Research on Mathematical Modeling and Kinematic Simulation of Elliptic Family Gears

2013 ◽  
Vol 579-580 ◽  
pp. 300-304 ◽  
Author(s):  
Lian Xia ◽  
Da Zhu Li ◽  
Jiang Han
Keyword(s):  
Performance Analysis ◽  
Tooth Profile ◽  
High Order ◽  
Numerical Control ◽  
Gear Pair ◽  
Cylindrical Gear ◽  
Kinematic Simulation ◽  
Cad Modeling ◽  
Pure Rolling ◽  
Pitch Curve

Elliptic family gears are commonly used in non-circular gears, which include elliptic gear, high-order gear, elliptic deformed gear and high-order deformed gear, thereinto high-order deformed gear can include the elliptic family gears through adjust its order and deformed coefficient. Because non-circular gear has different tooth profile in different position of pitch curve and there is difference in the left and right tooth profile of the same gear tooth, thus the CAD modeling of non-circular gear is difficult for these characteristics; but the precise model of non-circular gear has important significance to the realization of numerical control machining, kinematic simulation and relevant mechanical analysis. This paper deduce the corresponding pure rolling mathematical model based on the pure rolling contact theory that cylindrical gear and non-circular gear mesh in the end face, and realize the CAD modeling of non-circular straight and helical gears by letting the cylindrical gear and non-circular gear make solid geometry operation, which is suitable for pitch curve with convex and concave. The non-circular gear shaping methods with equal polar and equal arc length are simulated by setting different discrete polar angles, and the transmission ratio curve and the angular acceleration curve of driven gear are get through the kinematic simulation of gear pair, which realize the transmission performance analysis of elliptic family gear pair. The above research results can be applied to the modeling and kinematic performance analysis of other non-circular gears.

Algorithms for Involute and Octoidal Bevel-Gear Generation

2004 ◽  
Author(s):  
Giorgio Figliolini ◽  
Jorge Angeles
Keyword(s):  
Great Circle ◽  
Tooth Profile ◽  
Bevel Gear ◽  
Rolling Motion ◽  
Bevel Gears ◽  
Pure Rolling ◽  
Pitch Curve ◽  
Tooth Flank ◽  
Flank Surface

A suitable formulation and the implementing algorithms for involute and octoidal bevel-gear generation are proposed in this paper. In particular, the exact spherical involute tooth profile of bevel gears and their crown-rack is obtained through the pure-rolling motion of a great circle of the fundamental sphere on the base cone. Moreover, the tooth flank surface of octoidal bevel gears is obtained as the envelope of the tooth flat flank of the octoidal crown-rack during the pure-rolling motion of its flat pitch curve on the pitch cone. The proposed algorithms have been implemented in Matlab; several examples are included to illustrate their applicability.

Synthesis of Shaped Noncircular Gear Using a Three-Linkage Computer Numerical Control Shaping Machine

2017 ◽  
Vol 139 (7) ◽  
Author(s):  
Fangyan Zheng ◽  
Lin Hua ◽  
Xinghui Han ◽  
Bo Li ◽  
Dingfang Chen
Keyword(s):  
Manufacturing Process ◽  
Computer Numerical Control ◽  
Numerical Control ◽  
Cylindrical Gear ◽  
Manufacturing Method ◽  
Proposed Model ◽  
Gear Hobbing ◽  
Pitch Curve ◽  
Internal Gear ◽  
The Mathematical Model

Gear shaping, commonly regarded as the most widely used machining method for cylindrical gear, is in fact an ideal manufacturing method for noncircular gear due to its merit of not being restricted by gear type or pitch curve in contrast to gear hobbing. However, concerning researches are mainly focused on the generation of noncircular straight external gear, paying rare attention to noncircular internal gear and noncircular helix gear. Considering that this paper, through using a three-linkage computer numerical control (CNC) shaping machine, aims to synthesize shaped noncircular gear, covering external, internal, straight, and helix gear. The mathematical model, a three-linkage model, is first established. The corresponding manufacturing process in practice is subsequently discussed. Finally, with practical shaping experiments, the correctness of the proposed model and the appropriateness of the manufacturing process are verified.

Synthesis of N-Lobed Involute Modified Noncircular Bevel Gears

2014 ◽  
Author(s):  
Kan Shi ◽  
Jiqiang Xia ◽  
Chunjie Wang ◽  
Chunming Geng ◽  
Rui Wang
Keyword(s):  
3D Models ◽  
Tooth Profile ◽  
Transmission Ratio ◽  
Bevel Gears ◽  
Pure Rolling ◽  
Modification Method ◽  
Variable Transmission ◽  
Pitch Curve ◽  
Relationship Of ◽  
The Relationship

Noncircular bevel gears (NBG) can achieve variable transmission between intersecting axes. The transmission ratio function is obtained when giving the involute NBG’ pitch curve equations during their pure rolling. The relationship of least number of teeth avoiding undercutting and the pitch curve curvature of NBG is analyzed. Being directed against N-lobed involute NBG drive, a kind of varying-coefficient-profile-shift-modification method has been presented. The equations of the modified addendum and dedendum curves are implemented. The algorithm for generating tooth profile of NBG by shaper cutter under UG platform is proposed and significant examples are included. The 3D models of a pair of conjugate N-lobed involute modified NBG are demonstrated to verify the correctness of this modification method.

scholarly journals Design of Linear Functional Noncircular Gear with High Contact Ratio Used in Continuously Variable Transmission

2020 ◽  
Author(s):  
Ya-Nan Hu ◽  
Chao Lin ◽  
Chun-Jiang He ◽  
Yong-Quan Yu ◽  
Zhi-Qin Cai
Keyword(s):  
Bearing Capacity ◽  
Linear Functional ◽  
Continuously Variable Transmission ◽  
Tooth Profile ◽  
Gear Transmission ◽  
Gear Pair ◽  
Contact Ratio ◽  
Variable Transmission ◽  
Pitch Curve ◽  
Working Section

Abstract Continuously variable transmission (CVT) of noncircular gears has the technical advantages of large bearing capacity and high transmission efficiency. This paper focuses on the linear functional noncircular gear pair to realize continuously variable transmission. According to the required working section transmission ratio of noncircular gear pair, the transmission ratio function in the non-working section, was constructed by using polynomial, and then the influence of pitch curve parameters in working section on which in non-working section was also analyzed. Furthermore, the pitch curve of the linear functional noncircular gear pair, which are suitable for transmission, were obtained. In order to improve the stability and bearing capacity of gear transmission, the noncircular gear pair transmission with high contact ratio was designed, and an indirect method to verify the contact ratio by detecting the contact length error of tooth profile was proposed. The error was calculated by comparing the actual contact tooth profile length obtained from the rolling experiment with the accurate value of contact tooth profile length based on the principle of gear and the property of involute profile, and the noncircular gear transmission with high contact ratio was verified.

Numerical Algorithm of Tooth Profile of Noncircular Gear Pair

2012 ◽  
Vol 201-202 ◽  
pp. 194-197
Author(s):  
Ming Hui Fang
Keyword(s):  
Numerical Algorithm ◽  
Cutting Tool ◽  
Gear Tooth ◽  
Tooth Profile ◽  
Gear Pair ◽  
Curve Part ◽  
Generating Method ◽  
Curve Change ◽  
Pitch Curve ◽  
Real Machine

According to the generating method theory, noncircular gear with all convex pitch curve can be processed by generating, author designed noncircular gear tooth profile, described a method of precisely calculating noncircular gear tooth profile. And proposed automatically optimizes the noncircular gear tooth profile meshing curve part separately by the identical rack cutting tool different side envelope. Through rack cutting tool along two gear center of rotation segment deviation pitch curve change modified quantity to avoid tooth undercut, realization nonzero modified. The driving gear and the driven gear tooth profile meshing curve part separately by the identical rack cutting tool different side envelope, guarantees when meshing zero clearance. The noncircular gear machined by this method has been used in real machine, which verifies the feasibility of the method.

scholarly journals Research and Manufacturing of Oval Gear Pair Applied in Rotor of a Roots Type Compressor

2021 ◽  
Vol 31 (5) ◽  
pp. 31-38
Keyword(s):  
Mathematical Model ◽  
Theoretical Calculation ◽  
Tooth Profile ◽  
Transmission Ratio ◽  
Design Parameters ◽  
Gear Pair ◽  
Cylindrical Gear ◽  
Special Cases ◽  
Oval Gear

This study presents a method of shaping the tooth profile of an oval gear pair with the modified cycloid profile of the ellipse. The driving oval gear was shaped by a rack cutter with the profile of an improved cycloid, and the tooth profile of the driven oval gear was shaped by a shaper cutter to ensure that gear pair will mesh accurately with a function of transmission ratio. Mathematical model of the oval gear pair was established according to the theory of the non-circular gear with consideration to the conditions of the design parameters to avoid undercutting and tooth distribution of the oval gear. An improved pair of cycloid oval gears is designed and manufactured according to the results of the theoretical calculation. Furthermore, this gear pair is specifically applied to a Roots-type compressor to demonstrate its applicability in special cases where the traditional cylindrical gear pair cannot be replaced.

Computer Aided Design and Three-Dimensional Modeling of High-Order Deformed Elliptical Gear

2013 ◽  
Vol 391 ◽  
pp. 178-181
Author(s):  
Zhi Dong Huang ◽  
An Min Hui ◽  
Peng Chen ◽  
Yu Wang
Keyword(s):  
Computer Aided Design ◽  
Three Dimensional ◽  
High Order ◽  
Numerical Control ◽  
Element Analysis ◽  
Numerical Control Machining ◽  
Gear Teeth ◽  
Three Dimensional Modeling ◽  
Pitch Curve ◽  
Aided Design

The characteristics of high-order deformed elliptical gear is analyzed. The parameters of high-order deformed elliptical gear are chosen and calculated. The modeling method of high-order deformed elliptical gear is presented. The shape of pitch curve is determined. The position and orientation of gear teeth are clarified. The three-dimensional solid model of high-order deformed elliptical gear is achieved. The method and the result facilitate finite element analysis and numerical control machining simulation of high-order deformed elliptical gear.

The Design and Experiment of Oval Bevel Gear

2011 ◽  
Vol 86 ◽  
pp. 297-300 ◽  
Author(s):  
Chao Lin ◽  
Yu Jie Hou ◽  
Qing Long Zeng ◽  
Hai Gong ◽  
Ling Nie ◽  
...  
Keyword(s):  
Tooth Profile ◽  
Bevel Gear ◽  
Gear Transmission ◽  
Experimental Result ◽  
Engagement Theory ◽  
Generating Method ◽  
Solid Models ◽  
Complex Design ◽  
Pure Rolling ◽  
Pitch Curve

Aiming at the complex design and analysis methods of noncircular bevel gear, combining the gear space engagement theory and the generating method of tooth profile, the profile generating process was seen as the pure rolling between the pitch line of virtual processing tool and pitch curve of oval bevel gear based on the virtual generating method. The blank parameters and relation equation between tool parameters and procedure parameters of generating method were deduced. General parametric solid models of oval bevel gear were founded. Based on experimental result of oval bevel gear transmission, the design profile of oval bevel gear which was obtained according to generating method was verified.

scholarly journals μ-Synthesis for Fractional-Order Robust Controllers

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 911
Author(s):  
Vlad Mihaly ◽  
Mircea Şuşcă ◽  
Dora Morar ◽  
Mihai Stănese ◽  
Petru Dobra
Keyword(s):  
Control Problem ◽  
Fractional Order ◽  
Design Procedure ◽  
High Order ◽  
Numerical Control ◽  
Performance Criteria ◽  
Iteration Algorithm ◽  
Optimization Approach ◽  
Cnc Machine ◽  
Bee Colony

The current article presents a design procedure for obtaining robust multiple-input and multiple-output (MIMO) fractional-order controllers using a μ-synthesis design procedure with D–K iteration. μ-synthesis uses the generalized Robust Control framework in order to find a controller which meets the stability and performance criteria for a family of plants. Because this control problem is NP-hard, it is usually solved using an approximation, the most common being the D–K iteration algorithm, but, this approximation leads to high-order controllers, which are not practically feasible. If a desired structure is imposed to the controller, the corresponding K step is a non-convex problem. The novelty of the paper consists in an artificial bee colony swarm optimization approach to compute the nearly optimal controller parameters. Further, a mixed-sensitivity μ-synthesis control problem is solved with the proposed approach for a two-axis Computer Numerical Control (CNC) machine benchmark problem. The resulting controller using the described algorithm manages to ensure, with mathematical guarantee, both robust stability and robust performance, while the high-order controller obtained with the classical μ-synthesis approach in MATLAB does not offer this.

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