Time-varying Mesh Stiffness Calculation Research Articles

Calculation of Time-Varying Mesh Stiffness of Internal Gears based on Precise Tooth Profile and Dynamic Analysis of Planetary Systems with Root Cracks

The internal meshing spur gear pair is the research object, and the potential energy method is applied to calculate the time-varying meshing stiffness of the internal gear. The internal gear tooth profile is divided into two parts: involute and transition curves, and the gear tooth stiffness is calculated by numerical integration based on accurate tooth profile, which improves the calculation accuracy. Analyzed the influence of different crack size parameters on the stiffness of internal meshing gears. A coupled dynamic model of a planetary system with internal gear crack faults was established, and the influence of cracks on the dynamic response of the planetary system was studied using Zoom-FFT spectrum and cepstrum analysis methods. The simulation results show that as the crack parameter size increases, the mesh stiffness of the internal gear pair gradually weakens, and the periodic vibration impact of the planetary-internal gear pair is also more severe. Cepstrum analysis can easily capture weak fault characteristic frequency information in the system and discover their variation patterns. The research results can provide a theoretical basis for the state monitoring and fault diagnosis of gear systems with crack defects.

Time-varying mesh stiffness calculation of spiral bevel gear with spalling defect

Spiral bevel gears often operate at high speeds and under heavy loads, which can easily lead to the occurrence of spalling faults on the tooth surfaces, particularly in cases of inadequate lubrication. In this study, we introduce an efficient and accurate numerical calculation method to determine the time-varying meshing stiffness (TVMS) of a spiral bevel gear pair with actual spalling faults, taking into consideration the effects of multi-tooth meshing and the flexibility of the gear flank. In parallel, a finite element model is established to verify the accuracy of the proposed method across various spalling parameters. We delve into the impact of spalling length, width, location, and shape. The results demonstrate that the TVMS values obtained from our proposed method align closely with those obtained using the finite element method. The range of TVMS reduction caused by the spalling fault is primarily determined by the spalling location and width. Moreover, the magnitude of TVMS decline is strongly associated with the spalling length. Furthermore, the clearances between two conjugated tooth surfaces change due to the presence of the spalling fault, resulting in a redistribution of contact forces on the same contact surface and other meshing tooth surfaces.

Investigation of Effect of the Tooth Profile Modification on the Engagement Characteristics of the Gear Pair

As a vital component in the electric vehicle, the higher requirements for the gear precision are urgently needed in the service environment with high speed and heavy load. The tooth profile modification can greatly improve the accuracy and strength of the gear transmissions. Therefore, accurately obtaining the optimum modification parameters of the gear system under the action of specific loads is the premise of the research of gear vibration and noise reduction. The time-varying mesh stiffness calculation model is established in this paper considering the coupling effect of the tooth profile errors and structural coupling effect. The minimum variations of the loaded static transmission error are selected as the objective function to obtain the optimum modification parameters of gear pair. In addition, the effect of torque, modification amount and modification length on the engagement characteristics of the gear pair are studied. The research can provide some references for the design and optimization for the gear transmission system in the electric vehicle.

An improved time-varying mesh stiffness calculation method and dynamic characteristic analysis for helical gears under variable torque conditions

Based on the slice theory and numerical calculation methods, the time varying meshing stiffness (TVMS) calculation method of the gear pair was constructed for the helical gear pair of a electric continuously variable transmission (ECVT), and the TVMS of the helical gear were calculated under different constant and time-varying torque in this study. The obtained stiffness values were introduced into the established dynamics model of helical gear system, and the influence of changed TVMS, resulting from the variable speed and torque, on the nonlinear dynamic characteristics of gear pair was analyzed using the Runge-Kutta method. The results show that the proposed TVMS calculation method is very effective in computing the TVMS of helical gear pair under time-varying condition. The TVMS and dynamic transmission error (DTE) increase as the torque increase, and vice versa. Meanwhile, the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors at high speed and at low speed when torque is low, whereas not appear at steady high torque under low speed range. This study offers a method for gaining the TVMS of helical gear pair to analyze the dynamic characteristics of gear pair under actual working condition, and the influence of torque that change with the speed ratio on the dynamic performance of a gear system should be considered in the industrial applications.

Vibration-Based Fault Diagnosis of Helical Gear Pair With Tooth Surface Wear Considering Time-Varying Friction Coefficient Under Mixed Elastohydrodynamic Lubrication Condition

Abstract Based on “slice method,” the improved time-varying mesh stiffness (TVMS) calculation model of helical gear pair with tooth surface wear is proposed, in which the effect of friction force that obtained under mixed elastohydrodynamic lubrication (EHL) is considered in the model. Based on the improved TVMS calculation model, the dynamic model of helical gear system is established, then the influence of tooth wear parameters on the dynamic response is studied. The results illustrate that the varying reduction extents of mesh stiffness along tooth profile under tooth surface wear, in addition, the dynamic response in time-domain and frequency-domain present significant decline in amplitude under deteriorating wear condition.

TVMS calculation and dynamic analysis of carburized spur gear pair

Carburized gear tooth are widely used in industries due to their more resistance to wear and abrasion. The top surface of the gear becomes hard due to carburization and offers more resistance to fault penetration developed on the surface of the gear. In gear modeling, the time-varying mesh stiffness (TVMS) is the characteristic feature that indicates the health of the gears in the system. Much research has been carried out on the calculation of TVMS of uncarburized gear and their gear dynamics have been studied for different faults condition on the system. This paper presents first time the mathematical formulation of carburized spur gear pair including the cracked tooth at the root of the pinion for the calculation of TVMS through the potential energy approach. TVMS of carburized and uncarburized spur gear pair are studied and compared analytically. The TVMS results show that the carburized gear pair has more stiffness than the uncarburized gear pair at different crack depths. Furthermore, the gear modeling of a single-stage carburized spur gear pair is also developed to study the effect of carburization on the dynamic behavior of the system at different crack depths. The vibration responses for both carburized and uncarburized gear pairs are obtained and compared at different crack depths. The cepstrum analysis is also carried out on vibration responses to analyze the impact of tooth cracks on the characteristic frequencies and their sidebands for both cases. At the end of this work, both carburized and uncarburized responses are compared through spectral entropy. The obtained results provide important guidelines regarding the condition monitoring and fault diagnosis of carburized spur gear pair at different tooth crack depths.

Time-varying mesh stiffness calculation for gear pairs with misalignment errors in multiple degrees of freedom based on an analytical method

Misalignment errors (MEs) in multiple degrees of freedom (multi-DOFs) at the mesh position will lead to a change in the time-varying mesh stiffness (TVMS) and then affect the dynamic behaviour of gear pairs. Therefore, a new, more rapid, three-dimensional analytical model for TVMS calculation for gear pairs with three rotational and three translational MEs is established in this paper, and a new solution method based on potential energy theory is presented. In addition, the correctness of the new model is verified by the finite element method (FEM). Moreover, the effective contact line, uneven distribution of mesh force on the contact line, and mesh position change are taken into account. Finally, the TVMS under different ME conditions is calculated with the new analytical model. The results showed that the different MEs have dissimilar effects on the TVMS, and the relationship between the ME and TVMS is nonlinear. In addition, the region of single-pair and double-pair teeth in contact would also change with ME.

Time-varying mesh stiffness calculation of a planetary gear set with the spalling defect under sliding friction

Time-varying mesh stiffness (TVMS) is an important excitation source of a planetary gear set. Sliding friction and spalling defects have significant effect on the TVMS. Accurate evaluation of the TVMS can help obtain the vibration characteristics and further detect the spalling defects of the planetary gear set. In this paper, an improved analytical model is proposed to calculate the TVMS of the planetary gear set with sliding friction by considering the tooth profile beginning with the root circle. Then the impact of spalling defects on the TVMS is investigated. The results show that the improved analytical model can increase the precision of the TVMS. Besides, the effect of sliding friction and spalling defects on the TVMS is significant. This study offers a basis for dynamic performance analysis of a planetary gear set with spalling defects under sliding friction.

Time-varying Mesh Stiffness Calculation and Load Distribution among Teeth of Cycloid Internal Gear Pair with High Contact Ratio

Time-varying Mesh Stiffness Calculation and Load Distribution among Teeth of Cycloid Internal Gear Pair with High Contact Ratio

The Time-varying Mesh Stiffness Calculation for Gear Tooth Crack based on Analytical-finite Element Method

The Time-varying Mesh Stiffness Calculation for Gear Tooth Crack based on Analytical-finite Element Method

Time varying mesh stiffness calculation of spur gear pair considering sliding friction and spalling defects

Abstract Spalling is one of the common tooth surface failures of gear teeth and is defined as the formation of deeper cavities that are mainly developed from subsurface defects. The time varying mesh stiffness (TVMS) of gear pairs, gives significant information about the health of the system. The change indirection of time varying friction on both sides of the pitch line causes the change of gear mesh stiffness. This article proposes a computer simulation based approach to study the effect of time varying friction coefficient on the total effective mesh stiffness for the spur gear pair. An analytical method to calculate the TVMS of the spur gear for different spall shapes, size and location considering sliding friction is also proposed in this study. The results show that spall shape, size and location are very important parameters that need to be considered for calculation of TVMS and subsequently to know the dynamic response of the gear pair in the presence of a spall.

Time-varying mesh stiffness calculation of spur gears with spalling defect

Abstract Considering the effects of extended tooth contact (ETC), revised fillet-foundation stiffness under double-tooth engagement region, nonlinear contact stiffness and tooth spalling defect, an analytical model for time-varying mesh stiffness (TVMS) calculation of spur gears is established. In addition, the analytical model is also verified by comparing the TVMS under different spalling widths, lengths and locations with that obtained from finite element method. The results show that gear mesh stiffness decreases sharply with the increase of spalling width, especially during the single-tooth engagement; the spalling length only has an effect on the beginning and ending of gear mesh stiffness reduction; the spalling location can affect the range of gear mesh stiffness reduction, and the range will reduce when the spalling location is close to the addendum. This study can provide a theoretical basis for spalling defect diagnosis.

Time-varying mesh stiffness calculation of cracked spur gears

Considering the misalignment of gear root circle and base circle and accurate transition curve, an improved mesh stiffness model for a healthy gear pair is proposed and validated by the finite element method (FEM). Based on the improved method, three mesh stiffness calculation methods (method 1: straight lines for crack path and limiting line; method 2: straight line for crack path and parabolic curve for limiting line proposed in Ref. [1]; method 3: parabolic curves for crack path and limiting line) for cracked gear pair are presented and compared with FEM. The results show that there is a significant difference between method 1 and FEM under large crack condition and the results of methods 2 and 3 are quite close to FEM result, which also shows that the parabolic curve as a limiting line is appropriate. Mesh stiffness of method 2 is very close to that of method 3, which also shows that it is acceptable to assume the crack path to be a straight line.