Fillet Stresses Research Articles

Optimum Gear Tooth Geometry for Minimum Fillet Stress Using BEM and Experimental Verification With Photoelasticity

This paper introduces the concept of nondimensional gear teeth to be used in gear stress minimization problems. The proposed method of modeling reduces the computational time significantly when compared to other existing methods by essentially reducing the total number of design variables. Instead of modeling the loaded gear tooth and running BEA to calculate the maximum root stress at every iterative step of the optimization procedure, the stress is calculated by interpolation of tabulated values, which were calculated previously by applying the BEM on nondimensional models corresponding to different combinations of the design parameters. The complex algorithm is used for the optimization and the root stresses of the optimum gears are compared with the stresses of the standard gears for the same transmitted torque. Reduction in stress up to 36.5% can be achieved in this way. This reduction in stress has been confirmed experimentally with two-dimensional photoelasticity.

Helical Gears, Effects of Tooth Deviations and Tooth Modifications on Load Sharing and Fillet Stresses

Based on a few specific cases, this paper presents a comparative investigation of the effect of helix slope and form deviation tolerances as specified by grades 5 and 7 of the ANSI/AGMA ISO 1328-1 Standard for Cylindrical Gears. In addition, the consequences of longitudinal flank crowning and radial tip relief modifications are investigated, as applied on a misaligned helical gear set. For all simulations, the express model (Guilbault et al., 2005, ASME J. Mech. Des., 127(6), pp. 1161–1172) is employed. The bending deflection and fillet stresses are obtained from a combination of finite strip and finite difference meshes. The rolling-sliding motion of mating gear teeth is modeled with a cell discretization of the contact area, which offers fast and accurate results. Similar contact conditions arise from a helix slope deviation or a misalignment of the gear set: the first contact point is driven to a theoretical contact line endpoint. Such a condition produces a localized, and clearly impaired, contact area subject to overloading. Consequently, flank crowning and tip relief corrections must be carefully regarded in the design process. The presented results highlight that, if improperly combined, profile modifications can amplify the overloading condition.

低速ディーゼルエンジン用組立型クランク軸の高強度鋳鋼材料の開発及びクランク軸の応力評価と応力測定

High strength cast steel, Throw Grade 5, has been developed for semi-built-up crankshafts to cover all current types of low speed diesel engines. Material test results with test pieces taken from a full-scale crankthrow are reported. Manufacturing approvals for this material have been taken from major classification societies. Prior to the application of this cast steel to an actual crankshaft, the crank pin fillet stresses at several points were measured in the workshop. The dynamic stress of each point, as analyzed with the finite element method, corresponds well with the measured data. Finally, application of this material to the crankshaft of an engine in a severe vibratory condition is introduced to show effective use of the material.

Express Model for Load Sharing and Stress Analysis in Helical Gears

The performance of a gear set is strongly influenced by the manufacturing and assembly quality. Therefore, detailed analyses at the design stage, where the effects of expected assembly and manufacturing errors can be simulated, are crucial. At an early design stage, when contact conditions are addressed, the widely used finite element method (FEM) may still result in unwanted computing time. The paper presents an Express model developed to serve as a fast design tool offering fine simulation and a high precision level. The model establishes load sharing, fillet stresses and pressure distribution along the contacting surfaces of meshing helical gear teeth. The calculations combine the finite strip method with a pseudo-three-dimensional (3D) model of the tooth base solved with finite differences to calculate tooth bending deflexion and fillet stresses. The accuracy of the procedure is demonstrated through 3D FEM models. A contact cell discretization completes the model. This very fast and accurate approach gives the contact pressure distributions resulting from the roll-slide motion of mating teeth. An analysis of a helical gear set in two different assembly positions reveals the effects of edge contact, and exhibits the influence of tooth stiffness reduction near tooth corners.

Fast Three-Dimensional Quasi-Static Analysis of Helical Gears Using the Finite Prism Method

This paper describes a model for analyzing external and internal helical gears. The finite prism method is used to study the elastic behavior of the structure. Contact deformations are also included in the model. Load sharing, pressure distribution, meshing stiffness and three-dimensional tooth fillet stresses are calculated at each instant. An experimental validation of the numerical model is also presented.

Design of overlapped crankshafts. Part 3: Holes in crankpin and journal

In this paper, the boundary element (BE) technique is used to analyse the effects of 12 different holes in the crankpin and journal on the crankpin and journal fillet stress distributions. Significant reductions in journal fillet peak stresses can be achieved under radial bending, but crankpin fillet maxima are slightly increased. The angular position of the SCF is changed from a single peak in the longitudinal plane of symmetry for a solid crankshaft to two peaks symmetrically positioned relative to this longitudinal plane of symmetry for bored crankshafts. Under torsion, there is a change in fillet stress distribution, but little variation in SCF. Under both load cases, the stiffness of the crankshaft is slightly reduced.

The design of overlapped crankshafts. Part 2: Web shape

The boundary element (BE) technique was used to analyse nine widely different shapes to investigate the effects of crankweb shape on the peak stresses in the crankpin/web and journal/web fillets and on the crankshaft. Under radial three-point bending, pure radial bending and torsion, increasing the web thickness or increasing the overlap reduces the peak fillet stresses and stiffens the crankshaft. Empirical equations are presented for the maximum stresses in both types of bending, torsion, web spread and twist.

3D Mesh Generation for Static Stress Determination in Spiral Noncircular Gears Used for Torque Balancing

Standard procedures available for evaluating gear fillet stresses are targeted to circular gears where little information is available about noncircular gears. In this paper, a mesh generation algorithm is presented for the discretization of gear tooth profiles of noncircular gear elements for static stress evaluation using FEM (Finite Element Method). The procedure assumes that the noncircular gear tooth profile is known. Delimiting a tooth domain from the gear structure and subdividing it into sectors, the gear tooth mesh is generated by mapping pregenerated patterns into the sectors. The discretized tooth domain is used in conjunction with the FEM to obtain stresses within the tooth fillet region. For comparison, this procedure was applied to determine the stress distribution in circular gears and rectangular beams. The results were in complete agreement with existing analytical solutions. An illustrating example is presented where a noncircular gear pair is introduced to eliminate the speed and torque fluctuation that exists in an electrical generator when an I.C. engine is used as the power source for the generator. The maximum fillet Von Mises stress is calculated for each angular position of the noncircular gear pair.

Static and Dynamic Tooth Loading in Spur and Helical Geared Systems-Experiments and Model Validation

The primary objective of this study is to validate a specific finite element code aimed at simulating dynamic tooth loading in geared rotor systems. Experiments have been conducted on a high-precision single stage spur and helical gear reducer with flexible shafts mounted on hydrostatic or hydrodynamic bearings. The numerical model is based on classical elements (shaft, lumped stiffnesses, …) and on a gear element which accounts for non-linear time-varying mesh stiffness, gear errors and tooth shape modifications. External and parametric excitations are derived from the instantaneous contact conditions between the mating flanks by using an iterative contact algorithm inserted in a time-step integration scheme. First, experimental and numerical results at low speeds are compared and confirmed that the proposed tooth mesh interface model is valid. Comparisons were then extended to dynamic fillet stresses on both spur and helical gears between 50–6000 rpm on pinion shaft. Despite a localized problem in the case of spur gears with one particular bearing arrangement, the broad agreement between the experimental and numerical response curves demonstrated that the model is representative of the dynamic behavior of geared systems.

A comprehensive analysis of fillet and contact stresses in straight spur gears

Evaluating various stresses in gear teeth is of great importance to gear designers and manufacturers. In this study, fillet and surface contact stresses produced in straight spur gears were calculated using the standard AGMA equation and an analytical comprehensive formula derived from literature. A comparison between both methods was then carried out. Critical (maximum) values of stresses, and their locations along the profile contour, were determined. The procedure outlined in this study could be used to enhance design calcuations in spur gears.

DYNAMIC ANALYSIS OF A SPUR GEAR BY THE DYNAMIC STIFFNESS METHOD

This study treats a spur gear tooth as a variable cross-section Timoshenko beam to construct a dynamic model, being able to obtain transient response for spur gears of involute profiles. The dynamic responses of a single tooth and a gear pair are investigated. Firstly, polynomials are used to represent the gear blank and the tooth profile. The dynamic stiffness matrix and natural frequencies of the gear are in turn calculated. The forced response of a tooth subject to a shaft-driven transmission torque is calculated by performing modal analysis. This study takes into account time-varying stiffness and mass matrices and the gear meshing forces at moving meshing points. The forced response at arbitrary points in a gear tooth can be obtained. Calculation results of fillet stresses and strains are compared with those in the literature to verify the proposed 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.

Analysis of Stresses in Spur Gears by the Photoelastic Coating Method

Abstract This study uses a photoelastic coating method in spur gear stress analysis, and verifications of the results are made with two-dimensional boundary element analysis. The principal advantage of the photoelastic coating method comes from the fact that it is a full-field technique capable of showing the entire stress distribution over the surface of a part and thus highlighting the regions where the stresses are the largest. The general results of experimental work in which spur gears are analyzed, both in test fixtures and in loading conditions at the critical points, are presented in order to measure actual tooth root, tooth fillet, and blank stresses.

Centrifugal Lubrication of Spur Gears. 2nd Report, Influence of Cooling Hole on Tooth Fillet Stress.

We have recently proposed a new centrifugal lubrication method in which oil is injected through a hole drilled on the tooth crest of a spur gear. To adapt the method for wide face width spur gears, we set up a cooling hole parallel to the tooth trace on the center line of the tooth profile. The results of numerical analyses have been demonstrated that the recommended setting position of the hole is that where the hole inscribes to the root circle on the center line of the tooth profile; the addendum modification coefficient and rim thickness of the gear should be not less than zero and be not less than five times the module respectively. In such a spur gear, an increase in maximum tensile root fillet stress on the active flank side due to the hole of one module in diameter is less than 10% of the stress of the gear with no hole.

Stress Analysis of Cylindrical Webbed Spur Gears: Parametric Study

In this paper, a numerical computer software based on the Finite Prism Method, is proposed in order to design external cylindrical spur gears with a web. It enables computing load sharing, pressure distribution, meshing stiffness and 3D tooth fillet stresses. The software is generally used during the detailed design for optimizing gear meshing. The software is also used to quantify the influences of web design parameters. The process is based on a statistical method: experimental design, that permits studying the influence of parameters. Thus, a simple formula was found in order to estimate the maximum principal stress in the tooth root. The results of the formula were compared with those found in the bibliography. The formula can be useful during the preliminary design for predimensioning webbed spur gears in design department.

Differences in Dynamic Behavior between Straight and Skew Bevel Gears.

In this study, using two kinds of bevel gears (straight and skew) with a power-circulating-type bevel-gear testing machine, the fillet strains. the torsional and bending vibrations of the gear shaft and the vibrating displacement in the axial direction are measured under a constant transmitted torque. In comparison with these measured data, the difference in dynamic behavior between straight and skew bevel gears is investigated The conclusions are described as follows. When the mesh frequency increases, the fillet stress ratio of the straight bevel gears increases markedly, but that of the skew bevel gears increases in small increments. The amplitudes of the components of the torsional and bending vibrations of the gear shaft, and the vibrating displacement along the axial direction on a skew bevel gear are smaller than those of the straight bevel gears. The components of the higher frequency zone. such as 3 fz, and 4 fz, scarcely appear in the displacement of the skew bevel gear along the axial direction.

Prediction of Residual Stresses in Welded T- and I-Joints Using Inherent Strains

In order to develop a predicting method of residual stresses in fillet welded T- and I-joints, a concept of inherent strain, being regarded as a source of the residual stresses, was introduced. With the proposed method, the residual stress of an interested weldment may be predicted by performing an elastic analysis, in which the inherent strain is replaced to equivalent distributed loads. The inherent strain distributions in various welded T- and I-joints were investigated by numerical simulations. The results showed that the inherent strains distributing in flange side and in web side of the several joints are almost the same. The inherent strains vary not only with the average temperature rise due to welding, but with the geometric ratio of the joints. Being simplified by a trapezoid curve, the inherent strain distribution in a fillet weld was expressed by formulas, in which heat input, material properties, and geometric dimensions were taken into account. Welding residual stresses in T- and I-joints, predicted by the proposed method employing the derived formulas, were compared with those obtained by thermal elasto-plastic analysis, and good agreement was recognized. The validity of the proposed method was also confirmed by experiments.

Pseudo element method (PEM) for solving matrix with elementary operation and its application

A pseudo element method (PEM) is presented in this paper for solving the matrix with elementary operation arising from the static analysis of cyclic symmetric structures. Three kinds of pseudo elements are devised to consider the transformed terms in the stiffness matrix during the elementary operation. Thus, the overall stiffness matrix may not be assembled by using the frontal solution technique. This can need relatively small computer memory and relatively short CPU time. The centrifugal displacements and stresses of a ring structure are calculated by introducing the cyclic symmetric conditions. The calculated results show, in comparison with theoretical ones, that the present method and its program are effective and can be conveniently used on a personal computer. At the end of this paper, the centrifugal fillet stresses of left-handed helical gear structures with different rim and web sizes are evaluated.

Effect of Bearing on Dynamic Behaviors of Straight Bevel Gear.

In this study, using two kinds of bearings (tapered roller and angular contact bearings) with a straight bevel gear testing machine of power-circulating type, the fillet strains of gear tooth and the displacements of a straight bevel gear in three directions are measured under a constant transmitted torque. From these results, the influences of the tapered roller or angular contact bearings on the dynamic behaviors of bevel gears are investigated. The conclusions are described as follows. The tapered roller bearings produce a greater, effect of dynamic load on each position than the angular contact bearings. As for the meshing frequency components of higher zones, such as 3 fz and 4 fz, the displacements of the tapered roller bearing in the three directions are greater than those of the angular contact bearings. These results correspond to the tendency of the fillet stress ratio.

Analysis of the Influence of Crankshaft System Vibrations on the Crankshaft Bending Stresses in a V-10 Type Diesel Engine. Crankpin Fillet Stresses Induced by the Crankshaft Torsional-Bending Vibrations.

The characteristics of crankshaft stresses were investigated by experiments and calculations for a 10 cylinder V-engine. The three-dimensional vibrations of the crankshaft system were calculated by the dynamic stiffness matrix method. The dynamic bending stresses at the front and rear fillets of the No. 1 crankpin in the crankthrow plane and torsional vibrations at the front pulley were measured. From the experimental and calculated results we found that : ( 1 ) The maximum bending stresses are caused by the bending vibrations coupled with torsional vibrations, and the peaks of the bending stresses always appear at the resonant engine speeds of the torsional vibrations. ( 2 ) The bending stress at the front fillet is strongly affected by the moments of inertia of the front pulley about the axis orthogonal to the No. 1 crankthrow plane, whereas the stress at the rear fillet is little affected by them.