Non-circular Gear Research Articles

Research on the Tooth Surface Integrity of Non-Circular Gear WEDM Based on HPSO Optimization SVR

Non-circular gears have the characteristics of gear ratio accuracy, good dynamic performance, and wide application prospects but are difficult to manufacture. Wire electrical discharge machining (WEDM) can process almost all kinds of non-circular gear. In order to solve the problem that the process parameters are mainly adjusted using the operator’s experience and to improve the surface quality of non-circular gears machined using WEDM, this research took Pascal gears processed with a fast-walking WEDM machine as the object, conducted orthogonal tests, and used hybrid particle swarm optimization (HPSO) to optimize support vector regression (SVR) with different kernel functions, to predict various surface integrity indicators. The results showed that the rbf kernel function had a better performance in the prediction model of surface integrity indicators, which can provide a reference for the parameter selection of non-circular gear machining using WEDM. The final predicted results were R2 = 0.9978, MAPE = 0.4534 for surface roughness, R2 = 0.9483, MAPE = 3.1673 for surface residual stress, and R2 = 0.9786, MAPE = 0.4779 for surface microhardness.

Design and extraction of tooth profile of denatured elliptical gear

Elliptical gear is a common type of non-circular gear; it’s theory and design methods are mature. But the research on denatured elliptical gear was not enough, the design method of tooth profile was single, low precision, and there may occur tooth overlap at the splice when the denatured elliptical gear is generated by the envelope method. The denatured elliptical gear design in the existing literature did not include tooth profile point extraction, so the denatured elliptical gear design only exists in theory, it can’t obtain the necessary data for processing and production. Given this situation, a design method is proposed to design denatured elliptical gear, and the tooth profile data are extracted. The main steps are as follows: firstly, the arc length of gear was calculated, and the arc length corresponding to the rotation angle is obtained. The piecewise interpolation of the gear with equal arc length was spliced together to get a complete envelope image, then, the image was binary processed. Finally, using the designed program, the tooth profile pixels were extracted by the eight-neighborhood algorithm. This method provides a new way of designing denatured elliptical gear, this method is stable and has a fast operation speed. The tooth profile data can be directly used for 3D modeling or wire cutting machining.

A novel method for longitudinal modification and tooth contact analysis of non-circular cylindrical gears

A novel method for longitudinal modification and tooth contact analysis of non-circular cylindrical gears

Rigid-Body Guidance Synthesis of Noncircular Gear-Five-Bar Mechanisms and Its Application in a Knee Joint Rehabilitation Device

At present, the research objects of rigid-body guidance synthesis are mostly limited to pure linkages, and there is little research on the combined mechanisms of gears or cams and linkages. In order to expand the research objects of rigid-body guidance and improve the kinematic mapping theory, this paper proposes a rigid-body guidance method of noncircular gear-five-bar combined mechanisms. A noncircular gear-five-bar mechanism can be regarded as a combination of a 2R (two revolute joints) open chain, a 3R (three revolute joints) open chain and a pair of noncircular gears. Firstly, the circle point curves and circle center point curves of the 2R and 3R open chains are obtained by using kinematic mapping, and they are formed into a double crank five-bar linkage. Secondly, the B-spline curve is used to fit the rotation angle relationship of the gear pair to obtain the pitch curves of noncircular gears. Finally, aiming at correcting patients’ abnormal gait, a noncircular gear-five-bar exoskeleton knee joint rehabilitation device is designed based on four task poses. The prototype is developed and the wear test is carried out. The test results verify the correctness of the rigid-body guidance synthesis method and the effectiveness of rehabilitation training.

Approximate Motion Generation of an Epicyclic Gear Train With Noncircular Gears Based on Optimization–Homotopy Algorithm

Abstract This study aims to make approximate motion generation that has multiple solutions to facilitate the selection of the best mechanism parameters. Therefore, an approximate synthesis method for an epicyclic gear train (EGT) with noncircular gears based on optimization–homotopy algorithm is proposed in this paper. First, the multipose synthesis objective function of a simplified model of EGT (planar RR dyad) is established. The approximate motion generation is transformed into the extremum problem of multivariate function, and a variable is introduced to expand the solution space. A homotopy algorithm is used to solve the gradient equation of the objective function to obtain the finite range solution domain. Second, the feasible solution domain of EGT is obtained by identifying the total rotation of revolute joints of the RR dyad, and a mechanism error evaluation index is established. In accordance with actual constraints, the most suitable solution is selected in the feasible solution domain, and the relative angular displacement relationship of the selected RR dyad is used to determine the transmission ratio and pitch curve of noncircular gears to realize the design of an EGT. Lastly, a design example is provided to illustrate the feasibility of the method.

Design and Application of Non-Circular Gear with Cusp Pitch Curve

To solve the design problem of non-circular gears with cusp pitch curves, this paper proposed a new variable-involute and incomplete variable-cycloid composite tooth profile (VIIVC-CTF), deduced the new VIIVC-CTF mathematical model, and constructed the conjugate gear model based on the envelope method. The design software of the non-circular gear with a cusp pitch curve was developed based on MATLAB. The variation law of rolling radius on an incomplete cycloid profile and its characteristics such as pressure angle and radius of curvature were analyzed. The variation relationship of the rolling radius on the meshing line and the contact ratio of the VIIVC-CTF were studied. The variation relationship of incomplete variable-cycloid profile shape, pressure angle, and curvature radius corresponding to different elliptical eccentricities were analyzed. The meshing analysis of the non-circular gear transmission mechanism was carried out based on virtual software. A comparison of the consistency of the theoretical value and simulation value of the transmission ratio curve verified that the tooth profile design method was feasible, and the VIIVC-CTF was applied to the seedling pick-up mechanism of the non-circular gear planetary gear train. Through the seedling picking experiment of the seedling pick-up mechanism, the feasibility of the application of the VIIVC-CTF was verified.

Tooth surface Hobbing analysis for unconventional gears with ultrasonic-assisted motion

As an atypical conventional gear pair, non-circular gear has a variable transmission ratio, which can be used to improve the structure and transmission characteristics in some applications, such as variable speed grinding mechanism, wool blending machine guide mechanism, continuously variable transmission structure and so on. According to the friction theory of conjugate surfaces, the transmission characteristics will be affected by the micro topography of tooth surface, but the shaped tooth surface of non-circular gear with normal hobbing method is not uniform, because of its variable radius and flank orientations of tooth surface. The research on the relationship between micro topography and parameters of non-circular gear is necessary, thus an improved manufacturing method with ultrasonic-assisted motion was presented in this paper, the mathematical equations for the theoretical tooth surface of non-circular gear with and without ultrasonic-assisted motion have been derived in this paper, also the equations for cutting parameters (step displacement, step rotating angle) have been proposed, shown that all the parameters (initial parameters of gear and gear hob, cutting parameters in manufacturing process) will affect the formed micro surfaces. In detail, the height Hm and width Sm of micro undulating tooth surface periodically increase and decrease, getting the maximum value at θn=π, where is the maximum radius. The height Hm and width Sm of micro tooth surface will increase with any control parameter increase, the height Hm of most micro peaks is near 6.5um in normal hobbing process. While, the tooth surface with ultrasonic-assisted motion is more regular and clearer with the maximum height Hm decrease from 6.5 um to 2.7 um, getting more compact cutting tracks and lower surface roughness. The experimental results shows that ultrasonic-assisted hobbing has obviously positive influence for micro tooth surface, provides the theoretical support for a further analysis and better application of unconventional gear pair.

Design of piecewise deformed elliptical gear with closed pitch curve and its conjugate pair

The family of elliptical gears with closed pitch curves includes elliptical gears and deformed elliptical gears. Elliptical gears are not only the most widely used non-circular gears but also the research hotspot in the field of non-uniform transmission. However, adjusting the shape of its pitch curve is difficult, which seriously restricts its popularization. This article proposes a piecewise deformed elliptical gear with a closed pitch curve, which realizes the unity of all kinds of elliptical gears in the mathematical model, makes the shape of the pitch curve and gear ratio easier to adjust, and expands the connotation and application scope of the family of elliptical gears. Moreover, the design approach of transmission with piecewise deformed elliptical gear was studied, and the inspection method of transmission performance was provided. The internal relationship between piecewise deformed elliptical gears and the family of elliptical gears was then analyzed. Finally, the method of computer-aided design (CAD) system developed was also provided. Additionally, a case was provided to verify the above theories. The designed conjugate pair can realize a correct meshing and adjust the gear ratio more flexibly, laying a theoretical foundation to expand the application field of non-uniform transmission.

Processing Method of Gearbox with Non-Circular Gear Train and Its Application in Rice Potted Seedling Transplanting Mechanism

A non-circular gear train has the advantages of compact structure, easy to achieve dynamic balance, and is often used to drive an end-effector to achieve complex trajectory and attitude. The gearbox is the key component of non-circular gear train, which has complex structure, and requires high precision of each wheel shaft hole and its relative position. Conventional methods often use numerical control process tools to process the gearbox with non-circular gear train. The processing cost is too high, and there are some problems, such as shaft hole deflection and it can be difficult to ensure the phase position. This research has proposed a processing method of the gearbox with non-circular gear train and developed the corresponding combination tool. Taking the gearbox of a rice potted seedling transplanting mechanism as an example, a special cutting tool with five boring tools was designed which can process five holes at one clamping, and greatly improve the efficiency and accuracy of gearbox processing. The processed gearbox and other parts were assembled into a rice potted seedling transplanting mechanism and assembled into a rice potted seedling transplanting process to conduct a seedling transplanting performance experiment. The results show that the processing of the gearbox with non-circular gear train satisfies the requirement of processing accuracy, and the transplanting mechanism can accurately complete seedling picking, transporting, planting, and other series of actions. The processing method is suitable for the large-scale manufacturing of transplanting mechanism. In this research, aiming at the double arm transplanting mechanism of high-speed rice potted seedling transplanting, the gearbox was analyzed, and a processing method of the gearbox was proposed.

Screw analysis and experimental study of spatial noncircular gear compound transmission.

Curve face gear pair and noncircular bevel gear pair are collectively referred to as spatial noncircular gear pair (SNCGP). Different from the traditional gear transmission with fixed shafts, spatial noncircular gear (SNCG) compound transmission can realize compound motion with variable transmission ratio between intersecting axes, and has a broad engineering application prospect. It includes two categories named speed reduction and speed increase. Based on the gear meshing theory, screw theory and calculus, a screw analysis method of SNCG compound transmission was proposed, which directly and comprehensively reflected the compound motion characteristics of rotation and axial movement. Firstly, a unified coordinate system was established. The screw geometric characteristics of compound motion were discussed, including instant screw axis, axode, pitch surface and its normal equidistant surface. The screw principle of compound motion was revealed. Then, according to this method, combined with the motion process of generator with different tooth profiles, the tooth surface of each SNCG was obtained. Finally, gear machining, tooth surface measurement, transmission ratio measurement and axial displacement measurement were carried out, which verified the correctness of this screw analysis method for SNCG compound transmission, and laid a foundation for further study and application.

Optimal design of transplanting mechanism with differential internal engagement non-circular gear trains

This study aimed at the problems of unequal speed transmission ratio mutual restriction and side gap accumulation of the transplanting mechanism with single-degree-of-freedom K-H-V non-circular planetary gear train, which leads to poor trajectory and attitude, and poor precision of movement. This study has proposed a simple structure of transplanting mechanism with differential internal engagement non-circular planetary gear trains, which reconstructs the complex transplanting trajectory and attitude of the planting arm through single-stage unequal speed transmission. The working principle of the transplanting mechanism was analysed, and the kinematic theoretical model of the transplanting mechanism was established. The optimal design software for the transplanting mechanism was developed based on the visual platform. The dimensions optimisation on the transplanting mechanism was completed considering the requirements with strong coupling, and multi-objective and a set of superior mechanism parameters were obtained. The design theory and method of the internal engagement non-circular gear pair were proposed based on the generating principle. The correctness and accuracy of the trajectory and attitude of the transplanting mechanism were verified through virtual simulation experiments. The experiments show that the designed transplanting mechanism with internal engagement non-circular planetary gear trains was compact in structure, the trajectory meets the requirements of multi-objective transplanting, and the trajectory and attitude can be accurately reproduced, which provides a new feasible solution for the innovative design of the transplanting mechanism.

The methodology of design of satellite working mechanism of positive displacement machine

In this paper is described a methodology of design of satellite mechanism consisting of two non-circular gears (externally toothed rotor and internally toothed curvature) and circular gears (satellites). In the presented methodology is assumed that the rotor pitch line is known, and the curvature pitch line is necessary to designate. The presented methodology applies to mechanisms for which the number of the curvature humps is at least one greater than the number of rotor humps. The selection of the number of gears and the number of teeth in gear and rotor and curvature is also presented. The methodology of calculating the position of the satellite center and the angle of its rotation in order to shape the teeth on the rotor and curvature is presented. The article is also showed different types of satellite mechanisms—satellite mechanisms with the different numbers of humps on the rotor and curvature. The technical parameters of the mechanism for the rotor pitch line described by the cosine function are also presented.

Research on the varying-displacement-coefficient modification method for automobile steering noncircular sector gears

The local drum shape tooth profile is adopted for automobile steering noncircular sector gears, which can be obtained according to a piecewise linear function modification on CNC gear shaper. In the practical process, the actual amount of drum shape is always smaller than the theoretical amount, and we conclude that overcut of the tooth profile occurred. Because the modification coefficient is variable, we call it varying-displacement-coefficient modification(VDCM). Currently, researches on tooth profile of noncircular sector gears are primarily concerned with constant modification coefficient to improve gear transmission performance, while modification functions, overcut, and methods of reducing or eliminating the overcut are rarely touched upon.This paper focus on the VDCM method of automobile steering noncircular sector gears. On the basis of noncircular sector gear and rack pair meshing theory, the modifying motion mathematical model is first established, including kinematic relations, pith curves, track of relative instantaneous centre point, calculation formula for the modified tooth profile. And then, the reasons for the overcut are analyzed, and the local overlap and separation of the tooth profile are found. To reduce the overcut amount, a cosine function modification method is proposed. Furthermore, an inverse calculation method for the rack tool tooth profile is proposed to eliminate the overcut entirely. The VDCM modification methods are verified by simulation, trial cutting and parts measurement.

Precision Analysis of Noncircular Gears Based on CNC Machining Technology under Cloud Computing Platform

In order to solve the problem of machining error of noncircular gears, a method based on the analysis and research of the accuracy of noncircular gears based on CNC machining technology under the cloud computing platform is proposed to further improve the technical accuracy of noncircular gears. The error and error items are analyzed mainly, and a model framework based on cloud computing platform is proposed on this basis. The architecture of the system is deeply analyzed, focusing on the dynamic integration method of gear cloud measurement system resources based on cloud platform, building a gear cloud measurement system based on cloud computing platform, and testing needs to use Hadoop to install the software that needs to be prepared: virtual machine version, VMware Workstation-14.0.0 build-6661328, Linux version. The results show that the technical support of the cloud computing platform can effectively improve the accuracy of noncircular gears.

Dynamic characteristics of two-mass impact pipeline robot driven by non-circular gears

Considering small diameter pipes, special pipes conveying corrosive, and flammable medium, a two-mass impact pipeline robot driven by a non-circular gear is proposed to study the walking dynamics of the robot in a dry piping environment. The mechanism principle of the two-mass impact pipeline robot is given to illustrate the walking mechanism of the robot under the combined action of the internal force and the external isotropic coulomb friction force. A dynamic model of a two-mass impact pipeline robot driven by non-circular gear is established. The dynamic response of the robot calculated by the numerical method reveals the walking characteristics of the pipe-impact robot under nonharmonic inertial force. The results show that the moving speed of the robot fluctuating rises with the increase of the input speed. The moving speed of the robot can be effectively improved by increasing the eccentricity of the non-circular gear or setting a certain amount of compression of the spring which is between the internal oscillator and the housing.

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.

Design of two steering mechanisms for exact satisfaction of Ackermann equation

The common steering mechanisms approximately establish the Ackermann equation. The more the mechanism design parameters, the higher the accuracy can be achieved by using optimization. Also, some mechanisms have been developed to establish the Ackermann equation accurately. Most of these mechanisms contain non-circular gear or cam that have an infinite number of design parameters. In this paper, two novel mechanisms have been designed with a limited number of design parameters, but their geometric structure is such that the governing relationship is exactly the Ackermann equation. The proof of this claim has been given in detail and validated by GeoGebra software. Also, by using force analysis, it has been shown that the forces exerted on the joints in both mechanisms are uniform, and there are no vast forces on the joints.

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.

Research on non-circular helical gear NC hobbing linkage control scheme based on variable transmission ratio

Non-circular helical gears, having the advantages of cams and cylindrical gears, are widely used on various occasions, whose precision machining has always been a research focus. With the accuracy requirements increasing of non-circular gears, upgrading non-circular gear hobbing models is more and more important. This paper takes the non-circular helical gears hobbing with moving axis as the research object, and establishes the ellipse standard system for high-order non-circular gears, and derives the linkage mathematical model for machining helical non-circular gears with hobbing cutters. In this paper, a preliminary design for the processing of non-circular gear hobbing is carried out.

Pattern Classification and Gear Design of Spatial Noncircular Gear Continuously Variable Transmission

Noncircular gear and curve face gear are collectively referred to as spatial noncircular gear. Combined with the transmission characteristics of spatial noncircular gear, a new classification design method of spatial noncircular gear continuously variable transmission (CVT) pattern was proposed, including two categories named addition type and multiplication type, with a total of five subcategories. Through the combination of the transmission ratio changing mechanism and transmission selection mechanism, it can realize the CVT. The overall transmission ratio depends on the phase angle between the spatial noncircular gear pairs and is independent of the input rotation angle. Firstly, the CVT principle and transmission ratio characteristics of each pattern were analyzed. Then the tooth profile classification design of spatial noncircular gear was carried out, and the general parametric design equations of the tooth surface with different tooth profiles were obtained. Finally, through simulation, the overall transmission ratio of each pattern was compared between theory and simulation, which verified the correctness of the CVT classification design method. The advantages and disadvantages of the two categories were analyzed. Moreover, through the transmission experiments of curve face gear pairs with different tooth profiles, the transmission ratios were compared between theory and experiment. Considering the influencing factors such as machining error, assembly error and measurement error, the experimental error is within a reasonable error range, which verified the correctness of the tooth profile classification design method. This study provides a new idea for the further research and application of spatial noncircular gear CVT.