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

Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness

An expert system for fatigue testing of airframes: Gubarev, V.V., Grif, M.G., Metyolkin, N.G., Seryoznov, A.N. and Sklyarov, A.A. Conf. Fatigue and Fracture in Steel and Concrete Structures, ISFF '91, Madras, India, Dec 1991 Vol 2 (Oxford & IBH Publishing Co. Pvt. Ltd, New Delhi, India, 1991) 923–927

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.

Performance of spur gears largely depends on the magnitude and nature of variation of dynamic loads occuring between mating teeth. Variable tooth mesh stiffness is one of the primary sources causing parametric excitations resulting in dynamic loads. The usual method of varying the mesh stiffness to reduce dynamic loads is to use high contact ratio and profile modified gears. In this paper, a new type of tooth design to improve the dynamic performance of spur gears is presented. In this, a through hole is drilled in each tooth in a direction parallel to the gear axis. The diameter of the hole and its position on the tooth centre line are variable. Such a gear is called a hollow gear. Dynamic analysis is carried out for the mesh of hollow pinions mating with solid gears. The results are compared with solid pinions (no holes in teeth) meshing with solid gears. Finite element method is used for the analysis. For estimation of the dynamic load variation in hollow-solid and solid-solid gear meshes, a model incorporating the varying mesh stiffness and damping of gear teeth is used. Governing differential equations are solved using unconditionally stable Newmark-beta algorithm. The dynamic loads obtained are used as an input time varying loads for the determination of dynamic fillet and hole stress response of solid and hollow gear teeth whichever is applicable. Modal superposition technique is used for transient response analysis. The study shows that for the same damping ratio, dynamic loads in hollow-solid meshes are nearly the same as in a solid-solid mesh. In reality, the dynamic loads in a hollow-solid mesh are less than a solid-solid mesh due to its inherent higher material damping.

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

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.

An analytical solution to the dynamic transmission error of a helical gear pair is developed by using a single-degree-of-freedom model with piecewise stiffness functions that characterize the contact plane dynamics and capture the velocity reversal at the pitch line. By assuming a constant mesh stiffness density along the contact lines, a linear time-varying model (with parametric excitations) is obtained, where the effect of sliding friction is quantified by an effective mesh stiffness term. The Floquet theory is then used to obtain closed-form solutions to the dynamic transmission error, and responses are derived to both initial conditions and the forced periodic function under a nominal preload. Analytical models are validated by comparing predictions with numerical simulations, and the effect of viscous damping is examined. Stability analysis is also briefly conducted by using the state transition matrix. Overall, the sliding friction has a marginal effect on the dynamic transmission error of helical gears, as compared with spur gears, in the context of the torsional model.

This study analyzes the nonlinear static and dynamic response in spur gear pairs with tooth root crack damage. A finite element/contact mechanics (FE/CM) model is used that accurately captures the elastic deformations on the gear teeth due to kinematic motion, tooth and rim deformations, vibration, and localized increases in compliance due to a tooth root crack. The damage is modeled by releasing the connectivity of the finite element mesh at select nodes near a tooth crack. The sensitivity of the calculated static transmission errors and tooth mesh stiffnesses is determined for varying crack initial locations, final locations, and the path from the initial to final location. Gear tooth mesh stiffness is calculated for a wide range of tooth root crack lengths, including large cracks that extend through nearly all of the tooth. Mesh stiffnesses are meaningfully reduced due to tooth root crack damage. The dynamic response is calculated for cracks of varying length. Larger cracks result in increased peak dynamic transmission errors. For small tooth root cracks the spectrum of dynamic transmission error contains components near the natural frequency of the gear pair. The spectrum of dynamic transmission error has broadband frequency response for large tooth root cracks that extend further than one-half of the tooth’s thickness.

The parameters influencing high dynamic magnifications and instabilities in multiple-reduction geared systems are investigated in this study. These factors include the effect of external excitations, system inertia, variation in mesh stiffness, contact ratio and damping in the mesh. Design procedures and data are given to aid the designer in predicting system performance beforehand. The reported procedure and data can be used as a guide for safe design and application of geared systems with minimum calculations.

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.

A method for analysing the influence of profile errors upon load-sharing capabilities and mesh stiffness of spur gears is proposed. The analysis is based upon a static, two-dimensional finite element approach. The contacts between mating gear teeth are identified in the deformed state, and the nodal point density permits contact pressure distribution emulation for moderate to high load levels. In addition, artificial neural networks are employed for system identification in order to calculate load-sharing capabilities and mesh stiffness trends for low tooth load levels. The effect of tip relief on load-sharing properties is discussed. The contact force and the transmission error are simulated for two medium-quality spur gears where the profile and pitch errors are known. Experimental data validate the reliability of the approach. The contact ratio is shown to be highly load dependent due to profile and pitch errors. The discussion is focused upon the size of the area where the relation between torque and angular displacement is highly non-linear, i.e. load depending. Hence, the contact ratio is shown to be load depending due to geometrical errors, such as profile and pitch errors, in addition to material elasticity.

Dudley's Handbook of Practical Gear Design and Manufacture

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

<div class="htmlview paragraph">An investigation of transmission errors and bearing contact of spur, helical and spiral bevel gears was performed. Modified tooth surfaces for these gears have been proposed in order to absorb linear transmission errors caused by gear misalignment and localize the bearing contact. Numerical examples for spur, helical, and spiral bevel gears are presented to illustrate the behavior of the modified gear surfaces to misalignment and errors of assembly. The numerical results indicate that the modified surfaces will perform with a low level of transmission error in non-ideal operating environment.</div>

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.