Quasi-static Transmission Error Research Articles

Analytical model of meshing stiffness, load sharing, and transmission error for internal spur gears with profile modification

The influence of the teeth geometry on the load distribution, transmission error, and meshing stiffness of internal spur gears is very similar to that of external gears. Therefore, the formulation of the models for strength and dynamic calculations for internal and external gears is also similar. However, the internal gears have usually greater values of the contact ratio, frequently above 1.8, even 1.9. In these cases, the lengthening of the contact interval due to the elastic deflections of the teeth may result in an effective contact ratio greater than 2, with the subsequent changes in the load sharing and transmission error. These changes are not properly described by the theoretical models for high contact ratio spur gears. In this paper, a model of load sharing, quasi-static transmission error, and time-varying meshing stiffness for internal spur gears with profile modification is presented, which has been applied to low contact ratio internal gears which, due to the transmitted load, become high contact ratio internal gears. The influence of the variations of the transmitted load on the load capacity is also studied.

Accuracy characteristics of the testing bench for dynamic transmission error

A high-precision testing bench is required for accurately assessing the dynamic Transmission error (TE) of precision reducers. However, the relationship between the testing bench’s accuracy and deviations in testing conditions has not been determined, leading to an inaccurate evaluation of its metrological performance. In this study, we examined the accuracy characteristics of the testing bench. We analyzed the structure of the bench and identified the main error sources that affect its accuracy, including mechanical system deviation, sensor measurement deviation, and testing condition deviation. In addition, we studied the mechanism of these error sources and proposed a method to convert drive speed deviation and load torque deviation into angle deviation by using a three-dimensional characterization model of lost motion. Based on the expression of TE, we proposed an evaluation method for the accuracy of the testing bench. Furthermore, we presented the methods for calculating the error sources and proposed a method for determining the TE of adjacent units based on the specific coupling structure. Finally, we conducted practical tests to validate the effectiveness of the proposed method in evaluating the accuracy characteristics of the testing bench. We found that the relative test error between quasi-static TE and dynamic TE is small, and more accurate test results are obtained when applying an appropriate load.

Analysis of the tip interference in low gear ratio internal spur gears with profile modification

This paper presents a discussion on the influence of the tip relief on the tip interference and contact ratio, as well as a methodology to select proper values of the teeth heights, center distances, rack shift coefficients, and tip reliefs to ensure maximum values of the contact ratio while avoiding tip interference, root interference and pitch interference. The influence of the quasi-static transmission error on the tip interference is also considered in the analysis.

Low excitation spur gears with variable tip diameter

One important source of noise in drive trains are transmissions. In numerous applications, it is necessary to use helical instead of spur gear stages due to increased noise requirements. Besides a superior excitation behaviour, helical gears also show additional disadvantageous effects (e.g. axial forces and tilting moments), which have to be taken into account in the design process. Thus, a low noise spur gear stage could simplify design and meet the requirements of modern mechanical drive trains. The authors explore the possibility of combining the low noise properties of helical gears with the advantageous mechanical properties of spur gears by using spur gears with variable tip diameter along the tooth width. This allows the adjustment of the total length of active lines of action at the beginning and end of contact and acts as a mesh stiffness modification. For this reason, several spur gear designs are experimentally investigated and compared with regard to their excitation behaviour. The experiments are performed on a back-to-back test rig and include quasi-static transmission error measurements under load as well as dynamic torsional vibration measurements. The results show a significant improvement of the excitation behaviour for spur gears with variable tip diameter.

Research on the variation of mesh stiffness and transmission error for spur gear with tooth profile modification and wear fault

Tooth profile modification and wear are unavoidable, even intentional tooth profile deviations. Both of them have a significant influence on gear mesh stiffness and quasi-static transmission error (QSTE). In this paper, to efficiently and accurately obtain the two internal excitations of spur gear pairs with the above two kinds of deviations together, improved analytical mesh stiffness and QSTE models are proposed. And the effects of corner contact and structure coupling are considered together to overcome the defect of previous works. By comparing with the results of the finite element method, the established models are verified. Through the models, the mesh stiffness and QSTE of the gear pairs with the tooth profile modification and wear simultaneously can be obtained. The results show that both of the two kinds of deviations have significant effects on the limit value and distribution of the mesh stiffness and QSTE. Besides, it can be found that there is a competition between tooth profile modification and wear, and the influence extent of wear is gradually increasing that eventually can override the modification effect. Moreover, wear can affect the selection of tooth profile modification.

Control of transmission error of high contact ratio spur gears with symmetric profile modifications

This paper presents a study on the influence on the quasi-static transmission error of symmetric long profile modifications on high contact ratio spur gears. A previously developed model for the load sharing ratio and quasi-static transmission error, based on the hypothesis of equal delay interval in all the teeth in simultaneous contact, has been used. The simple formulation of the model allows to obtain the equation of the required profile modification to ensure a preestablished function of load sharing ratio or quasi-static transmission error. From this model, the tip reliefs for minimum peak-to-peak amplitude of quasi-static transmission error and for minimum dynamic load, have been obtained. In both cases, the optimum length of modification has been expressed as a function of the contact ratio.

Load sharing model for high contact ratio spur gears with long profile modifications

The start of contact between loaded involute gear teeth occurs before reaching the theoretical inner point of contact due to the load-induced deflections of previous tooth pairs in contact. This sooner contact occurs outside the pressure line and produces a shock between the driving tooth root and the driven tooth tip, which induces noise, vibrations and dynamic load. To avoid these undesirable effects profile modifications are often used, which through a suitable tip relief at the driven tooth delay the actual start of contact until locate it at the theoretical inner point of contact. However, the length and shape of profile modification have also influence on the curves of load sharing and quasi-static transmission error. Specifically, long tip relieves, beyond the interval of minimum tooth pair contact, which are unsuitable for standard contact ratio spur gears, may reduce drastically the load at the inner points of the path of contact of high contact ratio gears, though a peak of load arises at the outer interval of two pair tooth contact. Since the determinant contact stresses are usually located at the inner points of the contact interval and the determinant tooth-root stresses at the outer ones, long tip relieves can be used for balancing both determinant stresses and improving the load capacity.

Influence of profile modifications on meshing stiffness, load sharing, and trasnsmission error of involute spur gears

The meshing stiffness, load sharing ratio, and quasi-static transmission error of spur gear pairs are related to each other and are dependent on tooth geometry. In addition, profile modifications influence the contact conditions, because modified profile points may only contact under load conditions and owing to load-induced elastic deflections. However, this is not the case under no-load conditions, wherein a gap would exist between the teeth owing to the contact of other tooth pairs at unmodified points. In this paper, the contact conditions of modified teeth under load have been studied, and the influence of the amount, length, and shape of the profile modification on the load sharing and transmission error has been described. A simple, analytic formulation of all these relations is provided, based on a previously developed approximate equation for the meshing stiffness, which provides accurate results. The simplicity of the obtained equations facilitates the study of the influence of the profile modification on the load sharing and transmission error, and the required profile modification for predesigned functions of load sharing ratio or quasi-static transmission error was also obtained.

Reduction of the tonality of gear noise by application of topography scattering

In the design process of transmissions, one major criterion is the resulting noise emission of the powertrain due to the gear excitation. Within the past years, a lot of investigations have shown that the noise emission can be correlated to the quasi-static transmission error. Therefore, the transmission error can be used as a characteristic value for quality assurance by conducting experimental inspections as well as a tooth contact analysis in the design process.An optimization approach to solve gear whine issues is the reduction of the gear mesh excitation. In this report, a novel optimization approach for gears is presented. A targeted topography scatter from tooth to tooth is applied on the teeth of the gear set, in order to manipulate the composition of the gear mesh excitation. As a result of the slight deviations for each tooth mesh, the regularity of the transmission error excitation is broken up leading to a reduction of the tonality of the resulting gear whine.In the first step, different topography scatter variants are calculated in the tooth contact analysis. Based on the transmission error spectra, two variants with a topography scatter are chosen for manufacturing. The evaluation of the predicted excitation behavior is evaluated on a conventional bevel gear tester by means of single flank tests. A test fixture for the evaluation of the operational behavior under loaded and dynamic conditions is used to compare the measured and simulated dynamic excitation behavior. In order to simulate the dynamic behavior, a torsional multi-body-simulation model of the powertrain is developed that uses the stiffness of the gear mesh calculated by the tooth contact analysis as input data for the excitation forces. The force-coupling-element connects the drive and the driven train and forms the excitation of the gearing.Finally, it is the aim to evaluate the impact of the targeted topography scatter psychoacoustic parameters, such as loudness, sharpness, roughness and tonality. The potentials of the topography deviation for the optimization of ground bevel gears in terms of tonality reduction will be shown by means dynamic test rig trials.

Study of the tooth contact for high contact ratio spur gears with long tip relief

Profile modifications are commonly used to avoid shocks between meshing gear teeth produced by the delay of the driven gear, and the subsequent sooner start of contact, due to the teeth deflections. A suitable tip relief at the driven tooth shifts the start of contact to the proper location at the theoretical inner point of contact. The shape of the relief governs the loading curve of the tooth pair, while the length of relief determines the intervals in which this actual loading curve differs from the theoretical one of unmodified teeth. As at least one tooth pair should be in contact at the unmodified involute profile interval, the length of modification should be smaller than the length of the intervals of two pair tooth contact; otherwise, a shock at the end of contact of the previous pair is unavoidable. However this problem does not occur for high contact ratio spur gears, in which at least two couples of teeth are in contact at any moment. In this work, a study on the load sharing and the quasi-static transmission error for high contact ratio spur gears with long profile modification has been performed, and a model for the tooth contact has been developed.

Theoretical Method for Calculating the Unit Curve of Gear Integrated Error

A theoretical method is proposed in this paper to calculate the unit curve of gear integrated error (GIE). The calculated GIE unit curve includes the quasi-static transmission error (TE) curves of the approach stage, the involute stage, and the recession stage of the ZI worm and helical gear transmission. The misalignments between the two axes of the worm and gear, as well as the modifications or errors of the tooth flanks of the gear, are considered in the procedure of calculation. Optimization algorithm is introduced to replace the solving of implicit differential equations of the conventional tooth contact analysis (TCA) method. It is proved that the proposed method is clearer and more convenient than the conventional TCA methods in calculating the GIE unit curve. The correctness and merits of the proposed method are verified by two experiments.

On the analytical definition of profile modifications minimising transmission error variations in narrow-faced spur helical gears

Two original analytical formulations are presented which explicitly give the depths and extents of symmetric profile modifications minimising the fluctuations of quasi-static transmission error in narrow-faced spur and helical gears. Numerous comparisons with quasi-static and dynamic simulation results are presented which prove that the proposed theory is sound. The proposed formulae can therefore help define adapted reliefs in terms of transmission error and dynamic tooth loading with minimum effort.

Influence of indexing errors on dynamic response of spur gear pairs

In this study, a dynamic model of a spur gear pair is employed to investigate the influence of gear tooth indexing errors on the dynamic response. This transverse-torsional dynamic model includes periodically-time varying gear mesh stiffness and nonlinearities caused by tooth separations in resonance regions. With quasi-static transmission error time traces as the primary excitation, the model predicts frequency-domain dynamic mesh force and dynamic transmission error spectra. These long-period quasi-static transmission error time traces are measured using unity-ratio spur gear pairs having certain intentional indexing errors. A special test setup with dedicated instrumentation for the measurement of quasi-static transmission error is employed to perform a number of experiments with gears having deterministic spacing errors at one or two teeth of the test gear only and random spacing errors where all of the test gear teeth have a random distribution of errors as in a typical production gear.

New methodology to reduce the transmission error of the spiral bevel gears

New methods and tools have been developed the last years to improve the understanding of gear meshing. Mechanical industries attach growing attention to the dynamic behavior of mechanical transmissions, including vibration and noise that result. The transmission error of the gear, which measures the intensity of one of the main causes of dynamic phenomena, can be considered as a relevant indicator of gear performance.This paper presents a new design method of spiral bevel gears, the objective of this method being to reduce their quasi-static transmission error. The proposed approach is based on an optimization process including loaded meshing simulations. The simulation model has been evaluated using a helicopter tail gearbox as bench test. Measurement results are given, showing a good correlation with predictions.