Review Article | Open Access
Menglei Sun, Chihua Lu, Zhien Liu, Yi Sun, Hao Chen, Cunrui Shen, "Classifying, Predicting, and Reducing Strategies of the Mesh Excitations of Gear Whine Noise: A Survey", Shock and Vibration, vol. 2020, Article ID 9834939, 20 pages, 2020. https://doi.org/10.1155/2020/9834939
Classifying, Predicting, and Reducing Strategies of the Mesh Excitations of Gear Whine Noise: A Survey
Gear whine noise has attracted increasing attention from researchers in both the academe and the industry over the past two decades. The wide range of research topics demonstrates that there is a huge technical challenge in understanding the source-path-receiver mechanisms deeply and predicting the gear whine noise precisely. Thoroughly understanding the sources of gear whine noise is the first step to solving this issue. In this paper, the authors summarize a certain number of published articles regarding the sources of gear whine noise. The excitations of gear whine noise are classified into three groups: transmission error along the line of action direction, frictional excitations along the off-line of action direction, and shuttling excitation along the axial direction. The mechanisms, characteristics, predicting approaches, measuring methods, and decreasing strategies for these excitations are summarized. Current research characteristics and future research recommendations are presented at the end.
With the rapid development of technology in the automotive industry, electric vehicles fiercely shock the automobile market and are expected to occupy the market in a few years. The increasingly intense competition in the vehicle market and the rapid emergence of electric vehicles pose a series of challenges in improving interior acoustic comfort. Internal combustion engine noise, exhaust noise, and gearbox noise are major interior noise sources in traditional vehicles. However, for electric vehicles, gearbox noise can be perceived more easily, given the absence of the masking effect from the internal combustion engine and exhaust noise. Therefore, the key to reduce interior noise and improve interior acoustic comfort for electric vehicles is to reduce the noise from the gearbox.
According to the literature reviewed, there are two kinds of noise emitted from the automotive gearbox, namely, gear rattle noise and gear whine noise. Gear rattle noise is generally caused by fluctuations in the engine torque and speed [1–4], which usually lead to contact loss and impacts between lightly loaded mating gears. Gear rattle noise also has a close relationship with lubricant conditions [5–7]. Details of gear rattle noise for traditional and hybrid vehicles can be found in publications by Singh et al.  and Zhang et al. , respectively. Mesh excitations of gear rattle noise are beyond the scope of this survey. However, gear whine noise, with its pure tonal characteristic and high frequency, can be much more annoying in electric vehicles, where the mask effect of the engine noise is weak. Therefore, gear whine noise represents the main concern for acoustic comfort in an electric vehicle. This article focuses mainly on publications related to gear whine noise; limited space does not allow for publications related to gear rattle noise to be included in this paper. The objectives are limited to traditional cylinder involute gears. The objective gears discussed in this survey are restricted to those in parallel gearboxes.
In geared systems, if the loads transmitted by the gears were constant, the geometry of the gears were perfect, and the motions of the gears were smooth, there would not be any vibration. However, in real conditions, the profiles of gears are imperfect because of manufacturing errors and intentional modifications. In addition, the teeth will deflect significantly when they are subjected to transmitted torque. Moreover, supporting structures of gears are also deformable when subjected to operation loads, which induces inevitable misalignments. Deviations of the real gear profile from the ideal involute one, deflections of teeth under operating conditions, and inevitable misalignments all contribute to periodic displacement, which induces varying meshing forces along the line of action (LOA). The friction forces between gear surfaces, which act along the off-line of action (OLOA), change directions before and after the teeth passes through the pitch point. For helical gears and spur gears with severe misalignments, the centroid of the meshing forces shifts back and forth axially along the tooth face width. The displacement along the LOA and the friction forces along the OLOA, together with the axially back-and-forth shifts of the centroid of the meshing forces, change the amplitudes, action positions, and directions of the meshing forces between mating gears. The oscillating meshing forces are transmitted to supporting bearings by shafts, resulting in varying bearing forces. The varying bearing forces vibrate gearbox-housing plates, which finally radiate undesired gear whine noise.
The complexity of gear whine noise has inspired a large number of researchers to study this issue. Dating back to 1958, Harris  noticed the gear whine noise phenomenon and investigated the effect of the transmission error (TE) on gear whine noise. Because of computer technology breakthroughs in 1980s, a large number of studies on topics related to gear whine noise were conducted. Coming into the 21st century, the number of the publications in this area shows an exponential growth trend, as shown in Figure 1. Many outstanding researchers published review papers related to gear vibration and noise, such as gear system dynamic models , nonlinear dynamics of gear-driven systems , condition monitoring and fault diagnosis [13, 14], and planetary dynamics and vibration , respectively. Conversely, so far no one has made a comprehensive summary on the sources of gear whine noise. Therefore, there is an urgent need for a systematic literature review on the sources of gear whine noise.
During the generation of gear whine noise, excitations of gear meshes act as sources, shafts and bearings as vibration transfer paths, and the gearbox-housing plates as the receiver. The reduction of gear whine noise can be reached only via the reduction of the amplitudes of the housing plate vibrations. Both the strength of the sources and the propagation property of the transfer paths influence the magnitudes of the vibrations of the housing plates. In this review, the authors focus on the sources of gear whine noise.
This paper aims to summarize the excitations of gear whine noise from the literature, classify these excitations according to the three action directions of the meshing forces, present methods for predicting these excitations, conclude strategies to reduce these excitations, and propose new possibilities for gear whine noise reduction at its source. The remaining part of this paper is organized as follows. In Section 2, the excitation along the LOA direction, namely, TE, is summarized; the definition of TE, harmonic contributions from TE, strategies to calculate and measure TE, and approaches to reduce TE are also described. In Section 3, excitations along the OLOA are reviewed. The approaches to evaluate the frictional excitations and strategies to reduce them are presented. In Section 4, shuttling excitations in the axial direction and the properties of shuttling excitation are summarized. The potential of vibration reduction for lightweight gears is discussed in Section 5. Section 6 gives a summary. The authors also point out essential existing problems in reported research work and describe prospects of future research directions regarding excitations of gear whine noise.
2. Transmission Error
If the profiles of gears were geometrically perfect, the teeth on gears were perfectly rigid and correctly spaced, and the supporting structures were rigid and accurately installed, there would be no variance in meshing forces when meshing, which results in no vibration being generated. In reality, this ideal scenario does not occur for a variety of reasons such as manufacturing imperfections, intentional modifications, and inevitable teeth deflections. All of these imperfections contribute to a periodic displacement, namely, TE, which was first defined by Harris . As stated by Munro , the periodic variation of TE induces periodic advancement and retardation of the driven gear while the gears are rotating. In addition, if the rotation speed coincides with the frequency of one component of TE, then resonance will occur. The periodic motions of the driven gear and the potential resonance may give rise to large dynamic meshing loads and high noise levels. Transmission error is therefore a primary source of gear noise and vibration. Since these two studies, gear designers have gradually appreciated the importance of TE and tried to explore the relationship between TE and gear whine noise. Few researchers [17–20] showed that directionally reducing the transmission error should reduce noise as well. However, the direct relationship between TE and the level of gear whine noise remains unrevealed.
In this section, a detailed description of TE is presented. For researchers who have just started their studies in this area, this section will help them to quickly understand the relationship between TE and gear whine noise and the basic definition, components, prediction methods, and reduction strategies of TE.
2.1. The Definition of Static Transmission Error
Harris  proved that the periodic variations in the velocity ratio and the variance and nonlinearity of mesh stiffness, which all contribute to static measured relative displacement, were the main internal sources of vibration for spur gears. The statically measured relative displacement in that paper was the well-known TE that was subsequently defined , for any instantaneous angular position of one gear, as the angular displacement (given by equation (1)) of the mating gear from the position it would occupy if the teeth were perfect. To the authors’ knowledge, Harris was the first who defined TE and plotted the TE curves under different loads for rotating gears. These curves were known as the Harris map and were of great importance in understanding gear motions. Transmission error can also be described as linear displacement, as in equation (2), along the LOA direction. It is much more convenient to describe TE as a linear displacement than as an angular difference because the linear displacement of gear pairs along LOA closely relates to and triggers angular vibrations:in which is the rotating angular of gear as in Figure 2 and is the base radius of gear i, , in which p and represent the pinion and the gear, respectively.
Transmission error is widely recognized as the dominant excitation of gear whine noise [10, 11, 21–28]. A thorough research in TE is beneficial for deep understanding of gear whine noise. From the publications of Mark [29–32], TE can be expressed accurately using equation (3a) and (3b) as follows:in which is the rotating position, is the total time-varying mesh stiffness, is the mean component of mesh stiffness, is the varying component of mesh stiffness, is the transmitted torque, is the tooth number, , and is the deviation of the tooth surface for gear .
The first term in the right-hand side of equation (3a) is the elastic deformation, which also contains the mesh stiffness variation. The second term consists of deviations from perfect involute surfaces; these deviations include manufacturing errors, intentionally designed microgeometry modifications, and supporting misalignments. This analytical equation is suitable for both helical and spur-gear pairs. In a word, there are four primary sources of TE for a gear pair: gear teeth elastic deformation [33, 34] and stiffness variance [35–47], manufacturing errors [30, 48–50], misalignments [51–58] due to supporting structures, and intentional profile modifications [33, 53, 59–66]. In addition, gear surface roughness [67–70] also contributes to TE. It is worth mentioning that gear microgeometry correction is a significant approach for minimizing TE excitation; the details of the approach are described in the subsequent sections.
2.2. Harmonic Contributions for Transmission Error
Transmission error is generally analysed in the frequency domain. There are four sets of harmonics generated by the TE excitation of a meshing gear pair: tooth mesh harmonics, two sets of rotational harmonics, and the fundamental harmonics of the meshing gear pair. As described by Mark [31, 32], the mean deviations of the tooth working surfaces from equally spaced perfect involute surfaces include the average elastic deflections, mesh stiffness, mean component of manufacturing errors , and intentional modifications, all of which contribute to gear mesh harmonics. The tooth-to-tooth variations such as individual tooth manufacturing errors  contribute to rotational harmonics. In addition, shaft misalignment , gear plastic deformation [74, 75], and gear damages [76–79] also provide contributions to TE rotational harmonics. Variations associated with tooth surface contact points contribute to the fundamental harmonics of the gear pair.
Among the four sets mentioned above, the tooth meshing harmonics are the strongest excitation, the rotational ones are the second strongest, and the one associated with fundamental harmonics of the gear pair is the weakest. Because of frequency modulation, there are sidebands around each harmonic of the mesh frequency, as shown in Figure 3. For the analysis of gear whine noise, it is useful to examine how various TE sources are exhibiting in the frequency domain.
2.3. Transmission Error Calculation Methods
As we know, TE is the primary source of gear whine noise and serves as the forcing term in gear system dynamic analysis [80, 81]. Moreover, at the design stage, TE is the guideline to determine optimal tooth modifications . TE analysis is also an important tool for gear fault detection [74–79] at the early stage. Thus, the ability to accurately predict the TE of the target gears is essential in evaluating and minimizing gear whine noise. As described in equations (3a) and (3b), there are two parts of TE: tooth elastic deflections and surface deviations, respectively. Surface deviations, which consist of manufacturing errors determined by gear accuracy and intentional modifications designed by engineers , can be measured accurately. Therefore, TE calculation mainly focuses on the prediction of gear teeth elastic deflection and the varying mesh stiffness.
2.3.1. Analytical Method
When spur-gear teeth are considered as nonuniform cantilever beams, bending, shearing, and contact deformations all contribute to the teeth deflections. Mesh stiffness is the transmitted force divided by the elastic deformations. In 1963, Gregory et al.  developed a simple approximate TE formula, which was based on Weber’s deflection equation  and varied as a sinusoidal function. Based on the publications by Cornell  and Lin et al. [85, 86], there are three components in gear tooth deflection, which are bending and shearing, contact deformation, and deflections related to foundational effects. Subsequent TE calculations [87–90] considered corner contact and proposed an additional approximate equation for TE outside the normal path of contact. Another method for calculating the deflections of gear teeth [91, 92] was based on conformal mapping of complex variables in plane elasticity.
The teeth of helical gear pairs are considered as infinite cantilever plates when their deformations are calculated [93, 94]. For helical gears, the calculation of elastic deformation is more difficult than for spur gears because the geometry and load distribution are more complicated. Based on the cantilever plate theory and the mathematic programming approach, Conry and Seireg [95, 96] developed a load distribution program to determine the load distribution and elastic deformation of the gear tooth for both helical and spur gears. Many recent research studies [59, 97–100] related to TE prediction apply this computer program.
2.3.2. Numerical Method
Because of the vast assumptions and empirical parameters, the analytical equations for TE were not accurate enough to perform exact calculations. With recent computer hardware and software improvements, numerical simulation becomes increasingly popular in calculating TE. As long as the detailed 3D model of the teeth is established and the tooth surface is described precisely by high-quality elements, given the material property and boundary conditions, the finite element (FE) model can calculate the TE for a pair of gears accurately.
Finite element method is widely applied in calculating tooth deformation and mesh stiffness for spur [101, 102] and helical gears [103, 104]. The challenge is the modelling of contact deformation with its nonlinearity. For the improvement of computational efficiency, a properly simplified 2D FE model [105, 106] is popular for calculating TE for spur gears. When misalignment and nonuniform load distribution along contact lines are considered, a 3D FE model [107–111] of the spur gear should be applied. However, for helical gears [112, 113], a 3D FE model is essential for the complex geometry. Generally, the FE model of a spur or helical gear is specific for one gear pair, so parameter analysis based on FE models seems impractical. However, parametric FE modelling approaches [114–119] overcome this difficulty and offer an opportunity to study the effects of microgeometry parameters on TE.
For both spur and helical gears, the contact between two surfaces is localized in a very small region, so there is an urgent need for refined mesh  in the contact area. Accordingly, preparing gear element models with high quality is a time-consuming burden requiring high levels of skill. The microgeometry complexities of the tooth profile and tooth lead make it more difficult to model every detail on the tooth surface. In addition, too many details in FE models slow down the numerical simulation.
2.3.3. Analytical-Numerical Method
Due to too many assumptions, the analytical models are efficient but not accurate. Although the results of FE simulation are very accurate, its efficiency is not satisfactory. Therefore, researchers need to find a way to balance the efficiency and accuracy of the methodologies for calculating TE. A combination of analytical and FE methods is a compromise method offering quick simulation and high accuracy simultaneously.
For the gear teeth, the bending and shearing deflections are easy to calculate using the FE method, while the calculation of Hertzian contact deformations is more complicated due to the nonlinearity of the deformations. Therefore, Umeyama et al.  and Rincon et al.  estimated the deformations of a helical gear tooth by analytical Hertzian formulae for the contact deformation and by the FE method for the bending deflections. Simon [122, 123] calculated the tooth deflection of helical gears using analytical equations, which were obtained via regression analysis and interpolation of the results of the 3D FE model. Rao and Yoon  subsequently adopted the formulae proposed by Simon to calculate the deformations of helical gears. Another combined method is the integral equation method [62, 124–126, 129] consisting of the load distribution coefficient and mesh stiffness. The integral equations can calculate TE faster than the normal FE method. Normally, the mesh stiffness in the integral equations is obtained via the FE method, while the load distribution coefficient is determined analytically [127, 128]. Li [52, 53, 121] combined the mathematical approach with the FE method to calculate tooth load distribution and TE. Based on the surface response approach , the peak-to-peak TE derived from the FE model is represented and optimized via a mathematical formula. Many papers related to TE reduction via gear microgeometry modification adopt combined analytical-FE models. A summary of the analytical equations adopted in the reviewed publications is presented in Table 1.
2.4. Transmission Error Measurement
As described in equations (3a) and (3b), tooth surface deviations are the primary source of TE. Transmission error predictive approaches cannot accurately calculate the deviations due to manufacturing imperfections, which have a significant influence on the gear whine noise. Therefore, the prediction of TE mainly focuses on the calculations of deflections and deformations of gear teeth. The measurement strategies for TE make these calculations possible and can obtain tooth surface deviations and deflections simultaneously.
Transmission error can be measured statically or dynamically (under low or high speed) as described by Åkerblom . Gregory et al.  proposed the first static TE measurement equipment, but it is only suitable for 1 : 1 ratio gear pairs. Houser and Blankenship  summarized four methods for measuring static and dynamic TE. Transmission error measurements in the early stage were usually limited to isolated gears tested under extremely light-load and low-speed condition using the single-flank test rig. Traditionally, the measurements from single-flank tests were mainly for verifying gear manufacturing accuracy. More recently, measurements are conducted under quasistatic loaded conditions [136–141]. The measurements of loaded gears are the interactions of microgeometry deviations and tooth deflections. These experiments are generally conducted on a noncirculating power test rig , or on a power-circulating test rig . Optical encoders with proper resolution are generally installed at the shaft ends to record the rotating displacements of the gears. The challenge with TE measurement under quasistatic loaded conditions is that the magnitude of TE is extremely small and often in the micron order. Thus, selection of encoders with proper resolution is an important step in measuring TE.
Nowadays, the need for dynamic TE measurement grows faster because dynamic TE has a closer relationship with gear whine noise. There are three methods suitable for measuring dynamic TE: one based on optical encoders [143, 144], another based on vibrometers , and the last based on accelerations [142, 146–148].
Methods based on encoders are conducted on a power-circulating test rig under high-speed conditions. The challenge with this method is that it is difficult to eliminate the dynamic effects of the slave gearbox. In addition, the tested gears are generally mounted on two isolated shafts or in a specially designed gearbox. The boundary conditions of the gears are extremely different from the ones that gears are subject to in a real gearbox. Thus, there is an urgent need for technology that can successfully measure the static and dynamic TE in a real gearbox. Methods based on vibrometers provide a chance to measure the dynamic TE in a real gearbox . However, the outputs of this method are not as good as those produced by encoders and require an integration to obtain the dynamic TE. In addition, there should be several holes on the housing plates for laser access. Thus, the lubricant of the gearbox may be a great challenge for accurate measurement. Methods based on accelerations are conducted on a power-circulating test rig, and the accelerometers are installed either on the walls of the gearbox  or on the flanks of the gears . The outputs of this method are the second derivative of the TE, which means that there would be a large error when calculating TE. When the accelerations are measured, precise calibration, tight misalignment, and adequate mounting space are necessary conditions.
2.5. Methods to Reduce Transmission Error
As mentioned before, the meshing variations during the meshing process are the primary excitations of gear whine noise. A reduction in the level of excitations directly results in the decrease in the gear whine noise. One-micrometre reduction of TE results in 5 dB reduction of the gear whine noise . Gear tooth modification is such an approach for gear whine noise reduction by decreasing the meshing excitations. A smooth load transfer is crucial for reducing the gear whine noise and can be achieved by intentional tooth surface modifications. Microgeometries should be optimized carefully to obtain a quiet gearbox design.
For uniform transfer of motions, tip and root relieves are frequently implemented on the gear tooth profile. In the 1930s, Walker  first proposed the gear tip relief theory. Harris  observed that the amount of tip relief could be appropriately chosen so that TE can be constant at the design load. Since then, many researchers devote their efforts in clarifying the effects of linear [21, 59, 150, 151], parabolic [106, 152–156], cubic , and other complicated types [157, 158] of relief on TE reduction. For helical gears with bias modification , the TE is less sensitive to the transmitted torque. The common function of lead and profile crowning is the reduction of contact and root stress [160, 161]. To understand the influence of the combined profile and lead crown modifications on TE and contact stress for helical gears, Zhang et al.  conducted a numerical investigation, from which one could learn that each pair of modification parameters gives an optimum peak-to-peak TE at different torques.
These investigations mentioned above are limited to case studies and numerical analysis for specific gear applications. Although the outputs of the analysis are rather accurate, the global trend of TE when changing modification parameters is difficult to illustrate. The optimal modifications mentioned above are valid solely at the design load, while gears usually work under multiple load conditions. There is an urgent need for a robust modification optimization that can guarantee a relatively small excitation level under the whole operating range. Analytical methods seem to have more advantages in finding robust modification designs [82, 162] and allow for parameter analysis [129, 163–165]. Minimizing TE is the primary objective for gear-form modifications. Meanwhile, optimal TE achieves good performance only under static conditions. Reduction of meshing forces and bearing forces for gear systems is of great importance in the reduction of gear whine noise. Therefore, researchers should consider dynamic criteria [166–169] when the setup entails gear profile modifications.
3. Frictional Excitations
During the process of meshing, the teeth undergo pure rolling at the pitch point. The motions before and after the pitch point are known as approaching and recessing, respectively. The directions of the friction forces are opposite in these two periods. The sudden direction reversal of the friction forces near the pitch point has a significant influence on dynamic meshing forces and induces varying bearing forces. In addition, the arms of friction moments vary as the gears rotate. Time-varying sliding friction forces and friction moments between meshing teeth are significant excitations of gear whine noise. Friction forces and moments are usually referred as the secondary excitation when compared to TE.
3.1. Frictional Excitations
To the knowledge of the authors, Iida et al.  were the first researchers who investigated the vibrational characteristics of gears due to friction forces. Borner and Houser  quantitatively evaluated the influences of friction forces and reached the conclusion that friction forces should be considered as an excitation of gear whine noise when TE is low. Hochmann  proved via experimentation that friction force is a potential excitation. Vedmar and Henriksson  highlighted the importance of the motions along the OLOA direction on dynamic meshing forces. Velex and Cahouet  revealed that friction forces have a significant contribution on gear vibration and noise at low and medium speeds under high torque levels. Houser et al.  conducted a series of experiments to identify frictional excitation and to study the influences of tribology parameters. From the experimental results, one can learn that friction force is a dominant excitation for gear whine noise in higher harmonics of mesh frequencies. Vaishya and Singh [175, 176] proposed an analytical model to evaluate the contribution of friction torque, as an external excitation, on TE. Results show that frictional excitations have limited influence on lower harmonics of mesh frequency but have significant influence on higher harmonics. Velex and Sainsot  concluded that translational responses were sensitive to frictional excitations and that torsional responses were less sensitive to them. According to Gunda and Singh’s  observation, frictional excitations could not only change the shape of the TE curves but also increase the amplitude of the second harmonic of the mesh frequency. In Lundvall et al.’s investigation , the friction forces increased the peak-to-peak TE. In He et al.’s  study, friction forces induced large oscillations in bearing forces along the OLOA direction. The authors also emphasized the importance of friction forces when TE was small. He et al.  then proposed a 12-degree-of-freedom (DOF) helical gear model, which coupled rotation motions, translation motions, and axial motions. Simulation showed that friction forces are a potential excitation in the LOA direction, significant excitation in the OLOA direction, and insignificant excitation in the axial direction. According to another publication of He and Singh , motions along the LOA are insensitive to friction forces. Singh et al.  assessed the contributions of TE and friction forces on gear vibration and noise. The bearing force curves indicated that besides the excitation role along the OLOA, friction forces might influence the forces along the LOA significantly. He and Singh  stated that the coefficient of the friction forces affected only the first two harmonics of TE. Kahraman et al.  concluded that the motions along the OLOA were sensitive to friction forces, while the motions along the LOA were insensitive to these friction forces. He et al.  observed that the gear whine noise induced by friction forces was comparable to that induced by optimized TE. Liu and Parker  reported that tooth bending, induced by friction moments and forces, affected system dynamics significantly. Chen et al.  suggested that friction force might reduce the dynamic responses of gear systems in high-frequency and low-speed conditions. Wang et al.  stated that friction forces could change the motion stability of a gear system. Brethee et al.  discovered that the motions along the OLOA were sensitive to the variance of friction forces.
All the research studies mentioned above reach an agreement that friction forces and moments are indeed significant excitations of gear vibration and noise along the OLOA direction. Some investigators [171, 180, 186] emphasize the importance of frictional excitations when TE is low. However, there is no consistent conclusion about the role of frictional excitations in the LOA direction. A small number of scholars [175–177, 182, 185] believe that in the LOA direction, the influence of friction forces is ignorable. From other studies, the effects of frictional excitations on the amplitudes [179, 184] and curve shape  of TE, dynamic meshing forces [183, 188, 189], and gear bending  cannot be easily concluded. To the authors’ knowledge, He et al.  were the only researchers who studied the influence of frictional excitations on motions along the axial direction, indicating that this area needs further research. Frictional excitations have a significant influence on gear dynamic response, a fact that is recognized by many scholars. Meanwhile, Jiang et al.  reported that frictional excitations and gear vibration were interactive and coupled. Vibration affected the sliding velocities and directions of the frictional excitations and hence magnified vibrations along the OLOA direction.
3.2. Friction Force Prediction
Vaishya and Singh  summarized various strategies calculating friction forces and moments. Kar and Mohanty  proposed formulae of friction forces and moments based on time-varying contact length for helical gear pairs. The general expressions for frictional excitation are shown as follows:in which is the friction coefficient; is the normal tooth meshing load; is the sign function; is the arm of the friction moment.
The normal tooth load is the contact force along the LOA, which has a close relationship with the stiffness and deflections of the mating teeth. The general expression of the normal force in Ref  is illustrated byin which is the varying mesh stiffness; is the varying damping coefficient; is the displacement along the LOA direction; is the first derivation of the displacement.
The direction reversal is a nonlinear factor and of great importance in predicting frictional excitations. The formulae determining this nonlinear factor are summarized in Table 2. As can be seen from equations (4a) and (4b), besides the direction reversal, friction forces are the product of the normal tooth load and frictional coefficient. Friction moments are the product of friction forces and moment arms. Formulae for moment arms based on the gear geometry are described in Table 3. The coefficient of the frictional excitation is complicated and discussed in the following paragraph in detail.
As reviewed by Martin , the coefficient of friction force was mainly estimated empirically before the year of 2000, such as by Benedict and Kelley , Kelly and Lemanski , Johnson and Spence , and Rebbechi et al. . Four different frictional coefficient formulae, corresponding to the Coulomb friction , empirically estimated friction , smoothed Coulomb friction , and thermal non-Newtonian elastohydrodynamic lubricant (EHL) [200, 201], were then adopted into a spur-gear dynamic model by He et al. . The difference between motions along the OLOA and TE curves predicted by these models is not significant. A mixed EHL frictional coefficient formula  is accurate enough to predict the frictional excitation. Han et al.  proposed helical gear friction forces and torque formulae, considering nonuniform load distribution and the time-varying friction coefficient. Predictions indicated that effects of nonuniform load distribution were more significant than the variance of the friction coefficient. In addition, it has been proved that [173, 177] the exact form of the friction law is of secondary importance and the dominant factor is the reversal of the sliding directions at the pitch point. Afterwards, researchers understand that a constant friction coefficient is acceptable in gear dynamic analysis and whine noise prediction. Most scholars adopt the Coulomb friction model [205–210] and a user-defined constant friction coefficient. However, Liu et al.  observed that the variance of the frictional coefficient should not be neglected.
Among the reviewed frictional research studies, a vast majority of researchers adopted a Coulomb friction model, assumed a uniform distributed friction coefficient along the OLOA, and changed its value within an acceptable range to study the effects of the frictional excitations on gear vibration and noise. Some investigators utilized Benedict and Kelly’s empirical formulae to calculate the coefficient of frictional excitation. The empirical formulae indicate that the coefficient of frictional excitation is a function of many parameters such as sliding and rolling velocities, surface roughness, and load distribution. These formulae are more accurate than the constant coefficient, but being based on experimental results, they are expensive and not general. Based on experimentally validated simulations, Xu’s EHL formulae are accurate, general, and convenient. However, these formulae are widely used only in the area of gear system efficiency; the application of these formulae in gear dynamic, vibration, and noise areas should be more extensive in the future. Meanwhile, a mixed EHL formula is widely utilized in gear dynamic analysis. The frictional coefficient formulae from the abovementioned publications can be classified into four groups (listed in Table 4): Coulomb formula, empirical formula, EHL formula, and mixed EHL formula. Here, the ISO formula is cited for comparison.
The definition and value of each coefficient in the formula can be found in the corresponding reference.
3.3. Strategies to Reduce Frictional Excitations
Although the frictional coefficient is of secondary importance when calculating the frictional excitations, it is the most important term for reducing frictional excitation. From Table 4, one can learn that many parameters, such as gear surface roughness and property of the lubricant oil, have great influence on the coefficient of frictional excitation. This is because the gear whine noise increases with the surface roughness [214, 215]. Moreover, as described by Huang et al. , a smoother surface leads to a smaller frictional excitation. Therefore, by improving the surface finish of the gear teeth, the level of the gear whine noise induced by frictional excitations would be reduced significantly . Frictional excitation also has a close relationship with the viscosity of the lubricant oil. With a decrease in the viscosity of the lubricant oil, the frictional excitations decrease significantly . Sufficient lubricant oil would decrease the viscosity of the oil and hence reduce the frictional excitations for the gear whine noise. However, a possible alternative cause of noise in a spur gearbox is an overgenerous oil supply, if oil is trapped in the roots of the meshing teeth [5–7]. If the oil cannot escape fast through the backlash gap, it will be expelled forcibly axially from the tooth roots and, at once-per-tooth frequency, can affect the end walls of the gear case. This phenomenon induces another well-known annoying noise, gear rattle noise.
4. Shuttling Excitation
In the meshing process of helical gears, the centroid of the meshing force axially shifts back and forth along the tooth face width , which induces time-varying bearing forces and dynamic moments. In addition, angular misalignments may move the action position of the meshing force to one end of the face. The angular misalignments are also load dependent, which means that the action position of the meshing force would change as the load varies. Shuttling effect due to the helical angle acts along the LOA direction with a frequency equal to the gear mesh frequency. Shuttling effect due to angular misalignments results in a twisting moment that may push the gears to an aligned position.
Borner and Houser  quantitatively evaluated the influence of shuttling excitation and reported that the amplitude of the bearing forces induced by shuttling excitation might be three times as those induced by TE. Houser et al.  suggested that shuttling was one kind of excitation for gear whine noise. Nishino  proposed an excitation model in which both TE and shuttling excitation were considered. After evaluating the individual contributions of TE and shuttling excitation, the authors reached the conclusion that shuttling excitation had a significant influence on the total response of the housing plates. From the publications by Eritenel and Parker [216, 217], one can learn that shuttling motions can excite the twist mode of the geared system. According to the simulation by Palermo et al. [218, 219], shuttling excitation due to angular misalignment and helix angle both contribute to varying bearing forces that finally induce gear whine noise. Teja et al.  calculated the bearing forces induced by shuttling forces and TE for helical gears. The authors concluded that when TE was minimized by microgeometry modification, shuttling excitation becomes the dominant excitation for the gear whine noise.
Shuttling excitation is inherent for helical gear pairs with wide faces. As described by Eritenel and Parker [216, 217], for spur gears with misalignments and aligned helical gears, the shuttling excitation generates fluctuating moments with a mean value. For helical gears with misalignments, the fluctuating moments for gear and pinion are quite different. Helical angle and amplitude of misalignment both influence the strength of the shuttling excitation. The distance between supporting bearings also has a significant influence on the strength of the shuttling excitation : the longer the distance, the weaker the excitation. In addition, unequal length on either side of the gear increases the shuttling excitation. Teja et al.  also observed that helical gears with a higher contact ratio introduce stronger shuttling excitation.
Despite the fact that shuttling may be a nonnegligible excitation for the gear whine noise in the axial direction, there are a limited number of research studies focusing on shuttling excitation and its effect on gear whine noise. Shuttling is a 3D issue, so two-dimensional contact models of gear pairs cannot calculate the shuttling excitation. Detailed 3D contact models for gear pairs enable the inclusion of nonuniform load distribution and misalignments, and thus are suitable for predicting shuttling excitation. Analytical formulae for shuttling forces and moments in reviewed articles are summarized in Table 5. Meanwhile, strategies for decreasing the shuttling excitations are difficult to illustrate. Modelling strategies and the influence of shuttling excitation on gear whine noise should be investigated further. Approaches reducing shuttling excitations should also be proposed and verified as soon as possible. The relationship between shuttling excitation, TE, and frictional excitation should be studied further.
5. Lightweight Gears
Most of the strategies relating to gear whine noise reduction are gear tooth micromodification as mentioned in Section 2. However, with a precisely designed optimum profile, tooth meshing remains a vibration generator. Using lightweight gears originated in the aerospace industry and is recently extending to the vehicle industry. Because of the stricter regulation on emissions in common fuel vehicles and the population of the electric vehicles, lightweight gears are prior to solid steel gears in the gearbox. Therefore, the modern gear design should meet the need of weight reduction without compromising the requirements of NVH performance and reliability. From this point of view, several strategies have been proposed, which include the thin-rimmed gears, the material removal through holes, and the adoption of new materials.
Common lightweight strategies are based on material removal such as a thinner rim or manufacturing holes and sluts in the gear body. As mentioned by Li , although there is an increasing trend of adopting lightweight gears, design problems related to the dynamic strength and vibration of such gears are not fully solved yet. From the survey by Li, one can learn that the thickness of the rim and web significantly influences the bending modes and natural vibration behavior of the lightweight gears. Researchers [223–225] point out that, due to the nonuniform mass and stiffness distribution along the gear blank, the lightweight gears with holes produce additional harmonic components at a lower frequency range. The additional order of the static TE would increase the risk of exciting the rest of the system. System responses to the excitation due to the holes in the web are comparable to that of the mesh excitation. The lightweight gears with holes are more compliant than the solid gears, so the contact loss phenomenon would be eased. However, as described by Shweiki et al. , the peak-to-peak magnitude of the static TE and the magnitude of dynamic TE for the lightweight gears are larger than that of solid ones. The modelling strategies of the thin-rimmed gears and lightweight gears with holes are complicated than common solid gears as described by Guilbert et al. [226, 227] and Shweiki et al. , respectively. Only continuous flexible models can capture the particular dynamic properties of lightweight gears. Guilbert et al.  developed a mortar-based methodology that can connect 1D grids and 3D finite elements. Based on this approach, thin-rimmed gears can be precisely modelled in dynamic models. In order to calculate the load distribution and static TE of the lightweight gears precisely, a complex numerical model is required as demonstrated by Shweiki et al. . The remaining issue is the development of simulation models able to predict the impact of different design parameters on gear vibration behavior. Hou et al.  numerically and quantitatively investigated the impact of the rim and web thickness on static TE, meshing forces, bearing forces, and housing vibration. Gear rim and web thickness have a great influence on both the static and dynamic responses of the geared system of an electric vehicle. Optimization design of lightweight gears would at most reduce 68.5 percentage of the dynamic meshing force and 66.7 percentage of the housing vibration compared to solid gears.
Lightweight gears by material removal have many drawbacks such as the introduction of additional mechanical excitations, more harmonic components, greater fluctuations of mesh stiffness and static TE, severe stress concentration, lower load capacity, and higher magnitude of deformation. Meanwhile, thinner walls and induced holes simply do not reduce the vibration transmission from the toothed ring to other parts of the drive. Therefore, the dynamic responses and vibration level of these geared systems show no significant improvement compared to the solid ones.
Researchers develop innovative lightweight strategies that can maintain the load-carrying capability of the gears and improve the dynamic performance by the exploitation of new materials. Bimetallic gears and hybrid composite-metal gears are the most popular approaches. With steel ring and aluminum on the hub region, bimetallic gears  can provide up to 40 percentage of weight reduction without a decrease in the performance of the gears. Ring thickness has a great influence on the static performance of gears such as root stress, time-varying mesh stiffness, and static TE but has an ignorable influence on dynamic responses. However, in another publication , a hybrid gear with a carbon fiber-reinforced polymer web is superior to that with an aluminum alloy web. Composite materials are initially used in rotorcraft transmission and then are widely used in vehicles in recent years  for their outstanding damping property. Incorporating composite materials with metals into the structure of a gear has many promising benefits such as weight reduction and vibration dispassion. Carbon fiber-reinforced polymer material  is attractive for its capability of reducing vibrations and high damping capacity. Both experimental [233–235] and numerical  surveys showed that there are significant improvements in both weight reduction and NVH performance when adopting hybrid composite-metal gears. From the experiments conducted by Handschuh et al. , the hybrid gear pairs generally reduce both the vibration level and the sound pressure level of the gearbox due to their higher damping properties. From the report by Handschuh et al. , a hybrid-hybrid gear pair exhibits the lowest vibrations and noise level at high-speed ranges, while the hybrid gear driving the steel gear pair exhibits the highest vibrations. From the experiments conducted by Laberge et al. , the vibration and orbit magnitudes are comparable to steel gears. Catera et al.  compared the numerically calculated static TE of the thin-rimmed steel gear pair and the composite-steel hybrid one. The peak-to-peak and mean value of the static TE of the hybrid gear pair are lower than the thin-rimmed steel one, which indicates an enhancement in NVH performance for hybrid gears. Innovative web structures alter the vibration propagation path, which starts from the gear mesh through the gear body and shaft to the housing. Filling powders  and particle damper  with optimum diameter to the holes of the lightweight gears can effectively reduce the vibration since the holes are a vital place of vibration transfer. As described by Xiao et al. , tungsten alloy powder is the best, and the steel alloy and lead alloy are much better than that of magnesium alloy and aluminum alloy powder. Ramadani et al.  replaced the solid structure of the gear body by a lattice structure to reduce the total weight of the gear and to reduce the propagation of the vibration generated by the gear teeth. The addition of the polymer matrix to the cellular lattice structure may reduce the vibration significantly. Generally, the hybrid gear has three parts: steel rim, composite web, and steel hub, respectively. There are two composite-steel interfaces in a common hybrid gear, which increase the opportunity of stress concentration. To minimize the stress concentration phenomenon, Gauntt et al.  adopted composite shafts and a sinusoidal interface. Meanwhile, the accuracy of the FE model of the hybrid composite-metal gears strongly relies on the property estimations for interfaces and composite materials. The complicated property of the composites and interfaces poses difficulty in establishing predictive models for hybrid gears [236, 241–243]. There are two different joining technologies for metal-composite gears, namely, interface fitting and adhesive bonding, respectively. Cooley and Parker  established FE models with ply-by-ply and homogenized webs for hybrid composite-steel gears, in which the interface is an adhesive bounding one. Catera et al.  adopted an adhesive bounding interface in their multiscale model of the hybrid metal-composite gears. The FE models  of the hybrid gear with an adhesive bounding interface do not accurately predict the higher orders of the modes. Catera et al.  experimentally and numerically investigated two different joining technologies for metal-composite gears, namely, interface fitting and adhesive bonding. Both techniques are prior to the lightweight steel gears in terms of gear natural frequencies and damping properties. A hybrid metal-composite gear with the variable thickness web  is proposed to explore its dynamic properties, in which a film adhesive bounding method is adopted at the metal-composite interfaces.
Both material removal and composite-metal gears meet the requirement of weight reduction. However, the performance of lightweight gears by material removal in terms of load-carrying capacity and static and dynamic responses is not satisfactory. Hybrid composite-metal gears can reduce vibration not only by improving the gear body damping property but also by modifying the vibration propagation path. However, there are still several challenges to be faced. The manufacturing procedure of the hybrid gears is more complicated than solid and material removal gears. The properties of composite materials and hybrid interfaces are key parameters in establishing predictive models, which need further investigation. The modelling of the composite materials is at the microscale, while the modelling of the steel parts of the gear is at the macroscale, leaving the overall model of the hybrid gear at multiscales. Further effort should be devoted to the faster and more accurate modelling strategies of the hybrid gears. The author believes that innovative web structures in terms of modifying vibration propagation would be popular in the coming years.
6. Summary and Prospects
The authors reviewed more than 200 published papers related to the excitations of gear whine noise. The proportion of each excitation in the reviewed articles is calculated as can be seen in Figure 4. A considerable number of investigators considered TE as the primary excitation. Therefore, there are numerous papers focusing on prediction, measurement, and optimization of TE. A few researchers have mentioned that frictional excitation between mating gears is the secondary excitation of gear whine noise. Limited studies are related to shuttling excitation.
The main content of this article is summarized as follows.
Three kinds of excitations for gear whine noise are summarized: transmission error along the LOA direction, frictional excitations along the OLOA direction, and shuttling excitations along the axial direction. Transmission error is the primary excitation of gear whine noise, frictional excitations are the secondary excitations, and shuttling excitation is less important than the aforementioned two kinds of excitations.
Transmission error consists of two parts: deflections and deviations. Gear teeth deflections and variance of meshing stiffness contribute to the deflection part of TE. Manufacturing errors, intentional modifications, and misalignments due to supporting structures contribute to the deviation part of TE.
There are four sets of harmonics for TE in the frequency domain: mesh frequency harmonics, two sets of rotational harmonics related to the shaft rotating frequency, and fundamental harmonic of the meshing gear pairs. Among the four sets mentioned above, the tooth meshing harmonics are the strongest excitation, the rotational ones are the second strongest, and the one associated with fundamental harmonics is the weakest.
The deflection part of TE is generally predicted using analytical, numerical, or combined analytical and numerical methods. The advantages and disadvantages of these three methods are discussed. Among these methods, the analytical methods requiring many assumptions are efficient but not accurate, the numerical approach is precise but not efficient, and the combined methods offer a quick and accurate solution. The analytical formulae for predicting TE are summarized in Table 1. From this table, researchers who are new to this field can learn about many TE formulae without reading so many articles.
The deviation part of TE is generally measured using a specially designed test rig. Transmission error measured under light-load and slow-speed conditions mainly consists of the manufacturing error and intentional modifications. Transmission error measured under low-speed and varying-load conditions consists of deflections and deviations. Dynamic TE measured under varying speed and load conditions also consists of dynamic effects. The test rigs and instruments measuring TE under different operating conditions are summarized. Benefits and drawbacks of each measurement approaches are discussed. Methods based on encoders are accurate, while the boundary condition of the tested gears is quite different from that in a real gearbox. Methods based on vibrometers offer an opportunity to measure TE in a real gearbox. However, there should be holes in the housing plates for laser access, which poses a challenge for gearbox lubrication. Approaches based on accelerations require precise calibration, tight misalignment, and adequate mounting space. In spite of that, the output still requires a second integration to obtain the TE.
A reduction in the primary excitation directly leads to a reduction in the level of the gear whine noise. The most popular way to reduce TE is through microgeometry modifications. Tip and root relieves are widely applied to spur-gear pairs to reduce TE. For helical gears, bias modification may be beneficial.
For frictional excitations, it is clear that the friction forces are the main excitation of the gear whine noise along the OLOA direction. The friction forces and moments may become dominant excitations of the gear whine noise when the TE is minimized via microgeometry modifications. Meanwhile, the influences of frictional excitations on motions along LOA and axial directions are difficult to illustrate.
Frictional excitations are, in general, predicted analytically. Normal meshing forces, sign function, frictional coefficients, and moment arms are essential parameters for predicting frictional excitations. Among them, the sign function is the dominant factor, and the coefficient is of secondary importance. Analytical formulae for sign function, frictional coefficient, and moment arms are summarized in Tables 2–4.
Improving surface roughness and providing sufficient lubricant oil can reduce frictional excitations. Meanwhile, an overgenerous supply of lubricant oil will increase the chance of impacts and induce rattle noise.
Shuttling is an inherent excitation for helical gears; spur gears with misalignment should take shuttling into consideration as well. Parameters influencing the strength of shuttling excitation are summarized. Helical angle and amplitude of misalignment both have a close relationship with shuttling excitation. The distance between gears and bearings has a significant influence on the strength of the shuttling excitation. Analytical formulae for predicting the shuttling excitations are summarized in Table 5.
Lightweight gears based on material removal and adoption of new materials are popular in automotive transmission lately. The advantages and disadvantages of these gears are summarized. The author also points out the remaining challenges for lightweight gears in terms of manufacturing and modelling strategies.
Based on the topics covered in this review paper, some essential existing problems in the reported research work are presented and some prospects for future research directions are suggested.
Current TE prediction methods model two isolated gears under quasistatic condition, assuming rigid shaft and bearings and ignoring housing plates. A system-level model, containing multiple pairs of gears, flexible shafts, bearings, and housing plates, should be adopted in predicting TE and other kinds of excitations.
Once manufactured and mounted, any individual gear pair has a unique TE. When measuring TE, the tested gear pairs are mounted on two isolated shafts. The boundary conditions of the gears are extremely different from the ones the gear pairs are subject to in a real gearbox. Thus, the measured TE is quite different from the TE of the gear pair in the gearbox.
Microgeometry modification does reduce the gear whine noise to a certain extent. However, the amounts and the effects of modification both have a limitation especially for gears with a small module. Therefore, it is necessary to propose new strategies to reduce TE and hence gear whine noise. Using magnetic gears, gears with a small module, or gears with thin and tall teeth might be an effective strategy.
The effect of frictional excitations on motions along the OLOA is straightforward, while its influence on motions along LOA and axial directions is difficult to illustrate, which requires further study.
There are two strategies to reduce the frictional excitations: improving the surface roughness and providing sufficient lubricant oil. However, improving the surface roughness is expensive and providing sufficient lubricant oil may induce gear rattle noise. Other approaches to decrease the frictional excitations should be proposed.
A limited number of papers focus on the prediction and reduction of shuttling excitation. This does not mean shuttling excitation is not important; on the contrary, shuttling excitation may be dominant when TE and frictional excitations are minimized. Strategies for minimizing shuttling excitation should be proposed as soon as possible.
Reducing excitations is not the only way to minimize gear whine noise. Modifying vibration transmission paths and optimizing housing plates are also effective means to reduce radiated noise. Hybrid composite-metal gears have a great potential on vibration and noise reduction of the gearbox.
Conflicts of Interest
The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
This research was supported by the National Key Research and Development Program-Research on Application of Vibration, Noise, and Post-Processing of Medium Power Agricultural Diesel Engine (Grant no: 2016YFD0700704B), Innovative Research Team Development Program of the Ministry of Education of China (IRT_17R83), and 111 Project (B17034).
- S. Theodossiades, M. De la Cruz, and H. Rahnejat, “Prediction of airborne radiated noise from lightly loaded lubricated meshing gear teeth,” Applied Acoustics, vol. 100, pp. 79–86, 2015.
- R. Brancati, E. Rocca, D. Siano, and M. Viscardi, “Experimental vibro-acoustic analysis of the gear rattle induced by multi-harmonic excitation,” Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, vol. 232, no. 6, pp. 785–796, 2017.
- R. Brancati, E. Rocca, and D. Lauria, “Feasibility study of the Hilbert transform in detecting the gear rattle phenomenon of automotive transmissions,” Journal of Vibration and Control, vol. 24, no. 12, pp. 2631–2641, 2017.
- A. Fernandez-Del-Rincon, A. Diez-Ibarbia, and S. Theodossiades, “Gear transmission rattle: assessment of meshing forces under hydrodynamic lubrication,” Applied Acoustics, vol. 144, pp. 85–95, 2019.
- R. Brancati, E. Rocca, and R. Russo, “An analysis of the automotive driveline dynamic behaviour focusing on the influence of the oil squeeze effect on the idle rattle phenomenon,” Journal of Sound and Vibration, vol. 303, no. 3–5, pp. 858–872, 2007.
- C. Gill-Jeong, “Analysis of the nonlinear behavior of gear pairs considering hydrodynamic lubrication and sliding friction,” Journal of Mechanical Science and Technology, vol. 23, no. 8, pp. 2125–2137, 2009.
- R. Russo, R. Brancati, and E. Rocca, “Experimental investigations about the influence of oil lubricant between teeth on the gear rattle phenomenon,” Journal of Sound and Vibration, vol. 321, no. 3–5, pp. 647–661, 2009.
- R. Singh, H. Xie, and R. J. Comparin, “Analysis of automotive neutral grear rattle,” Journal of Sound and Vibration, vol. 131, no. 2, pp. 177–196, 1989.
- J. Zhang, D. Liu, and H. Yu, “Experimental and numerical analysis for the transmission gear rattle in a power-split hybrid electric vehicle,” International Journal of Vehicle Design, vol. 74, no. 1, pp. 1–18, 2017.
- S. L. Harris, “Dynamic loads on the teeth of spur gears,” Proceedings of the Institution of Mechanical Engineers, vol. 172, no. 1, pp. 87–112, 1958.
- H. N. Özgüven and D. R. Houser, “Mathematical models used in gear dynamics—a review,” Journal of Sound and Vibration, vol. 121, no. 3, pp. 383–411, 1988.
- J. Wang, R. Li, and X. Peng, “Survey of nonlinear vibration of gear transmission systems,” Applied Mechanics Reviews, vol. 56, no. 3, pp. 309–329, 2003.
- Y. Lei, J. Lin, M. J. Zuo, and Z. He, “Condition monitoring and fault diagnosis of planetary gearboxes: a review,” Measurement, vol. 48, pp. 292–305, 2014.
- X. Liang, M. J. Zuo, and Z. Feng, “Dynamic modeling of gearbox faults: a review,” Mechanical Systems and Signal Processing, vol. 98, pp. 852–876, 2018.
- C. G. Cooley and R. G. Parker, “A review of planetary and epicyclic gear dynamics and vibrations research,” Applied Mechanics Reviews, vol. 66, no. 4, p. 40804, 2014.
- R. G. Munro, “Paper 10: effect of geometrical errors on the transmission of motion between gears,” Proceedings of the Institution of Mechanical Engineers, Conference Proceedings, vol. 184, no. 15, pp. 79–84, 1969.
- H. K. Kohler, A. Pratt, and A. M. Thompson, “Paper 14: dynamics and noise of parallel-axis gearing,” Proceedings of the Institution of Mechanical Engineers, Conference Proceedings, vol. 184, no. 15, pp. 111–121, 1969.
- D. R. Houser, F. B. Oswald, M. J. Valco, R. J. Drago, and J. W. Lenski, “Comparison of transmission error predictions with noise measurements for several spur and helical gears,” in Proceedings of the 30th Joint Propulsion Conference, Indianapolis, IN, USA, June 1994.
- D. Palmer and R. G. Munro, “Measurements of transmission error, vibration and noise in spur gears,” in Proceedings of the 1995 British Gear Association Technical Congress, Birmingham, UK, November 1995.
- M. Åkerblom, Gearbox Noise Correlation with Transmission Error and Influence of Bearing Preload, Royal Institute of Technology, Stockholm, Sweden, 2008.
- R. W. Gregory, S. L. Harris, and R. G. Munro, “Dynamic behaviour of spur gears,” Proceedings of the Institution of Mechanical Engineers, vol. 178, no. 1, pp. 207–218, 1963.
- D. B. Welbourn, “Fundamental knowledge of gear noise: a survey,” in Noise and Vibrations of Engines and Transmissions, pp. 9–14, Cranfield Institute of Technology, Cranfield, UK, 1979.
- H. N. Özgüven and D. R. Houser, “Dynamic analysis of high speed gears by using loaded static transmission error,” Journal of Sound and Vibration, vol. 125, no. 1, pp. 71–83, 1988.
- R. G. Munro, “A review of the theory and measurement of gear transmission error,” Gearbox Noise and Vibration, Marcel Dekker, New York, NY, USA, 1990.
- J. D. Smith, Gear Noise and Vibration, Marcel Dekker, New York, NY, USA, 2nd edition, 2003.
- P. Velex and M. Ajmi, “On the modelling of excitations in geared systems by transmission errors,” Journal of Sound and Vibration, vol. 290, no. 3–5, pp. 882–909, 2006.
- P. Davoli, C. Gorla, F. Rosa, F. Rossi, and G. Boni, “Transmission error and noise emission of spur gears: a theoretical and experimental approach,” in Proceedings of the 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 7: 10th International Power Transmission and Gearing Conference, pp. 443–449, Las Vegas, NV, USA, 2007.
- A. Palermo, D. Mundo, A. S. Lentini, R. Hadjit, P. Mas, and W. Desmet, “Gear noise evaluation through multibody TE-based simulations,” in Proceedings of ISMA-International Conference on Noise and Vibration Engineering, pp. 3033–3046, Leuven, Belgium, 2010.
- W. D. Mark, “Analysis of the vibratory excitation of gear systems: basic theory,” The Journal of the Acoustical Society of America, vol. 63, no. 5, pp. 1409–1430, 1978.
- W. D. Mark, “Analysis of the vibratory excitation of gear systems. II. Tooth error representations, approximations, and application,” The Journal of the Acoustical Society of America, vol. 66, no. 6, pp. 1758–1787, 1979.
- W. D. Mark, Performance-Based Gear Metrology: Kinematic-Transmission-Error Computation and Diagnosis, John Wiley & Sons, Chichester, UK, 2012.
- W. D. Mark, “Tooth-meshing-harmonic static-transmission-error amplitudes of helical gears,” Mechanical Systems and Signal Processing, vol. 98, pp. 506–533, 2018.
- H. Walker, “Gear tooth deflection and profile modification,” Engineer, vol. 166, pp. 409–412, 1938.
- C. Weber, The Deformation of Loaded Gears and the Effect on Their Load Carrying Capacity, British Dept. of Scientific and Industrial Research, London, UK, 1949.
- K. Umezawa, T. Suzuki, and T. Sato, “Vibration of power transmission helical gears. Approximate equation of stiffness,” Transactions of the Japan Society of Mechanical Engineers Series C, vol. 51, no. 469, pp. 2316–2323, 1985.
- Y. Cai, “Simulation on the rotational vibration of helical gears in consideration of the tooth separation phenomenon (a new stiffness function of helical involute tooth pair),” Journal of Mechanical Design, vol. 117, no. 3, pp. 460–469, 1995.
- T. Kiekbusch and I. Howard, “A common formula for the combined torsional mesh stiffness of spur gears,” in Proceedings of the 5th Australasian Congress on Applied Mechanics, pp. 710–716, Brisbane, Australia, 2007.
- V. Skrickij and M. Bogdevičius, “Vehicle gearbox dynamics: centre distance influence on mesh stiffness and spur gear dynamics,” Transport, vol. 25, no. 3, pp. 278–286, 2010.
- Z. Chen and Y. Shao, “Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack,” Mechanism and Machine Theory, vol. 62, pp. 63–74, 2013.
- A. F. D. Rincon, F. Viadero, M. Iglesias, P. García, A. De-Juan, and R. Sancibrian, “A model for the study of meshing stiffness in spur gear transmissions,” Mechanism and Machine Theory, vol. 61, pp. 30–58, 2013.
- N. L. Pedersen and M. F. Jørgensen, “On gear tooth stiffness evaluation,” Computers & Structures, vol. 135, no. 3, pp. 109–117, 2014.
- Z. Wan, H. Cao, Y. Zi, W. He, and Z. He, “An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack,” Engineering Failure Analysis, vol. 42, pp. 157–177, 2014.
- X. Gu, P. Velex, P. Sainsot, and J. Bruyère, “Analytical investigations on the mesh stiffness function of solid spur and helical gears,” Journal of Mechanical Design, vol. 137, no. 6, p. 63301, 2015.
- L. Chang, G. Liu, and L. Wu, “A robust model for determining the mesh stiffness of cylindrical gears,” Mechanism and Machine Theory, vol. 87, pp. 93–114, 2015.
- Z. Wan, H. Cao, Y. Zi, W. He, and Y. Chen, “Mesh stiffness calculation using an accumulated integral potential energy method and dynamic analysis of helical gears,” Mechanism and Machine Theory, vol. 92, pp. 447–463, 2015.
- C. G. Cooley, C. Liu, X. Dai, and R. G. Parker, “Gear tooth mesh stiffness: a comparison of calculation approaches,” Mechanism and Machine Theory, vol. 105, pp. 540–553, 2016.
- H. Ma, M. Feng, Z. Li, R. Feng, and B. Wen, “Time-varying mesh characteristics of a spur gear pair considering the tip-fillet and friction,” Meccanica, vol. 52, no. 7, pp. 1695–1709, 2017.
- C. I. Park and J. M. Lee, “Experimental investigation of the effect of lead errors on helical gear and bearing vibration transmission characteristics,” KSME International Journal, vol. 16, no. 11, pp. 1395–1403, 2002.
- A. Fernández-del-Rincón, M. Iglesias, A. De-Juan, A. Diez-Ibarbia, P. García, and F. Viadero, “Gear transmission dynamics: effects of index and run out errors,” Applied Acoustics, vol. 108, no. 1-2, pp. 63–83, 2016.
- D. Talbot, A. Sun, and A. Kahraman, “Impact of tooth indexing errors on dynamic factors of spur gears: experiments and model simulations,” Journal of Mechanical Design, vol. 138, no. 9, p. 93302, 2016.
- E. Rigaud and D. Barday, “Modelling and analysis of static transmission error. Effect of wheel body deformation and interactions between adjacent loaded teeth,” in Proceedings of the 4th World Congress on Gearing and Power Transmission, pp. 1961–1972, Paris, France, 1999.
- S. Li, “Effects of machining errors, assembly errors and tooth modifications on loading capacity, load-sharing ratio and transmission error of a pair of spur gears,” Mechanism and Machine Theory, vol. 42, no. 6, pp. 698–726, 2007.
- S. Li, “Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors, assembly errors and tooth modifications,” Mechanism and Machine Theory, vol. 42, no. 1, pp. 88–114, 2007.
- Y. Guo, T. Eritenel, T. M. Ericson, and R. G. Parker, “Vibro-acoustic propagation of gear dynamics in a gear-bearing-housing system,” Journal of Sound and Vibration, vol. 333, no. 22, pp. 5762–5785, 2014.
- H. Jiang, Y. Shao, C. K. Mechefske, and X. Chen, “The influence of mesh misalignment on the dynamic characteristics of helical gears including sliding friction,” Journal of Mechanical Science and Technology, vol. 29, no. 11, pp. 4563–4573, 2015.
- S. Li, “Effects of misalignment error, tooth modifications and transmitted torque on tooth engagements of a pair of spur gears,” Mechanism and Machine Theory, vol. 83, pp. 125–136, 2015.
- A. Saxena, A. Parey, and M. Chouksey, “Effect of shaft misalignment and friction force on time varying mesh stiffness of spur gear pair,” Engineering Failure Analysis, vol. 49, pp. 79–91, 2015.
- Y. Guo, S. Lambert, R. Wallen, R. Errichello, and J. Keller, “Theoretical and experimental study on gear-coupling contact and loads considering misalignment, torque, and friction influences,” Mechanism and Machine Theory, vol. 98, pp. 242–262, 2016.
- M. S. Tavakoli and D. R. Houser, “Optimum profile modifications for the minimization of static transmission errors of spur gears,” Journal of Mechanisms, Transmissions, and Automation in Design, vol. 108, no. 1, pp. 86–94, 1986.
- F. L. Litvin, A. Fuentes, I. Gonzalez-Perez, L. Carvenali, K. Kawasaki, and R. F. Handschuh, “Modified involute helical gears: computerized design, simulation of meshing and stress analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 33-34, pp. 3619–3655, 2003.
- D. R. Houser and J. Harianto, “The effect of micro-geometry and load on helical gear noise excitations,” in Proceedings of the 2005 Noise and Vibration Conference and Exhibition, Traverse City, MI, USA, 2005.
- E. N. Mohamad, M. Komori, H. Murakami, A. Kubo, and S. Fang, “Effect of convex tooth flank form deviation on the characteristics of transmission error of gears considering elastic deformation,” Journal of Mechanical Design, vol. 132, no. 10, Article ID 101005, 2010.
- A. Artoni, M. Guiggiani, A. Kahraman, and J. Harianto, “Robust optimization of cylindrical gear tooth surface modifications within ranges of torque and misalignments,” Journal of Mechanical Design, vol. 135, no. 12, Article ID 121005, 2013.
- J. Jiang and Z. Fang, “Design and analysis of modified cylindrical gears with a higher-order transmission error,” Mechanism and Machine Theory, vol. 88, pp. 141–152, 2015.
- W. Yu and C. K. Mechefske, “Analytical modeling of spur gear corner contact effects,” Mechanism and Machine Theory, vol. 96, pp. 146–164, 2016.
- B. Yu and K.-l. Ting, “Compensated conjugation and gear tooth modification design,” Journal of Mechanical Design, vol. 138, no. 7, p. 73301, 2016.
- W. D. Mark, “Contributions to the vibratory excitation of gear systems from periodic undulations on tooth running surfaces,” The Journal of the Acoustical Society of America, vol. 91, no. 1, pp. 166–186, 1992.
- J. Yoo, K. Kang, J. Huh, and C. Choi, “The mechanism and solution of harmonic gear whine noise in automotive transmission systems,” in Proceedings of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 7: 10th International Power Transmission and Gearing Conference, pp. 569–575, Las Vegas, NV, USA, 2007.
- S. Sundar, R. Singh, K. Jayasankaran, and S. Kim, “Effect of the tooth surface waviness on the dynamics and structure-borne noise of a spur gear pair,” SAE International Journal of Passenger Cars—Mechanical Systems, vol. 6, no. 2, pp. 1087–1093, 2013.
- S. Jolivet, S. Mezghani, J. Isselin, and M. El Mansori, “Experimental and numerical study of tooth finishing processes contribution to gear noise,” Tribology International, vol. 102, pp. 436–443, 2016.
- G. Bonori and F. Pellicano, “Non-smooth dynamics of spur gears with manufacturing errors,” Journal of Sound and Vibration, vol. 306, no. 1-2, pp. 271–283, 2007.
- M. Inalpolat, M. Handschuh, and A. Kahraman, “Influence of indexing errors on dynamic response of spur gear pairs,” Mechanical Systems and Signal Processing, vol. 60-61, pp. 391–405, 2015.
- S. A. Hambric, A. D. Hanford, M. R. Shepherd, R. L. Campbell, and E. C. Smith, Rotorcraft Transmission Noise Path Model, Including Distributed Fluid Film Bearing Impedance Modeling, NASA, Washington, DC, USA, 2010.
- W. D. Mark, C. P. Reagor, and D. R. McPherson, “Assessing the role of plastic deformation in gear-health monitoring by precision measurement of failed gears,” Mechanical Systems and Signal Processing, vol. 21, no. 1, pp. 177–192, 2007.
- W. D. Mark and C. P. Reagor, “Static-transmission-error vibratory-excitation contributions from plastically deformed gear teeth caused by tooth bending-fatigue damage,” Mechanical Systems and Signal Processing, vol. 21, no. 2, pp. 885–905, 2007.
- W. D. Mark, H. Lee, R. Patrick, and J. D. Coker, “A simple frequency-domain algorithm for early detection of damaged gear teeth,” Mechanical Systems and Signal Processing, vol. 24, no. 8, pp. 2807–2823, 2010.
- W. D. Mark, “Time-synchronous-averaging of gear-meshing-vibration transducer responses for elimination of harmonic contributions from the mating gear and the gear pair,” Mechanical Systems and Signal Processing, vol. 62-63, pp. 21–29, 2015.
- W. D. Mark, “Analytical approximations to damaged gear tooth transmission-error contributions for gear-health monitoring,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 230, no. 7-8, pp. 1157–1182, 2016.
- W. D. Mark, A. C. Isaacson, and M. E. Wagner, “Transmission-error frequency-domain-behavior of failing gears,” Mechanical Systems and Signal Processing, vol. 115, pp. 102–119, 2019.
- A. Kahraman and G. W. Blankenship, “Effect of involute tip relief on dynamic response of spur gear pairs,” Journal of Mechanical Design, vol. 121, no. 2, pp. 313–315, 1999.
- M. Kubur, A. Kahraman, D. M. Zini, and K. Kienzle, “Dynamic analysis of a multi-shaft helical gear transmission by finite elements: model and experiment,” Journal of Vibration and Acoustics, vol. 126, no. 3, pp. 398–406, 2004.
- C. I. Park, “Multi-objective optimization of the tooth surface in helical gears using design of experiment and the response surface method,” Journal of Mechanical Science and Technology, vol. 24, no. 3, pp. 823–829, 2010.
- C. Zhou, Z. Wang, and S. Chen, “Coupling of the 2D microtopography of tooth surface and transmission error,” Journal of Mechanical Science and Technology, vol. 32, no. 2, pp. 723–730, 2018.
- R. W. Cornell, “Compliance and stress sensitivity of spur gear teeth,” Journal of Mechanical Design, vol. 103, no. 2, pp. 447–459, 1981.
- H. Lin and R. L. Huston, Dynamic Loading on Parallel Shaft Gears, NASA, Washington, DC, USA, 1986.
- H.-H. Lin, R. L. Huston, and J. J. Coy, “On dynamic loads in parallel shaft transmissions: part I-modelling and analysis,” Journal of Mechanisms, Transmissions, and Automation in Design, vol. 110, no. 2, pp. 221–225, 1988.
- C.-H. Liou, H. H. Lin, F. B. Oswald, and D. P. Townsend, “Effect of contact ratio on spur gear dynamic load with no tooth profile modifications,” Journal of Mechanical Design, vol. 118, no. 3, pp. 439–443, 1996.
- K. Y. Yoon and S. S. Rao, “Dynamic load analysis of spur gears using a new tooth profile,” Journal of Mechanical Design, vol. 118, no. 1, pp. 1–6, 1996.
- R. G. Munro, L. Morrish, and D. Palmer, “Gear transmission error outside the normal path of contact due to corner and top contact,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 213, no. 4, pp. 389–400, 1999.
- H. H. Lin, J. Wang, F. B. Oswald, and J. J. Coy, “Effect of extended tooth contact on the modeling of spur gear transmissions,” in Proceedings of the 29th Joint Propulsion Conference and Exhibit, Joint Propulsion Conferences, Monterey, CA, USA, 1993.
- Y. Terauchi and K. Nagamura, “Study on deflection of spur gear teeth: 1st report, calculation of tooth deflection by two-dimensional elastic theory,” Bulletin of JSME, vol. 23, no. 184, pp. 1682–1688, 1980.
- Y. Terauchi and K. Nagamura, “Study on deflection of spur gear teeth: 2nd report, calculation of tooth deflection for spur gears with various tooth profiles,” Bulletin of JSME, vol. 24, no. 188, pp. 447–452, 1981.
- C. W. MacGregor, “Deflection of a long helical gear tooth,” Mechanical Engineering, vol. 57, pp. 225–227, 1935.
- E. J. Wellauer and A. Seireg, “Bending strength of gear teeth by cantilever-plate theory,” Journal of Engineering for Industry, vol. 82, no. 3, pp. 213–220, 1960.
- T. F. Conry and A. Seireg, “A mathematical programming method for design of elastic bodies in contact,” Journal of Applied Mechanics, vol. 38, no. 2, pp. 387–392, 1971.
- T. F. Conry and A. Seireg, “A mathematical programming technique for the evaluation of load distribution and optimal modifications for gear systems,” Journal of Engineering for Industry, vol. 95, no. 4, pp. 1115–1122, 1973.
- J. Harianto and D. R. Houser, “A methodology for obtaining optimum gear tooth micro-topographies for noise and stress minimization over a broad operating torque range,” in Proceedings of the 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 7: 10th International Power Transmission and Gearing Conference, pp. 289–303, Las Vegas, NV, USA, 2007.
- Q. Zhang, J. Kang, W. Dong, and S. Lyu, “A study on tooth modification and radiation noise of a manual transaxle,” International Journal of Precision Engineering and Manufacturing, vol. 13, no. 6, pp. 1013–1020, 2012.
- H. S. Kwon, A. Kahraman, H. K. Lee, and H. S. Suh, “An automated design search for single and double-planet planetary gear sets,” Journal of Mechanical Design, vol. 136, no. 6, p. 61004, 2014.
- B. Yuan, S. Chang, G. Liu, L. Chang, and L. Liu, “Optimization of bias modification and dynamic behavior analysis of helical gear system,” Advances in Mechanical Engineering, vol. 9, no. 11, Article ID 168781401773325, 2017.
- L. Vedmar and B. Henriksson, “A general approach for determining dynamic forces in spur gears,” Journal of Mechanical Design, vol. 120, no. 4, pp. 593–598, 1998.
- J. Wang and I. Howard, “Finite element analysis of high contact ratio spur gears in mesh,” Journal of Tribology, vol. 127, no. 3, pp. 469–483, 2005.
- A. Andersson and L. Vedmar, “A dynamic model to determine vibrations in involute helical gears,” Journal of Sound and Vibration, vol. 260, no. 2, pp. 195–212, 2003.
- T. Lin, H. Ou, and R. Li, “A finite element method for 3D static and dynamic contact/impact analysis of gear drives,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 9–12, pp. 1716–1728, 2007.
- K. Mao, “An approach for powertrain gear transmission error prediction using the non-linear finite element method,” Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, vol. 220, no. 10, pp. 1455–1463, 2006.
- Y. A. Tesfahunegn, F. Rosa, and C. Gorla, “The effects of the shape of tooth profile modifications on the transmission error, bending, and contact stress of spur gears,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 224, no. 8, pp. 1749–1758, 2010.
- Y.-j. Wu, J.-j. Wang, and Q.-k. Han, “Contact finite element method for dynamic meshing characteristics analysis of continuous engaged gear drives,” Journal of Mechanical Science and Technology, vol. 26, no. 6, pp. 1671–1685, 2012.
- J. S.-S. Wu, S.-L. Xu, Y.-T. Lin, W.-H. Chen, and Y.-L. Lai, “3D contact analysis of conjugate spur gears by a complete mating process,” Journal of Mechanical Science and Technology, vol. 27, no. 12, pp. 3787–3795, 2013.
- X. Deng, L. Hua, and X. Han, “Research on the design and modification of asymmetric spur gear,” Mathematical Problems in Engineering, vol. 2015, Article ID 897257, 13 pages, 2015.
- N. K. Raghuwanshi and A. Parey, “Effect of back-side contact on mesh stiffness of spur gear pair by finite element method,” Procedia Engineering, vol. 173, pp. 1538–1543, 2017.
- N. Wang, X. Li, K. Wang, Q. Zeng, and X. Shen, “A novel axial modification and simulation analysis of involute spur gear,” Strojniški Vestnik—Journal of Mechanical Engineering, vol. 63, no. 12, 2017.
- M. Barbieri, A. Zippo, and F. Pellicano, “Adaptive grid-size finite element modeling of helical gear pairs,” Mechanism and Machine Theory, vol. 82, pp. 17–32, 2014.
- Y. Wang, Y. Liu, W. Tang, and P. Liu, “Parametric finite element modeling and tooth contact analysis of spur and helical gears including profile and lead modifications,” Engineering Computations, vol. 34, no. 8, pp. 2877–2898, 2017.
- J. Brauer, “A general finite element model of involute gears,” Finite Elements in Analysis and Design, vol. 40, no. 13-14, pp. 1857–1872, 2004.
- K. Jao Huang and H. Wei Su, “Approaches to parametric element constructions and dynamic analyses of spur/helical gears including modifications and undercutting,” Finite Elements in Analysis and Design, vol. 46, no. 12, pp. 1106–1113, 2010.
- J. Wei, W. Sun, and L. Wang, “Effects of flank deviation on load distributions for helical gear,” Journal of Mechanical Science and Technology, vol. 25, no. 7, pp. 1781–1789, 2011.
- Z. He, T. Lin, T. Luo, T. Deng, and Q. Hu, “Parametric modeling and contact analysis of helical gears with modifications,” Journal of Mechanical Science and Technology, vol. 30, no. 11, pp. 4859–4867, 2016.
- X. Li, C. Li, B. Chen, and D. Liang, “Design and investigation of a cycloid helical gear drive,” Journal of Mechanical Science and Technology, vol. 31, no. 9, pp. 4329–4336, 2017.
- T. Lin and Z. He, “Analytical method for coupled transmission error of helical gear system with machining errors, assembly errors and tooth modifications,” Mechanical Systems and Signal Processing, vol. 91, pp. 167–182, 2017.
- I. Gonzalez-Perez and A. Fuentes-Aznar, “Implementation of a finite element model for stress analysis of gear drives based on multi-point constraints,” Mechanism and Machine Theory, vol. 117, pp. 35–47, 2017.
- S. Li, “Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear,” Journal of Mechanical Design, vol. 124, no. 3, pp. 511–517, 2002.
- V. Simon, “Load and stress distributions in spur and helical gears,” Journal of Mechanisms, Transmissions, and Automation in Design, vol. 110, no. 2, pp. 197–202, 1988.
- V. Simon, “Optimal tooth modifications for spur and helical gears,” Journal of Mechanisms, Transmissions, and Automation in Design, vol. 111, no. 4, pp. 611–615, 1989.
- T. Nishino, “Vibration analysis of the helical gear system using the integrated excitation model,” Journal of Advanced Mechanical Design, Systems, and Manufacturing, vol. 1, no. 4, pp. 541–552, 2007.
- J. S. Kang and Y.-S. Choi, “Optimization of helix angle for helical gear system,” Journal of Mechanical Science and Technology, vol. 22, no. 12, pp. 2393–2402, 2008.
- E. N. Mohamad, M. Komori, H. Murakami, A. Kubo, and S. Fang, “Analysis of general characteristics of transmission error of gears with convex modification of tooth flank form considering elastic deformation under load,” Journal of Mechanical Design, vol. 131, no. 6, p. 61015, 2009.
- R. Guilbault, C. Gosselin, and L. Cloutier, “Express model for load sharing and stress analysis in helical gears,” Journal of Mechanical Design, vol. 127, no. 6, pp. 1161–1172, 2005.
- Y. Miyoshi, K. Tobisawa, and K. Saiki, “Composite analysis method of tooth contact load distribution of helical gear,” in Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 7: 10th International Power Transmission and Gearing Conference, pp. 173–180, Las Vegas, NV, USA, 2007.
- P. Velex, J. Bruyère, and D. R. Houser, “Some analytical results on transmission errors in narrow-faced spur and helical gears: influence of profile modifications,” Journal of Mechanical Design, vol. 133, no. 3, p. 31010, 2011.
- M. Umeyama, M. Kato, and K. Inoue, “Effects of gear dimensions and tooth surface modifications on the loaded transmission error of a helical gear pair,” Journal of Mechanical Design, vol. 120, no. 1, pp. 119–125, 1998.
- S. S. Rao and K. Y. Yoon, “Minimization of transmission error in helical gears,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 215, no. 4, pp. 447–459, 2001.
- J. A. Korta and D. Mundo, “Multi-objective micro-geometry optimization of gear tooth supported by response surface methodology,” Mechanism and Machine Theory, vol. 109, pp. 278–295, 2017.
- M. Åkerblom, Gear Noise and Vibration—A Literature Survey, KTH Royal Institute of Technology, Stockholm, Sweden, 2001.
- R. W. Gregory, S. L. Harris, and R. G. Munro, “A method of measuring transmission error in spur gears of 1 : 1 ratio,” Journal of Scientific Instruments, vol. 40, no. 1, pp. 5–9, 1963.
- D. R. Houser and G. W. Blankenship, “Methods for measuring gear transmission error under load and at operating speeds,” in Proceedings of the International Off-Highway and Powerplant Congress and Exposition, Warrendale, PA, USA, 1989.
- S. Sasaoka, “Measurement technique for loaded gear transmission error,” in Proceedings of the 1997 International Congress and Exposition, Detroit, MI, USA, 1997.
- C. Gosselin, T. Guertin, D. Remond, and Y. Jean, “Simulation and experimental measurement of the transmission error of real hypoid gears under load,” Journal of Mechanical Design, vol. 122, no. 1, p. 109, 2000.
- R. G. Munro, D. Palmer, and L. Morrish, “An experimental method to measure gear tooth stiffness throughout and beyond the path of contact,” Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science, vol. 215, no. 7, pp. 793–803, 2001.
- S. Kurokawa, Y. Ariura, Y. Matsukawa, and T. Doi, “Evaluation of gear engagement accuracy by transmission error with sub-microradian resolution,” International Journal of Surface Science and Engineering, vol. 3, no. 3, pp. 160–177, 2009.
- J. H. Yoon, B. J. Choi, I. H. Yang, and J. E. Oh, “Deflection test and transmission error measurement to identify hypoid gear whine noise,” International Journal of Automotive Technology, vol. 12, no. 1, pp. 59–66, 2011.
- S. Park, S. Kim, and J.-H. Choi, “Gear fault diagnosis using transmission error and ensemble empirical mode decomposition,” Mechanical Systems and Signal Processing, vol. 108, pp. 262–275, 2018.
- M. R. Kang and A. Kahraman, “Measurement of vibratory motions of gears supported by compliant shafts,” Mechanical Systems and Signal Processing, vol. 29, pp. 391–403, 2012.
- D. Remond and J. Mahfoudh, “From transmission error measurements to angular sampling in rotating machines with discrete geometry,” Shock and Vibration, vol. 12, no. 2, pp. 149–161, 2005.
- A. Palermo, L. Britte, K. Janssens, D. Mundo, and W. Desmet, “The measurement of gear transmission error as an NVH indicator: theoretical discussion and industrial application via low-cost digital encoders to an all-electric vehicle gearbox,” Mechanical Systems and Signal Processing, vol. 110, pp. 368–389, 2018.
- R. White and V. Palan, “Measurement of transmission error using rotational laser vibrometers,” in Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 7: 10th International Power Transmission and Gearing Conference, pp. 527–545, Las Vegas, NV, USA, 2007.
- V. K. Tamminana, A. Kahraman, and S. Vijayakar, “A study of the relationship between the dynamic factors and the dynamic transmission error of spur gear pairs,” Journal of Mechanical Design, vol. 129, no. 1, pp. 75–84, 2007.
- C. Brecher, C. Gorgels, J. Hesse, and M. Hellmann, “Dynamic transmission error measurements of a drive train,” Production Engineering, vol. 5, no. 3, pp. 321–327, 2011.
- B. Anichowski, A. Kahraman, and D. Talbot, “Dynamic transmission error measurements from spur gear pairs having tooth indexing errors,” in Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 10: 2017 ASME International Power Transmission and Gearing Conference, Cleveland, OH, USA, 2017.
- H. Guo, J. Zhang, and H. Yu, “Dynamic modelling and parametric optimization of a full hybrid transmission,” Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-Body Dynamics, vol. 233, no. 1, pp. 17–29, 2019.
- Y. Teranchi, H. Nanano, and M. Nohara, “On the effect of the tooth profile modification on the dynamic load and the sound level of the spur gear,” Bulletin of JSME, vol. 25, no. 207, pp. 1474–1481, 1982.
- H. Ma, J. Yang, R. Song, S. Zhang, and B. Wen, “Effects of tip relief on vibration responses of a geared rotor system,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 228, no. 7, pp. 1132–1154, 2014.
- H. H. Lin, D. P. Townsend, and F. B. Oswald, “Profile modification to minimize spur gear dynamic loading,” in Proceedings of the Design Engineering Technical Conference, Orlando, FL, USA, 1987.
- H. H. Lin, F. B. Oswald, and D. P. Townsend, “Dynamic loading of spur gears with linear or parabolic tooth profile modifications,” Mechanism and Machine Theory, vol. 29, no. 8, pp. 1115–1129, 1994.
- G. Bonori, M. Barbieri, and F. Pellicano, “Optimum profile modifications of spur gears by means of genetic algorithms,” Journal of Sound and Vibration, vol. 313, no. 3–5, pp. 603–616, 2008.
- M. Barbieri, G. Bonori, and F. Pellicano, “Corrigendum to: optimum profile modifications of spur gears by means of genetic algorithms,” Journal of Sound and Vibration, vol. 331, no. 21, pp. 4825–4829, 2012.
- Y.-j. Wu, J.-j. Wang, and Q.-k. Han, “Static/dynamic contact FEA and experimental study for tooth profile modification of helical gears,” Journal of Mechanical Science and Technology, vol. 26, no. 5, pp. 1409–1417, 2012.
- N. Yaldirim and R. G. Munro, “A systematic approach to profile relief design of low and high contact ratio spur gears,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 213, no. 6, pp. 551–562, 1999.
- N. Yaldirim and R. G. Munro, “A new type of profile relief for high contact ratio spur gears,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 213, no. 6, pp. 563–568, 1999.
- P. Wagaj and A. Kahraman, “Impact of tooth profile modifications on the transmission error excitation of helical gear pairs,” in Proceedings of ESDA2002: 6th Biennial Conference on Engineering Systems Design and Analysis, Istanbul, Turkey, 2002.
- S. Oh, S. Oh, J. Kang, I. Lee, and S. Lyu, “A study on modeling and optimization of tooth microgeometry for a helical gear pair,” International Journal of Precision Engineering and Manufacturing, vol. 14, no. 3, pp. 423–427, 2013.
- C. Jia, Z. Fang, and Y. Zhang, “Topography of modified surfaces based on compensated conjugation for the minimization of transmission errors of cylindrical gears,” Mechanism and Machine Theory, vol. 116, pp. 145–161, 2017.
- D. Ghribi, J. Bruyère, P. Velex, M. Octrue, and M. Haddar, “A contribution to the design of robust profile modifications in spur and helical gears by combining analytical results and numerical simulations,” Journal of Mechanical Design, vol. 134, no. 6, p. 61011, 2012.
- J. Bruyère and P. Velex, “Derivation of optimum profile modifications in narrow-faced spur and helical gears using a perturbation method,” Journal of Mechanical Design, vol. 135, no. 7, p. 71009, 2013.
- J. Bruyère and P. Velex, “A simplified multi-objective analysis of optimum profile modifications in spur and helical gears,” Mechanism and Machine Theory, vol. 80, pp. 70–83, 2014.
- J. Bruyère, X. Gu, and P. Velex, “On the analytical definition of profile modifications minimising transmission error variations in narrow-faced spur helical gears,” Mechanism and Machine Theory, vol. 92, pp. 257–272, 2015.
- A. Fernández, M. Iglesias, A. De-Juan, P. García, R. Sancibrián, and F. Viadero, “Gear transmission dynamic: effects of tooth profile deviations and support flexibility,” Applied Acoustics, vol. 77, no. 3, pp. 138–149, 2014.
- H. Liu, C. Zhang, C. L. Xiang, and C. Wang, “Tooth profile modification based on lateral- torsional-rocking coupled nonlinear dynamic model of gear system,” Mechanism and Machine Theory, vol. 105, pp. 606–619, 2016.
- S. S. Ghosh and G. Chakraborty, “On optimal tooth profile modification for reduction of vibration and noise in spur gear pairs,” Mechanism and Machine Theory, vol. 105, pp. 145–163, 2016.
- H. Ma, X. Pang, R. Feng, and B. Wen, “Evaluation of optimum profile modification curves of profile shifted spur gears based on vibration responses,” Mechanical Systems and Signal Processing, vol. 70-71, pp. 1131–1149, 2016.
- H. Iida, A. Tamura, and Y. Yamada, “Vibrational characteristics of friction between gear teeth,” Bulletin of JSME, vol. 28, no. 241, pp. 1512–1519, 1985.
- J. Borner and D. R. Houser, “Friction and bending moments as gear noise excitations,” in Proceedings of the International Off-Highway and Powerplant Congress and Exposition, Indianapolis, IN, USA, 1996.
- D. Hochmann, Friction Force Excitations in Spur and Helical Involute Parallel Axis Gearing, The Ohio State University, Columbus, OH, USA, 1997.
- P. Velex and V. Cahouet, “Experimental and numerical investigations on the influence of tooth friction in spur and helical gear dynamics,” Journal of Mechanical Design, vol. 122, no. 4, pp. 515–522, 2000.
- D. R. Houser, M. Vaishya, and J. D. Sorenson, “Vibro-acoustic effects of friction in gears: an experimental investigation,” in Proceedings of the Noise and Vibration Conference and Exposition, Traverse City, MI, USA, 2001.
- M. Vaishya and R. Singh, “Analysis of periodically varying gear mesh systems with coulomb friction using Floquet theory,” Journal of Sound and Vibration, vol. 243, no. 3, pp. 525–545, 2001.
- M. Vaishya and R. Singh, “Sliding friction-induced non-linearity and parametric effects in gear dynamics,” Journal of Sound and Vibration, vol. 248, no. 4, pp. 671–694, 2001.
- P. Velex and P. Sainsot, “An analytical study of tooth friction excitations in errorless spur and helical gears,” Mechanism and Machine Theory, vol. 37, no. 7, pp. 641–658, 2002.
- R. Gunda and R. Singh, “Dynamic analysis of sliding friction in a gear pair,” in Proceedings of the Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 4: 9th International Power Transmission and Gearing Conference, Parts A and B, pp. 2–6, Chicago, IL, USA, 2003.
- O. Lundvall, N. Strömberg, and A. Klarbring, “A flexible multi-body approach for frictional contact in spur gears,” Journal of Sound and Vibration, vol. 278, no. 3, pp. 479–499, 2004.
- S. He, R. Gunda, and R. Singh, “Effect of sliding friction on the dynamics of spur gear pair with realistic time-varying stiffness,” Journal of Sound and Vibration, vol. 301, no. 3–5, pp. 927–949, 2007.
- S. He, R. Gunda, and R. Singh, “Inclusion of sliding friction in contact dynamics model for helical gears,” Journal of Mechanical Design, vol. 129, no. 1, pp. 48–57, 2007.
- S. He and R. Singh, “Analytical study of helical gear dynamics with sliding friction using Floquet theory,” in Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 7: 10th International Power Transmission and Gearing Conference, pp. 433–442, Las Vegas, NV, USA, 2007.
- R. Singh, A. Lake, V. Asnani, and S. He, “Vibro-acoustic model of a geared system including friction excitation,” in Proceedings of the 2007 Inter-Noise and Noise-Con Congress and Conference, pp. 2921–2930, Istanbul, Turkey, 2007.
- S. He and R. Singh, “Dynamic transmission error prediction of helical gear pair under sliding friction using Floquet theory,” Journal of Mechanical Design, vol. 130, no. 5, p. 52603, 2008.
- A. Kahraman, J. Lim, and H. Ding, “A dynamic model of a spur gear pair with friction,” in Proceedings of the 12th IFToMM World Congress, Besancon, France, 2007.
- S. He, R. Singh, and G. Pavić, “Effect of sliding friction on gear noise based on a refined vibro-acoustic formulation,” Noise Control Engineering Journal, vol. 56, no. 3, pp. 164–175, 2008.
- G. Liu and R. G. Parker, “Impact of tooth friction and its bending effect on gear dynamics,” Journal of Sound and Vibration, vol. 320, no. 4-5, pp. 1039–1063, 2009.
- S. Chen, J. Tang, C. Luo, and Q. Wang, “Nonlinear dynamic characteristics of geared rotor bearing systems with dynamic backlash and friction,” Mechanism and Machine Theory, vol. 46, no. 4, pp. 466–478, 2011.
- J. Wang, J. Zheng, and A. Yang, “An analytical study of bifurcation and chaos in a spur gear pair with sliding friction,” Procedia Engineering, vol. 31, pp. 563–570, 2012.
- K. F. Brethee, J. Gao, F. Gu, and A. D. Ball, “Analysis of frictional effects on the dynamic response of gear systems and the implications for diagnostics,” in Proceedings of the 21st International Conference on Automation and Computing, Glasgow, UK, 2015.
- H. Jiang and F. Liu, “Dynamic modeling and analysis of spur gears considering friction-vibration interactions,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 39, no. 12, pp. 4911–4920, 2017.
- M. Vaishya and R. Singh, “Strategies for modeling friction in gear dynamics,” Journal of Mechanical Design, vol. 125, no. 2, pp. 383–393, 2003.
- C. Kar and A. R. Mohanty, “An algorithm for determination of time-varying frictional force and torque in a helical gear system,” Mechanism and Machine Theory, vol. 42, no. 4, pp. 482–496, 2007.
- K. F. Martin, “A review of friction predictions in gear teeth,” Wear, vol. 49, no. 2, pp. 201–238, 1978.
- G. H. Benedict and B. W. Kelley, “Instantaneous coefficients of gear tooth friction,” ASLE Transactions, vol. 4, no. 1, pp. 59–70, 1961.
- B. W. Kelley and A. J. Lemanski, “Paper 11: lubrication of involute gearing,” Proceedings of the Institution of Mechanical Engineers, Conference Proceedings, vol. 182, no. 1, pp. 173–184, 1967.
- K. L. Johnson and D. I. Spence, “Determination of gear tooth friction by disc machine,” Tribology International, vol. 24, no. 5, pp. 269–275, 1991.
- B. Rebbechi, F. B. Oswald, and D. P. Townsend, “Measurement of gear tooth dynamic friction,” in Proceedings of the 7th International Power Transmission and Gearing Conference, San Diego, CA, USA, 1996.
- C. Duan and R. Singh, “Dynamics of a 3DOF torsional system with a dry friction controlled path,” Journal of Sound and Vibration, vol. 289, no. 4-5, pp. 657–688, 2006.
- H. Xu, Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs, The Ohio State University, Columbus, OH, USA, 2005.
- H. Xu, A. Kahraman, N. E. Anderson, and D. G. Maddock, “Prediction of mechanical efficiency of parallel-axis gear pairs,” Journal of Mechanical Design, vol. 129, no. 1, pp. 58–68, 2007.
- S. He, S. Cho, and R. Singh, “Prediction of dynamic friction forces in spur gears using alternate sliding friction formulations,” Journal of Sound and Vibration, vol. 309, no. 3–5, pp. 843–851, 2008.
- S. Hou, J. Wei, A. Zhang, T. C. Lim, and C. Zhang, “Study of dynamic model of helical/herringbone planetary gear system with friction excitation,” Journal of Computational and Nonlinear Dynamics, vol. 13, no. 12, p. 121007, 2018.
- L. Han, W. Niu, D. Zhang, and F. Wang, “An improved algorithm for calculating friction force and torque in involute helical gears,” Mathematical Problems in Engineering, vol. 2013, Article ID 575302, 13 pages, 2013.
- L. Wang, Z. Chen, Y. Shao, and X. Wang, “Dynamic features of gear system with tooth crack and tooth surface sliding due to speed variation,” in Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 1: 24th Conference on Mechanical Vibration and Noise, Parts A and B, pp. 171–175, Chicago, IL, USA, 2012.
- H. Jiang, Y. Shao, and C. K. Mechefske, “Dynamic characteristics of helical gears under sliding friction with spalling defect,” Engineering Failure Analysis, vol. 39, pp. 92–107, 2014.
- A. Guerine, A. El Hami, L. Walha, T. Fakhfakh, and M. Haddar, “A polynomial chaos method for the analysis of the dynamic behavior of uncertain gear friction system,” European Journal of Mechanics—A/Solids, vol. 59, pp. 76–84, 2016.
- F. Liu, H. Jiang, S. Liu, and X. Yu, “Dynamic behavior analysis of spur gears with constant & variable excitations considering sliding friction influence,” Journal of Mechanical Science and Technology, vol. 30, no. 12, pp. 5363–5370, 2016.
- A. Guerine, A. E. Hami, L. Walha, T. Fakhfakh, and M. Haddar, “Dynamic response of a spur gear system with uncertain friction coefficient,” Advances in Engineering Software, vol. 120, pp. 45–54, 2018.
- I. Howard, S. Jia, and J. Wang, “The dynamic modelling of a spur gear in mesh including friction and a crack,” Mechanical Systems and Signal Processing, vol. 15, no. 5, pp. 831–853, 2001.
- C. Liu, D. Qin, and Y. Liao, “Dynamic model of variable speed process for herringbone gears including friction calculated by variable friction coefficient,” Journal of Mechanical Design, vol. 136, no. 4, p. 41006, 2014.
- K. Huang, Y. Xiong, T. Wang, and Q. Chen, “Research on the dynamic response of high-contact-ratio spur gears influenced by surface roughness under EHL condition,” Applied Surface Science, vol. 392, pp. 8–18, 2017.
- ISO, Calculation of Scuffing Load Capacity of Cylindrical, Bevel and Hypoid Gears, ISO, Geneva, Switzerland, 2000.
- S. Kim and R. Singh, “Gear surface roughness induced noise prediction based on a linear time-varying model with sliding friction,” Journal of Vibration and Control, vol. 13, no. 7, pp. 1045–1063, 2007.
- K. Jayasankaran, Structure-borne Noise Model of a Spur Gear Pair with Surface Undulation and Sliding Friction as Excitations, The Ohio State University, Columbus, OH, USA, 2010.
- T. Eritenel and R. G. Parker, “An investigation of tooth mesh nonlinearity and partial contact loss in gear pairs using a lumped-parameter model,” Mechanism and Machine Theory, vol. 56, pp. 28–51, 2012.
- T. Eritenel and R. G. Parker, “Three-dimensional nonlinear vibration of gear pairs,” Journal of Sound and Vibration, vol. 331, no. 15, pp. 3628–3648, 2012.
- A. Palermo, D. Mundo, R. Hadjit, P. Mas, and W. Desmet, “Multibody modelling of shuttling excitation in spur and helical geared transmissions,” in Proceedings of ISMA 2012-USD 2012, pp. 4005–4016, Leuven, Belgium, 2012.
- A. Palermo, D. Mundo, R. Hadjit, and W. Desmet, “Multibody element for spur and helical gear meshing based on detailed three-dimensional contact calculations,” Mechanism and Machine Theory, vol. 62, pp. 13–30, 2013.
- R. Teja, T. R. Milind, R. C. Glover, and S. Sonawane, “Dynamic analysis of helical gear pair due to TE and shuttling moment excitations,” in Proceedings of the Noise and Vibration Conference and Exhibition, Warrendale, PA, USA, 2017.
- D. R. Houser, J. Harianto, and Y. Ueda, “Determining the source of gear whine noise,” Gear Solutions, pp. 17–22, 2004.
- S. Li, “Experimental investigation and FEM analysis of resonance frequency behavior of three-dimensional, thin-walled spur gears with a power-circulating test rig,” Mechanism and Machine Theory, vol. 43, no. 8, pp. 934–963, 2008.
- A. Toso, F. van Wermeskerken, N. Cappellini, and G. Heirman, On the Effect of Lightweight Gear Blank Topology on Transmission Dynamics, ASME, New York, NY, USA, 2015.
- S. Shweiki, A. Palermo, and D. Mundo, “A study on the dynamic behaviour of lightweight gears,” Shock and Vibration, vol. 2017, Article ID 7982170, 12 pages, 2017.
- S. Shweiki, A. Rezayat, T. Tamarozzi, and D. Mundo, “Transmission error and strain analysis of lightweight gears by using a hybrid FE-analytical gear contact model,” Mechanical Systems and Signal Processing, vol. 123, pp. 573–590, 2019.
- B. Guilbert, P. Velex, D. Dureisseix, and P. Cutuli, “A mortar-based mesh interface for hybrid finite-element/lumped-parameter gear dynamic models-applications to thin-rimmed geared systems,” Journal of Mechanical Design, vol. 138, no. 12, Article ID 123301, 2016.
- B. Guilbert, P. Velex, D. Dureisseix, and P. Cutuli, “Modular hybrid models to simulate the static and dynamic behaviour of high-speed thin-rimmed gears,” Journal of Sound and Vibration, vol. 438, pp. 353–380, 2019.
- L. Hou, Y. Lei, Y. Fu, and J. Hu, “Effects of lightweight gear blank on noise, vibration and harshness for electric drive system in electric vehicles,” Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-Body Dynamics, vol. 234, no. 3, pp. 447–464, 2020.
- T. G. Yılmaz, O. Doğan, and F. Karpat, “A comparative numerical study of forged bi-metal gears: bending strength and dynamic response,” Mechanism and Machine Theory, vol. 141, pp. 117–135, 2019.
- F. Karpat, T. G. Yılmaz, O. Doğan, and O. C. Kalay, “Stress and mesh stiffness evaluation of bimaterial spur gears,” in Proceedings of the ASME 2019 International Mechanical Engineering Congress and Exposition, Salt Lake City, UT, USA, 2019.
- J.-H. Bae, K.-C. Jung, S.-H. Yoo, S.-H. Chang, M. Kim, and T. Lim, “Design and fabrication of a metal-composite hybrid wheel with a friction damping layer for enhancement of ride comfort,” Composite Structures, vol. 133, pp. 576–584, 2015.
- A. Treviso, B. Van Genechten, D. Mundo, and M. Tournour, “Damping in composite materials: properties and models,” Composites Part B: Engineering, vol. 78, pp. 144–152, 2015.
- R. F. Handschuh, G. D. Roberts, R. R. Sinnamon, D. B. Stringer, B. D. Dykas, and L. W. Kohlman, “Hybrid gear preliminary results—application of composites to dynamic mechanical components,” Gear Technology, vol. 30, 2013.
- R. F. Handschuh, K. E. LaBerge, S. DeLuca, and R. Pelagalli, Vibration and Operational Characteristics of a Composite-Steel (Hybrid) Gear, NASA, Washington, DC, USA, 2014.
- K. E. Laberge, R. F. Handschuh, G. Roberts, and S. Thorp, “Performance investigation of a full-scale hybrid composite bull gear,” in Proceedings of the 72nd AHS 2016 Forum, West Palm Beach, FL, USA, 2016.
- P. G. Catera, D. Mundo, A. Treviso et al., “On the design and simulation of hybrid metal-composite gears,” Applied Composite Materials, vol. 26, no. 3, pp. 817–833, 2019.
- W. Xiao, Y. Huang, H. Jiang, and L. Jin, “Effect of powder material on vibration reduction of gear system in centrifugal field,” Powder Technology, vol. 294, pp. 146–158, 2016.
- W. Xiao, J. Li, S. Wang, and X. Fang, “Study on vibration suppression based on particle damping in centrifugal field of gear transmission,” Journal of Sound and Vibration, vol. 366, pp. 62–80, 2016.
- R. Ramadani, A. Belsak, M. Kegl, J. Predan, and S. Pehan, “Topology optimization based design of lightweight and low vibration gear bodies,” International Journal of Simulation Modelling, vol. 17, no. 1, pp. 92–104, 2018.
- S. Gauntt, R. Campbell, and S. Mcintyre, “Design optimization of a hybrid spur gear,” Vertical Flight Society Forum, vol. 75, 2019.
- N. Contartese, P. G. Catera, and D. Mundo, “Static mesh stiffness decomposition in hybrid metal-composite spur gears,” Advances in Mechanism and Machine Science, vol. 73, pp. 977–985, 2019.
- P. G. Catera, F. Gagliardi, D. Mundo, L. De Napoli, A. Matveeva, and L. Farkas, “Multi-scale modeling of triaxial braided composites for FE-based modal analysis of hybrid metal-composite gears,” Composite Structures, vol. 182, pp. 116–123, 2017.
- S. M. Gauntt and R. L. Campbell, “Characterization of a hybrid (steel-composite) gear with various composite materials and layups,” in Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 2019.
- P. G. Catera, D. Mundo, F. Gagliardi, and A. Treviso, “A comparative analysis of adhesive bonding and interference fitting as joining technologies for hybrid metal-composite gear manufacturing,” International Journal on Interactive Design and Manufacturing (IJIDeM), vol. 14, no. 2, pp. 535–550, 2020.
- K. E. LaBerge, J. P. Johnston, R. F. Handschuh, and G. D. Roberts, “Evaluation of a variable thickness hybrid composite bull gear,” in Proceedings of the AHS International 74th Annual Forum & Technology Display, Phoenix, AZ, USA, 2018.
Copyright © 2020 Menglei Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.