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The quasi-static behaviors of a double-helical gear pair is investigated both experimentally and theoretically with the main focus on the influence of the key design and manufacturing parameters associated with double-helical gears, including nominal right-to-left stagger angle, the stagger angle deviation (error) from the nominal stagger angle, and axial gear supporting conditions. On the experimental side, a double-helical gear test setup proposed earlier (Kang, M. R., and Kahraman, A., 2015, “An Experimental and Theoretical Study of Dynamic Behavior of Double-Helical Gear Sets,” J. Sound Vib., 350, pp. 11–29). for studying dynamics of the same system is employed that allows adjustable right-to-left stagger angles, intentional stagger errors, and axial support conditions. Specific measurement systems are developed and implemented simultaneously to measure the static motion transmission error and axial motions of the gears under low-speed conditions, as well as gear root strains to determine right-to-left load-sharing factors. A test matrix that covers wide ranges of stagger angles, intentional stagger error, and axial support conditions is executed within a range of torque transmitted to establish an extensive database. On the modeling side, the measured quasi-static behavior of double-helical gear pairs is simulated by using an existing quasi-static double-helical load distribution model (Thomas, J., and Houser, D. R., 1992, “A Procedure for Predicting the Load Distribution and Transmission Error Characteristics of Double Helical Gears,” World Congress-Gear and Power Transmission, The 3rd World Congress—Gear and Power Transmission, Paris.). Direct comparison of the measurements and predictions of loaded static transmission error, axial play, root stresses, and right-to-left load-sharing factors are used to validate the quasi-static model as well as describing the measured behavior.

High-cycle gear tooth bending fatigue lives of spur gears under fully reversed and fully released loading conditions are compared in this experimental study. The experimental methodology described in an earlier publication, (Hong et al. 2020, “A Rotating Gear Test Methodology for Evaluation of High-Cycle Tooth Bending Fatigue Lives Under Fully Reversed and Fully Released Loading Conditions,” Int. J. Fatigue, 133, p. 105432. 10.1016/j.ijfatigue.2019.105432), is employed to perform two sets of rotating, gear tooth bending fatigue tests. Statistical analyses are performed to regress stress versus life (S–N) curves under both loading conditions. These curves indicate that a gear under fully reversed loads has a shorter bending life at the same maximum tooth root stress as a gear under fully released loads. Various planetary gear set kinematic conditions with different stationary members are considered to determine the equivalent number of tooth loading cycles per revolution of the sun gear. They are combined with established S–N curves under both loading conditions to determine the ratios of allowable maximum tooth root stresses amongst the gear components of a P-planet gear set such that each gear in the set has the same bending fatigue life. A “stress-balanced” gear set designed to these stress ratios is expected to have the same bending fatigue life for its sun, ring, and planet gears, ensuring that the planetary gear set life is the longest.

In order to accurately predict ring gear deformations and to investigate the effects of ring gear flexibility on quasi-static behaviors of planetary gear sets, a complete load distribution model of planetary gear sets having flexible ring gears will be formulated here based on the baseline model proposed by the same authors (Hu, Y., Talbot, D., and Kahraman, A., 2018, “A Load Distribution Model for Planetary Gear Sets,” ASME J. Mech. Des., 140(5), p. 053302). Direct comparisons to published experiments are provided to assess the accuracy of the proposed load distribution methodology. Example analyses with flexible ring gear rims are performed indicating that ring gear flexibility could influence gear mesh-level and planetary gear set system-level behaviors. Influence of spline supporting a ring gear is also investigated revealing that positions of planet branches with respect to external splines could influence ring deflections and resultant gear mesh load distributions.

Planet-to-planet load sharing is a major design and manufacturing tolerancing issue in planetary gear sets. Planetary gear sets are advantageous over their countershaft alternatives in many aspects, provided that each planet branch carries a reasonable, preferably equal, share of the torque transmitted. In practice, the load shared among the planets is typically not equal due to the presence of various manufacturing errors. This study aims at enhancing the models for planet load sharing through a three-dimensional (3D) formulation of N planet helical planetary gear sets. Apart from previous models, the proposed model employs a gear mesh load distribution model to capture load and time dependency of the gear meshes iteratively. It includes all the three types of manufacturing errors, namely, constant errors such as carrier pinhole position errors and pinhole diameter errors, constant but assembly dependent errors such as nominal planet tooth thickness errors, planet bore diameter errors, and rotation, and assembly dependent errors such as gear eccentricities and run-outs. At the end, the model is used to show combined influence of these errors on planet load sharing to aid designers on how to account for manufacturing tolerances in the design of the gears of a planetary gear set.

This paper investigates the influences of tooth indexing errors on dynamic factors of spur gears. An experimental study is performed using root strain measurements to (i) establish baseline dynamic behavior of gears having negligible indexing errors and (ii) characterize changes caused by tightly controlled intentional indexing errors to this baseline dynamic behavior. For this, test gears having different forms of indexing errors are paired with an instrumented gear having negligible indexing error. Dynamic root strains of teeth in the neighborhood of teeth with indexing error are measured. A dynamic gear load distribution model is employed to simulate these experiments. Both measurements and predictions indicate clearly that the baseline dynamic response, dominated by well-defined resonance peaks, is altered significantly by transient vibrations induced by indexing errors, in the process increasing dynamic factors significantly in comparison to the case of negligible indexing error.

In this paper, influences of tooth indexing errors on load distribution and tooth load sharing of spline joints are investigated by modifying an existing semi-analytical load distribution model for side-fit involute splines. Two commonly observed loading conditions, namely (i) combined torsion and radial loads representative of a spline joint of a spur gear with shaft and (ii) combined torsion, radial loads, and tilting moment representative of a spline joint of a helical gear with shaft are considered in this study. Numerical results of an example spline having (i) no tooth indexing error, (ii) a single tooth with indexing error, and (iii) a random sequence of tooth indexing errors under these two loading conditions are presented to demonstrate the effects of tooth indexing errors. In addition, a practical study of the robustness to manufacturing tolerances is also presented where probability distributions of load sharing factor of the critical tooth of an example spline designed to certain manufacturing tolerance classes are obtained with a large number of randomly generated indexing error sequences.

A computational model is proposed for the prediction of friction-related mechanical efficiency losses of parallel-axis gear pairs. The model incorporates a gear load distribution model, a friction model, and a mechanical efficiency formulation to predict the instantaneous mechanical efficiency of a gear pair under typical operating, surface, and lubrication conditions. The friction model uses a new friction coefficient formula obtained by using a validated non-Newtonian thermal elastohydrodynamic lubrication (EHL) model in conjunction with a multiple linear regression analysis. The load and friction coefficient distribution predictions are used to compute instantaneous torque/power losses and the mechanical efficiency of a gear pair at any given rotational position. Efficiency measurements from gear pairs having various gear designs and surface treatments are compared to model predictions. Mechanical efficiency predictions are shown to be within 0.1% of the measured values, indicating that the proposed efficiency model is accurate. Results of a parametric study are presented at the end to highlight the influence of key basic gear geometric parameters, tooth modifications, operating conditions, surface finish, and lubricant properties on mechanical efficiency losses.

In this study, two different dynamic models, a finite-element-based deformable-body model and a simplified discrete model, are developed to predict dynamic behavior of spur gear pairs. Dynamic transmission error (DTE) and dynamic factors (DF) defined based on the gear mesh loads, tooth loads and bending stresses are computed for a number of unmodified and modified spur gears within a wide range of rotational speed for different involute contact ratios and torque values. Although similar models were proposed in the past, they were neither fully validated nor equipped to predict both DTE and different forms of DF. Accordingly, this study focuses on (i) validation of both models through an extensive set of experimental data obtained from a set of tests using spur gear having unmodified and modified tooth profiles, and (ii) establishment of a direct link between DTE and different forms of DF, especially the ones based on tooth forces and the root stresses. The predicted DF and DTE values are related to each other through simplified formulas. Impact of nonlinear behavior, such as tooth separations and jump discontinuities on DF, is also quantified.

A generalized formulation for analyzing speeds and forces of the gear components of planetary automatic transmissions is proposed. The formulation is capable of analyzing any typical one-degree-of-freedom automatic transmission gear train containing any number of simple, compound or complex-compound planetary gear sets. The formulation consists of three components: a kinematic analysis formulation to compute rotational speeds of gears and carriers, a gear ratio and kinematic configuration search algorithm, and a power flow analysis formulation. The kinematics component computes rotational speeds of gears and carriers. Given the type and number of planetary gear sets, the search algorithm determines all possible kinematic configurations and gear tooth count combinations that result in a required set of gear ratios while eliminating all kinematic redundancies and unfavorable clutching sequences. The third component, the power-flow analysis formulation, performs a complete static force (power-flow) analysis to determine all gear and bearing forces and moments, and clutch and connection torque values. A five-speed transmission example is considered to highlight the capabilities of the proposed formulation.

In this study, a surface wear prediction model for helical gears pairs is employed to investigate the influence of tooth profile deviations in the form of intentional tooth profile modifications or manufacturing errors on gear tooth surface wear. The wear model combines a finite-element-based gear contact mechanics model that predicts contact pressures, a sliding distance computation algorithm, and Archard’s wear formulation to predict wear of the contacting tooth surfaces. Typical helical gear tooth modifications are parameterized by an involute crown, a lead crown, and an involute slope. The influence of these parameters on surface wear are studied within typical tolerance ranges achievable using hob/shave process. The results indicate that wear is related to the combined modification parameters of a gear pair rather than individual gear parameters. At the end, a design formula is proposed that relates the mismatch of contacting surface slopes to the maximum initial wear rate.

A nonlinear finite element contact mechanics model of a helical gear pair was used to study the effect of intentional tooth profile modifications on durability of helical gear pairs. Both two-dimensional (2D) and three-dimensional (3D) modifications were considered. A detailed parametric study was performed to quantify the changes in the contact and bending stresses as a function of tooth profile modification parameters as compared to an unmodified gear pair baseline. The combined influence of modification parameters and torque transmitted on the maximum stresses is described. Results indicate that both bending and contact stresses of a helical gear pair are increased significantly when 2D modifications are applied, potentially causing an underestimation of the actual stress values when the modifications are not included in stress calculations. When properly selected, 3D modifications cause significantly less stress increases.

In this study, experimental and theoretical investigations of the effect of tooth spacing errors on the root stresses of spur gear pairs are performed. A test setup with instrumentation for the measurement of root stresses has been developed. A number of experiments were performed with gears having deterministic (at one or two teeth only) and random spacing errors (all teeth having a random distribution of errors as in a typical production gear). These experiments were simulated using a deformable-body gear contact model to describe the empirical trends physically. A methodology is proposed to relate increases in root stresses to the spacing error magnitudes directly. Closed-form expressions resulting from this methodology permit the determination of the stress amplification factors due to a certain range of spacing error tolerances as well as quantifying how much spacing error can be tolerated within a user defined stress limit.

Design of planetary gear sets is more involved than the design of their counter-shaft counterparts as it involves simultaneous design of a set of internal and external gear meshes while complying with a large number of systems and gear mesh related requirements for assembly, durability, noise, and efficiency. A manual iterative design process often results in suboptimal designs that fail to meet all these requirements simultaneously. In this paper, a methodology for an automated design search of single and double-planet planetary gear sets is proposed. With the input of a number of system-level constraints associated with the spacing and phasing of the planets, and acceptable ranges of basic geometric design parameters, this methodology defines a large design space in that a large number of geometric design concepts are identified and checked for any interferences. The external and internal meshes of these concepts are evaluated by using computationally efficient loaded gear tooth contact analysis model to predict their performance metrics such as transmission error amplitudes and contact and root stresses. They are then rank-ordered based on their performance metrics to identify balanced planetary gear set designs meeting all requirements equally well. At the end, results of an example design search were presented to demonstrate the effectiveness of the proposed methodology in defining a balanced solution that is acceptable in terms of all of its requirements.

Effect of flexibility of an internal gear on the quasi-static behavior of a planetary gear set is investigated. A state-of-the-art finite elements/semi-analytical nonlinear contact mechanics formulation is employed to model a typical automotive automatic transmission planetary unit. The model considers each gear as deformable bodies and meshes them to predict loads, stresses and deformations of the gears. Actual support and spline conditions are included in the model. The rim thickness of the internal gear is varied relative to the tooth height and gear deflections and bending stresses are quantified as a function of rim thickness. Influence of rim thickness on the load sharing amongst the planets is also investigated with and without floating sun gear condition. The results are discussed in detail and guidelines regarding the design of a planetary internal gear are presented.

The influence of involute contact ratio on the torsional vibration behavior of a spur gear pair is investigated experimentally by measuring the dynamic transmission error of several gear pairs using a specially designed gear test rig. Measured forced response curves are presented, and harmonic amplitudes of dynamic transmission error are compared above and below gear mesh resonances for both unmodified and modified gears having various involute contact ratio values. The influence of involute contact ratio on dynamic transmission error is quantified and a set of generalized, experimentally validated design guidelines for the proper selection of involute contact ratio to achieve quite gear systems is presented. A simplified analytical model is also proposed which accurately describes the effects of involute contact ratio on dynamic transmission error.

A modeling methodology is proposed to predict load-dependent (mechanical) power loss of cylindrical roller bearings under combined radial and moment loading with focus on planetary gear set planet bearings. This methodology relies on two models. The first model is a bearing load distribution model to predict load intensities along rolling element contacts due to combined force–moment loading. This model takes into account planet bearing macrogeometry as well as micromodifications to the roller and race surfaces. The second model is an elastohydrodynamic lubrication (EHL) model employed to predict rolling power losses of bearing contacts with load intensities predicted by the load distribution model. The bearing mechanical power loss methodology is applied to bearings of an automotive planetary gear set to quantify the sensitivity of mechanical power loss to key bearing, lubrication and surface parameters as well as operating speed, load and temperature conditions.

A new method of measuring planet load sharing of planetary gear sets is proposed in this paper. The method uses strain gauges mounted directly on the planet pins to measure continuously the loads carried by the planets assembled in a fixed carrier. Example 4-planet gear sets of different planet phasing conditions are procured and tested with a family of planet carriers having various levels and combinations of planet pinhole position errors to demonstrate the measurement method. As the radial floating capability of the central members is critical to planet-to-planet load sharing, a companion proximity-based measurement system is also implemented to measure the radial motions of the floating sun gear under various planet phasing and carrier pinhole error conditions. It is shown that both measurement methods are effective in characterizing the loads carried by planets.Copyright © 2011 by ASME

This paper presents a model to predict the crack formation fatigue lives of spur gear contacts operating under mixed lubrication conditions where surface roughnesses introduce intermittent metal-to-metal contacts and severe stress concentrations. The proposed model consists of several submodels, including (i) a gear load distribution model to determine the normal tooth force distribution along the tooth surface, incorporating any profile modifications and manufacturing deviations, (ii) a mixed elastohydrodynamic lubrication model customized to handle transient contact conditions of gears, (iii) a stress formulation that assumes the plane strain condition to compute the transient elastic stress fields on and below the tooth surface induced by the mixed lubrication surface pressure and shear stress distributions, and (iv) a multi-axial fatigue model to predict the crack nucleation life distribution. The proposed spur gear fatigue model is used to simulate the contacts of gear pairs having different surface roughness amplitudes. The predictions are compared to the measured gear fatigue stress-life data for each surface condition to assess the model accuracy in the prediction of the crack nucleation fatigue lives as well as the location of the critical failure sites.

A model to simulate the dynamic behavior of a single-stage planetary gear train with helical gears is developed. The three-dimensional dynamic model includes all six rigid body motions of the gears and the carrier. The generic nature of the formulation allows the analysis of a planetary gear set with any number of planets. Planets can be arbitrarily spaced (equally or unequally) around the sun gear. The model is also capable of handling different planet meshing conditions which are functions of number of gear teeth and planet positions. The linear time-invariant equations of motion are solved to obtain the natural modes and the forced vibration response due to static transmission errors. The proposed model is employed to describe the effects of the planet mesh phasing conditions on the dynamic behavior of a four-planet system.

Microgeometry optimization has become an important phase of gear design that can remarkably enhance gear performance. For spiral bevel and hypoid gears, microgeometry is typically represented by ease-off topography. The optimal ease-off shape can be defined as the outcome of a process where generally conflicting objective functions are simultaneously minimized (or maximized), in the presence of constraints. This matter naturally lends itself to be framed as a multi-objective optimization problem. This paper proposes a general algorithmic framework for ease-off multi-objective optimization, with special attention given to computational efficiency. Its implementation is fully detailed. A simulation model for loaded tooth contact analysis is assumed to be available. The proposed method is demonstrated on a face-hobbed hypoid gear set. Three objectives are defined: maximization of gear mesh mechanical efficiency, minimization of loaded transmission error, minimization of maximum contact pressure. Bound constraints on the design variables are imposed, as well as a nonlinear constraint aimed at keeping the loaded contact pattern inside a predefined allowable contact region. The results show that the proposed method can obtain optimal ease-off topographies that significantly improve the basic design performances. It is also evident that the method is general enough to handle geometry optimization of any gear type.

Determining the locations and orientation of potential instantaneous contact lines between two contacting tooth surfaces of gears is one major step in the prediction of unloaded and loaded gear tooth contacts. This study defines and utilizes new general concept based on the construction of a surface of roll angle to simplify the task of locating instantaneous contact lines of any type of gearing. The position and normal vectors of points on one of the mating surfaces and axes of both gears are used in conjunction with the equation of meshing to define the surface of action and the surface of roll angle. With the surface of roll angle defined, instantaneous contact lines are determined by a novel approach based on an analogy to parallel-axes gears. At the end, the proposed method is applied to various parallel- and cross-axis gear pairs to demonstrate its effectiveness.

Manufacturing errors typically cause real (measured) spiral bevel and hypoid gear surfaces to deviate from the theoretical ones globally. Tooth surface wear patterns accumulated through the life span of the gear set are typically local deviations that are aggravated especially in case of edge contact conditions. An accurate and practical methodology based on ease-off topography is proposed in this study to perform loaded tooth contact analysis of spiral bevel and hypoid gears having both types of local and global deviations. It starts with definition of the theoretical pinion and gear tooth surfaces from the machine settings and cutter parameters, and constructs the theoretical ease-off and roll angle surfaces to compute unloaded contact analysis. Manufacturing errors and localized surface wear deviations are considered to update the theoretical ease-off to form a new ease-off surface that is used to perform a loaded tooth contact analysis according to the semi-analytical method proposed earlier. At the end, a numerical example with locally deviated surfaces is analyzed to demonstrate the effectiveness of the proposed methodology as well as quantifying the effect of such deviations on load distribution and the loaded motion transmission error.Copyright © 2009 by ASME

A simplified discrete model to predict load sharing among the planets of a planetary gear set having carrier planet position errors is presented in this study. The model proposes a translational representation of the torsional system and includes any number of planets positioned at any spacing configuration. The discrete model predictions are validated by comparing them to (i) the predictions of a deformable-body planetary gear set model and (ii) planet load sharing measurements from planetary gear sets having three to six planets. A set of closed-form planet load sharing formulas are derived from the discrete model for gear sets having equally-spaced planets for conditions when all of the planets are loaded. These formulas allow, in an accurate and direct way, calculation of planet loads as a function of position errors associated with each planet.

The influence of gear tooth flank modifications in the form of linear involute tip relief on the torsional vibration behavior of a spur gear pair is investigated by using an experimental test stand. Measured dynamic transmission error (DTE) values are compared and a family of forced response curves is presented. Guidelines for the design of quiet spur gear sets are also given.

This paper presents results of a comprehensive experimental and theoretical study to determine the influence of certain key factors in planetary transmissions on gear stresses and planetary load sharing. A series of tests are conducted on a family of planetary gear sets, and strains are recorded at various locations on the outer diameter and gear tooth fillet of the ring gear. Pinion position errors are introduced as a representative key manufacturing tolerance, and the resultant changes in the planetary behavior are observed. The experimental data is compared to the predictions of a state-of-the-art multi-body contact analysis model — ‘Gear System Analysis Modules’ (GSAM). This model is capable of including the influences of a number of system-level variables and quantifying their impact on gear strains. The model predictions are shown to compare well with the measured strain at the ring gear outer diameter and tooth fillet. GSAM predictions of planet load sharing are then used to quantify the influence of pin hole position errors on the 3, 4, 5, and 6 planet test gear sets. These predictions also agree well with the planet load sharing experiments presented in a companion paper [20].Copyright © 2007 by ASME and General Motors

In this study, the results of an experimental parametric study of the combined influence of shaft misalignments and gear lead modifications on the load distribution and tooth bending stresses of helical gear pairs are presented. A set of helical gear pairs having various amounts of total lead crown was operated under loaded, low-speed conditions with varying amounts of tightly controlled shaft misalignments. Gear teeth were instrumented through strips of strain gages along the face width of gears at the tooth fillet region at a roll angle that is near the start of the active profile. Variations of root strains along the face width were quantified for different levels of shaft misalignments and gear lead crown. The results presented demonstrate the direct link between the lead crown and gear misalignments as well as the effectiveness of the lead crown in preventing edge loading conditions due to misalignment. The results presented here form a database that should be available for a validation of gear contact models in terms of their ability to simulate misalignments.

In this paper, results of an experimental study are presented to describe the impact of certain types of manufacturing errors on gear stresses and the individual planet loads of an n-planet planetary gear set (n=3–6). The experimental setup includes a specialized test apparatus to operate a planetary gear set under typical speed and load conditions and gear sets having tightly controlled intentional manufacturing errors. The instrumentation system consists of multiple strain gauges mounted on the ring gear and a multichannel data collection and analysis system. A method for computing the planet load-sharing factors from root strain-time histories is proposed. Influence of carrier pinhole position errors on gear root stresses is quantified for various error and torque values applied to gear sets having three to six planets. The results clearly indicate that manufacturing errors influence gear stresses and planet load sharing significantly. Gear sets having larger number of planets are more sensitive to manufacturing errors in terms of planet load-sharing behavior.