Calculation Of Load Distribution Research Articles

A Practical Method for Measuring Rolling Element Load Distribution of Rolling Bearings Using a Fiber Optical Sensor (2nd Report, Dynamic Measurement)

An external load applied to a rolling bearing is distributed among the rolling elements. The conventional calculation of load distribution among rolling elements already established assumes that the bearing races, shaft and housing are rigid except for elastic deformations at the individual contacts between rolling elements and raceways. In many applications, the bearing rings as well as the supporting structures can not be considered rigid, so that the actual load distribution is different from that predicted on the above assumption. Several experimental methods to determine the load distribution of rolling bearings have been proposed so far. Their methods, however, can not provide dynamic load distributions while rolling elements make one revolution along the raceway. The objective of this study is to establish a more direct method than the other ones for determining rolling element load distributions not only statically but also dynamically. In the method herein proposed, the strains of rollers developed under an applied load are measured by using an optical fiber strain sensor mounted on a roller of a measuring cylindrical roller bearing. The obtained strain data provides rolling element load distributions. This paper presents the method for determination of dynamic load distributions, and shows examples of dynamic measurements using the method. The dynamic behaviors of the load distributions and the correspondence of the experimental results to the conventional theory have been investigated. In addition, the comparison of the dynamic load distributions obtained here and the static ones presented in Part I of this study has been made.

光ファイバセンサによる転がり軸受の転動体荷重測定法 : 第1報,静的測定(機械要素,潤滑,設計,生産加工,生産システムなど)

An external load applied to a rolling bearing is distributed among the rolling elements. The conventional calculation of load distribution among rolling element already established assumes that the bearing races, shaft and housing are rigid except for elastic contact deformations at the individual contacts between rolling elements and raceways. In many applications, the bearing rings as well as the supporting structures can not be considered rigid, so that the load distribution is different from that predicted on the above assumption. It is known that a bearing fatigue life is affected by a load distribution. Thus accurate prediction of the bearing life requires the precise determination of actual load distributions. Several experimental methods to determine the load distribution of rolling bearings have been proposed so far. Their methods, however, can not provide dynamic load distributions while rolling elements make one revolution along the raceway. The objective of this study is to establish a more direct method than the other ones for determining rolling element load distributions not only statically but also dynamically. The method herein proposed provides the measurement of strains of rollers developed under an applied load by using an optical fiber strain sensor mounted on a roller of a measuring cylindrical roller bearing. The obtained strain data provides rolling element load distributions. This paper presents the method for determination of static load distributions.

Head-cutter for optimal tooth modifications in spiral bevel gears

Abstract A method for the determination of optimal tooth modifications in spiral bevel gears based on improved load distribution and reduced maximum tooth contact pressure and transmission errors is presented. The modifications are introduced into the pinion tooth-surface by using a head-cutter with bicircular profile and with optimal diameter. In the optimization of tool parameters the influence of relative position errors of the mating gears is included. By using the computer program developed for load distribution calculation, the influence of head-cutter parameters and relative position errors of the mating gears on tooth contact and transmission errors is investigated. Based on the results obtained, the radii and the position of the circular tool profile arcs and the cutter diameter for pinion teeth finishing were optimized. By applying the optimal tool parameters, the maximum tooth contact pressure is reduced by 45% and the maximum angular position error of the driven gear by 60%, in regard to the spiral bevel gear pair with a pinion manufactured by a cutter of straight-sided profile and of diameter determined by the commonly used methods.

3D Simplified Finite Elements Analysis of Load and Contact Angle in a Slewing Ball Bearing

Bolted bearing connections are one of the most important connections in some industrial structures, and manufacturers are always looking for a quick calculation model for a safe design. In this context, all the analytical and numerical models reduce the global study to the study of the most critical sector. Therefore, the main inputs for these models are the maximal equivalent contact load and the corresponding contact angle. Thus, a load distribution calculation model that takes all the important parameters, such as the stiffness of the supporting structure and the variation in the contact angle, into consideration is needed. This paper presents a 3D finite element (FE) simplified analysis of load distribution and contact angle variation in a slewing ball bearing. The key element of this methodology, which is based on the Hertz theory, is modeling the rolling elements under compression by nonlinear traction springs between the centers of curvature of the raceways. The contact zones are modeled by rigid shells to avoid numerical singularities. Each raceway curvature center is coupled to the corresponding contact zone by rigid shells. The main contribution of this method is not only the evaluation of the contact loads with a relatively reduced calculation time but also the variation in the contact angle from the deformed coordinates of the curvature centers. Results are presented for several loading cases: axial loading, turnover moment, and a combined loading of axial force and turnover moment. The influence of the most important parameters such as the contact angle, the stiffness of the bearings, and the supporting structure is discussed. Finally, a preliminary experimental validation is conducted on a standard ball bearing. The results presented in this paper seem encouraging. The FE study shows an important influence of several parameters and a good correlation with experimental results. Consequently, this model can be extended to other types of slewing bearings such as roller bearings. Moreover, it can be implemented in complex industrial structures such as cranes and lifting devices to determine the corresponding load distributions and contact angles and, consequently, the most critical sector.

A combined contact elasticity and finite element-based model for contact load and pressure distribution calculation in a frictional workpiece-fixture system

Contact forces between workpiece and fixture define fixture stability during clamping and influence workpiece accuracy during machining. In particular, forces acting in the contact region are important for understanding deformation of the workpiece at the contact region. This paper presents a model that combines contact elasticity with finite element methods to predict the contact load and pressure distribution at the contact region in a workpiece-fixture system. The objective is to determine how much clamp forces can be applied to generate adequate contact forces to keep the workpiece in position during machining. The model is able to predict the normal and tangential contact forces as well as the pressure distribution at each workpiece-fixture contact in the fixturing system. Model prediction is shown to be in good agreement with known industry practice on clamp force determination. The presented method has no limits on the types of materials that can be analyzed.

Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears

The influence of tooth errors and shaft misalignments on loaded tooth contact in different types of cylindrical worm gears is investigated. The geometry and kinematics of worm gears based on the generation of gear teeth by a hob is presented. A method for loaded tooth contact analysis of such worm gear pairs is developed. The method includes the determination of contact lines in fully conjugated worm gears with theoretical line contact, the determination of path of contact, potential contact lines and separations of mating teeth surfaces along these contact lines in modified worm gears with theoretical point contact, and the calculation of load distribution and transmission errors for different types of modified and non-modified worm gears. A computer program based on the theoretical background presented has been developed. Using this program the influence of tooth errors and shaft misalignments of mating members on tooth contact is investigated and discussed.

Load Distribution in Cylindrical Worm Gears

A method for the determination of load sharing between the instantaneously engaged worm threads and gear teeth, for the calculation of load distribution along the teeth and transmission errors in different types of cylindrical worm gears is presented. The method covers both cases—that of the theoretical line and point contact. The bending and shearing deflections of worm thread and gear tooth, the local contact deformations of the mating surfaces, the axial deformations of worm body, gear body bending and torsion, deflections of the supporting shafts, and the manufacturing and alignment errors of worm and gear are included. Based on the real load distribution the tooth contact pressure is calculated, in the case of point contact in two different ways, and the obtained results are compared. Also, the total transmission error, consisting of the kinematical transmission error due to the mismatch of the worm gear drive and of the transmission error caused by the deflections of worm thread and gear teeth, is calculated. The method is implemented by a computer program. By using this program the influence of the type of worm gear drive and of design and manufacturing parameters on load distribution and transmission errors is investigated and discussed.

2636 コンフォーマル接触状態での接触圧力の計算

A method of calculation for contact pressure in conformal contacts was studied. In elastic contact analysis, the Hertzian theory is well known. But the Hertzian theory is hardly applicable to a problem of an extreme conformal contact condition, such as ball joints or journal bearings. Providing a relation between contact pressure and interference depth of the original configurations, contact pressure was calculated and compared with the FEM analysis. This method is analogous to the calculation of load distribution in radial ball bearings. The accuracy of the method was numerically evaluated by simple problems. It was confirmed that the accuracy of this method for contact pressure was enough for practical use.

Optimal Machine Tool Setting for Hypoid Gears Improving Load Distribution

A method for the determination of optimal machine tool setting for manufacturing modified (mismatched) hypoid gears based on improved load distribution and reduced transmission errors is presented. The applied load distribution calculation is based on the conditions that the total angular position errors of the gear teeth being instantaneously in contact under load must be the same, and along the contact line of every tooth pair instantaneously in contact, the composite displacements of tooth surface points—as the sums of tooth deformations, geometrical surface separations, gear body bending and torsion, deflections of the supporting shafts, misalignments, and composite tooth errors—should correspond to the angular position of the gear. The tooth deformations consists of the bending and shearing deflections of gear teeth and of the local contact deformations of the mating surfaces. The tooth deflections are calculated by the finite element method. As the equations governing the load sharing and load distribution are nonlinear, an approximate and iterative technique is used to solve this system of equations. The method is implemented by a computer program. Using the program that was developed the influence of machine tool setting parameters for pinion manufacture on maximum tooth contact pressure, load distribution factor, and transmission errors is investigated. By successively choosing the optimal value for every machine tool setting parameter, and by applying the optimal set of these parameters, the maximum tooth contact pressure is reduced by 5.8%, the load distribution factor by 5.9%, and the angular position error of the driven gear by 65.4%, in regard to the hypoid gear pair manufactured by the machine tool setting determined by the commonly used method.

FEM stress analysis in hypoid gears

Stress analysis in hypoid gears by using finite element method is performed in order to develop simple equations for the calculation of tooth deflections and stresses. A displacement type finite element method is applied with curved, twenty-node isoparametric elements. A method is developed for the automatic finite element discretization of the pinion and the gear. The full theory of mismatched hypoid gears is applied for the determination of the nodal point coordinates on the teeth surfaces. The corresponding computer program is developed. By using this program the influence of design parameters and load position on tooth deflections and fillet stresses is investigated. On the basis of the results, obtained by performing a big number of computer runs, by using regression analysis and interpolation functions, equations for the calculation of tooth deflections and fillet stresses are derived. The advantages of the use of these equations in load distribution calculation in hypoid gears are shown.

Load Distribution in Hypoid Gears

A new approach for the computerized simulation of load distribution in mismatched hypoid gears with point contact is presented. The load distribution calculation is based on the bending and shearing deflections of gear teeth, on the local contact deformations of the mating surfaces, on gear body bending and torsion, on the deflections of the supporting shafts, and on the manufacturing and alignment errors of the mating members. The tooth deflections of the pinion and gear teeth are calculated by FEM, and the tooth contact is treated in a special way: it is assumed that the point contact under load spreads over a surface along the “potential” contact line, which line is made up of the points of the mating tooth surfaces in which the separations of these surfaces are minimal, instead of assuming an elliptical contact pattern. The system of governing equations is solved by approximations and by using the successive-over-relaxation method. The corresponding computer program is developed. The calculations, performed by this program, show that in the case of hypoid gears, the new approach gives a more realistic contact pattern and contact pressure than the usually assumed and applied elliptical contact approach, especially for the tooth pairs contacting on the toe and on the heel of teeth, and in the case of load distribution calculations made in misaligned gear pairs. By using this program the influence of design data on load distribution parameters is investigated and discussed. [S1050-0472(00)00504-3]

Characteristics of a New Type of Cylindrical Worm-Gear Drive

A new type of cylindrical worm-gear drive is presented. The worm is ground by a grinding wheel whose profile consists of two circular arcs, and the obtained worm profile is concave. The teeth of the gear are processed by a hob whose generator surface is identical to the worm surface. The aim of this paper is to introduce this new type of worm gearing and to compare the operating characteristics of the new type and of the commonly used worm-gear drives on the basis of tooth contact analysis, load distribution calculation, and thermal elastohydrodynamic lubrication analyses. The results obtained show the advantages of the new type of worm gear. The results of the load distribution calculation and EHD lubrication analysis are also used for the optimization of the design parameters of the new type of worm-gear drive.

A Factorial Design Approach to Investigate the Effect of Geometry in Drill String Screw Connectors

This paper describes a parametric study of drill string threaded connector design based on a “factorial design” method. The study is facilitated by a hybrid model which has been developed, validated, and reported previously, enabling efficient calculation of load and stress distribution along threaded connectors subjected to both axial and bending modes of loading. A parametric equation is produced where stress concentration factor is defined in terms of various geometrical variables. The equation is then utilized to carry out a constrained optimization within the feasible parameter space, and hence produce an “optimum” thread and connector design.

Comparison of three-dimensional neutronics calculations for a fusion breeder with large channels

Abstract Neutronics calculations for the blanket of a tokamak engineering test breeder were compared. Although the agreement is quite good for one-dimensional calculations using different codes and data libraries, a discrepancy appears for three-dimensional calculations. To clarify this discrepancy, a series of calculations was done with the conclusion that for a blanket with large openings such as divertor channels, one-dimensional calculation using the toroidal cylindrical model in addition to a fusion neutron load distribution calculation will result in over-estimation of the neutronics parameters, and detailed three-dimensional calculations are necessary.

Load Distribution in Double Enveloping Worm Gears

A method for the determination of load sharing among the instantaneously engaged worm threads and gear teeth of double enveloping worm gears and for the calculation of load distribution along their instantaneous contact lines is presented. The bending and shearing deflection of worm thread and gear tooth, the contact deformation, the axial deformation of worm body, and the manufacturing and alignment errors of worm and gear are included. The obtained system of integral equations is solved by using approximations and an iterative technique. The corresponding computer program is developed. By using this program, the load distribution in the classical and in a new type of double enveloping worm gear drives is calculated. The influence of design parameters on load distribution factor and on maximum tooth pressure is investigated and discussed.

Distribution of Tooth Load Along a Pinion

Abstract The distribution of load along a spur gear and pinion or among the teeth of a helical gear and pinion is of great interest in the design and performance of gears. If the load is unevenly distributed, certain parts are not carrying their due share of load, while the resulting concentration of load elsewhere may result in local failure. This paper is concerned with the calculation of load distribution, and the analysis of the factors responsible for the character of this distribution.