Theory Of Gearing Research Articles

Theoretical and experimental investigation on gearbox vibration signal separation of planetary gear set

The gearbox vibration signal of a planetary gear set contains multiple components caused by different excitations. Separating the gearbox vibration signal to decouple its components can provide core information for operational status monitoring. In this paper, we propose an indirect gearbox vibration signal separation method based on parameter identification. A novel vibration signal model is established using the modal shape of ring gear. A parameter identification method is adopted to determine the parameters of the vibration signal model. The vibration signal components are reconstructed based on the vibration signal model and identified parameters. To evaluate the performance of the proposed method, a ring gear response calculation method derived from the elastic theory of ring gear is presented. The influences of unequal load sharing and meshing position errors on vibration signals are studied through numerical simulations using the calculation method. Experiments are conducted in a planetary gear set test rig. Comparisons between the experimental results obtained by the separation and calculation methods indicate the effectiveness of the proposed separation method.

A coaxial multi-ring detection method for measuring the pitch and thickness accuracy of cylindrical gears

Performance of gear transmissions affects the performance of mechanical equipment. It is necessary to develop more reliable gear pitch and tooth thickness accuracy detection methods in order to evaluate gear transmission performance and detect gear pitch and tooth thickness accuracy more accurately. Based on the basic theory of gears, binocular vision technology, and statistical principles, a new method that measures gear pitch and tooth thickness using machine vision is proposed: coaxial multi-ring detection (MCD). There is no contact, no damage, and a high degree of efficiency with this method. Using this method, we are able to detect the pitch and thickness accuracy of each gear tooth multiple times within the tooth width range and in multiple directions perpendicular to the gear axis. We statistically analyze the measurement results to determine the gear’s most accurate detection results. The measurement method for gear machining accuracy is investigated using the coaxial multi-ring detection method. The statistical analysis of multiple measurement results is carried out, and the measurement results obtained are highly consistent with those of the gear detection center. In conclusion, the measurement results of this method are highly reliable, and they can be used as a reliable basis for evaluating gear transmission performance.

Mathematical modeling and simulation of generation milling process to manufacture complex face gear surface through multi-axis machine kinematics

Manufacturing of face gear is a complex process, due to its varying geometry, hence the application is not wide as conventional gear geometry. The developments in analytical tools and kinematics of the CNC machine tools motivate researchers to explore the manufacturing process and associated applications. This article presents an efficient method for the generation milling of spur face gear utilizing a nutating head multi-axis CNC machine tool where the tool swings about an arbitrary axis in an inclined plane. The mathematical model of the disk milling cutter and face gear geometry were derived from the shaper geometry by applying the theory of gearing, equation of meshing and coordinate transformation. The cutter location (CL) data was evaluated by aligning the tool orthogonal unit vector and the tooth surface normal vector at the contact point by calculating the appropriate transformation matrices. The evaluation of machine coordinates on a multi-axis CNC milling machine was analyzed with respect to the machine pivot point and the axes movement during generation milling for the face gear drive was studied on the nutating head machine. The NC data is derived from the CL data using the inverse kinematics and the machining simulation was carried out using suitable software. The tooth surface deviation of a maximum of 43.5 μm was observed between the simulated model and the theoretical model, which is well within the requirement of roughing. The generation milling process with the proposed machine configuration is concluded as a profound face gear manufacturing technique.

CORRECTION OF THE CYLINDRICAL HELICAL-GEAR WHEEL PAIR FOR WEAR DECREASE OF TEETH’ INVOLUTE SURFACES

Purpose. It is to create a method for determination of profile shift coefficients, such that during the operation of a cylindrical helical-gear pair the sum of the wear rates of involute surfaces on the contacting sections at the highest and lowest active points of teeth’ profiles of driving and driven wheels will be a minimum.
 Research methods are based on integrated application of the involute gear theory and tribological laws. The secant method that is known in computational mathematics is used to find a root of nonlinear equation.
 Results. The method is created for determination of rational values of correction parameters for cylindrical helical-gear wheels, which maximize the service life of this gear pair until the maximum sum of worn-layer thicknesses on the contacting sections of the teeth’ active surfaces reaches limiting permissible value. An example of application of this method in design calculation of the cylindrical helical-gear pair is given.
 Scientific novelty. Calculation model is elaborated for determination of the sum of worn-layer thicknesses on the most wearing contacting sections of the teeth’ active surfaces of a cylindrical helical-gear wheel pair. It is derived an equation of the linkage between such values of profile shift coefficients for the driving and driven helical-gear wheels which minimize the greatest sum of wear rates on mentioned sections of teeth’ surfaces. Solution of this equation by numerical method for the helical-gear pair having standard distance between axes allows determination of rational values of correction parameters for these wheels.
 Practical value. Application of the created method for determination of profile shift coefficients is expedient in design of cylindrical helical-gear pairs for machines and equipment, which operate in conditions when it is impossible to protect the engagement region from ingress of abrasive particles.

Ellipsoid contact analysis and application in the surface conjugate theory of face gears

Abstract. The face gear tooth surface is a high-order variable curvature surface, and the curvature of the surface is complicated, so it is difficult to describe the characteristics of the face gear tooth surface by a specific equation. In this paper, the discrete curvature relation of the surface is used instead of an analytical equation to describe a spatial meshing surface, and the traditional meshing theory is neutralized to analyze the characteristics of the face gear tooth surface. Firstly, according to the structural characteristics of the face gear, the sampling method of the face gear tooth surface is analyzed, and the mathematical model of fitting the tooth surface contact point is established. Then, the discrete asymptotic surface development analysis method is studied, and the ellipsoid contact analysis method of the face gear pair is established by simplifying the conjugate surface and its region. Finally, the contact analysis method of the discrete tooth surface is studied, and the instantaneous contact ellipse of the face gear tooth surface is calculated, which formed a new numerical meshing method of space curved surface.

Gear skiving with minimum twist errors

Over the last 15 years, gear skiving has established itself as a highly productive gear cutting process for the production of internal gears and gears with near interference contours. As with all processes with crossed axes, gear skiving generally results in a pronounced natural twist when gears with lead crowning or other flank modifications are produced. In practical applications, the unintended profile angle changes over the tooth width resulting from the twist leading to unwanted contact patterns and unfavorable NVH behavior. In this work, a contact line-based method for tool profile calculation for gear skiving is developed based on conical-screw gear theory. The relationship between contact line and natural twist errors is worked out. The process and tool design strategies for minimizing the twist are elaborated and finally, an adaptive process kinematics for low-twist error gear skiving is presented.

Meshing theory of face worm gear drive with hardened cylindrical worm

This article proposes a novel linear contact face worm gear drive. The novel face worm gear drive consists of a face worm gear and a cylindrical worm enveloped by a tapered surface. The meshing theory for the novel face worm gear drive is systematically built up. The meshing theory on the base of the reference point is further developed aimed at the worm pair where the indexing surfaces are planar and cylindrical. The design methods of the rough size for the face worm gear are introduced. The computing method for the worm limit axial mounting distance is presented by employing the meshing limit line function. The asymmetry of the tooth surfaces for the worm pair is evaluated using the helix angle at the midpoint on the line of section for the worm gear tooth surface and the tooth profile angle for the worm axial section. Numerical results show that the geometric characteristics and the meshing properties for the tooth surface on both flanks of the novel face worm gear drive are asymmetrical. However, the meshing properties for the novel face worm gear drive are satisfactory.

Non-Linear Dynamic Modeling and Analysis of Split-Torque Face-Gear Drive Systems

Abstract A non-linear dynamic model of a multi-mesh involute spur pinion-driven face-gear split-torque drive system is developed. The lumped mass model consists of five pinions and two face-gears with seven rotational degrees-of-freedom. Model includes two inputs, two outputs, and three idler gears meshing with two identical face-gears at the bottom and top of the assembly. Face-gear tooth form and corresponding spur gear teeth are established by differential geometry and the theory of gearing. Face-gear drive system mesh stiffnesses are established by utilizing finite strip method for the first time in the literature. The mesh stiffness and damping are time-variant. The model includes clearance-type non-linearity. The harmonic balance method is utilized to solve the non-linear differential equations of motion. The results are compared with the time simulation results. The stability is checked with Floquet theory and bifurcation diagrams from Poincare sections. The model is fully capable of generating the tooth geometries, mesh stiffnesses, and gear train dynamics without reliance on any package programs. Hence, the model provides flexibility and fast computation for parametric studies compared to existing literature. The effect of the pinions’ orientations on the face-gear is investigated to evaluate the mesh phasing effects among system dynamic response. The effect of sub-harmonic motion on the dynamic response of the split-torque face-gear system is also demonstrated for the first time in the literature. The case studies demonstrate that sub-harmonic resonance peaks are avoided.

The designing and modeling of equal base circle herringbone curved bevel gears

Based on the theory of equal base circle bevel gears, a new type of gear, the equal base circle herringbone curved bevel gear, is studied further to reduce the axial force problem of curved bevel gears. Using the principle of equal basis circle herringbone curved bevel gear as a basis, the tooth trace and the tool design of this gear were studied. Based on the principle of space meshing, the process of finger milling cutter enveloping machining gear was analyzed, and a tooth surface equation was calculated; This paper proposes two modeling methods: tooth trace modeling and tooth surface modeling. The gear model is consistent through the analysis and comparison of the two modeling methods. The axial force of the gear contact between the equal base circle herringbone curved bevel gear and Gleason are analyzed and compared, which verified the proper reduction of axial force.

Analysis of flow characteristics of cylindrical and helical type multi-lobe roots blower

Roots blower is a positive displacement machine that has the advantage of a larger flow than conventional blowers. Roots blowers are widely used in industrial production such as chemicals, food, medical, etc. However, during actual operation, this type of machine often achieves low performance. One of the issues that greatly affect performance is the flow characteristics of the blower. Flow characteristics include factors related to flow rate, pressure, and flow phenomena in the blower chamber. Flow characteristic analysis is a complex problem in hydraulic machines. Flow analysis helps to investigate the motion of the flow to design high-performance machines. This study uses a mathematical model of gear theory to design the rotor profile with cylindrical and helical lobes of the multi-lobe Roots blower. The rotor profile is formed on the principle that the ellipse rolls without slipping on the base circle. On the basis of the mathematical model of the rotor profile, the paper compares the flow rate and pressure characteristics of the two blowers. The fluid dynamics analysis model was built on ANSYS software. The structural grid model is also built to increase the computational efficiency of the mathematical model. The lobes are embedded and rotated in the blower chamber. The results show that with the same radial and axial dimensions, the cylindrical lobe has a larger flow. However, the helical lobe has a more stable flow quality than the cylindrical lobe (15.2 % less flow fluctuation). In terms of pressure, the helical lobe type has a higher pressure than the cylindrical lobe type. In addition, the helical lobe type also reduces the influence of eddy currents acting on the blower chamber walls and rotors. That results in increased blower efficiency. The results of the paper will be a reliable basis for reducing time in the development of multi-lobe Roots blowers with high performance

Contact stresses and bending stresses analysis of curvilinear cylindrical gears generated by a face-milling cutter with parabolic profile

Curvilinear cylindrical gear drives are a type of parallel-axis gear set, and the longitudinal geometrical shape of the gear teeth is a part of the circular arc. The main purpose of this paper is to explore the influence of modification amount and assembly error on con-tact stress and bending stress. This study proposes a curvilinear cylindrical gear drive with a parabolic profile. The gear tooth surfaces are generated by a face-milling cutter with a straight-line profile, and the matching gear tooth surfaces are generated by a face-milling cutter with the parabolic profile. Initially, mathematical models of the gear set are described based on the theory of gearing. The finite element method and tooth contact analysis algorithms were applied to contact and bending stress analyses. Finally, four numerical examples are provided to illustrate the mechanical performance of the curvilinear cylindrical gear drive with the parabolic profile. The study results show that curvilinear cylindrical gear drives have superior mechanical performance compared to those with a straight-line profile.

Расчет геометро-кинематических характеристик зацепления арочных зубьев цилиндрической передачи при погрешностях взаимного расположения

The use of cylindrical gears with arched teeth in the drives of energy-saturated machines (locomotives, final drives of tractors, top drives of drilling rigs, helicopters) makes it possible to multiply the durability of gears compared to spur and helical gears. To date, a dozen and a half different techniques and methods for designing and manufacturing arched teeth of cylindrical wheels have been proposed. Among them, the most promising is the method of cutting arched teeth on CNC machines with circular cutter heads using the running-in method with a single division. The purpose of the article is to implement the construction of mathematical models for the formation of arched teeth on the basis of the theory of spatial gearing and to study the geometric and kinematic characteristics of their gearing under conditions of non-zero errors in the relative position of the wheel and gear in the transmission. The object of study is a cylindrical gear with arched teeth. In the article, dependencies for radius-vectors and unit vectors of the normals of the surfaces of arched teeth of cylindrical wheels are obtained, a mathematical model is developed for calculating the points of contact of arched teeth in engagement in the presence of a misalignment angle, a non-parallelism angle and wheel displacement. On the example of the final transmission of a locomotive, for the first time, the influence of the density of fit of the surfaces of the teeth in engagement on the displacement of the working lines from the average section of the tooth due to errors in the position of the transmission elements was studied for the first time. By calculating the error of the position function in the transmission, the procedure for quantifying its nonconjugacy is considered. It has been established that the geometry of the arched teeth of the method under study, under conditions of deviation of the positions of the transmission elements from the nominal ones, leads to a mismatch of the engagement, the position error function has a sawtooth shape, as a result of which impacts are inevitable when the teeth enter the engagement. Ways are proposed to achieve, within the framework of the considered method of manufacturing arched teeth, an approximate nature of the engagement of the teeth in the gear

Development of Harmonic Drive Combining Four Arcs for Conventional Kinematic Application

This paper proposes a modified harmonic cam, which is used to design a harmonic drive. Conventional harmonic drives using a sinusoidal cam type pertain to an elastic body concept and are thus difficult to design and evaluate. Unlike the conventional sinusoidal cam, the proposed harmonic cam involves a combination of four arcs. Such cams can be designed based on a rigid body concept, and thus, the tooth profile can be designed and analyzed using the established gear theory. The novel harmonic cam is designed by exploiting four-arc combinations, and guides for designing harmonic drives with involute and cycloid angular teeth are presented. A harmonic drive with a difference of two teeth between the circular spline and flex spline is designed. The operation, tooth engagement, and interference are validated through a mock-up production using a 3D printer and gear simulation program. The findings can facilitate the facile design and manufacturing of harmonic drives.

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.

A Novel Curve for the Generation of the Non-circular Gear Tooth Profile

This paper presents a novel curve generated by a point attached to an ellipse as it rolls without slipping along a datum line of rack cutter. A mathematical model of the non-circular gear profile has been developed based on the theory of gearing. The effect of an axial ratio l (the ratio of the lengths of the major and minor axes of the ellipse) and the position of the point KR, at which the novel curve starts to generate on the tooth shape and the undercut of the non-circular gear pair is also taken into consideration. A numerical program developed from the mathematical model has been proposed for the calculation and design of the non-circular gears (NCGs). Case studies are presented to show the steps of tooth shape design and to examine the geometrical profile of the NCGs in relation to design parameters of the rack cutter etc. From that, the axial ration l and position of the point KR on the generating ellipse ƩE can be selected for each specific case in order to design the appropriate profiles of the NCGs. On that basis, an experiment to determine the gear ratio of the NCGs pair based on the meshing between gears has manufactured.

Investigation on the generation of bevel gear by disk tool with curved tooth surface of trapezoidal tooth profile

A bevel gear pair with curved tooth surface is widely used for power transmission of intersecting shafts. The bevel gear pair with curved tooth surface has higher load bearing capacity than straight bevel gears, longer line length, quieter, center distance assembly error, and does not change the speed ratio. This paper proposes a disk cutter with curved tooth surface of trapezoidal tooth profile to generate a bevel gear pair. One continuously distributed a trapezoidal tooth profile with curved tooth surface on a disc into an imaginary disc cutter to generate a bevel gear pair with curved tooth surface. This method has never appeared in the literature. Therefore, a mathematical model of a disc cutter with curved tooth surface of trapezoidal tooth profile was established and used to generate the mathematical model of the bevel gear pair via the theory of gearing. Through the undercutting analysis, explore the undercutting conditions of the proposed disc cutter and determine the parameter size limit. Based on tooth contact analysis (TCA), the transmission error and the contact pattern of the bevel gear pair with curvilinear teeth is calculated under the assembly error of the bevel gear. The advantage of the proposed disc cutter is that the transmission error of the parabolic type without modification of the tooth surface of the bevel gear pair can be obtained.

Rattle Dynamic Model and Vibration Behaviors of Noncircular Face Gear

Abstract Noncircular face gear (NFG) is a new type of variable transmission ratio gear mechanism. Based on nonlinear dynamic theory of gear, the rattling behavior of the NFG under multifrequency excitation was studied. The generalized transmission ratio function of the NFG with closed pitch curve was constructed by Fourier series. The nonlinear relative displacement between teeth was described by an analytical expression during the process of time-varying transmission. Considering the factors of time-varying instantaneous center, meshing stiffness, transmission error, and backlash, a nonlinear dynamic model of the gear was established with the rotational angle as the independent variable. The dynamical responses were calculated by Runge–Kutta method to analyze the effects of the input speed, load torque, transmission ratio coefficient, and error on rattling behaviors. The results show that under the action of combined excitation of time-varying instantaneous center and meshing stiffness, the rattle of noncircular gear is more likely to occur. As the increase in input speed, error, and range of transmission ratio variation or the decrease of load, the NFG appears no rattling, unilateral, and bilateral rattle successively and the rattling intensity increases accordingly. The rattling threshold speed can be improved by increasing the accuracy and load of gear or decreasing the range of the transmission ratio.

Comparison and Verification of Dynamic Simulations and Experiments for a Modified Spur Gear Pair

In this study, static and dynamic analyses of a spur gear pair with linear tip relief and longitudinal crowning were performed. Dynamic experiments were conducted to compare and verify the dynamic analysis results. On the basis of the theory of gearing, the mathematical model of a modified spur gear pair was established. In addition, a finite element mesh generation program was developed according to the mathematical model. Finite element analysis was applied to calculate the contact stress, contact patterns, and mesh stiffness of the gear pair. Moreover, a dynamic analysis was performed to predict the dynamic transmission error (DTE) at various speeds and loads. Dynamic experiments were performed on the spur gear pair by using a gear dynamic tester. Accelerometers were mounted on the gear pair to record the vibrations during gear meshing. The experimental DTE was calculated and compared with the simulated DTE to verify the prediction of the vibration performance of the spur gear pair under different rotational conditions.

Substantiations of the beginnings of a unified theory of planetary gears

Multi-satellite planetary mechanisms are analyzed. A new approach to the creation of planetary gearboxes on the principle of L. V. Assur's mechanisms is proposed, according to which the satellites add a group of links of zero mobility to the kinematic chain. Schemes of patented multi-satellite planetary gears are shown. The basis for the development of a new theory of design and manufacture of planetary gearboxes is presented. Keywords: planetary gear, satellite, multi-satellite transmission, uniform load distribution, gear transmission. tmmiok@yandex.ru, tmmiok@yandex.ru

Meshing theory of toroidal surface enveloping cylindrical worm drive with arc-toothed worm

This paper aims to establish a rich and complete meshing theory of the toroidal surface enveloping cylindrical worm drive with arc-toothed worm and obtain the meshing performance of the proposed worm drive. The arc-toothed worm is machined by a disc-shaped grinding wheel whose formulae of the installation position parameters are determined according to differential geometry. The characteristic of the contact line on the worm helicoid during cutting engagement is analyzed by establishing the Frenet frame. Then the relative motion model and some important results of the worm pair are acquired using the meshing theory of gearing. Based on this, the locus features of the knot of meshing are analyzed in different coordinate systems. In addition, the meshing points which determine the conjugated zone and instantaneous contact lines are classified according to their different positions on the tooth surface of the worm gear. Utilizing the methods of eliminating and geometric construction, the nonlinear equations are solved. Finally, a numerical example is implemented and the result shows the proposed worm drive has favorable meshing performance.