Application of Isogeometric Analysis Method in Three-Dimensional Gear Contact Analysis

Gears are pivotal in mechanical drives,and gear contact analysis is a typically difficult problem to solve.Emerging isogeometric analysis(IGA)methods have developed new ideas to solve this problem.In this paper,a threedimensional body parametric gear model of IGA is established,and a theoretical formula is derived to realize single-tooth contact analysis.Results were benchmarked against those obtained from commercial software utilizing the finite element analysis(FEA)method to validate the accuracy of our approach.Our findings indicate that the IGA-based contact algorithmsuccessfullymet theHertz contact test.When juxtaposed with the FEA approach,the IGAmethod demonstrated fewer node degrees of freedomand reduced computational units,all whilemaintaining comparable accuracy.Notably,the IGA method appeared to exhibit consistency in analysis accuracy irrespective of computational unit density,and also significantlymitigated non-physical oscillations in contact stress across the tooth width.This underscores the prowess of IGA in contact analysis.In conclusion,IGA emerges as a potent tool for addressing contact analysis challenges and holds significant promise for 3D gear modeling,simulation,and optimization of various mechanical components.

Geometry Design and Tooth Contact Analysis of Crossed Beveloid Gears for Marine Transmissions

Beveloid gears,also known as conical gears,gain more and more importance in industry practice due to their abilities for power transmission between parallel,intersected and crossed axis.However,this type of gearing with crossed axes has no common plane of action which results in a point contact and low tooth durability.Therefore,a geometry design approach assuming line contact is developed to analyze the tooth engagement process of crossed beveloid gears with small shaft angle for marine transmission applications.The loaded gear tooth contact behavior is simulated by applying a quasi-static analysis to study the effects of gearing parameters on mesh characteristics.Using the proposed method,a series of sensitivity analyses to examine the effects of critical gearing parameters such as shaft angle,cone angle,helix angle and profile-shift coefficient on the theoretical gear mesh is performed.The parametric analysis of pitch cone design shows that the dominant design parameters represented by the angle between the first principle directions（FPD） and normal angular factor are more sensitive to the shaft and cone angles than they are to the helix angle.The theoretical contact path is highly sensitive to the profile-shift coefficient,which is determined from the theoretical tooth contact analysis.The FPD angle is found to change the distribution of contact pattern,contact pressure and root stress as well as the translational transmission error and the variation of the mesh stiffness significantly.The contact pattern is clearly different between the drive and coast sides due to different designed FPD angles.Finally,a practical experimental setup for marine transmission is performed and tooth bearing test is conducted to demonstrate the proposed design procedure.The experimental result compared well with the simulation.Results of this study yield a better understanding of the geometry design and loaded gear mesh characteristics for crossed beveloid gears used in marine transmission.

TOOTH CONTACT ANALYSIS OF CONICAL INVOLUTE GEARS

The mathematical model of conical involute gears is developed based on the theory of gearing and the generating mechanism. Tooth contact analysis （TCA） is performed to examine the meshing and bearing contact of the conical involute gear pairs with intersected and crossed axes. In addition, the principal directions and curvatures of the gear surfaces are investigated and the contact ellipses of the mating tooth surfaces are also studied. Finally, the numerical illustrative examples are provided to demonstrate the computational results, test gears are made for tooth-bearing tests, and the conclusion is verified that the theory has the applicability.

An efficient formulation based on the Lagrangian method for contact–impact analysis of flexible multi-body system

In this paper,an efficien formulation based on the Lagrangian method is presented to investigate the contact–impact problems of f exible multi-body systems.Generally,the penalty method and the Hertz contact law are the most commonly used methods in engineering applications.However,these methods are highly dependent on various non-physical parameters,which have great effects on the simulation results.Moreover,a tremendous number of degrees of freedom in the contact–impact problems will influenc thenumericalefficien ysignificantl.Withtheconsideration of these two problems,a formulation combining the component mode synthesis method and the Lagrangian method is presented to investigate the contact–impact problems in fl xible multi-body system numerically.Meanwhile,the finit element meshing laws of the contact bodies will be studied preliminarily.A numerical example with experimental verificatio will certify the reliability of the presented formulationincontact–impactanalysis.Furthermore,aseries of numerical investigations explain how great the influenc of the finit element meshing has on the simulation results.Finally the limitations of the element size in different regions are summarized to satisfy both the accuracy and efficien y.

A NOTE FOR ANALYSIS OF THERMOMECHANICAL CONTACT PROBLEMS

A discussion about the bifurcation and non-uniqueness of solutions in theanalysis of thermo-mechanical contact problems with initial gap is given. Without loss ofgenerality, a mechanical contact problem coupled with steady heat transfer is studied and an exampleof non-uniqueness of solutions caused by the thermo-mechanical mechanism is presented. Theimportant work is that the non-uniqueness of solutions, which is different from that found in theanalysis of the traditional frictional contact problems, is studied in detail. The possibleoscillation and non-convergence problems in the iteration process of the numerical computation arediscussed, and an enhanced algorithm is put forward to overcome the difficulties.

Finite Element Analysis of 6300 Deep Groove Ball Bearing

Rolling bearing is widely used in mechanical support, its general components are the inner ring, outer ring, the ball, retainer etc.. Now many companies in developed countries and university in the rolling bearing as the research object, and has made great progress in design theory, the experiment method and production technology etc. We will use the finite element ANSYS to establish the model of deep groove ball bearing. Through the contact analysis, we can get the contact stress between the rings and balls, strain, contact state, penetration, sliding distance and the friction stress distribution. These values are compared to the theoretical values with Hertz theory, and they have better consistency, provide the good theoretical basis for the optimization design of rolling bearings.

Dynamics of a Dual Power-split Transmission Based on Loaded Tooth Contact Analysis

In order to implement the dynamic characteristic of a dual power-split transmission, a dynamic me- chanics model is built. Firstly, according to the method of theoretical analysis of the tooth contact analysis （TCA） and loaded tooth contact analysis （LTCA）, the actual meshing process of each gear pairs is simulated, and the time-varying mesh stiffness excitations are obtained, which can improve the numerical precision. Second- ly, by using the lumped mass method, the bending-torsional coupling three dimensional dynamic model of the dual power-split transmission is established, and the identical dimensionless equations are deduced by elimina- ting the effect of rigid displacement and the method of dimensional normalization. Finally, by the method of the fourth order Runge-Kutta algorithm with variable step lengths, the responses of this system in a frequency domain and time domain are obtained, and the dynamic load change characteristics of each gear pairs are analyzed. The results show that the establishment, solution and analysis of the system dynamics model could provide a basis for the dynamic design, and have an important significance for the dynamic efficiency analysis and dynamic perform- ance optimization design of the dual power-split transmission.

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 .

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.

Fissure evolution and evaluation of pressure-relief gas drainage in the exploitation of super-remote protected seams

Based on nonlinearity contact theory and the geological structure of the Xieqiao Coal Mine in the newly developed Huainan coal field,rock movements,mining fissures and deformation of overlying strata were simulated by using the interface unit of FLAC3D to evaluate the pressure-relief gas drainage in the exploitation of super-remote protected seams.The simulation indicates that the height of the water flowing fractured zone is 54 m in the overlying strata above the protective layer.The maximum relative swelling deformation of the C13 coal seam is 0.232%,while the mining height is 3.0 m and the distance from the B8 roof to the C13 floor is 129 m,which provides good agreement with a similar experiment and in situ results.The feasibility of exploitation of a super-remote protective coal seam and the performance of the pressure-relief gas drainage in a super-remote protected layer are evaluated by comparisons with practice projects.It demonstrates that the relieved gas in the super-remote protected layers could be better drained and it is feasible to exploit the B8 coal seam before the C13 super-remote protected coal seam.The method is applicable for the study of rock movements,mining fissures and deformation of the overburden,using the interface unit to analyze the contact problems in coal mines.

Reverse Engineering of Machine-tool Settings with Modified Roll for Spiral Bevel Pinions

Although a great deal of research has been dedicated to the synthesis of spiral bevel gears, little related to reverse engineering can be found. An approach is proposed to reverse the machine-tool settings of the pinion of a spiral bevel gear drive on the basis of the blank and tooth surface data obtained by a coordinate measuring machine(CMM). Real tooth contact analysis(RTCA) is performed to preliminary ascertain the contact pattern, the motion curve, as well as the position of the mean contact point. And then the tangent to the contact path and the motion curve are interpolated in the sense of the least square method to extract the initial values of the bias angle and the higher order coefficients(HOC) in modified roll motion. A trial tooth surface is generated by machine-tool settings derived from the local synthesis relating to the initial meshing performances and modified roll motion. An optimization objective is formed which equals the tooth surface deviation between the real tooth surface and the trial tooth surface. The design variables are the parameters describing the meshing performances at the mean contact point in addition to the HOC. When the objective is optimized within an arbitrarily given convergence tolerance, the machine-tool settings together with the HOC are obtained. The proposed approach is verified by a spiral bevel pinion used in the accessory gear box of an aviation engine. The trial tooth surfaces approach to the real tooth surface on the whole in the example. The results show that the convergent tooth surface deviation for the concave side on the average is less than 0.5 μm, and is less than 1.3 μm for the convex side. The biggest tooth surface deviation is 6.7 μm which is located at the corner of the grid on the convex side. Those nodes with relative bigger tooth surface deviations are all located at the boundary of the grid. An approach is proposed to figure out the machine-tool settings of a spiral bevel pinion by way of reverse engineering without having known the theoretical tooth surfaces and the corresponding machine-tool settings.

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.

DIRECT DIGITAL DESIGN AND SIMULATION OF MESHING IN WORM-GEAR DRIVE

A direct digital design method （DDDM） of worm-gear drive is proposed. It is directly based on the simulation of manufacturing process and completely different from the conventional modeling method. The loaded tooth contact analysis （LTCA） method is analyzed, in which the advanced surface to surface searching technique is included. The influence of misalignment errors and contact deformations on contact zone and transmission error （TE） is discussed. Combined modification approach on worm tooth surface is presented. By means of DDDM and LTCA, it is very conven- ient to verify the effect of worm-gear drive＇s modification approach. The analysis results show that, the modification in profile direction reduces the sensitivity of worm-gear drive to misalignment errors and the modification in longitudinal direction decreases the TE. Thus the optimization design of worm-gear drive can be achieved prior to the actual manufacturing process.

Indentation of a compressible soft electroactive half-space: Some theoretical aspects

We theoretically study the indentation response of a compressible soft electroactive material by a rigid punch. The half-space material is assumed to be initially subjected to a finite deformation and an electric biasing field. By adopting the linearized theory for incremental fields, which is established on the basis of a general nonlinear theory for electroelasticity, the appropriate equations governing the perturbed infinitesimal elastic and electric fields are derived particularly when the material is subjected to a uniform equibiaxial stretch and a uniform electric displacement. A general solution to the governing equations is presented, which is concisely expressed in terms of four quasi-harmonic functions. By adopting the potential theory method, exact contact solutions for three common perfectly conducting rigid indenters of fiat-ended circular, conical and spherical geometries can be derived, and some explicit relations that are of practical importance are outlined.

Seismic and stress qualification of LMFR fuel rod and simple method for the determination of LBE added mass effect

In this study, two different designs of liquid metal fast reactor(LMFR) fuel rods wire-wrapped and nonwire-wrapped(bare) are compared with respect to different parameters as a means of considering the optimum fuel design. Nuclear seismic rules require that systems and components that are important for safety must be capable of bearing earthquake effects, and that their integrity and functionality should be guaranteed. Mode shapes, natural frequencies, stresses on cladding, and seismic aspects are considered for comparison using ANSYS. Modal analysis is compared in a vacuum and in lead–bismuth eutectic(LBE) using potential flow theory by considering the added mass effect. A simple and accurate approach is suggested for the determination of the LBE added mass effect and is verified by a manually calculated added mass, which further proved the usefulness of potential flow theory for the accurate estimation of the added mass effect. The verification of the hydrodynamic function(τ) over the entire frequency range further validated the finite element method(FEM) modal analysis results. Stresses obtained for fuel rods against different loading combinations revealed that they were within the allowable limits with maximum stress ratios of 0.25(bare) and 0.74(wire-wrapped). In order to verify the structural integrity of cladding tubes, stresses along the cladding length were determined during different transients and were also calculated manually for static pressure. The manual calculations could be roughly compared with the ANSYS results, and the two showed a close agreement. Contact analysis methodology was selected,and the most appropriate analysis options were suggested for establishing contact between the wire and cladding for the wire-wrapped design grid independence analysis,which proved the accuracy of the results, confirmed the selection of the appropriate procedure, and validated the use of the ANSYS mechanical APDL code for LMFR fuel rod analysis. The results provided detailed insight into the structural design of LMFR fuel rods by considering different structural configurations(i.e., bare and wire-wrapped) in the seismic loading;this not only provides a FEM procedure for LMFR fuel with complex configuration, but also guides the reference design of LMFR fuel rods.

Effect of wheelset flexibility on wheel–rail contact behavior and a specific coupling of wheel–rail contact to flexible wheelset

The fexibility of a train＇s wheelset can have a large effect on vehicle–track dynamic responses in the medium to high frequency range.To investigate the effects of wheelset bending and axial deformation of the wheel web,a specifi coupling of wheel–rail contact with a fexible wheelset is presented and integrated into a conventional vehicle–track dynamic system model.Both conventional and the proposed dynamic system models are used to carry out numerical analyses on the effects of wheelset bending and axial deformation of the wheel web on wheel–rail rolling contact behaviors.Excitations with various irregularities and speeds were considered.The irregularities included measured track irregularity and harmonic irregularities with two different wavelengths.The speeds ranged from 200 to400km/h.The results show that the proposed model can characterize the effects of fexible wheelset deformation on the wheel–rail rolling contact behavior very well.

Local buckling of thin plate on tensionless elastic foundations under interactive uniaxial compression and shear

This paper uses a mathematical method to develop an analytical solution to the local buckling behaviour of long rectangular plates resting on tensionless elastic Winkler foundations and under combined uniform longitudinal uniaxial compressive and uniform in-plane shear loads. Fitted formulas are derived for plates with clamped edges and simplified supported edges. Two examples are given to demonstrate the application of the current method： one is a plate on tensionless spring foundations and the other is the contact between the steel sheet and elastic solid foundation. Finite element （FE） analysis is also conducted to validate the analytical results. Good agreement is obtained between the current method and FE analysis.

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.

A FINITE ELEMENT PARALLEL ALGORITHM OF THE PARAMETRIC VARIATIONAL PRINCIPLE FOR ELASTIC CONTACT PROBLEMS

Basetl on the finite element solution of the parametric varialional principle of elastic con/del problem, a corresponding parallel algorithm has been created bv utilizing the specialities of parallel computer and the architecture of concurrent processing in this paper. In this algorithm. the parallelisms have heen realized in the processes of creation and assembly of stiffness matrix, of the static condensation, of the solution of stresses and in many other aspects. The programme of this algorithm has been realized on ELXSI-6400 parallel computer of Xi'an Jiaotong University. The results of computation show that the computational time can be saved efficiently and it is an effective parallel algorithm for the analyses of contact problems.

NUMERICAL APPROACH TO DETERMINING INSTANTANEOUS CONTACT REGION FOR CONJUGATE SURFACES

According to the defect of traditional method of determining instantaneous contact regions for conjugate surfaces, a numerical approach to the determination is proposed. A local coordinate system is built by using the surface unit tangent and unit normal at the contact point. Considering that the gap forming the boundary of instantaneous contact region in the direction of the common normal vectors is given, a system of nonlinear equations is built to find the instantaneous contact boundary in local coordinate system, a modified Powell＇s hybrid algorithm of finite-difference approximation to the Jacobian used to solve the system. The new method simplifies the task of determining instantaneous contact regions without calculating curvatm＇e and relative curvature. The validity of the proposed approach is verified by an example of hypoid gears.