Loaded Transmission Error Research Articles

Research on Multiobjective Optimization Design of Meshing Performance and Dynamic Characteristics of Herringbone Gear Pair Under 3D Modification

Abstract Herringbone gear transmission system is widely used in high-speed overloaded fields such as ships. Tooth deformation and installation error, which can cause meshing impact, load mutation, and load uneven, will seriously affect dynamic performance of the whole transmission system. Modification technology is the most effective way to achieve damping and noise reduction. In this article, we propose a three-dimensional (3D) modified tooth surface by grinding wheel along axial to the spiral rise and along radial to the parabolic movement based on the grinding principle and a multiobjective ant lion optimizer model for optimizing meshing performance and dynamic characteristics of herringbone gear transmission system and analyze the impact of optimized modifications of tooth profile, axial, and three dimension on meshing performance, loaded transmission error, load distribution coefficient and meshing impact, meshing stiffness, and vibration acceleration by the example. The results show that the 3D modification of optimization can not only eliminate the contact between the tooth side edge and tooth top edge but also eliminate the influence of installation error on contact performance. The root mean square values of the relative comprehensive vibration acceleration of tooth profile modification, axial modification, and 3D modification are reduced by 30.11%, 49.24%, and 61.41% compared with the standard, respectively. The 3D modification can greatly reduce tooth vibration, reduce resonance peak, and achieve the goal of noise reduction.

Optimization design for dynamic characteristics of face gear drive with surface-active modification

The surface-active modification(SAM) method is proposed to decrease the vibration of the face gear drive. Firstly, the unloading performances such as the fourth-order transmission error(4-TE), the length of the contact ellipse, and the direction of the contact path are preset to modify the face gear pair. Secondly, a dynamic model is developed to attain the normal vibration acceleration(NVA) which is chosen as the optimization objective. The genetic algorithm(GA) is applied to solve the optimization model for searching for the optimal modification parameters. Thirdly, a numerical calculation is performed to investigate the influence of the modification parameters and the meshing stiffness on vibration. Besides, the bifurcation characteristics of the system are also explored. The results show that the amplitude of the loaded transmission error(ALTE) and the vibration are reduced after the optimization. Not only is vibration linked to mean meshing stiffness, but also the harmonic amplitude of meshing stiffness impacts it significantly. SAM lessens the range of chaotic motion, which helps to improve system stability. This study provides a theoretical basis for the vibration reduction of face gear pairs design.

Computerized design, simulation of meshing and stress analysis of pure rolling internal helical gear drives with combined tooth profiles

The computerized design of non-generated pure rolling internal helical gear drives with combined transverse tooth profiles based on the active design of the meshing line function is presented. The application of different combinations of curves, namely circular arcs and involutes, to form the gear active tooth profiles is discussed. The parametric equations for tooth surfaces and the basic design parameters and equations of geometric sizing are proposed. The meshing performance and mechanical behavior are studied in terms of the contact patterns, maximum contact and bending stresses and peak-to-peak level of loaded transmission errors compared with a design of internal helical gear drive with corrected fillet. The combination of a circular arc and an involute connected at the pitch point to form the active tooth profile for both the pinion and the internal gear shows reduced maximum contact and bending stresses and lower peak-to-peak level of the loaded function of transmission errors. The proposed design of combined tooth profiles will contribute to lower the noise and vibration of the proposed gear drives, increase the endurance and life of the gear drive, and lower the heat generation due to pure rolling conditions of the achieved bearing contact.

Computerized Design, Simulation of Meshing and Stress Analysis of External Helical Gear Drives Based on Critical Control Points

Helical gears are widely used in powertrain systems. The computerized design of a new type of non-generated external helical gears based on critical control points at the transverse tooth profile is presented. The entire tooth profile is divided into different parts including the active tooth profile and fillet by control points. Involutes, circular arcs and Hermite curves are defined between two critical control points and smoothly connected with each other at those control points. The parametric equations for the tooth surfaces are derived considering the position of the mentioned control points. The basic design parameters and equations of the geometric sizing are proposed. The contact patterns, variation of the maximum stresses and peak-to-peak level of loaded transmission errors for six cases of design of the proposed new geometry of helical gear drives are studied with two cases of traditional helical gear drives as a reference, including gears with and without micro-geometry modifications. One case of an external helical gear drive designed with a combination of a circular arc and an involute to form the active tooth profile for both the pinion and the gear shows a much lower maximum bending stress and similar lower peak-to-peak level of loaded transmission errors with respect to the other cases of design. The proposed design method of tooth profiles based on critical control points lays the foundation for the topological optimization of helical gear drives.

A novel calculation method of long period pinion axial displacement and meshing impact force for double helical gear considering asymmetry error

The cumulative pitch errors of the left and right helical gears are the main source of the asymmetry error for the double helical gear. In this paper, a new three-dimensional loaded tooth contact analysis (3-DLTCA) model of double helical gear considering long period asymmetry error is proposed. The desired error (modification), undesired error (cumulative pitch error) and the change of normal clearance increment of left and right helical gear pairs caused by the axial shift of the driving pinion are considered in the model. Based on the new 3-DLTCA model, combined with the equivalent gap of the current meshing tooth pair, the meshing impact force (MIF) of the current meshing tooth pair in each meshing period is calculated. The results show that the proposed model can quickly and accurately calculate the long period end loaded transmission error (ELTE) and pinion axial displacement (PAD) of double helical gear with asymmetry error. Moreover, the long period asymmetry error of double helical gear has great effect on the MIF for left and right helical gears.

Design and Analysis for Hypoid Gears with Ease-Off Flank Modification

An approach of ease-off flank modification for hypoid gears was proposed to improve the meshing performance of automobile drive axle. Firstly, a conjugate pinion matching with gear globally was developed based on gear meshing theory. Secondly, a modified pinion was represented by a sum of two vector functions determining the conjugate pinion and the normal ease-off deviations expressed by both predesigned transmission error function and tooth profile modification curves to change the initial contact clearance of the tooth. Thirdly, the best ease-off deviations were determined by optimizing the minimum amplitude of loaded transmission error (ALTE) based on tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA). Finally, the results show that effective contact ratios (εe) are established by clearances both teeth space and of contact elliptical, and greatly affect ALTE. The εe is a variable value with increasing loads for the tooth with modification. ALTE decreases with increasing εe. After εe reaches the maximum, ALTE increases with increasing loads. The mismatch of the best ease-off tooth is minimal, which contributes to effective reduction in ALTE, thus significantly improving drive performance.

Optimization of Topological Modification Parameters for Heavy-Duty Helical Gears

To improve the meshing state of gear pairs, the abrasion of tooth surfaces must be reduced and on the other hand, the functional requirement of transmission stability for different conditions of heavy-duty helical gears must be increased. To do this requires a topological modification of the tooth profile and tooth alignment. The topological modification parameter optimization model is established based on Hertz theory, which takes the functional requirement as a goal, and the tooth surface topological modification parameters as the design variables. The constraint condition equations in the tooth contact analysis process are derived, and the improved genetic algorithm is used to solve the numerical conditions of the tooth surface microgeometry parameter optimization model. Then, the modification parameters optimized by different modification schemes are obtained. The loaded transmission errors, the mesh misalignment movements, and the distribution characteristics of the tooth surface contact stress are studied. The parameter optimization model of the tooth surface topological modification for the functional requirements is established. With different modification parameters heavy-duty helical gear pairs under various loads, the research results provide a theoretical basis for tooth surface micro design of heavy-duty helical gears in practical application and in determining the reasonable modification parameters.

Experimental Evaluation of Semi-Aromatic Polyamides as a Gear Material

Remarkable improvements in load capacity and transmission error have recently expanded the application fields of plastic gears. POM is one of the most suitable materials for them but has a limit on working temperature. Although PA46 has good mechanical properties even under relatively high temperatures, it also has high water absorbency. The water absorption changes the micro geometries of teeth and decreases the strength, which limits the application fields of PA46. The present study describes a semi-aromatic polyamide as a novel candidate for gear materials. It also has superior mechanical properties under high temperatures but shows little water absorption. Running tests following JIS B 1759, which specifies the calculation method of cylindrical-gear load capacity, examined gears of various semi-aromatic polyamides, which include modifiers and fortifiers. The results showed that a semi-aromatic polyamide could be suitable for gear material, including a proper modifier and fortifier. Further running tests will define applications in which semi-aromatic polyamides have more advantages than other plastics.

ASPECTS REGARDING OPTIMAL DESIGN FOR THE WORM-GEAR DRIVE

It is known that, from the point of view of the accuracy of a machine-tool, at its design, the dynamic behaviour of each element of the kinematic chains prevails. Worm-gear drives are widely used in the different machine-tools and robots. Therefore, it is important that during meshing, as far as possible, there are no vibrations, shocks, power losses, noise and low durability. These requirements can be met if, for example, the gear ratio is constant during meshing, without transmission errors, which means that the worm-gear drive should have a high accuracy. The accuracy improvement of the worm-gear drive has long been a focus of attention for machine-tools designers. Thus, this paper presents various approaches to solving such problems, based on modelling and simulation, such as: estimating the load share of worm-gear drives and to calculate the instantaneous tooth meshing stiffness and loaded transmission errors; the desired worm-gear drive design configuration by altering the optimum set of worm-gear drive design parameters which are suitable for the required performance by associating it with SVM (Support Vector Machine); optimization approach for design of worm-gear drive based on Genetic Algorithm; design optimization of worm-gear drive with reduced power loss; etc. The optimization of the worm-gear design is an important problem for the research because the design variables are correlated to each other. An optimal design algorithm developed by the authors of this paper, for worm-gear drive, is also presented.

Digital twin modeling for tooth surface grinding considering low-risk transmission performance of non-orthogonal aviation spiral bevel gears

Low-risk transmission performance including elastic deformation, loaded contact pattern, load distribution and loaded transmission error is of paramount significance to the actual manufacturing for non-orthogonal aviation spiral bevel gears. The advanced digital twin technology is introduced into tooth flank grinding. A new digital twin modeling considering low-risk transmission performances is proposed. In the modeling, low-risk transmission performance driven simulation, sensitivity analysis and robust control are developed, respectively Firstly, data-driven tooth surface modeling is developed by simulating free-form tooth surface grinding including gear tilt method and pinion double helical method. With local geometric boundary setup meshing stiffness is determined by using local Rayleigh–Ritz solution. Then, to deal with the sensitivity of gear assembly, an improved tooth contact analysis (TCA) is developed. Moreover, numerical loaded tooth contact analysis (NLTCA) is performed to build a bridge between of low-risk performances and hypoid generator parameters. The low-risk transmission performance driven control model is established by using hypoid generator parameters modification. Finally, sensitivity analysis strategy-based robust control model is solved by using Levenberg–Marquardt method for accurate hypoid generator parameters having modification amount. The provided numerical instance can verify the proposed method.

Adaptive data-driven modeling, prediction and optimal control for loaded transmission error of helicopter zerol spiral bevel gear transmission system

In complex multi-stage transmission system especially for the helicopter zerol spiral bevel gears, the loaded transmission error (LTE) is a primary source of noise and vibration, and the misalignment is always an important noise factor affecting geometric and physical performances. An innovative adaptive data-driven LTE modeling, prediction and optimal control are proposed by modifying misalignment evaluations. Firstly, misalignment evaluations on helicopter transmission system are used as the data-driven design variables for tooth contact analysis with error (eTCA). Then, numerical loaded tooth contact analysis (NLTCA) considering flank elastic contact deformation, loaded contact pattern and load distribution is proposed for data-driven LTE modeling. To distinguish with the conventional minimizing LTE, an adaptive data-driven prediction and control system is provided. By prescribing the design target relating to the actual requirements, an adaptive data-driven functional relationship with respect to misalignment evaluations is established. For the input signal namely initial misalignment and its predicted LTE, by predicting its gap from the prescribed design target, this result is used to establish data-driven control objective function. Finally, after giving misalignment optimal operation, adaptive control strategy and solution are proposed to get a robust control result. The numerical instance is provided to verify the proposed methodology.

Investigation of Load Sharing and Dynamic Load Characteristics of a Split Torque Transmission System with Double-Helical Gear Modification

A new dynamic model for a two-input two-path split torque transmission system which considers meshing error, time-varying meshing stiffness, and meshing-in impact is proposed. Time-varying meshing stiffness and meshing-in impact of each gear pair are accurately calculated based on tooth contact analysis and loaded tooth contact analysis. Equivalent displacements of eccentricity error and installation error along the meshing line of second- and third-stages gears are derived. The modified tooth surface of a third-stage double-helical gear is obtained by optimizing the amplitude of static loaded transmission error and meshing-in impact via nondominated sorting genetic algorithm-II (NSGA-II). Influence of modification on load sharing and dynamic load characteristics of split torque transmission system is investigated. The results indicate that the system’s dynamic meshing force increases when meshing-in impact is accounted for, which is unfavorable for the transmission. Following the modification of a double-helical gear, the dynamic load characteristics of the split torque transmission system are significantly improved, while its load sharing characteristics are improved to a certain extent.

Analytical model for meshing stiffness, load sharing, and transmission error for spur gears with profile modification under non-nominal load conditions

The loaded tooth deflections, manufacturing deviations, and assembly errors on spur gears induce a delay on the driven gear respect to the driving gear, which is called loaded transmission error. This delay causes an earlier start of contact outside the pressure line, resulting in a mesh-in impact, which is an important source of noise and vibrations. To avoid or mitigate this mesh-in impact, profile modifications on the driven tooth tip are frequently used. The additional gap due to the profile relief delays the effective start of contact, and a suitable amount of relief sets the mesh-in point back to the theoretical inner point of contact, which prevents the mesh-in impact and ensures a smooth contact. However, as the loaded transmission error depends on the transmitted load, a specific amount of relief will be suitable only for a specific input torque. For higher transmitted load mesh-in impact will occur again, while for lower transmitted load the effective contact ratio will be reduced due to an excessive delay of the start of contact. Therefore, under variable load conditions it is not possible to adjust the start of contact to the theoretical inner point of contact, so mesh-in impact, smooth contact and reduction of contact ratio may alternatively occur. In this paper, a simple analytical model for the meshing stiffness, load sharing ratio and transmission error is presented for spur gears with profile modification under non nominal -or variable- load conditions.

An innovative ease-off flank modification method based on the dynamic performance for high-speed spiral bevel gear with high-contact-ratio

To reduce the running vibration of spiral bevel gear, an innovative ease-off flank modification method based on the dynamic performance for spiral bevel gear to minimize loaded transmission error (main vibration excitation of gear transmission in low speed range) and meshing impact (main vibration excitation of gear transmission in high speed range) is proposed. First, an innovative wave flank modification method related to high contact ratio, where the concave transmission error is used to replace the parabola transmission error, is proposed to minimize the loaded transmission error of high-contact-ratio spiral bevel gear. Secondly, meshing impact is calculated by considering the base pitch difference generated from loaded deformation. Finally, based on the advanced ease-off technology and wave flank modification method, an optimization model is established to reduce the loaded transmission error and meshing impact of high-speed spiral bevel gear with high-contact-ratio. The numerical results show that compared with optimized second-order spiral bevel gear transmission, the innovative ease-off flank modification method can further reduce the loaded transmission error and meshing impact of high-speed spiral bevel gear with high-contact-ratio, which greatly improves the dynamic performance of high-speed spiral bevel gear in the entire speed range.

Design and dynamic performance analysis of high-contact-ratio spiral bevel gear based on the higher-order tooth surface modification

To reduce the running vibration of high-contact-ratio (HCR) spiral bevel gear transmissions, an innovative higher-order tooth surface modification (HTSM) method is proposed to reduce the vibration excitation of gear transmission. First, according to the meshing characteristic of HCR spiral bevel gear transmissions, an innovative modification curve is proposed. Second, the auxiliary tooth surface modification (ATSM) method is used to generate the higher-order modified spiral bevel gear. Third, the loaded transmission error (LTE) and meshing impact (MI) of higher-order modified spiral bevel gear transmissions are calculated based on the LTCA method and the impact theory while considering tooth deformation. Last but not least, based on the ease-off technology, an optimization model is built to reduce the LTE and MI of HCR spiral bevel gear transmissions. Through simulation analysis, it is found that compared with the second-order modified spiral bevel gear, the high-order tooth surface modification method can not only effectively reduce the LTE, MI and dynamic load factor (DLF) of HCR spiral bevel gear, but also improve its dynamic performance within full speed range. This study provides an effective way for improving the dynamic performance of HCR spiral bevel gear transmissions within full speed range.

Meshing impact analysis of planetary transmission system considering the influence of multiple errors and its effect on the load sharing and dynamic load factor characteristics of the system

The meshing impact on transmission system and internal meshing gear pair and its impact on the load sharing and dynamic characteristics of the system are not well understood yet. In this paper, the meshing impact models of internal gear pairs and planetary transmission system were successfully constructed, and the meshing impact point, meshing impact time and meshing impact force were accurately obtained. Meshing impact in gear transmission system is obviously affected by eccentricity error, installation error, and other errors. Due to the difference in error of each component, the internal and external gears of each branch lead to different meshing positions, which causes the constant change in meshing impact point, meshing impact time and meshing impact force. This creates difficulties in the analysis of meshing impact characteristics of gear transmission system. Load Tooth Contact Analysis (LTCA) method can be used to accurately analyse the change in position of gear tooth under load condition. Through the dynamic model of planetary transmission system, the influence in component errors on the contact position of tooth surface is obtained. Combining the loaded transmission error of the tooth surface under load and the geometric transmission errors under the influence of component errors, the model of meshing impact for accurately solving the system is deduced, and the influence of meshing impact on the system's load sharing coefficient and dynamic load factor coefficient is analysed. By comparing the planetary transmission system before and after considering the meshing impact of the system, it is found that the system's load-sharing coefficient increases slightly, dynamic load factor coefficient fluctuates significantly, and meshing force becomes more clutter after considering meshing impact.

Tooth flank modification to reduce transmission error and mesh-in impact force in consideration of contact ratio for helical gears

In this paper, a new tooth modification method considering the contact ratio of gears and a new method for calculating the mesh-in impact force of modified helical gears are proposed. The new method for calculating the mesh-in impact force is based on tooth contact analysis and loaded tooth contact analysis. The mesh-in impact position can be calculated accurately via the new method. First, the procedures for creating the new tooth modification and the details of calculation method of the mesh-in impact force are exhibited. Second, the optimal modification of the tooth flank is achieved by solving the optimization problem. Third, a dynamic model of the gear system considering the loaded transmission error and the mesh-in impact force is used to study the dynamic characteristics. Ultimately, numerical examples are presented and the simulation results suggest that the amplitude of the loaded transmission error and the mesh-in impact force can be reduced more effectively based on the introduced new tooth modification method. And the mesh-in impact effects should not be neglected in gear dynamic analysis, regardless of whether the tooth modified or not, especially for high-speed gears.

A calculation method of sliding friction coefficient on tooth surface for helical gear pair based on loaded tooth contact analysis and elastohydrodynamic lubrication theory

The sliding friction coefficient on tooth surface is related to power loss, carry capacity and transmission performance of gear. Reasonable transmission analysis of gear pair is the premise of accurate calculation of sliding friction coefficient on tooth surface. However, for helical gear pair, the line contact without considering machining error/installation error/modification of gear is usually adopted to replace the major axis of ellipse caused by contact load. Therefore, in this paper, contact path on tooth surface, length of contact line, load distribution on tooth surface and loaded transmission errors are accurately calculated by loaded tooth contact analysis (LTCA). Combing with elastohydrodynamic lubrication (EHL) theory, a calculation method of sliding friction coefficient on tooth surface for helical gear pair is proposed.

An Experimental and Theoretical Study of Quasi-Static Behavior of Double-Helical Gear Sets

AbstractThe 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.

An innovative semi-analytical determination approach to numerical loaded tooth contact analysis (NLTCA) for spiral bevel and hypoid gears

Loaded tooth contact analysis (LTCA) has always been a significant contact mechanical performance optimization technique for spiral bevel and hypoid gears. To distinguish with the conventional simulated loaded tooth contact analysis (SLTCA) based on economical software package, numerical loaded tooth contact analysis (NLTCA) and its innovative semi-analytical determination approach is developed. In full consideration of tooth flank flexural behavior characteristic, double-curved shell finite element modeling is performed and then improved tooth contact analysis (TCA) is applied for data-driven determination of loaded tooth contact points. Then, in additional to the global boundary condition, the accurate local boundary conditions is also investigated. In particular, local Rayleigh-Ritz method is used to get new semi-analytically tooth compliance determination by considering computational accuracy and efficiency. Furthermore, NLTCA model is established in considerations of the accurate contact point matching, compatibility and equilibrium conditions. Finally, the main NLTCA evaluations including load distribution, loaded contact pattern, loaded contact pressure and distribution, and loaded transmission error in consideration of time-varying meshing characteristics, respectively. The provided numerical instance can verify the proposed methodology.