Claude Gosselin

A general formulation for the calculation of the load sharing and transmission error under load of spiral bevel and hypoid gears

Abstract This paper presents the basis of a Loaded Tooth Contact Analysis (LTCA) program predicting the motion error of spiral bevel gear sets under load, and explores some of the influences of the unloaded motion error curve shape and amplitude over the kinematical behaviour under load. The effects of tooth composite deflection caused by bending and shearing, tooth contact deformation and initial profile separation due to profile mismatch are considered in the development. Due to the complex geometry of spiral bevel gear teeth, the tooth bending stiffness is calculated by finite elements. Classical Hertz theory is used to calculate the contact deformation. Numerical examples are presented to illustrate the behaviour of spiral bevel gear motion error under load as the unloaded motion error is modified. Results show that under general circumstances, low contact ratio spiral bevel gears with parabolic unloaded motion error may produce undesirable kinematics under load, while parabolic motion error high contact ratio spiral bevel gears are likely to produce acceptable kinematics under a larger load range. It is also shown that the governing factors, in loaded motion error, are the contact ratio, thus combined mesh stiffness, and the amplitude of unloaded motion error curve which is linked to load sharing between adjacent tooth pairs.

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.

Simulation and Experimental Measurement of the Transmission Error of Real Hypoid Gears Under Load

The Transmission Error and Bearing Pattern of a gear set are fundamental aspects of its meshing behavior. To assess the validity of gear simulation models, the Transmission Error and Bearing Pattern of a Formate Hypoid gear set are measured under a variety of operating positions and applied loads. Measurement data are compared to simulation results of Tooth Contact Analysis and Loaded Tooth Contact Analysis models, and show excellent agreement for the considered test gear set.

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.

Experimental and numerical investigation of the meshing cycle and contact ratio in spiral bevel gears

The load carrying capacity of spiral bevel gears is governed by the way load is distributed between meshing tooth pairs, which is highly dependent on the actual contact ratio as the gears mesh. Previous experimental studies have shown that the AGMA calculated contact ratio is substantially different from that found in practice, because of the shape and dimensions of the bearing pattern produced by the load transmitted between meshing gear teeth. Given the geometrical complexity of spiral bevel gear tooth profiles, a Tooth Contact Analysis computer program is used to determine the location of the path of contact, the dimensions of the bearing pattern under load and the associated contact ratio. The objectives of this paper are to experimentally and numerically study the relationships between the bearing pattern of selected spiral bevel gear sets and their load meshing cycles, to evaluate the actual contact ratio, and to calculate and compare the corresponding AGMA contact ratio to measurements and values calculated by a Loaded Tooth Contact Analysis computer program.

Determination of a cholesterol oxide mixture by a single-run high-performance liquid chromatographic analysis using benzoylation

Abstract A rapid, single-run high-performance liquid chromatographic method has been developed to separate and quantify benzoate esters of cholesterol oxides, providing high sensitivity via ultraviolet detection. The mobile phase was 85% isopropanol-water (v/v). Analyses of 7-ketocholesterol, cholestane-triol, epoxy-cholesterol, 7-hydroxycholesterol and 25-hydroxycholesterol were done using a 30 cm × 3.9 mm I.D. Novapak C18 column and a variable-wavelength ultraviolet detector (set at 230 nm). Linearity was excellent since a good correlation was observed. As low as 500 ng of cholesterol benzoate per 20 μl of solution can be detected by this method.

Selected ion monitoring of tert-butyldimethylsilyl cholesterol ethers for determination of total cholesterol content in foods

Abstract A gas chromatography/mass spectrometry (GC/MS) method, using tert-butyldimethylsilyl (tBDMS) derivatives of sterols for detection in the selected ion monitoring (SIM) mode has been applied to the determination of total cholesterol in food (eggs, dairy products). Generally, comparison of results with literature data shows agreement with the amount of cholesterol determined by other chromatographic techniques, and slight underestimation if compared to the amount of cholesterol estimated by colorimetric techniques. The saponification and extraction procedure allowed for 98.6% recovery of spiked cholesterol in milk samples with a coefficient of variation of 2.1%. Amounts as low as 5 ng per 100 g food can be detected using external standards with 95% accuracy.

Contamination problems with 63Ni electron capture detectors

Abstract A study was performed on two 63Ni electron capture detectors operating in d.c. mode. Several parameters have been examined. It appeared that the temperature difference between the detector and the column is important. Scavenger gas temperature and detector isolation are not less important. Furthermore the detector polarization switch must always be off when no chromatogram is being run. These modifications help to keep the detector sensitivity much longer, with a consequent reduced number of washings.