Study on Gear Contact Stiffness and Backlash of Harmonic Drive Based on Fractal Theory

Contact stiffness and backlash model of harmonic reducer is related to robot’s positioning accuracy and vibration characteristics. Harmonic reducer tooth pair height is typically less than 1 mm. Thus, backlash and contact stiffness measurement and modeling are relatively complex. In this paper, contact stiffness and backlash model is proposed by establishing a relationship between fractal parameters and tooth contact load. Non-contact optical profiler and RMS method are combined to obtain fractal roughness parameters of real machined tooth surface. Finally, the effect of rough tooth surface and contact force fractal parameters on contact stiffness and gear backlash is studied. The results indicate that surface topography parameters and contact force have significant effects on contact stiffness and backlash. By increasing the fractal dimension, a decrease of gear backlash and contact stiffness is observed. However, the opposite is true for the fractal roughness parameter. Lastly, an increase in contact force improves the contact stiffness.

Behavior of the average-log-ratio ALR gear-damage detection algorithm in cases of a single damage form

A mathematical model of static-transmission-error frequency-domain contributions caused by a single generic form of gear-tooth damage is used to explain observed behavior of the average-log-ratio (ALR) gear-damage detection algorithm applied to a case of tooth-bending-fatigue damage. The periodic behavior of rotational-harmonic frequency spectra resulting from tooth damage is explained and experimentally verified. Monotonic increases in ALR contributions in the rotational-harmonic region below the tooth-meshing fundamental harmonic are unambiguously related to increasing gear damage by use of Parseval’s theorem for the discrete Fourier transform. Computation of ALR using rotational-harmonic bands between adjacent tooth-meshing harmonics is suggested for early detection of gear damage. Large high-frequency ALR contributions are explained by transmission-error jump (step) discontinuities caused by large tooth-pair deformations, indicating a severe state of damage.

A Detailed Investigation of Gear Body-Induced Tooth Deflections and Development of an Improved Analytical Solution

Many researchers have developed analytical methods to evaluate gear meshing stiffness. Some of these methods ignored the effect of the gear body while others used a simplified model to consider its effect. Until now, a detailed investigation of gear body-induced tooth deflections has been rare, especially for the double-tooth-pair meshing period. In this study, we present a detailed investigation of gear body-induced tooth deflections. To be specific, we will discuss how to accurately evaluate gear body-induced tooth deflections using the finite element analysis, and what are the effect of parameters such as loading and gear parameters on gear body-induced tooth deflections. Then, an improved solution is developed for evaluating the body-induced tooth deflection. In the single-tooth-pair meshing period, the improved formula is developed based on a popular formula proposed by Sainsot and Velex. This is achieved by optimizing the coefficients used in their formula to make the formula more accurate to evaluate gear body-induced tooth deflections. Meanwhile, we introduce a new term called affiliated body stiffness to evaluate the body-induced tooth deflections in the double-tooth-pair meshing period. The improved method can give higher accuracy in evaluating gear body-induced tooth deflections of spur gears with a pressure angle of 20°.

Comparison on torsional mesh stiffness and contact ratio of involute internal gear and high contact ratio internal gear

The mesh stiffness and contact ratio of gear drive are very important factors which have a great impact on the dynamic load. Contact ratio also affects the fluctuation and the mode of change of the mesh stiffness. In this research, a novel high contact ratio internal gear with a circular arc contact path is introduced. However, the irregular tooth profile of non-involute gear usually causes the numerical calculation to be more complex. To get the torsional mesh stiffness of a pair of internal spur gear, the two-dimensional finite element models of involute internal gear and high contact ratio internal gear are presented and compared. In addition, the influence of input torque on torsional mesh stiffness and contact ratio are analyzed. The mesh stiffness of a single tooth pair and the effect of different engagement positions on mesh stiffness are obtained and compared. Finally, experimental measurement of contact ratio is established by strain gauge technique. It is shown that the torsional mesh stiffness increases with the increase of input torque, and the greater the contact ratio, the smoother the gear drive.

On Dynamic Mesh Force Evaluation of Spiral Bevel Gears

The mesh model and mesh stiffness representation are the two main factors affecting the calculation method and the results of the dynamic mesh force. Comparative studies considering the two factors are performed to explore appropriate approaches to estimate the dynamic meshing load on each contacting tooth flank of spiral bevel gears. First, a tooth pair mesh model is proposed to better describe the mesh characteristics of individual tooth pairs in contact. The mesh parameters including the mesh vector, transmission error, and mesh stiffness are compared with those of the extensively applied single-point mesh model of a gear pair. Dynamic results from the proposed model indicate that it can reveal a more realistic and pronounced dynamic behavior of each engaged tooth pair. Second, dynamic mesh force calculations from three different approaches are compared to further investigate the effect of mesh stiffness representations. One method uses the mesh stiffness estimated by the commonly used average slope approach, the second method applies the mesh stiffness evaluated by the local slope approach, and the third approach utilizes a quasistatically defined interpolation function indexed by mesh deflection and mesh position.

Flash Temperature Analysis Method for Polymer Gears With Consideration of Deviations in Meshing Kinematics

The presented work describes a computational method for carrying out a detailed and thorough examination of the flash temperature rise (i.e. the local ‘instantaneous’ temperature increase on a contact interface, due to frictional effects) present on the tooth flanks of a polymer gear pair, composed of a combination of POM and PA66 thermoplastics, during a given meshing cycle. The method involves a decoupled sequential procedure, where first the mechanical response of the gear teeth during a whole meshing cycle is analyzed using finite element analysis and, subsequently, a semi-analytical thermal analysis procedure is employed, with which the local flash temperature rise under a given tooth-pair contact can be evaluated. The method provides an accurate reproduction of the actual thermo-mechanical processes taking place at the gear teeth contact interfaces and allows for an investigation of the influence of deviations in the gear flank geometry and gear tolerances, while retaining a manageable enough form for application with moderate computational resources.

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.

Planar Tooth Profile Synthesis for Relative Curvature

This paper is the first that uses the new conjugation curvature theory [1] to directly synthesize conjugate tooth profiles with the given relative curvature that determines the Hertzian contact stress. Conjugation curvature theory offers a systematic methodology to synthesize the relative curvature for a tooth pair. For any given relative curvature between the contact tooth profiles, a generating point can be located on an auxiliary body. Under the rolling motion among the pinion pitch, the gear pitch and the pitch on the auxiliary body, the generating point will trace fully conjugate profiles on the pinion and gear bodies with the given relative curvature at the instant of the contact. Full conjugation throughout the contact of the profiles is guaranteed with the three instant centers remaining coincident [1]. The methodology is demonstrated with a planar tooth profile synthesis with given relative curvature. One may find that the Wildhaber-Novikov tooth profile, which is known to have low relative curvature and Hertzian contact stress, and its variations become special cases under such methodology.

The Generation Principle and Mathematical Models of Double-Enveloping Internal Gear Pairs

In this paper, the planar double-enveloping method is presented for the generation of tooth profiles of the internal gear pair for various applications, such as gerotors and gear reducers. The main characteristic of this method is the existence of double contact between one tooth pair such that the sealing property, the load capacity and the transmission precision can be significantly improved as compared to the conventional configuration by the single-enveloping theory. Firstly, the generation principle of the planar double-enveloping method is introduced. Based on the coordinate transformation and the envelope theory, the general mathematical model of the double-enveloping internal gear pair is presented. By using this model, users can directly design different geometrical shape profiles to obtain a double-enveloping internal gear pair with better meshing characteristics. Secondly, to validate the effectiveness of the proposed model, specific mathematical formulations of three double-enveloping internal gear pairs which apply circular, parabolic and elliptical curves as the generating curves are given. The equations of tooth profiles and meshing are derived and the composition of tooth profiles is analyzed. Finally, numerical examples are provided for an illustration.