Time-Varying Mesh Stiffness Calculation and Dynamic Modeling of Spiral Bevel Gear with Spalling Defects

Time-varying mesh stiffness(TVMS)is a vital internal excitation source for the spiral bevel gear(SBG)transmission system.Spalling defect often causes decrease in gear mesh stiffness and changes the dynamic characteristics of the gear system,which further increases noise and vibration.This paper aims to calculate the TVMS and establish dynamic model of SBG with spalling defect.In this study,a novel analytical model based on slice method is proposed to calculate the TVMS of SBG considering spalling defect.Subsequently,the influence of spalling defect on the TVMS is studied through a numerical simulation,and the proposed analytical model is verified by a finite element model.Besides,an 8-degrees-of-freedom dynamic model is established for SBG transmission system.Incorporating the spalling defect into TVMS,the dynamic responses of spalled SBG are analyzed.The numerical results indicate that spalling defect would cause periodic impact in time domain.Finally,an experiment is designed to verify the proposed dynamic model.The experimental results show that the spalling defect makes the response characterized by periodic impact with the rotating frequency of spalled pinion.

Design of Pinion Machine Tool-settings for Spiral Bevel Gears by Controlling Contact Path and Transmission Errors

This paper proposes a new approach to design pinion machine tool-settings for spiral bevel gears by controlling contact path and transmission errors. It is based on the satisfaction of contact condition of three given control points on the tooth surface. The three meshing points are controlled to be on a predesigned straight contact path that meets the pre-designed parabolic function of transmission errors. Designed separately, the magnitude of transmission errors and the orientation of the contact path are subjected to precise control. In addition, in order to meet the manufacturing requirements, we suggest to modify the values of blank offset, one of the pinion machine tool-settings, and redesign pinion ma- chine tool-settings to ensure that the magnitude and the geometry of transmission errors should not be influenced apart from minor effects on the predesigned straight contact path. The proposed approach together with its ideas has been proven by a numerical example and the manufacturing practice of a pair of spiral bevel gears.

GEOMETRIC MODELING OF GENERATING-MANUFACTURED SPIRAL BEVEL GEARS BASED ON CUTTING SIMULATION

A geometric modeling method for generating-manufactured spiral bevel gears（SBGs） is proposed. It consists of two steps： （1） creating a reference model by simulating the process of cutting spiral bevel gear,（2） reconstructing the final solid model by collecting data points from the reference model and fitting these points into NURBS surfaces. In this method,cutting simulation avoids abstruse mathematical theories and complex methods,thus making it convenient to obtain data points on the complex tooth surface before the gear is manufactured and efficient to increase the accuracy of the solid model. Also,the representations of tooth surfaces of the final model is unified as a NURBS surface function. The NURBS surface is continuous and smooth,thus it is available for wide applications in CAD/CAE. The experiment proves that the method can be used to establish an accurate pair of SBG models,thus providing a feasible and effective way for CAD/CAE modeling.

ALGORITHMS OF GENERATION MOTION AND DIRECT INTERPOLATION FOR SPIRAL BEVEL GEAR NC MACHINING

The generating motion of the generating gear and the work gear on spiral bevel gear NC machining is analyzed. The mathematical model of the tooth surface of spiral bevel gear is presented. A direct interpolation algorithm of spiral bevel gear NC machining is proposed, thus establishing the relationship between the motion of the cutter-head center and the rotation of the work gear. The interpolation algorithm is implemented to control the gear cutting on self-developed spiral bevel gear NC cutting machine. An example is given to verify the mathematical model and the interpolation algorithm.

Pinion Tooth Surface Generation Strategy of Spiral Bevel Gears

Aviation spiral bevel gears are often generated by spiral generated modified（SGM） roll method.In this style,pinion tooth surface modified generation strategy has an important influence on the meshing and contact performances.For the optimal contact pattern and transmission error function,local synthesis is applied to obtain the machine-tool settings of pinion.For digitized machine,four tooth surface generation styles of pinion are proposed.For every style,tooth contact analysis（TCA） is applied to obtain contact pattern and transmission error function.For the difference between TCA transmission error function and design objective curve,the degree of symmetry and agreement are defined and the corresponding sub-objective functions are established.Linear weighted combination method is applied to get an equivalent objective function to evaluate the shape of transmission error function.The computer programs for the process above are developed to analyze the meshing performances of the four pinion tooth surface generation styles for a pair of aviation spiral bevel gears with 38/43 teeth numbers.The four analytical results are compared with each other and show that the incomplete modified roll is optimal for this gear pair.This study is an expansion to generation strategy of spiral bevel gears,and offers new alternatives to computer numerical control（CNC） manufacture of spiral bevel gears.

Measurement and Compensation of Deviations of Real Tooth Surface of Spiral Bevel Gear

This paper presents a method for measurement of deviation of the real gear tooth surface from the theoretical one with a coordinate measurement machine and compensation of repeatable parts. By investigation of characteristics of distortion of the gear tooth surface along the circle direction, the deviation is derived from distortion, and the definition of deviation with the geometrical invariability is proposed. Then the approach for determination of the location and orientation of the gear with respect to the coordinate measurement machine and the measurement way are developed. The deviation is represented with a difference surface, and an algorithm for derivation of parameters of global form deviations from the discrete points has been provided. Finally, the compensation approach is discussed.

Effect of static transmission error on dynamic responses of spiral bevel gears

The effect of static transmission error on nonlinear dynamic response of the spiral bevel gear system combining with time-varying stiffness and backlash was investigated.Firstly,two different control equations of the spiral bevel gear model were adopted,where the static transmission error was expressed in two patterns as predesigned parabolic function and sine function of transmission errors.The dynamic response,bifurcation map,time domain response,phase curve and Poincare map were obtained by applying the explicit Runge-Kutta integration routine with variable-step.A comparative study was carried out and some profound phenomena were detected.The results show that there are many different kinds of tooth rattling phenomena at low speed.With the increase of speed,the system enters into stable motion without any rattling in the region(0.72,1.64),which indicates that the system with predesigned parabolic function of transmission error has preferable capability at high speed.

Analysis and modeling of error of spiral bevel gear grinder based on multi-body system theory

Six-axis numerical control spiral bevel gear grinder was taken as the object, multi-body system theory and Denavit-Hartenberg homogeneous transformed matrix （HTM） were utilized to establish the grinder synthesis error model, and the validity of model was confirmed by the experiment. Additionally, in grinding wheel tool point coordinate system, the errors of six degrees of freedom were simulated when the grinding wheel revolving around C-axis, moving along X-axis and Y-axis. The influence of these six errors on teeth space, helix angle, pitch, teeth profile was discussed. The simulation results show that the angle error is in the range from -0.148 4 tad to -0.241 9 rad when grinding wheel moving along X, Y-axis; the translation error is in the range from 0.866 0 μm to 3.605 3μm when grinding wheel moving along X-axis. These angle and translation errors have a great influence on the helix angle, pitch, teeth thickness and tooth socket.

Influence of cutter diameter on meshing performance in spiral bevel gears

Playing a critical role in transmitting movement and power, the meshing performance of spiral bevel gears has a significant effect on products' operational performance. To evaluate the meshing performance, the accurate three-dimensional(3D) spiral bevel gear models are established through the Pro/E and MATLAB softwares, and the finite element analysis(FEA) methods are applied to the theoretical investigation of the influence of cutter diameter on meshing performance in spiral bevel gears. The results obtained show that the cutter diameter has a significant influence on spiral bevel gears' meshing performance, such as the contact area, contact pressure, bending stress, torsional stiffness and transmission error.

NC Machining of Spiral Bevel Gear and Hypoid Gear Based on Unity Transformation Model

A unity transformation model (UTM) was presented for flexible NC machining of spiral bevel gears and hypoid gears. The model can support various machining methods for Gleason spiral bevel gears and hypoid gears, including generation machining and formation machining for wheel or pinion on a universal five-axis machining center, and then directly produce NC codes for the selected machining method. Wheel machining and pinion machining under UTM were simulated in Vericut 6.0 and tested on a five-axis machining center TDNC-W2000 with NC unit TDNC-H8. The results from simulation and real-cut verify the feasibility of gear machining under UTM as well as the correctness of NC codes.

Research on global form deviations for spiral bevel gear based on latticed measurement

In order to improve the machining ac cu racy of spiral bevel gear,difference surface was adopted to characterize its gl obal form deviations quantifiably and correct its deviations.The theoretical to oth surface model of spiral bevel gear was built,and the actual tooth surface o f spiral bevel gear had been got by using latticed measurement.The equation of difference surface which can characterize the actual tooth surface deviation s was built by means of mathematical method in combination with measurement prin ciple.The quantitative mathematical relationship between the actual tooth surfa ce deviations of spiral bevel gear and the corrected values of the machine-sett ing parameters had been referred,and the theoretical correction formula of the global form deviations had been got by the least square method.Finally,the pinion of spiral bevel gear in the automobile rear axle has been set for an exam ple to account for the effectiveness of the deviation correction by use of the d ifference surface method.

METHOD FOR PRECISE CALCULATION OF ROOT STRESS OF SPIRAL BEVEL GEARS

A method for precise calculation of tooth root stress of spiral bevel gears is presentedand developed. On the basis of the machine settings analysis, tooth geometry anaysis and loadedtooth contact analysis, by using the tooth surface distribution load from tooth load analysis, thecalculation model is established and the root stress is calculated by means of finite element meth-od. The method mentioned is verified by a tested gears example.

STUDY ON DOUBLE CIRCULAR ARC PROFILE SPIRAL BEVEL GEAR TRANSMISSION

A stepped double arc spiral bevel gear tooth profile is examined. The development ofconjugate tooth surfaee of double are profile spiral bevel gears is described. The design and man-ufacture of the cutter is also discussed. Experimental results show that these gears have high loadcarrying capacity and that they are suitable for heavily loaded transmission. It is asserted thatsuch gears have higher power-to-weight ratios, are longer-lived, and provide more reliable per-formance than ordinary spiral bevel gears.

Design and analysis of spiral bevel gears with seventh-order function of transmission error

This paper proposes a new approach to mial function of transmission error （TE） for spiral design and implement a seventh-order polyno- bevel gears with an aim to reduce the running vibration and noise of gear drive and improve the loaded distribution of the tooth. Based on the constraint conditions of predesigned seventh-order polynomial function curve and the theory of linear algebra, the polynomial coefficients of the seventh-order polynomial function of transmission error can be obtained. By applying a method named reverse tooth contact analysis, the modified roll coefficients as well as parts of machine-tool settings for the face-milling of spiral bevel gears can be individually determined. Therefore, a predesigned seventh-order polynomial function of transmission error for spiral bevel gears can be obtained by the modified roll with high-order coef- ficients, and comparisons of the seventh-order polynomial and parabolic functions of transmission error are also performed. The achievement of spiral bevel gears with the seventh-order function of transmission error can be accomplished on a universal Cartesian-type hypoid gear generator or a numerically controlled cradle-style hypoid gear generator due to its simple generating motion of axes of the cradle and the work piece. The results of a numerical example show that the bending stresses of the tooth of seventh-order are less than those of a parabolic one, while the contact stresses remain almost eouivalent.

Design of face-hobbed spiral bevel gears with reduced maximum tooth contact pressure and transmission errors

The aim of this study is to define optimal tooth modifications, introduced by appropriately chosen head-cutter geometry and machine tool setting, to simultaneously minimize tooth contact pressure and angular displacement error of the driven gear （transmission error） of face-hobbed spiral bevel gears. As a result of these modifications, the gear pair becomes mismatched, and a point contact replaces the theoretical line contact. In the applied loaded tooth contact analysis it is assumed that the point contact under load is spreading over a surface along the whole or part of the ‘‘potential’’ contact line. A computer program was developed to implement the formulation provided above. By using this program the influence of tooth modifications introduced by the variation in machine tool settings and in head cutter data on load and pressure distributions, transmission errors, and fillet stresses is investigated and discussed. The correlation between the ease-off obtained by pinion tooth modifications and the corresponding tooth contact pressure distribution is investigated and the obtained results are presented.

Dynamics of lubricated spiral bevel gears under different contact paths

To assess the meshing quality of spiral bevel gears,the static meshing characteristics are usually checked under different contact paths to simulate the deviation in the footprint from the design point to the heel or toe of the gear flank caused by the assembly error of two gear axes.However,the effect of the contact path on gear dynamics under lubricated conditions has not been reported.In addition,most studies regarding spiral bevel gears disregard the lubricated condition because of the complicated solutions of mixed elastohydrodynamic lubrication(EHL).Hence,an analytical friction model with a highly efficient solution,whose friction coefficient and film thickness predictions agree well with the results from a well-validated mixed EHL model for spiral bevel gears,is established in the present study to facilitate the study of the dynamics of lubricated spiral bevel gears.The obtained results reveal the significant effect of the contact path on the dynamic response and meshing efficiency of gear systems.Finally,a comparison of the numerical transmission efficiency under different contact paths with experimental measurements indicates good agreement.

Mathematic model and tooth contact analysis of a new spiral non-circular bevel gear

A novel spiral non-circular bevel gear that could be applied to variable-speed driving in intersecting axes was proposed by combining the design principles of non-circular bevel gears and the manufacturing principles of face-milling spiral bevel gears.Unlike straight non-circular bevel gears,spiral non-circular bevel gears have numerous advantages,such as a high contact ratio,high intensity,good dynamic performance,and an adjustable contact region.In addition,while manufacturing straight non-circular bevel gears is difficult,spiral non-circular bevel gears can be efficiently and precisely fabricated with a 6-axis bevel gear cutting machine.First,the generating principles of spiral non-circular bevel gears were introduced.Next,a mathematical model,including a generating tooth profile,tooth spiral,pressure angle,and generated tooth profile for this gear type was established.Then the precision of the model was verified by a tooth contact analysis using FEA,and the contact patterns and stress distributions of the spiral non-circular bevel gears were investigated.

Contact stiffness and damping of spiral bevel gears under transient mixed lubrication conditions

Existing studies primarily focus on stiffness and damping under full-film lubrication or dry contact conditions.However,most lubricated transmission components operate in the mixed lubrication region,indicating that both the asperity contact and film lubrication exist on the rubbing surfaces.Herein,a novel method is proposed to evaluate the time-varying contact stiffness and damping of spiral bevel gears under transient mixed lubrication conditions.This method is sufficiently robust for addressing any mixed lubrication state regardless of the severity of the asperity contact.Based on this method,the transient mixed contact stiffness and damping of spiral bevel gears are investigated systematically.The results show a significant difference between the transient mixed contact stiffness and damping and the results from Hertz(dry)contact.In addition,the roughness significantly changes the contact stiffness and damping,indicating the importance of film lubrication and asperity contact.The transient mixed contact stiffness and damping change significantly along the meshing path from an engaging-in to an engaging-out point,and both of them are affected by the applied torque and rotational speed.In addition,the middle contact path is recommended because of its comprehensive high stiffness and damping,which maintained the stability of spiral bevel gear transmission.

Microstress cycle and contact fatigue of spiral bevel gears by rolling-sliding of asperity contact

The rolling contact fatigue(RCF)model is commonly used to predict the contact fatigue life when the sliding is insignificant in contact surfaces.However,many studies reveal that the sliding,compared to the rolling state,can lead to a considerable reduction of the fatigue life and an excessive increase of the pitting area,which result from the microscopic stress cycle growth caused by the sliding of the asperity contact.This suggests that fatigue life in the rolling-sliding condition can be overestimated based only on the RCF model.The rubbing surfaces of spiral bevel gears are subject to typical rolling-sliding motion.This paper aims to study the mechanism of the micro stress cycle along the meshing path and provide a reasonable method for predicting the fatigue life in spiral bevel gears.The microscopic stress cycle equation is derived with the consideration of gear meshing parameters.The combination of the RCF model and asperity stress cycle is developed to calculate the fatigue life in spiral bevel gears.We find that the contact fatigue life decreases significantly compared with that obtained from the RCF model.There is strong evidence that the microscopic stress cycle is remarkably increased by the rolling-sliding motion of the asperity contact,which is consistent with the experimental data in previous literature.In addition,the fatigue life under different assembling misalignments are investigated and the results demonstrate the important role of misalignments on fatigue life.

TOOTH CONTACT FINITE ELEMENT ANALYSIS OF SPIRAL BEVEL AND HYPOID GEARS

A very useful new method of tooth contact finite element analysis(TCFEA) for spiralbevel and hypoid gears is presented, combines 3-d finite element contact stress analysis withLTCA (Loaded Tooth Contact Analysis). The TCFEA uses mixed finite element method to ana-lyze the 3-d contact stress. The related formulas are derived and an efficient analyzing method asseveral pairs of teeth in contact occurs is presented, which greatly reduce the computationalamount. It is of great significance that the tooth stress. geometry. contact condition and load areall considered in the same model. Finally the related experimental results are used to verify thesolution of TCFEA .