Parabolic Function Of Transmission Errors Research Articles

Mathematical model and algorithm for contact stress analysis of gears with multi-pair contact

The problem of contact pressure evaluation in gears with predesigned parabolic function of transmission errors and with localized contact is considered. It is proposed that several tooth pairs are in contact simultaneously. We construct an algorithm to define the torque transmitted by each pair in contact. Number of pairs in simultaneous contact depends on applied load and phase of meshing. This number is not known in advance. The feature of the algorithm is an automatic detection of the tooth pair number that is in contact in any phase of meshing under load. The algorithm allows you to define forces and torques that are transmitted on each of the pairs of teeth being in contact. Moreover, it allows determining the contact pressure and tooth-to-tooth accuracy of transmission. The proposed algorithm is universal and can be used to study the multi-pair contact in any gears.An application of the algorithm for a spiral bevel gear has been demonstrated. It is shown that tooth overlap factor, dimensions of instant contact patch, tooth bearing contact and tooth-to-tooth accuracy of gears increase with growth of load.

A Precise and Efficient Method to Manipulate the Amplitude of Parabolic Function of Transmission Errors

This paper proposes a system of nonlinear equations to manipulate the amplitude of parabolic function of transmission errors. Firstly, the characteristics of parabolic function of transmission errors are defined. Then, a system of nonlinear equations for manipulating the amplitude of parabolic function of transmission errors is created based on both the conditions of contact and the constraint on the amplitude of function of transmission errors. As the number of independent scalar equations in the system minus the number of unknown parameters is one, one extra design parameter can be applied to manipulate the amplitude of parabolic function of transmission errors. The solution to the extra design variable is automatically, precisely, and efficiently determined by the computer program which is created based on the Newton’s root finding method. The time-consuming manual iterations for trying the value of design variable are eliminated. The proposed method can be applied to both two-and three-dimensional gearing problems. At last, a pair of meshing gears composed of a circular-arc spur gear and an involute spur gear is presented to verify the methodology proposed in this paper.

Effect of static transmission error on dynamic responses of spiral bevel gears

The effect of static transmission error on nonlinear dynamic response of the spiral bevel gear system combining with time-varying stiffness and backlash was investigated. Firstly, two different control equations of the spiral bevel gear model were adopted, where the static transmission error was expressed in two patterns as predesigned parabolic function and sine function of transmission errors. The dynamic response, bifurcation map, time domain response, phase curve and Poincare map were obtained by applying the explicit Runge-Kutta integration routine with variable-step. A comparative study was carried out and some profound phenomena were detected. The results show that there are many different kinds of tooth rattling phenomena at low speed. With the increase of speed, the system enters into stable motion without any rattling in the region (0.72, 1.64), which indicates that the system with predesigned parabolic function of transmission error has preferable capability at high speed.

Research on Meshing Characteristics for Face Gear with Arcuate Tooth

The generating of face gear with arcuate tooth has been proposed in this paper, and the meshing characteristics are investigated. Based on the concept of imaginary gear cutter, tooth surface equation has been derived, flank modification has also been considered. The transmission errors and bearing contacts of the face gear drive with arcuate tooth under different assembly conditions are investigated by applying the tooth contact analysis. The numerical results reveal that the bearing contacts are not sensitive to the errors of misalignments, and a more favorable type parabolic function of transmission errors with better symmetry and reduced amplitude may be obtained according to the modification of the face gear.

Design of Pinion Machine Tool-settings for Spiral Bevel Gears by Controlling Contact Path and Transmission Errors

This paper proposes a new approach to design pinion machine tool-settings for spiral bevel gears by controlling contact path and transmission errors. It is based on the satisfaction of contact condition of three given control points on the tooth surface. The three meshing points are controlled to be on a predesigned straight contact path that meets the pre-designed parabolic function of transmission errors. Designed separately, the magnitude of transmission errors and the orientation of the contact path are subjected to precise control. In addition, in order to meet the manufacturing requirements, we suggest to modify the values of blank offset, one of the pinion machine tool-settings, and redesign pinion machine tool-settings to ensure that the magnitude and the geometry of transmission errors should not be influenced apart from minor effects on the predesigned straight contact path. The proposed approach together with its ideas has been proven by a numerical example and the manufacturing practice of a pair of spiral bevel gears.

Reduction of noise of loaded and unloaded misaligned gear drives

Transmission errors are considered as the main source of vibration and noise of gear drives. The impact of two main functions of transmission errors on noise is investigated: (i) a linear one, caused by errors of alignment, and (ii) a predesigned parabolic function of transmission errors, applied for reduction of noise. It is shown that a linear function of transmission errors is accompanied with edge contact, and then inside the cycle of meshing, the meshing becomes a mixed one: (i) as surface-to-surface tangency, and (ii) surface-to-curve meshing when edge contact starts. Application of a predesigned parabolic function of transmission errors enables to absorb the linear functions of transmission errors caused by errors of alignment, reduce noise, and avoid edge contact. The influence of the load on the function of transmission errors is investigated. Elastic deformations of teeth enable to reduce the maximal transmission errors in loaded gear drives. Computerized simulation of meshing and contact is developed for loaded and unloaded gear drives. Numerical examples for illustration of the developed theory are provided.

Computerized Developments in Design, Generation, Simulation of Meshing, and Stress Analysis of Gear Drives

The paper represents new computerized developments in design, generation, simulation of meshing, and stress analysis of gear drives. The main contents of the paper are: (i) application of a predesigned parabolic function of transmission errors for reduction of noise, (ii) computerized simulation of noise caused by transmission errors, (iii) modification of the basic algorithm of tooth contact analysis, and (iv) application of approaches developed for enhanced design and simulation of meshing of the following gear drives: (a) spiral bevel gear drives, (b) face-gear drives (including an approach for grinding), and (c) modified helical gear drives. The developed theory is illustrated with numerical examples.

Design, manufacture, stress analysis, and experimental tests of low-noise high endurance spiral bevel gears

The contents of the paper cover the design, manufacturing, stress analysis, and results of experimental tests of prototypes of spiral bevel gears with low levels of noise and vibration and increased endurance. The main goals are the improvement of the bearing contact, the achievement of a predesigned parabolic function of transmission errors, reduction of the magnitude of transmission errors as the precondition of reduction of noise and vibration, and avoidance of areas of severe contact stresses for the increase of the endurance of the gear drives. The goals are achieved by application of local synthesis algorithm, tooth contact analysis, and stress analysis by application of finite element method. The proposed ideas have been proven by the manufacturing and test of prototypes of spiral bevel gear drives. An example of design and optimization of a spiral bevel gear drive for the illustration of the developed theory is represented.

Design of One Stage Planetary Gear Train With Improved Conditions of Load Distribution and Reduced Transmission Errors

The authors propose an approach for the design of one-stage planetary gear train with reduced transmission errors, localized bearing contact and improved conditions of distribution of load between the planetary gears. The planetary gear train is considered as a multi-body mechanical system of rigid bodies. The proposed approach is based: (i) on modification of geometry of gear tooth surfaces, and (ii) minimization and equalization of the backlash between the contacting gear tooth surfaces. The modification of geometry is accomplished: (i) by double-crowning of planetary gears, and (ii) by application of screw involute surfaces of the same direction of screws for the sun and the ring gears. The proposed geometry enables: (i) predesign of parabolic function of transmission errors for reduction of noise and vibration, and (ii) a simple method of regulation and equalization of the backlash between the gear tooth.

Design and Stress Analysis of Low-Noise Adjusted Bearing Contact Spiral Bevel Gears

An integrated computerized approach for design and stress analysis of low-noise spiral bevel gear drives with adjusted bearing contact has been developed. The computational procedure is an iterative process requiring four separate steps that provide: (a) a parabolic function of transmission errors that is able to reduce the effect of errors of alignment, and (b) reduction of the shift of bearing contact caused by misalignment. Application of finite element analysis permits the contact and bending stresses to be determined and the formation of the bearing contact to be investigated. The design of finite element models and boundary conditions is automated and does not require intermediate CAD computer programs. A commercially available finite element analysis computer program with contact capability is used to conduct the stress analysis. The theory developed is illustrated with numerical examples.

New geometry of face worm gear drives with conical and cylindrical worms: generation, simulation of meshing, and stress analysis

New geometry of face worm gear drives with conical and cylindrical worms is proposed. The generation of the face worm gear is based on application of a tilted head-cutter (grinding tool) instead of application of a hob applied at present. The generation of a conjugated worm is based on application of a tilted head-cutter (grinding tool) as well. The bearing contact of the gear drive is localized and is oriented longitudinally. A predesigned parabolic function of transmission errors for reduction of noise and vibration is provided. The stress analysis of the gear drive is performed, the contacting model is designed automatically. The developed theory is illustrated with numerical examples.

Computerized design, simulation of meshing, and contact and stress analysis of face-milled formate generated spiral bevel gears

A new approach for design, tooth contact analysis (TCA) and stress analysis of formate generated spiral bevel gears is proposed. The advantage of formate generation is the higher productivity. The purposes of the proposed approach are to overcome difficulties of surface conjugation caused by formate generation, develop a low noise and stabilized bearing contact, and perform stress analysis. The approach proposed is based on application of four procedures that enable in sequence to provide a predesigned parabolic function of transmission errors with limited magnitude of maximal transmission errors, a bearing contact with reduced shift of contact caused by misalignment, and perform stress analysis based on application of Finite Element Method. The advantage of the approach developed for finite element analysis (FEA) is the automatic generation of finite element models with multi-pairs of teeth. The stress analysis is accomplished by direct application of ABAQUS. Intermediate auxiliary CAD computer programs for development of solid models are not required. The theory developed is illustrated with an example of design and computation.

New developments in the design and generation of gear drives

Design, generation and simulation of the meshing and contact of gear drives with favourable bearing contact and reduced noise are considered. The proposed approach is based on replacement of the instantaneous line of contact of tooth surfaces by point contact and on application of a predesigned parabolic function of transmission errors that is able to absorb linear discontinuous functions of transmission errors caused by misalignment. Basic algorithms for analysis and synthesis of gear drives are presented. The developed theory is applied for design and generation of the following gear drives with modified geometry: (a) spur and helical gears, (b) a new version of Novikov-Wildhaber (N-W) helical gears, (c) asymmetric face gear drives with a spur pinion, (d) formate-cut spiral bevel gears. Generation of the tooth surface of a worm gear is presented as the formation of a two-branch envelope. The discussed topics are illustrated with examples.

Computer program in Visual Basic language for simulation of meshing and contact of gear drives and its application for design of worm gear drive

The authors propose: (i) enhanced computer program for simulation of meshing and contact of gear drives; (ii) application of computer program for analysis of worm gear drive; and (iii) advanced design of worm gear drive with reduced transmission errors and favorable bearing contact. The output of Tooth Contact Analysis (TCA) program are: function of transmission errors, path of contact and bearing contact. The program is written in Visual Basic language and enables to combine numerical computation and graphical illustration. The improved design of worm gear drive is based on application of an oversized hob and modified worm generated by varied plunging of the generating tool. It is discovered that worm generation without tool plunging may cause positive transmission errors, unacceptable for favorable conditions of force transmission. Positive transmission errors are the herald of possible surface interference. A predesigned parabolic function of transmission errors is provided in order to absorb transmission errors caused by errors of alignment and reduce the level of vibrations, especially in the case of application of multi-thread worms. The investigation is accomplished for a worm-gear drive with the Klingelnberg type of the worm that is ground by a circular cone, but the proposed approach may be applied for other types of worm gear drives with cylindrical worms.

Computerized Design, Generation and Simulation of Meshing of a Spiroid Worm-Gear Drive with a Ground Double-Crowned Worm

The authors propose methods of computerized design and analysis of a spiroid worm-gear drive with ground worm based on the following considerations: (1) The theoretical thread surface of the hob is generated by a cone surface. (2) The worm surface is crowned in profile and longitudinal directions in comparison with the hob thread surface. (3) The double crowning of the worm enables to localize the bearing contact and obtain a predesigned parabolic function of transmission errors of an assigned level. Computerized design of the worm-gear drive enables to discover and avoid singularities of the generated worm face-gear surface and pointing of teeth. The meshing and contact of the double-crowned worm and the worm face-gear is simulated to determine the influence of misalignment on the shift of bearing contact and transmission errors. Computer program for numerical solution is developed and applied. A numerical example that illustrates the developed theory is provided.

Computerized Design and Generation of Low-Noise Helical Gears with Modified Surface Topology

An approach for the design and generation of low-noise helical gears with localized bearing contact is proposed. The approach is applied to double circular arc helical gears and modified involute helical gears. The reduction of noise and vibration is achieved by application of a predesigned parabolic function of transmission errors that is able to absorb a discontinuous linear function of transmission errors caused by misalignment. The localization of the bearing contact is achieved by the mismatch of pinion-gear tooth surfaces. Computerized simulation of meshing and contact of the designed gears demonstrated that the proposed approach will produce a pair of gears that has a parabolic transmission error function even when misalignment is present. Numerical examples for illustration of the developed approach are given.