A distributed dynamic mesh model of a helical gear pair with tooth profile errors

A dynamic model of a helical gear rotor system is proposed.Firstly,a generally distributed dynamic model of a helical gear pair with tooth profile errors is developed.The gear mesh is represented by a pair of cylinders connected by a series of springs and the stiffness of each spring is equal to the effective mesh stiffness.Combining the gear dynamic model with the rotor-bearing system model,the gear-rotor-bearing dynamic model is developed.Then three cases are presented to analyze the dynamic responses of gear systems.The results reveal that the gear dynamic model is effective and advanced for general gear systems,narrow-faced gear,wide-faced gear and gear with tooth profile errors.Finally,the responses of an example helical gear system are also studied to demonstrate the influence of the lead crown reliefs and misalignments.The results show that both of the lead crown relief and misalignment soften the gear mesh stiffness and the responses of the gear system increase with the increasing lead crown reliefs and misalignments.

CNC FLEXIBLE GENERATING SPECIFIC GEAR TOOTH PROFILE BASED ON STANDARD INVOLUTE GEAR HOB

A new method named orthogonal two-way link-shift here has been proposed.Based on the method and a standard involute gear hob, a specific gear tooth profile (including anarbitrary gear tooth profile and a modified involute gear tooth profile) can be generated on aCNC(computer numerical control) bobbing machine. Computer simulation has been carried out, and theresults prove that the method is right and practicable. So, the fabrication costs can be greatlydecreased than before. The new method has momentous significance to realize gear's optimizedmodification under different work conditions.

Influence of Setting Error of Tool on Tooth Profile and Contact Point of Face Gear Drive

In order to analyze the influence of setting error of tool on both tooth profile and contact characteristic of orthogonal face gear drive,the coordinate systems with and without setting error are established.Moreover,the equations of tooth profile and contact points of face gear drive are derived by envelope principle.According to the equations,the change of tooth profile and the contact points position on face gear are analyzed.The tooth surface and contact points are obtained by numerical simulation.Results show that the tooth profile and contact characteristic of face gear drive are not sensitive to the setting error of tool.

Geometric design and simulation of tooth profile using elliptical segments as its line of action

A mathematical model of gear tooth profiles using the ellipse curve, whose curvature is convenient to control by changing the mathematical parameters as its line of action, was built based on the meshing theory. The equation of undercutting condition was derived from the model. A special epicycloidal tooth profile was also presented. An example gear drive with variation of the ellipse parameters was taken to illustrate the proposed method. The contact ratio of the gear drive designed by the proposed method was analyzed. A comparison of the property of the gear drive designed with the involute gear drive was also carried out. The results confirm that the proposed gear drive has higher contact ratio in comparison with the involute gear drive.

Generation Principle of Helix Tooth Profile in Hypocycloid Pinw heel Transmission and Its Contact Characteristics

The transmission characteristics of gear drive can be improved with the use of novel tooth profiles.A theoretical study on tooth profile of the hypocycloid pinwheel transmission and contact analysis of gear pair based on finite element method(FEM) are carried out,respectively.The line contact between mated tooth surfaces becomes point contact according to a plus movement.Through loaded tooth contact analysis(LTCA),the contact stress and load distributions for the proposed hypocycloid pinwheel transmission and the traditional one are discussed.The analysis results show that the developed tooth surfaces have anticipatory point contact characteristics under loads and contact fatigue dangerous area locates around the ultimate contact position.

Accurate Modeling of the Spiral Bevel and Hypoid Gear with a New Tooth Profile

Distinguishing with traditional tooth profile of spiral bevel and hypoid gear, it proposed a new tooth profile namely the spherical involute. Firstly, a new theory of forming the spherical involute tooth profile was proposed. Then, this theory was applied to complete parametric derivation of each part of its tooth profile. For enhancing the precision, the SWEEP method used for formation of each part of tooth surface and G1 stitching schema for obtaining a unified tooth surface are put forward and made the application in the accurate modeling. Lastly, owing to the higher accuracy of tooth surface of outputted model, it gave some optimization approaches. Given numerical example about the model can show that this designed gear with spherical involute tooth profile can achieve fast and accurate parametric modeling and provide a foundation for tooth contact analysis （TCA） in digitized design and manufacture.

THEORETICAL TOOTH PROFILE AND ITS SUBSTITUTE CURVE OF THE PINION CUTTER FOR PINHOLES

The formula for the engaging curve of the pinion cutter with pitholes and the formula forthe theoretical tooth profile of the tinion cutter for the generating gearing of pinhoes are developedbased on the engaging theory. The substitute curve of the theoretical tooth profile of the pinion cut-ter is determined by the least square method.

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

Continuously variable transmission(CVT)of noncircular gear has the technical advantages of large bearing capacity and high transmission efficiency.The key technology of CVT with noncircular gear has been broken through some countries,and is in the stage of deep application research.Although the characteristics and design methods of noncircular gear pairs have been continuously studied in China,the noncircular gear CVT is still in the preliminary exploration and research stage.The linear functional noncircular gear pair,whose transmission ratio is a linear function in the working section,to realize continuously variable transmission was the research object in this paper.According to the required transmission ratio in the working section,the transmission ratio function in the non-working section was constructed by using a polynomial.And then the influence of pitch curve parameters in the working section on which in the non-working section was also analyzed to obtain the pitch curve suitable for transmission of this gear pair.In addition,for improving the stability and bearing capacity of gear transmission,the noncircular gear pair transmission with high contact ratio was designed.Furthermore,the accurate value of the contact tooth length was calculated based on the gear principle and the characteristics of the involute tooth profile,from this the contact tooth length error was calculated by comparing the accurate value with its actual value obtained by the rolling experiment.Finally,an indirect method to verify the contact ratio by detecting the contact length error of the tooth profile was proposed.

Compensation modification of plastic gear tooth profile considering meshing deformation

The plastic gear is widely used in agricultural equipment,electronic products,aircraft,and other fields because of its light weight,corrosion resistance,and self-lubrication ability.However,it has a limited range of working conditions due to the low modulus and thermal deformation of the material,especially in high-speed and heavy-duty situations.A compensation modification method(CMM)is proposed in this paper to restrain the heat production of the plastic gear tooth surface by considering the meshing deformation,and the corresponding modification formulas are derived.Improving the position of the maximum contact pressure(CP)and the relative sliding velocity(RSV)of the tooth surface resulted in a 30%lower steady-state temperature rise of the modified plastic gear tooth surface than that of the unmodified plastic gear.Meanwhile,the temperature rise of plastic gear with CMM is reduced by 19%compared with the traditional modification of removal material.Then,the influences of modification index and the segment number of modification on the meshing characteristics of plastic gear with CMM are discussed,such as maximum CP and steadystate temperature rise,RSV,transmission error,meshing angle,and contact ratio.A smaller segment number and modification index are beneficial to reduce the temperature rise of plastic gear with CMM.Finally,an experiment is carried out to verify the theoretical analysis model.

WEDM oriented micro gear design and mesh simulation

For the design of gears manufactured with wire electrical discharge machining （WEDM） technology, determination of the primary gear parameters is discussed considering the characteristics of the machining method. Some constraint conditions on gear parameters are abnegated, which makes micro gear design more flexible. Based on gear mesh theory, the algorithm of generating gear tooth profiles is studied, which includes involute and non-involute curve segments. The phenomena of tooth profile interferences during gear mesh are analyzed, and a gear mesh simulation algorithm is designed. Based on ACIS, the WEDM oriented software for the design and mesh simulation of micro gears is developed, by which the modeling, mesh simulation and interference check can be implemented. An experiment is carried out to design and manufacture a pair of micro involute gears, and the proposed method is proved feasible.

Modification Design Method for an Enveloping Hourglass Worm Gear with Consideration of Machining and Misalignment Errors

The influences of machining and misalignment errors play a very critical role in the performance of the anti-backlash double-roller enveloping hourglass worm gear(ADEHWG).However,a corresponding efficient method for eliminating or reducing these errors on the tooth profile of the ADEHWG is seldom reported.The gear engagement equation and tooth profile equation for considering six different errors that could arise from the machining and gear misalignment are derived from the theories of differential geometry and gear meshing.Also,the tooth contact analysis(TCA) is used to systematically investigate the influence of the machining and misalignment errors on the contact curves and the tooth profile by means of numerical analysis and three-dimensional solid modeling.The research results show that vertical angular misalignment of the worm wheel(Δβ) has the strongest influences while the tooth angle error(Δα) has the weakest influences on the contact curves and the tooth profile.A novel efficient approach is proposed and used to minimize the effect of the errors in manufacturing by changing the radius of the grinding wheel and the approaching point of contact.The results from the TCA and the experiment demonstrate that this tooth profile design modification method can indeed reduce the machining and misalignment errors.This modification design method is helpful in understanding the manufacturing technology of the ADEHWG.

Investigation of a Design Modification for Double Helical Gears Reducing Vibration and Noise

To reduce vibration and noise and increase transmission efficiency, a three segment method for modifying the pinion profile was proposed. Cutter surface equations were deduced by changing the shape of the cutter-edge, substituting three segment parabolas for the line. The influence of longitudinal tooth modifications on tooth surface load distributions was discussed. Transmission error minimization and uniformity of tooth surface load distribution were chosen as optimization goals and the modified parameters were obtained by applying the complex method. Finally, an experiment comparing the loaded transmission error, vibration, and noise both before and after modifications was carried out. The results indicate that the modified design is reliable.

DESIGN OF NONCIRCULAR GEARS WITH DISCONTINUOUS PITCH CURVE

When the noncircular gear pair is applied to the continuously variable transmission （CVT） with gear, the transmission ratio function is discontinuous. In accordance with this unique characteristic, a new approach to design and analyze noncircular gears with discontinuous pitch curve is proposed. The design courses of various noncircular gear pairs with discontinuous pitch curve are unified based on the numerical algorithm of spline fitting and ＂fairing boundary condition＂. According to the particularity of discontinuous pitch curve, the rules and procedures for teeth distribution are recommended. It is explained in detail why the undercut is formed and how to manage the undercut based on meshing principle.In addition, the calculation formulas for each tooth profile segment are also derived. If the tooth profile data are calculated, the measurement and the incision process for noncircular gear can be conducted and the CAD simulation can be achieved easily. To ensure the continuity of the transmission, the transmission interference of the tooth which is located at the pitch curve joint point is managed by utilizing Bezier curve with CAD software. And the contact ratio of gear pair is obtained. The case study shows that this approach is successful and opens up a new way for the design of noncircular gear.

Non-undercutting condition and parameter optimization for cycloid ball planetary transmission

The transmission performance of cycloid ball planetary transmission(CBPT) is affected by cycloid tooth undercutting directly, and the design of CBPT can be optimized by the non-undercutting condition. Firstly, the theoretical equation of cycloid tooth is given, and the curvature radius of cycloid tooth profile is derived. Secondly, according to the relationship between the curvature radius and the distribution circle of balls, the non-undercutting condition of cycloid tooth profile is established, and the non-undercutting critical condition is deduced. Finally, the validity of the non-undercutting critical condition is verified by simulation. The result shows that the non-undercutting critical condition can be used to optimize the design of CBPT.

Research on Design Method of Non-Circular Gear Pair with Double Generating Angles

In order to enhance the bearing capacity of non-circular gear pair, the non-circular gear pair with double generating angles is proposed based on the design idea of unsymmetrical gear with double pressure angles. The tooth profile is designed by generating cutting theory, the pure rolling mathematic model that the center line of unsymmetrical rack roll along non-circular pitch curve is built, the digital model of non-circular gear with double generating angles is created through the second development method of CAD software, and then the drive characteristic and tooth strength are analyzed. The results show that the design method for double generating angles non-circular gear proposed in this paper is feasible, which is significant to improve the bearing capacity of non-circular gear pair.

Mesh stiffness evaluation of an internal spur gear pair with tooth profile shift

Tooth profile shift will change the thickness of gear teeth and a part of geometrical parameters of a gear pair, thus influencing its mesh stiffness and consequently the dynamic performances. In this paper, an analytical mesh stiffness calculation model for an internal gear pair in mesh considering the tooth profile shift is developed based on the potential energy principle. Geometrical representations of the tooth profile shift are firstly derived, and then fitted into the analytical tooth stiffness model of gears. This model could supply a convenient way for mesh stiffness calculation of profile shifted spur gears. Then, simulation studies are conducted based on the developed model to demonstrate the effects of tooth profile shift coefficient on the tooth compliances and the mesh stiffness of the internal spur gear pair. The results show that tooth profile shift has an obvious influence on the mean value, amplitude variation and phase of the mesh stiffness, from which it can be predicted that the dynamic response of an internal gear transmission system will be affected by the tooth profile shift.

Modeling and experiment of grinding wheel axial profiles based on gear hobs

Hobbing is one of the crucial gear manufacturing processes with high precision and high efficiency.In order to improve the accuracy of hobbing and hob relief grinding,more precise grinding wheels are demanded for relief grinding in the production of gear hobs.In this paper,a new and accurate mathematical method for calculating the axial profile of grinding wheel based on the corresponding gear hob to be ground is proposed.Considering that most researches have been conducted to focus on the optimization of hob geometric feature and only can be applied to a specific type of hand of helix and rake angle of gear hobs.The method in this paper can be served as a general algorithm for generating axial profile of grinding wheels on the basis of the spatial envelope theory,the principle of coordinate system transformation,and studies on normal profile of single-tooth.The accuracy of grinding hobs is based on applying the approach in practical manufacturing.Taking two typical different kinds of pre-grinding gear hobs as examples for calculation and experiment,the results of tooth profile errors strengthen the position that the validity and feasibility of this method,it can be employed for gear hob design and grinding processing.

Geometrical design of variable ratio tooth profile based on Boolean subtraction operation and a novel modification method

Variable transmission ratio racks show great potential in rice transplanters as a key component of variable transmission ratio steering to balance steering portability and sensitivity.The objective of this study was to develop a novel geometrical design method to achieve quick,high-quality modeling of the free curvilinear tooth profile of a variable transmission ratio rack.First,a discrete envelope motion 3D model was established between the pinion-sector and the variable transmission ratio rack blank based on the mapping relationship between the rotation angle of the pinion-sector and the displacement of the rack,according to the variable transmission ratio function.Based on the loop Boolean subtraction operation,which removed the pinion-sector from the rack blank during all moments of the discrete motion process,the final complex changing tooth shape of the variable transmission ratio rack was enveloped.Then,since Boolean cutting residues made the variable ratio tooth surface fluctuant and eventually affected the precision of the model,this study proposed a modification method for establishing a smooth and continuous tooth profile.First,a novel fitting algorithm used approximate variable ratio tooth profile points extracted from the Boolean cutting marks and generated a series of variable ratio tooth profiles by utilizing B-spline with different orders.Next,based on a transmission stability simulation,the variable ratio tooth profile with optimal dynamic performance was selected as the final design.Finally,tests contrasting the transmission stability of the machining samples of the initial variable ratio tooth profile and the final variable ratio tooth profile were conducted.The results indicated that the final variable ratio tooth profile is more effective than the initial variable ratio tooth profile.Therefore,the proposed variable ratio tooth profile modeling and modification method for eliminating Boolean cutting residues and improving surface accuracy is proved to be feasible.

Evaluation of measurement uncertainty based on grey system theory for small samples from an unknown distribution

To evaluate measurement uncertainty for small sample size and measurement data from an unknown distribution, we propose a grey evaluation method of measurement uncertainty based on the grey relation coefficient. The uncertainty of measurement is analyzed using grey system theory, and the defects of the grey evaluation model of measurement uncertainty (GEMU) are studied. We then establish an improved grey evaluation model of measurement uncertainty (IGEMU). Simulations show that the precision of IGEMU is greater than that of GEMU, and that sample size has only a small effect on the precision of IGEVU. In particular, IGEMU is applied to evaluating measurement uncertainty for small sample size and measurement data from an unknown distribution. The measurement uncertainty of total profile deviation, which is measured by the CNC gear measuring center, can be evaluated by a combination of IGEMU and the Monte Carlo method.

Method for generating non-circular gear with addendum modification and its application in transplanting mechanism

With a focus on the design of tooth profiles of non-circular gears with addendum modifications that conform to the given complex transmission ratio curves and based on a series of discrete point sets on the pitch curves,the criteria for tooth profile distortion and undercut in non-circular gears were proposed.With the constraint of no tooth profile distortion and undercut,the addendum modification coefficient was introduced into the calculation of the distorted and undercut segments of the tooth profile of a non-circular gear,and the addendum modification coefficient of each contact point of the non-circular gear was obtained.Based on the generating principle for gear shaping with gear cutters,a theoretical model for calculating the tooth profile of non-circular gears with addendum modification was derived.The cutting and radial feeding motions of the pinion cutter were ignored,and only the meshing motion was considered.The involute tooth profiles of the pinion cutter enveloped the tooth profile of the non-circular gear.3D automatic non-circular gear model generation software was developed,which was a secondary development product of the software UG.A non-circular gear in a rice potted seedling transplanting mechanism was designed,and the transplanter was developed and tested.The test results showed that the designed tooth profile of non-circular gear achieved the variable transmission ratios required by the transplanting mechanism.When the transplanting efficiencies were 140 plants/min,160 plants/min and 180 plants/min,the transplanter completed the rice potted seedling transplanting operation with high quality.