Analytical model for meshing stiffness, load sharing, and transmission error for helical gears with profile modification

Effects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system

Dynamic analysis of high speed gears by using loaded static transmission error

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.

The measurement of the meshing stiffness in gear pairs is a technological problem. Many studies have been conducted, but a few results are available. A tailored test bench was designed and realized to measure the Static Transmission Error in two mating gears to address this issue. The bench is capable of testing several kinds of gears, e.g., spur, helical, conical, and internal, and it measures the transmission error concerning the applied torque. The Static Transmission Error is due to the variable stiffness of the gear teeth during a mesh cycle. In this paper, a dynamical method for measuring gear mesh stiffness is presented. The tooth stiffness is estimated from the torsional modal behavior of the rotating parts of the test bench. The dynamics of the system are acquired using accelerometers and very precise encoders to measure the angular accelerations and displacements of rotating parts. The torsional mode shapes are identified; those that show a vibrational behavior of the gears that do not follow the transmission ratio’s sign of the mating kinematic condition are selected because they depend on the flexibility of the teeth. In such a way, the engagement stiffness is estimated from the natural frequencies of the selected mode-shapes and the known inertia of gears and shafts. The experimentally identified results are also compared with numerical values computed with a commercial software for mutual validation.

In this paper, the harmonic balance method has been extended to investigate the nonlinear dynamics of a compound planetary gear sets. The lumped parameter dynamic model, incorporating parametric gear mesh stiffness fluctuations, transmission errors and gear backlash, is established. The responses of the system, mesh stiffness, transmission error, and gear backlash have all been represented by harmonic functions, which have been utilized to derive the algebraic equations via Galerkin process. The nonlinear dynamic characteristics of the gear sets are researched, and the effect of nonlinearities on the frequency response characteristic has been investigated by changing the value of the non-dimensional backlash, mesh stiffness, and amplitude of error excitations. The multiple value, jump and discontinuities phenomena have been revealed in the presented frequency response curves. The formulation derived here can also be extended to incorporate incremental harmonic balance method and parameter continuation scheme for further nonlinear dynamics investigation.

This paper investigates the nonlinear dynamics of the wheel-side planetary reducer, considering the tooth wear effect. The tooth wear model based on the Archard adhesion wear theory is established, and the impact of tooth wear on meshing stiffness and piecewise-linear backlash of the planetary gear system is discussed. Then, the torsional vibration model and dimensionless differential equations considering tooth wear for the wheel-side planetary reducer are established, in which meshing excitations include time-varying mesh stiffness (TVMS), piecewise-linear backlash, and transmission error. The dynamic responses are numerically solved using the fourth-order Runge–Kutta method. On this basis, the nonlinear dynamics, such as the bifurcation and chaos properties of the wheel-side planetary reducer with tooth wear, are analyzed. Simulation results demonstrate that the existence of tooth wear reduces meshing stiffness and increases backlash. The reduction in the meshing stiffness changes the bifurcation path and chaotic amplitude of the system, inducing chaotic phenomena more easily. The increase in the gear backlash causes a higher amplitude of the relative displacement and more severe vibration.

Angular misalignment is a common error in gear transmission systems, significantly altering the contact and dynamic behaviors of the gear system. Conventionally, the time-varying mesh stiffness (TVMS) of helical gear pairs with angular misalignment is estimated under the assumption of a uniform load distribution along a fixed contact line. However, this simplification neglects the variations in contact line length and load distribution due to angular misalignment, thereby compromising the accuracy of TVMS predictions. To address this limitation, we propose a novel dual-iterative model for TVMS calculation in helical gears, accounting for the angular misalignment-induced contact line variation. This model combines traditional iterative calculations for gear slices with an additional iterative loop for the misalignment-induced contact line changes. Furthermore, it considers both axial and tangential meshing stiffnesses. The proposed dual-iterative TVMS model is verified by using the finite element (FE) method. Moreover, a six degree-of-freedom (DOF) dynamic model for misaligned helical gears is established to investigate the effects of misalignments on the vibration characteristics of the gear pair. An experimental setup utilizing a back-to-back helical gear test rig with adjustable angular misalignment is constructed to validate the proposed model. The experimental results exhibit a close alignment with the simulation predictions. It is concluded that the proposed model is suitable for estimating the TVMS and dynamic analysis of helical gear pairs with angular misalignment errors.

Misalignment of gear shafts, due to manufacturing and assembly errors, has an adverse influence on gear tooth stresses, load sharing and smooth transmission. In practice, gear tooth modifications, such as crowning, are often resorted to. In this work, investigations of simultaneous effect of both misalignment and crowning on transmission error and mesh stiffness variation have been carried out. Necessary three-dimensional model is created using in-house developed code for generation of involute and trochoidal profiles. The three-dimensional stress analysis is carried out using general purpose FEM software ANSYS 12 for contact stresses. The effect of misalignment on the contact stress for different angular rotations during meshing cycle is analysed. A method to obtain contact stress pattern on the basis of a threshold value is proposed in this work. Mesh stiffness, load sharing ratio and transmission error are also evaluated over a complete mesh cycle. The results for uncrowned and crowned spur gear with misalignment errors are compared and discussed in this paper.

The stiffness of the shaft of transmission has an important influence on the meshing quality of gear pair. Taking the 1st helical gears and shaft of an automatic transmission as the research object, two rigid-flexible coupling models for gearbox with different supporting span were presented, and the accuracy of the models was verified by comparing the contact pattern results of experiment and simulation. Based on this, the meshing misalignment and transmission error of the gear pair were calculated, and the influence laws of supporting stiffness on meshing characteristics of helical gear were analyzed under different torques and speeds. It is shown that when the supporting span is large, the meshing misalignment and transmission error are more sensitive to the change of torque. The thickness of the oil film on the meshing tooth surface is larger when the stiffness of the shaft is smaller. However, the meshing misalignment, transmission error and sliding velocity increase. Further more, the distribution uniformity of load along the direction of tooth width is even worse, and the contact temperature of tooth surface increases. By reasonably increasing the supporting stiffness of the shaft, the stability and reliability of the gear pair can be effectively improved.

Nonlinear characteristics of a multi-degree-of-freedom spur gear system with bending-torsional coupling vibration

Analytical modeling of spur gear corner contact effects

Comparison of meshing characteristics of helical gears with spalling fault using analytical and finite-element methods

In every power transmission system, it is necessary to transmit the power effectively and smoothly. Any variations in transmission of power or torque will results in unwanted noise or vibrations development. Transmission Error (TE) is one of the factors which leads to noise or vibrations in gear transmission system. TE arises due to the various types of errors such as alignment error, tooth geometry error, backlash, cracks, profile error etc. in a gear pair. TE is the difference in the position of the output shaft from its actual position in a true gear pair. In this work the consequences occurring due to the change in number of teeth and pitch error on TE is observed using interference volume method. The spur gear pair of 1:1 gear ratio having involute tooth profile of standard and stub tooth proportions are taken for the work. The gears for the analyses are modelled and meshed/assembled in CAD software SOLIDWORKS. The interference volume values obtained during meshing are recorded for one mesh cycle of the gear pair. Then various graphs are plotted between the angle of rotation for one mesh cycle and interference volume values for different number of teeth and amount of pitch error. From the analyses it is found that the TE is affected when the number of teeth and amount of pitch error are varied. Also, the effects of variation of pitch error and number of teeth on TE is more in standard tooth proportion than stub tooth proportion.

Investigation on crack failure of helical gear system of the gearbox in wind turbine: Mesh stiffness calculation and vibration characteristics recognition

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.