microgeometry; Vijayakar showed that the changes were not insignificant [212]. Very rarely is the involute equation solved iteratively [275], so the geometry has to get discretized to some degree. Some smoothing is commonly assumed to prevent singularities at very sharp corners, such as the tip chamfer. Lastly, the geometry is rarely refined enough to account for deviations in the tooth surface from the manufacturing process, such as unintentional bias, feed rate marks, etc.; SPARTApro, developed at WZL, is an example of advanced manufacturing software that can calculate chip geometries [310].
• Assumptions on Contact: Contact itself is inherently challenging to model because of the dependency on geometry at the nano level that is stochastic and highly non-uniform, causing the need for finer discretization. In addition, coupling fluid, thermal, and structural mechanics in the time domain is no easy task. Because of these difficulties, many assumptions are made in analytical contact formulations, such as Weber’s assumptions in his Hertzian formulation, which assume a small contact area and no friction. When friction is included, it can either be constant or calculated at each time step, assuming a single lubrication state or mixed lubrication states; thermal effects can be added for further fidelity. Lastly, small sliding is often assumed, which greatly simplifies the contact search.
• Geometry Included in Contact Search: Similar to the stiffness, the contact must also be discretized. This discretization sometimes omits areas that come into contact with the gear set. It is common to ignore discretizing the non-loaded flank, but this prevents the modeling of double-flank contact if the gears are ever forced into tight mesh, either intentionally or unintentionally [344]. The fillets and roots of the gears are often omitted, but they can come into contact under these tight-mesh conditions if the gears are not properly designed. In the past, not all teeth in the mesh were included in the contact zone, meaning that extended-tip contact at higher loads could not be captured [245]. Lastly, the top lands of the gear teeth are rarely modeled, as they should never come into contact in a properly designed gear set.
• Modeling Away from Nominal Geometry: Most analysis performed in the design of a gear set is done at nominal conditions; limited analysis is performed at minimum and maximum tolerances, and instead, those are set based on experience. A safety factor is then applied to the results to account for worsening results due to geometry that deviates from the nominal and other effects, such as dynamics. The geometry of the tips and tooth thickness can include the impact of radial and pitch errors. The geometry of the flank can include any profile and helix errors, most commonly caused by manufacturing, wear, and fatigue. The geometry of the fillet and root can include the effects of cracks. The actual geometry of a measured gear can be used to capture each tooth flank if enough time is available. Most software providers offer some level of manufacturing error modeling, ranging from simple analytical models to models of measured flank surfaces.
• Implementation of Chamfering: Just like in manufacturing, chamfers are often an undervalued part of the process. Including chamfers, particularly tip-edge chamfers, is not available in all software implementations. In addition, at the edge of the chamfer, some assumption needs to be made on the radius of curvature, as it cannot be modeled as infinitely sharp, as previously mentioned.
• Correction/Tuning/Correlation Factors: Some software includes correction factors, which allow param-eters such as the stiffness of the tooth to be increased or decreased. The ISO mesh stiffness uses a default correction factor of 0.80 to reduce the tooth stiffness and better match experimental results [278]. It is common for the coupling stiffness or influence coefficients to be based on a correlation factor determined by comparing them with results from the finite element method.
• Convergence Criteria: While often never discussed, the convergence criteria used for different solutions in the algorithm or the global algorithm itself have to be chosen to a level that achieves the required level of accuracy, but without drastically increasing the computation time.
• Included Forces: While the contact forces dominate most loaded tooth contact analyses, other forces from gravity, centripetal motion, thermal effects, and fluid interaction can cause minor effects on the gear distor-tion. These are most often excluded to reduce computation time. Gear pumps exhibit a unique fluid pressure distribution that has been documented in the literature [312, 313, 355].
• Mesh Stiffness Calculation: Gear mesh stiffness is highly nonlinear due to tooth contact, which creates two different calculation methods to quantify it [362]. Figure 20 shows an illustration of the two methods to calculate gear mesh stiffness: tangent slope approach (aka local slope) and secant slope approach (aka average slope). According to Cooley et al. [362], the tangent slope approach is best for dynamics, and the secant slope approach is best for quasi-statics.
• Constant vs. Non-Constant Misalignment: In reality, the misalignment changes across the face width of the gear. Some software providers assume the misalignment is constant. This effect becomes more significant for larger face width gears.
• Solution Procedure: While the solution speed is often less talked about, it plays a crucial role when extensive DoEs or optimizations are performed. The solution algorithm can significantly affect solution speed, particularly if it can be implemented in parallel. Efficient contact search algorithms, matrix operations, and nonlinear solvers are critical to reducing runtime. The choice of the core programming language is also crucial in reducing the runtime. Finally, it is also up to the user to make sure the fastest computing solution is used in terms of processor speed, number of processors, random access memory (RAM), and the use of solid-state disks (SSD) over hard disk drives (HDD).
• Automated Standard LTCA Outputs: Not all software providers automate all aspects of the post-processing to produce results similar to those shown in Figure 38 of the online Appendix, which are vital for fast assessments of the LTCA results that can lead to critical design decisions.

The field of gear design and analysis has been heavily published for more than 100 years. Modern journals with the most publications on the subject include Mechanism and Machine Theory, the Journal of Mechanical Design, and the Journal of Sound and Vibration. Even with the thousands of papers written on the subject over the years, there are still areas for research. The following areas for research in the field of gear system analysis, on the topic of gear mesh stiffness and loaded tooth contact analysis, were identified as lacking from the literature review:

• Standardized LTCA Verification Models: Currently, no standardized gear designs are used to verify a new LTCA method or compare two LTCA methods against one another. Instead, software providers each do their own independent validation with models most convenient to them. To change this would require creating a series of gear designs designed to test the LTCA models’ fidelity at extreme conditions that all LTCA algorithms should be able to handle. The models would require every aspect of the geometry to be in the public domain to allow the same modeling in LTCA models from any software provider, and ideally, be highly manufacturable to allow experimental correlation. These models would need to be created for all major gear geometry families.

• Refined Finite Element Method Correlation Models: There is a lack of comparison of models to the refined finite element method, which would include contact at such a small size that a computer cluster would likely be required. This is partially because supercomputing has only recently taken off with the capabilities to perform such an analysis. If a standard set of models encompassing enough unique conditions to validate a loaded tooth contact analysis algorithm were solved in this manner, it would create a baseline to compare different solutions on the market. Both low and high contact pressure scenarios need to be investigated, with the low contact pressure cases being more challenging to solve due to the small contact patch width, which requires a small discretization size (especially for large face widths).
• Published Quantified Comparison of Commercial LTCA Methodologies: There is yet to be a bombshell paper that compares all the different commercial LTCA methodologies against one another in regards to both speed and fidelity, to determine what the true strengths and weaknesses of each solver are. Some attempts at comparisons between commercial software can be found in the literature [433, 463, 335], which have shown variations between software packages. The comparison would ideally need a third party to monitor the work (i.e., OSU or FZG) while directly contacting all software providers to ensure the integrity of the results and using the same hardware for all calculations. A sub-objective would be to study the influence of choosing a normal vs transverse slice method on the results of an LTCA.

It should be noted there are no “bad” methodologies in gear SDA and LTCA software; there are just different compromises between speed and fidelity. To properly compare specialized gear software against one another, it is crucial to include both a parameter to characterize speed and a parameter to represent fidelity. Once complete, the different methods can be compared against one another on a 2D scatter plot where the X-axis is normalized fidelity, and the Y-axis is normalized speed. A single software could have multiple points on the chart. A single methodology could yield numerous points on the chart when the discretization size or convergence tolerance is changed. Multiple charts must be included for different geometry and loading scenarios, such as thin-rimmed, wide-faced gears, and narrow-faced gears in low and high torque and misalignment conditions.

• Improvements to the Coupled-Slice Method (CSM): The CSM is still a large area of research, as it can be implemented in many different ways. Methods in the past have failed to come up with an analytical way to determine the coupling stiffness between mating slices. In addition, plane strain formulations have been heavily focused on as opposed to plane stress. Figure 42 of the online Appendix shows the history of the coupled-slice method, along with improvements the author plans on investigating.
• Optimum Discretization Sizes: Each method inherently involves some level of discretization, whether of the contact, stiffness, or mesh cycle. Comprehensive studies have not been published outlining the optimum discretization sizes to use for varying levels of accuracy. This is particularly important for large DoEs or optimizations. Discretization sizes are often part of companies’ confidential best practices.
• Implementation of Machine Learning to Gear Stiffness: Machine learning is an ever-growing field that has yet to be applied to predict gear stiffness and transmission error. While it may not be ready for critical analyses where precision is vital, it could be used for lower-fidelity optimizations at the beginning of the design stage. The technique has been used to predict gear stress and for other parts of the gear design process [461].
• Parameters to Assess Contact Patch Results: A two-dimensional color plot of a gear pair’s contact patch over the full mesh cycle is one of the most valuable indicators of good or bad microgeometry at a macro level. Saving images of color plots during microgeometry optimizations can be very time-consuming, depending on the selected resolution. It becomes impractical to save these images in terms of both space and run time when running large DoEs. A set of parameters that can characterize contact patch results is required to be more practical in the optimization process. The parameters must capture the contact patch size, shape, and location.
• Using LTCA for Nonlinear Life Analysis: Using AGMA, DIN, or ISO standards to predict gear life is standard practice. Computer-aided engineering (CAE) is used to help predict the effect of misalignment on the load distribution. The current Achilles’ heel of the process is that only a single life number is output for the entire flank. In reality, similar to a color plot of stress, you can achieve a color plot of life, predicting the life for every discretized element on the flank by using loaded tooth contact analysis. Without using loaded tooth contact analysis, there is an inherent assumption that the worst-case loading occurs at the same spot on the tooth flank. This can create a conservative approach, especially for helical gears with significant misalignment. While this more advanced analysis is not commonly seen in commercial software, it was first published in 2011 [318] (see Figure 21). In addition, using linear life methods does not capture effects stemming from the loading order [279, 358]. It can only be resolved through a nonlinear life analysis [467]. Implementing LTCA-based nonlinear life calculations is the next step toward predicting both the time and the specific location of pitting failure on the flank.
• Minimum Number of DoF to Use for Microgeometry Optimization: When running a DoE, as the number of microgeometry variables increases, it exponentially increases the number of design points that must be run to capture the entire design space. For example, a simple barrelling and crowning strategy would be a two-DoF optimization, while a more modern strategy would be to add tip and root relief for the profile and crowning and slope to the helix, which would be a six-DoF optimization. The six-DoF optimization algorithm would take substantially longer than the two-DoF optimization algorithm if the whole design space were to be explored. Finding the minimum number of DoF to achieve a desired noise target would be a welcome set of research.
• Multi-Objective Optimization Algorithm for e-Mobility Gears with Normalization Factors and Weights: All modern gear microgeometry optimizations run in industry must be capable of looking at bending, pitting, scuffing, and NVH metrics to find the optimum microgeometry to move forward with. As this becomes a multi-objective function, defining normalization factors and weights for a given strategy must be presented as part of the objective function. This objective function is greatly complicated for e-mobility applications because a minimum of 3 load cases (minimum load, optimum load, and maximum load) in both DRIVE and REGEN (six cases total) must be included to represent the entire operating range of the gear set in some form. While general normalization factors might be agreed upon, weights are unique to each gear designer and company.
• Comparison of Optimized Microgeometry in the Quasi-Static Domain and the Dynamic Domain with Experimental Validation: It would be interesting to optimize the microgeometry of a gear set both with a Tier 2 method and a Tier 3 method, with a fixed time constraint, and compare the results both numerically and experimentally. The investigations could start with a simple parallel-axis gear set and work towards more complicated configurations like planetary gear sets. While the dynamic domain will offer higher fidelity, it will suffer in computation time. While the quasi-static domain will offer a larger view of the design space, it will come at a lower fidelity. Future work could also investigate hybrid approaches where the quasi-static domain is used to scout the design space, and the dynamic domain is used to find the local optimum.
• Blind-Robin Gear LTCA Challenges: In 2014, Sandia National Laboratories created a blind-robin challenge that was sent to a majority of crack-propagation software providers to predict where they think the crack would propagate from a set crack problem [338]. In parallel, tests were conducted to determine where cracks prop-agated in an experimental environment for comparison with the numerical results. This challenge would be exciting to create for the field of gear design and analysis, with separate papers for bending, contact, NVH, and scuffing. It would also be interesting to put a twist on this by having designers compete in FZG tests for the longest bending life, longest pitting life, highest scuffing load, and lowest NVH, based solely on changing the microgeometry of the design, with nothing else. This challenge would be expensive to compete in since prototype gears made in small quantities are costly. In addition, because of the nature of fatigue, multiple gears with the same design would be required to account for the inherent statistical variation.
• Comparison of Gearbox NVH Metrics: Multiple calculated NVH metrics can be used to assess the NVH characteristics of a gearbox, including peak-to-peak transmission error (PPTE) at the gear mesh from a loaded tooth contact analysis (LTCA), peak surface vibration of the gearbox housing from a surface vibration analy-sis (SVA), estimated sound power based on the integral of the surface vibration over the area of the gearbox from an SVA using ISO/TS 7849-1 [301], acoustic sound pressure at some distance from the gearbox from a vibroacoustic analysis (VAA), and finally acoustic sound power from a VAA. Each calculation has pros and cons regarding fidelity, speed, and practical use as a system-level characterization parameter. A study has not been performed to assess the ideal gearbox NVH metric for use in gear microgeometry design optimizations with the perfect compromise between speed and fidelity. If the gearbox NVH characteristics could be evaluated with enough confidence not to perform an entirely separate acoustic simulation and instead stop at the SVA, it could significantly decrease the computation time required for each design point. The study’s outcome would quantify the gain from using a VAA with one-way coupling rather than the simpler approach of using an NVH metric from the SVA.

To end this discussion, it should be noted that gear LTCA methodologies are highly tied to advances in computing hardware, which in the past had followed Moore’s law [66, 90] but has started to deviate over the years to the point it no longer holds [374, 220, 414]. The entire point of specialized gear software is to provide faster options than standard FEM and MBD implementations, thereby speeding up the design process and exploring the design space as much as possible. One day, advances in hardware might make methods such as FAMs impractical due to substantially reduced run times, so there is no need to make the compromises found in FAMs. That being said, even with advances in hardware, it will also take advances in preprocessing automation to reduce the time required to build and train models (particularly large models for system deflection analysis). Software providers’ ability to leverage parallel processing and to easily integrate their solutions with computer clusters (making the software compatible with Linux) could determine who holds the competitive edge in the future.

7 Conclusion

The author highlighted that gear software largely emerged to meet the defense industry’s need to design the most compact gearboxes possible (particularly for the aerospace industry) and now serves a dual-use purpose for commercial applications, such as e-mobility. It was also highlighted that for years now, NASA has been the main driver for fundamental gear research leading to more compact gearboxes; the author hopes this valuable research will be continued with the proper amount of funding to keep making gearboxes more and more compact, primarily for critical defense applications such as the infamous Bell-Boeing V-22 Osprey. A new definition, gear system analysis (GSA), was introduced as the combination of system deflection analysis (SDA) and loaded tooth contact analysis (LTCA) in an integrated, coupled approach. GSA software is complex to develop, and it’s estimated that it would take roughly 10 years to develop software from scratch with enough functionality to compete in the current market, and another 10 years to fine-tune it enough to start gaining market share. Around 2010, GSA became a standard job in the gear industry and plays a crucial role in today’s e-mobility designs.

It was highlighted that there is currently no, or very limited, academic or industrial courses offered on GSA; typically, the only way to get trained is to join a company that performs GSA and either learn internally from subject matter experts or externally through the software provider. It should also be noted that the material taught in today’s universities on gear design and analysis is often rudimentary, which may be another factor contributing to the lack of interest in this critical design field. The author hopes this work makes the field of gear design and analysis a little less intimidating for the younger generation entering the workforce, particularly in the field of gear system analysis (history covered in Section 2), in an attempt to alleviate the current gear engineer shortage seen in industry (including the critical defense industry).

8 Acknowledgments

The authors would like to acknowledge the University Library Interlibrary Loan Departments from both Indiana University Indianapolis and Purdue University West Lafayette for providing a substantial amount of the cited references. Additional acknowledgments are due to all software providers who assisted in verifying the software tables and provided valuable insights and guidance (please contact the software providers directly to confirm that the details in the tables are up to date). These additional acknowledgments also go out to numerous gear experts in the field who provided valuable guidance. Finally, the authors would like to acknowledge AGMA for reviewing, editing, and publishing the first version of the paper presented at the 2024 AGMA FTM, as well as for providing figures for reference.

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492. KIMOS. https://klingelnberg.com/en/business-divisions/bevel-gear-technology/software.
493. KISSsoft. https://www.kisssoft.com/en.
494. MASTA. https://www.smartmt.com/.
495. MDESIGN LVR. https://www.mdesign.de/en/mdesign-gear/.
496. RIKOR. https://www.fzg.mw.tum.de/en/research/simulation-and-computer-programs/.
497. Romax. https://hexagon.com/products/product-groups/computer-aided-engineering-software/romax.
498. SIMDRIVE 3D. https://www.contecs-engineering.com/our-company.
499. SIMPACK. https://www.3ds.com/products-services/simulia/products/simpack/.
500. STIRAK / ZaKo3D. https://www.wzl.rwth-aachen.de/.
501. Transmission3D / CALYX. http://ansol.us/Products.
502. Windows LDP. https://mae.osu.edu/gearlab/research.
503. eAssistant SystemManager. https://www.eassistant.eu/en/systemmanager.html.

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