Computerized Analysis of Gear Meshing Based on Coordinate Measurement Data

To inspect cam profile errors, a coordinate measuring machine (CMM) is frequently employed. The coordinate data of discrete points at the cam surface are measured by a CMM in order to further evaluate the actual cam profile with complicated interpolation algorithms, and then the cam contour approximated by these algorithms may be compared with the theoretical one. In this paper, a direct and simple analytical method, instead of a numerical interpolation algorithm, is proposed for inspecting the profile deviations of conjugate disk cams with coordinate measurement data. The method is based on the derived correlation between the radial-dimension errors and the normal-direction errors of conjugate cam profiles. To verify the proposed method, an experiment of inspecting a pair of conjugate disk cam profiles was conducted. The experimental results obtained from the proposed method were compared with those obtained by using the Hermite interpolation method. It shows that this method is accurate and more efficient for inspecting the profile errors of conjugate disk cams.

<div class="htmlview paragraph">The spur timing gears in Eaton superchargers operate at low torque loads and the supercharger system is especially sensitive to gear whine noise created by minute differences in the spur gear tooth profile quality. This has necessitated the grinding of very high quality profiles on high-contact-ratio spur gears. The manufacturing operation has used subjective evaluation of profile and lead measurements to qualify grinder diamonds and audit gear quality related to noise. They have also relied on supercharger end-of-line-testers to provide a direct measurement of gear noise as the primary quality feedback to the gear manufacturing process. Since the difference in the inspection plots of very high quality profiles is difficult to determine subjectively, the inspection process assessments have been difficult to correlate to the resultant gear noise measurements. The transmission error (TE) calculated from these profile measurements provided and objective assessment that was shown to correlate well with the actual measured gear whine noise. This study resulted in the creation of TE monitoring software that was an add-on to the inspection machine. This software, which uses a version of the LDP code (by The Ohio State Gear Dynamics Laboratory) as the core calculation routine, provides an objective metric of the gear mesh frequency TE and several harmonics calculated from the measured involute profile data. This objective TE metric provides a traceable standard for ensuring low vibration excitation in the gear mesh directly after grinding the gears.</div>

There are two types of curvilinear cylindrical gear drives: those with point contact and those with line contact. In this paper, a mathematical model, tooth contact analysis and stress analysis are investigated for curvilinear cylindrical gears with line contact. Here, fixed-setting face-milling cutters are used for the generation of the tooth surfaces of the gears and their matching gears. Based on the theory of gearing, a mathematical model of curvilinear gears is derived, and the instantaneous contact lines on the gear tooth surfaces are examined under ideal assembly conditions. Meanwhile, the instantaneous contact points on the gear tooth surfaces and transmission errors under assembly error conditions are investigated by a tooth contact analysis algorithm. Finally, contact and bending stresses for curvilinear cylindrical gears with tip relief and without tip relief are explored by the finite element method. This research result shows that (1) under ideal assembly conditions, the length of the instantaneous contact lines is greater than the tooth face width, but the effect is very weak. (2) In terms of the transmission errors, crossing shaft angle errors are more sensitive than intersecting shaft angle errors; axial displacement errors are more sensitive than centre distance errors. (3) Severe contact stresses caused by edge contact are eliminated by the tip relief. (4) The maximum bending stress at every contact position is always distributed in both ends of the gear tooth surfaces.

Evaluation of circularity deviation from coordinate measurement data using an improved area hunting method

The traditional one-component transmission error of parallel-axis helical gears is generalized to a three-component transmission error which characterizes the composite displacement in the plane-of-contact resulting from arbitrary small deviations in the positions of both gears of a meshing pair from the positions of their rigid perfect involute counterparts. A set of linear algebraic equations is derived for the contribution to the three generalized transmission error components arising from elastic deformations of the teeth and gear bodies and deviations of the tooth running surfaces from equispaced perfect involute surfaces. It is shown how to combine this set of equations with the generalized transmission error definition and the equations of motion of a gear system to predict the dynamic response of gear elements in the system. For the case of negligible gearbody and bearing/bearing support inertial forces, an additional set of algebraic equations that includes the effects of bearing flexibility and misalignment is derived. Combining the solution of this set of equations with the above-mentioned generalized transmission error equations yields the three-component generalized static transmission error.

The transmission error of gears is an important factor for power transmissions, particularly automotive. Consequently, a lot of research has been conducted on the gear transmission. However, in contrast, there remains relatively little research clarifying the characteristics of gear transmission error and its relationship to the tooth flank form. The authors have proposed a general model for tooth meshing between gears. This expresses the transmission error theoretically from a quasi-infinite elastic model, which is composed of springs with stiffness specific to the gears. In this report, the influence of tooth flank form deviation on transmission error is further investigated by using this model. It is shown that the form of the ridge curve of the tooth flank form deviation greatly influences the actual contact ratio at the minimum point of peak-to-peak value of transmission error. The peak-to-peak value of transmission error is affected by the amplitude and the form of the ridge curve.

Loaded tooth contact analysis (LTCA), deformation, and stress calculations of three types of thin-rimmed spur gears with inclined webs at the left side, the center, and the right side of the tooth are conducted in this paper with finite element methods (FEM) combined with a mathematical programming method when these gears are engaged with a solid mating gear. The contact stresses on the tooth surfaces, the bending stresses at the tooth roots, and the joint stresses of the rim and the web are analyzed as the web position and the web angle are changed. It was found that the web position and the web angle of the thin-rimmed gears have a significant effect on the tooth contact stresses, the root bending stresses, and the joint stresses. The maximum contact stress of the right web gear becomes very small when the web is inclined from 0 to 30 deg, and it increases again when the web angles are changed from 45 to 60 deg. This interesting relationship between the web angle and the tooth contact stresses of thin-rimmed right inclined web gears is used to absorb the misalignment errors in the gears and improve the partial tooth contact pattern resulting from the misalignment errors. A calculation example is provided to illustrate how to improve the partial tooth contact patterns in gears with misalignment errors.

Towards a haptic virtual coordinate measuring machine

Spur gears for many applications make use of profile shapes that are modifications of the mathematically defined involute to avoid premature tooth contact and unfavourable transmission error. It is well known that specification of the involute modification is important as it relates to the resulting performance, vibration and gear noise. In this research, a hybrid analysis method using finite elements and contact mechanics was used to investigate the effects of linear, parabolic and exponential tooth profile modifications on performance of an involute spur gear pair. For each type of modification, 10 different magnitudes of tip relief were used to study its influence on static transmission error (STE), tooth loads, root stresses and contact stresses. Peak-to-peak transmission error, root mean square transmission error and total area per cycle transmission error were used to characterise STE. Multiple output responses were calculated including the STE characteristics, tooth loads, tooth root stresses, contact stresses, gear moments and bearing forces to compare and contrast the type and magnitude of the tooth modification. Results show that linear modification produced low root stresses; parabolic modification produced low STE, contact stress, gear moment variations, and bearing force variations; exponential modification produced nearly the same STE characteristics over a wide range of tip modification magnitudes.

The measurement of micro gears with a touch-trigger probe will be introduced in this paper. Due to the small size and complex geometry of micro gears, it is difficult to measure them by traditional instruments like coordinate measuring machines (CMM) or laser displacement sensors. This study focuses on this topic and proposes an approach to measure the three dimensional profile of micro gears. The methodology is based on coordinate measurement. A special touch-trigger probe which detects the moment of contact is employed in the measuring system. The diameter of the probe tip is smaller than 2µm and therefore small enough for measuring micro structures. All the calibration and error compensation for the measuring system are discussed in the view of micro metrology. The comparison of measured profiles from both coordinate and image measurement is discussed. Results of experiments show that the proposed approach is worth of further development in the future.

The traditional definition of the transmission error of parallel-axis gear pairs is reviewed and shown to be unsuitable for characterizing the deviation from conjugate action of bevel gear pairs for vibration excitation characterization purposes. This situation is rectified by generalizing the concept of the transmission error of parallel-axis gears to a three-component transmission error for spiral bevel gears of nominal spherical involute design. A general relationship is derived which expresses the contributions to the three-component transmission error from each gear of a meshing spiral bevel pair as a linear transformation of the six coordinates that describe the deviation of the shaft centerline position of each gear of the pair from the position of its rigid perfect involute counterpart.

Purpose Crossed helical gears have point contact and their surfaces are subject to high surface stress. Contact stress and root tooth stress are the most common sources of failure in crossed helical gear. This paper aims to study the load-carrying capacity and performance of crossed helical gear teeth with different gear tooth profiles. To overcome defects and reduce the sensitivity to small shaft angle changes. The combined tooth profile is designed to reduce the bending stresses, contact stresses and tooth deflection, and prevent pinion failure in the gearbox. Design/methodology/approach The principle of the method is a line contact is introduced instead of a point contact between two teeth in mesh with each other. The tooth surface of the helical gear is designed and cut by a modified tool. Higher normal pressure angles like 25° and 35° are used. The modified involute is accomplished to eliminate interference between the teeth. Engineering software packages have been applied to generate all crossed helical gears gear profiles. The modification is compounding curves consisting of an epicycloid-involute-hypocycloid gear teeth profile generated by the cutter modified. Findings The stresses in the crossed helical gear teeth profile were reduced by increasing the normal pressure angle values. Using a 35° pressure angle the enhancement percentage in contact stress and teeth fillet region will be about 29.345% and 15.421%, respectively. The best enhancement in a gear’s resistance is the epicycloid-involute-hypocycloid gear teeth profile. The enhancement was 32.610% and 18.588%. The skew in line of action in skewed helical gear will be sensitive when the crossing angles are small. Their teeth surface tends to be easily worn out; however, the wearing process will be reduced by using a proposed gear teeth profile. Practical implications The gear teeth to be modified are cut by a shaper process. The modifying rack cutter of this study can be used as a reference for creating a different helical gears sample. The helical teeth surface is modified to become an envelope of the other. This makes an original point contact change into a line contact. The epicycloid-involute-hypocycloid gear teeth profile is preferred for a higher contact ratio and a large load capacity. This work explicitly introduces a new method of kinematic consideration to improve the load capacity of crossed helical gear. Originality/value This paper showed some novel results by the unique shape of the rack-cutter designed to generate different helical angles and different gear positions. In the future, it will make valuable contributions to the further development of the dynamic performance of a crossed helical gear system through the study in the field of using asymmetric teeth profiles of helical gears with tip relief as the manner to enhance the crossed helical gear performance. Investigation of crossed helical gear by applying a predesigned parabolic function of transmission error enables the absorption of linear discontinuous functions caused by misalignment.

Transmission error is typically understood to act as the primary source of gearbox noise and vibration. This paper investigates the effect of sliding friction on the transmission error and tooth load of spur gears. To do so, the kinematic relation for the sliding mechanism of spur gears and mesh stiffness was calculated. The relationship between tooth load, tooth errors and mesh compliance as well as the moment balance equation in consideration of the teeth friction force are derived. Transmission error, tooth load, and the teeth friction force of gears with/without modification were investigated. As the results, friction caused an increase in tooth load and transmission error in gear approach and a decrease in tooth load and transmission error in gear recess.

In this paper, we present a method for the global optimization of the tooth contact pattern and transmission error of spiral bevel and hypoid gears, which includes three optimization objectives, three control parameters, and a complex-constrain genetic algorithm solving method. A new set of fundamental equations for pitch cone parameters of hypoid gear drives are established, as well as the relationships between pitch cone and curvature parameters. Based on this theory, three control parameters are selected to determine the pinion tooth surface. A hypoid gear drive is chosen for case studies. The results verify that the optimization methodology can achieve the expected optimization objectives and has good convergence. Correlations between optimization objectives and control parameters are discussed. Furthermore, a finite element model of a simplified hypoid gear drive system is established and its quasi-static meshing characteristics analyzed. The results again confirm the correctness of the optimization method. The effects of torque load on the contact pattern and transmission error are discussed. The results provide a theoretical reference for geometric calculations, quasi-static analysis, and optimal design of spiral bevel and hypoid gears.

The contact between tooth surfaces of the conical involute gear is a point contact. Therefore, tooth surface strength of the conical involute gear is smaller than the cylindrical gear. In the design of the conical involute gear, the evaluation of tooth surface strength is more important than bending strength. Tooth surface strength is evaluated at the allowable contact stress of tooth surface. A fatigue test is performed in order to evaluate this allowable contact stress. But, tooth surface is worn out during this test. Therefore, it is difficult to calculate the contact stress. Wear of tooth surface is influenced slip. Slip is evaluated the spesific sliding. But, there is no theoretical formula of the conical involute gear's spesific sliding now, so this theoretical formula is asked on an approximation target in this research. The relation between wear of the test gear and the spesific sliding is considered.