Longitudinal Crowning Research Articles

Numerical-experimental analysis of highly crowned spherical gear couplings working at high misalignment angles

Spherical gear couplings are characterized by their high capacity to transfer power between rotating shafts in highly misaligned conditions. They are usually small because of space restrictions and require a high amount of longitudinal crowning, which sometimes leads to undercutting profiles. These issues have been solved recently with analytical models for the generation of tooth surfaces that consider the hob thread surface, however, they are not experimentally validated. Likewise, it has been numerically concluded that the number of teeth in contact in high misalignment conditions decreases, and thus may fail by tooth root breakage. Nevertheless, published experimental tests for such high misalignment angles are scarce. This paper presents test techniques to study highly crowned spherical gear couplings, together with a test rig design adapted to high misalignment angles. The tooth surface geometry, maximum misalignment, stiffness, and contact pattern are compared. The results demonstrate that the analytical and numerical models employed accurately represent the mechanical behavior of spherical gear couplings working at high misalignment applications.

Load distribution and tooth root stress of highly crowned spherical gear couplings working at high misalignment angles

Spherical gear couplings are commonly used mechanical components to transmit power between highly misaligned rotating shafts. For that, gear couplings are manufactured with high longitudinal crowning and are usually small due to space restrictions, with the probability to contain undercutting sections. High misalignment angles cause the number of teeth in contact to decrease drastically, resulting in the failure of the component by tooth root fatigue breakage.This paper investigates the load distribution and the tooth root stress of highly crowned spherical gear couplings working at high misalignment angles using a finite element model. Moreover, a deep understanding of the bending tooth root stresses in terms of the operating conditions is presented, which is novel for such high misalignment angles (γ≥3∘). Results show that different mechanical behaviors are observed at low or high misalignment angles since teeth in the pivoting position lose contact. This results in a tooth root stress history change from a sinusoidal cycle to a pulsating cycle. Finally, this study shows evidence that current sizing methods are not suitable, and underlines the need for further research to determine the spherical gear coupling life to tooth root bending fatigue.

A general mathematical modeling and finite element analysis of a strain wave gear possessing a double-circular-arc-with-a-common-tangent profile

In this study, the characteristics of a strain wave gear (SWG) with a double-circular-arc with a common tangent (DCACT) profile were simulated using two-dimensional (2D) and three-dimensional (3D) finite element analysis (FEA). First, theoretically generated tooth profiles for the flexible spline (FS) and circular spline (CS) of an SWG were developed according to the theory of gearing. An automated mesh-generating computer program was developed in Visual C++ to establish 2D and 3D finite element models of the SWG based on its derived geometry. An FEA was then performed to explore the meshing characteristics, including the stress distribution, torsional angle, torsional stiffness, and number of engaged teeth, of the SWG. Continuous meshing of multiple tooth pairs between the FS and CS from the 2D FEA verified the conjugation of tooth profiles. The coning effect on the FS during assembly of the SWG was also predicted and discussed from the 3D FEA results. The 3D FEA produced more accurate and practical results than the 2D FEA did. Additionally, 3D FEAs of a modified (with longitudinal crowning) and unmodified (without longitudinal crowning) FS were performed and compared. Finally, a 3D FEA of a modified FS was successfully conducted to predict cyclic contact stress, root stress, and transmission error.

Approximation model for longitudinal-crowned involute helical gears with flank twist in continuous generating grinding

Longitudinal-crowned involute helical gears perform better in load capacity and can improve transmission quality. When longitudinal crowning of helical gears is processed with continuous generating grinding, the flank twist occurs and reduces the transmission stability of gears. In the paper, an innovative approximation model is proposed to demonstrate the relationships between the flank twist and parameters (including gear parameters and modification parameters) in a simple mathematical form. The flank twist calculated by the approximation model is compared with that calculated based on the gear meshing principle. The comparison results confirm that the model is right. Based on the new model, the influences of gear parameters on the flank twist are analyzed. The model can reduce the flank twist in the gear design.

Principle and Method for Machining Face Gear with Developable Ruled Surface

The tooth surface of a face gear is recognized as un-developable ruled surface in tradition, which result in the problem that face gear generation using simple cutter with high efficiency and accuracy puzzled people for a long time. Applying the traditional shaping method for face gear generation in derivation of the tooth surface position vector, a space straight line equation was established by using this vector, to illustrate that the tooth surface of the traditional face gear could be considered as developable ruled surface constituted by space lines family, the position vectors of the developable ruled surface were established respectively based on involute, rack and considering the spatial position relationship in meshing. Elaboration and proposition of the principle and method that were used to generate the face gear with the developable ruled surface by straight edged cutter in line contact, and representation of the generated surface model. Furthermore the developable ruled surface was compared to the traditional surface, the normal difference between the two kinds of surfaces is on micro-scale. Numerical examples shows that the normal distance is no more than 1/70 of modulus, the developable ruled surface to which was performed simple longitudinal crowning is able to mesh with the involute surface of the pinion correctly, and demonstrates acceptable geometric meshing performance.

Generation of the Anti-Twist Crowned Helical Gear by Modifying the Gear Rotation Angle in the Hobbing Process

In the gear-hobbing process, the work gear tooth flank is usually longitudinally crowned by varying the center distance between the hob and the work gear. Without crossed angle compensation, however, this center distance variation produces a twisted tooth flank on the work gear. This paper therefore proposes a methodology to reduce this tooth flank twist and achieve anti-twist in longitudinal crowning by modifying the gear rotation angle in the hobbing process which is practiced using a CNC hobbing machine with three synchronous axes.

Generation of a double-crowned involute helical gear with twist-free tooth flanks by a CNC hobbing machine with three synchronous axes

In the conventional hobbing process, a double-crowned involute helical gear is generated by the hob cutter with parabolic-curve tooth profiles for the cross-profile crowning and varied the center distance between the hob and work gear for the longitudinal crowning. Therefore, to cut a double-crowned helical gear not only requires at least four synchronous axes and hob cutter regrinding (which increases production costs) but also induces twisted tooth flanks on the generated work gear. In this paper, I propose a hobbing method by applying a modified work gear rotation angle that enables double-crowning of involute helical gear's tooth flanks using a standard hob cutter and a computer numerical control (CNC) hobbing machine with only three synchronous axes. The proposed method has also verified by using two computer simulation examples to compare the meshing-conditions, contact ellipses, and transmission errors of the double-crowned gear pairs with that produced by applying the conventional hobbing method. Computer simulation results reveal the advantages of the proposed novel hobbing method.

LONGITUDINAL CROWNING OF THE GEAR TOOTH SURFACE OF STRAIGHT BEVEL GEARS WITH A SMALL SHAFT ANGLE WITHNON-GENERATED GEAR AND GENERATED PINION

The aim of this thesis is to develop computer program shows generating process of the longitudinal-crowned gear for straight bevel gears with a small shaft angle with non-generated gear and generated pinion. The purpose of computer program is the simulation of meshing and contact of the gear tooth surface with longitudinal crowning and straight pinion tooth surface with localized bearing contact.

A Methodology for Longitudinal Tooth Flank Crowning of the Helical Gear on a CNC Honing Machine

The gear honing is the most economical way for hard finishing an involute helical gear after hobbing and heat treatment or after shaving and heat treatment. The gear honing can also be applied to the modification of gear tooth surfaces to compensate for the distortions that occur during heat treatment process. Most published papers on the technology of gear honing describes on the principle of generated gear surface. However, the longitudinal tooth flank crowning of a helical gear with honing has not been investigated yet. Therefore, in this paper, we proposed a novel method for longitudinal tooth flank crowning of work gear surfaces by setting a crossed angle between the honing cutter and work gear axes as a linear function of honing cutter's traverse feed in the honing process. A mathematical model for the tooth profile of work gear honed by a standard honing cutter is also established. Three numeral examples are presented to illustrate and verify the merits of the proposed gear honing method in longitudinal crowning.

Tooth contact analysis for helical gear pairs generated by a modified hob with variable tooth thickness

Conventionally, the longitudinal crowning of helical gears can be accomplished by varying the center distance between the hob and work gear in gear hobbing process. Without a crossed angle compensation, however, this center distance variation produces twisted tooth flanks on the work gear. A methodology is thus proposed herein to reduce the tooth flank twisting of a longitudinal crowning gear by applying a modified variable tooth thickness hob and having a diagonal feed without varying the center distance. This study also investigates the gear tooth surface topologies, contact ellipses and transmission errors of the work gear pairs, generated by a conventional hob and the proposed modified hob with variable tooth thickness, respectively. Three numeral examples are presented to illustrate and verify the merits of the proposed gear hobbing method with longitudinal crowning.

Load Tooth Contact Analysis for Double-Crowned Face Gear

In order to further reduce sensitivity to misalignments not only double crowned pinion but also profile and longitudinal geometry modified face gear have been studied. The profile geometry modification were realized by application of rack cutters, and applied parabolic motion of the center of grinding disk for the pinion and face gear longitudinal crowning. Tooth contact analysis (TAC) and Load Tooth contact analysis (LTCA) have been performed to simulate the meshing and contact of double-crowned face gear driver. The further avoidance of edge contact in presence of big magnitude misalignment was illustrated with numerical examples.

Generation and TCA of Straight Bevel Gear Drive with Modified Geometry

In order to reduce the sensitivity of straight bevel gear drives to misalignment, a new geometry of such gear drives is proposed in longitudinal direction. Point contact instead of line contact of tooth surfaces is achieved by longitudinal crowning of pinion tooth surface. The tooth surface modeling and tooth contact analysis (TCA) of straight bevel gear drives have been established. TCA program of a pair of straight bevel gears was performed in MATLAB and tooth bearing contact and transmission errors were obtained.

Novel variable-tooth-thickness hob for longitudinal crowning in the gear-hobbing process

In the gear-hobbing process, the work gear tooth flank is usually longitudinally crowned by varying the center distance between the hob and the work gear. Without crossed angle compensation, however, this center distance variation produces a twisted tooth flank on the workgear. This paper therefore proposes a methodology to reduce this tooth flank twist in longitudinal crowning by modifying the hob tooth thickness and hob tangential feed without varying the center distance. ► The longitudinal crowning is usually caused a twisted tooth flank on a hobbed gear. ► This paper proposes a methodology to reduce the tooth flank twist on the gear. ► The methodology is using a variable tooth thickness (VVT) hob in gear hobbing. ► Besides, the VVT hob should be shifted diagonally in the hobbing process. ► This methodology will reduce the tooth flank twist on the gear effectively.

Topology of modified surfaces of involute helical gears with line contact developed for improvement of bearing contact, reduction of transmission errors, and stress analysis

The main defects of misaligned helical gear drives with parallel axes are: edge contact, noise, and not favorable conditions of bearing contact. One of the greatest concern of manufacturing of helical gear drives (with parallel axes) is the edge contact of tooth surfaces that is caused by misalignment. At present, attempts to avoid edge contact are based on providing chamfers of the tooth surfaces of the gears that may be obtained by modification of the profile of the cutting hobs. The zones of tooth surfaces with chamfers and zones with conventional screw involute surfaces are not connected smoothly, in many cases the magnitude of required chamfers is not determined analytically, and the edge contact is not avoided. The finishing process of helical gears with chamfers is a complicated one. The existing design has to be complemented with TCA (tooth contact analysis). These are the reasons why a new topology of modified helical gear tooth surfaces with involute and crowned zones is proposed. An involute zone is provided in the central area of gear tooth surfaces that will allow line contact if misalignment does not occur. Zones at the top, bottom, front, and back sides of tooth surfaces are crowned and allows localization of the bearing contact when misalignments occur. Crowned zones with smooth connections to the involute zone are obtained as the result of profile and longitudinal crowning. The function of transmission errors is provided at each cycle of meshing as the sum of three branches. Two branches of a parabolic function at the extremes of the cycle of meshing are in tangency with the middle branch of zero transmission errors. The generation of gear tooth surfaces is accomplished by a grinding worm. This has required the solution of two problems, (i) determination of the worm thread surface that is conjugated to the theoretical pinion tooth surface, and (ii) generation by the grinding worm of a crowned pinion tooth surface. Simulation of meshing of misaligned gear drives is accomplished by application of TCA program developed by the authors. The formation of the bearing contact is analyzed considering more than one cycle of meshing. Existence of areas of severe contact stresses is avoided. The developed approach is illustrated with numerical examples.

On Tooth Contact Marking and Gear Performance : 1st Report, Fundamental Characteristics of Tooth Contact Marking

A calculating method of tooth contact marking of involute helical gears is shown with the following results: 1) When the shape of composite gear error surface of a pair of involute helical gears is mathematically similar and the ratio between amplitude and the transmitting load is kept constant, the pattern of tooth contact marking does not change and the state of tooth loading distribution is similar. 2) When a pattern of tooth contact marking is the same, the ratio of stress increase owing to gear error is almost the same. Some restricting conditions for these two theorems are indicated. Measured and calculated tooth contact markings and tooth fillet stresses of gears which have concave tooth form error and simultaneous longitudinal crowning of tooth trace correction verify the propriety of these two theorems.