Static and Dynamic Tooth Loading in Spur and Helical Geared Systems: Experiments and Code Validation

2000 ◽  
Author(s):  
Sébastien Baud ◽  
Philippe Velex
Keyword(s):  
Linear Time ◽  
Primary Objective ◽  
Integration Scheme ◽  
Spur Gears ◽  
Mesh Stiffness ◽  
Response Curves ◽  
Time Step ◽  
Contact Conditions ◽  
Rotor Systems ◽  
Contact Algorithm

Abstract The primary objective of this study is to validate a specific finite element code aimed at simulated dynamic tooth loading in geared rotor systems. Experiments have been conducted on a high-precision single stage spur and helical gear reducer with flexible shafts mounted on hydrostatic or hydrodynamic bearings. The numerical model is based on classical elements (shaft, lumped stiffnesses, ...) and on an original gear element which accounts for non-linear time-varying mesh stiffness, gear errors and tooth shape modifications. External and parametric excitations are derived from the instantaneous contact conditions between the mating flanks by using an iterative contact algorithm inserted in a time-step integration scheme. In a first step, experimental and numerical results at low speeds are compared and it is demonstrated that the proposed tooth mesh interface model is valid. Comparisons are then extended to dynamic fillet stresses on both spur and helical gears between 50–6000 rpm on pinion shaft. Despite a localized problem in the case of spur gears with one particular bearing arrangement, the broad agreement between the experimental and numerical response curves proves that the model is representative of the dynamic behavior of geared systems.

Static and Dynamic Tooth Loading in Spur and Helical Geared Systems-Experiments and Model Validation

2002 ◽  
Vol 124 (2) ◽  
pp. 334-346 ◽  
Author(s):  
S. Baud ◽  
P. Velex
Keyword(s):  
Linear Time ◽  
Primary Objective ◽  
Integration Scheme ◽  
Spur Gears ◽  
Mesh Stiffness ◽  
Response Curves ◽  
Time Step ◽  
Contact Conditions ◽  
Rotor Systems ◽  
Contact Algorithm

The primary objective of this study is to validate a specific finite element code aimed at simulating dynamic tooth loading in geared rotor systems. Experiments have been conducted on a high-precision single stage spur and helical gear reducer with flexible shafts mounted on hydrostatic or hydrodynamic bearings. The numerical model is based on classical elements (shaft, lumped stiffnesses, …) and on a gear element which accounts for non-linear time-varying mesh stiffness, gear errors and tooth shape modifications. External and parametric excitations are derived from the instantaneous contact conditions between the mating flanks by using an iterative contact algorithm inserted in a time-step integration scheme. First, experimental and numerical results at low speeds are compared and confirmed that the proposed tooth mesh interface model is valid. Comparisons were then extended to dynamic fillet stresses on both spur and helical gears between 50–6000 rpm on pinion shaft. Despite a localized problem in the case of spur gears with one particular bearing arrangement, the broad agreement between the experimental and numerical response curves demonstrated that the model is representative of the dynamic behavior of geared systems.

Optimization of Profile Modifications With Regard to Dynamic Tooth Loads in Planetary Gears With Flexible Ring-Gears

2015 ◽  
Author(s):  
Matthieu Chapron ◽  
Philippe Velex ◽  
Jérôme Bruyère ◽  
Samuel Becquerelle
Keyword(s):  
Equations Of Motion ◽  
Integration Scheme ◽  
Mesh Stiffness ◽  
Time Step ◽  
Planetary Gears ◽  
Contact Algorithm ◽  
Beam Elements ◽  
Internal Gear ◽  
Gear Meshes ◽  
Optimum Profile

This paper deals with the optimization of tooth profile modifications in planetary gears. A dynamic model is proposed based on 3D two-node gear elements connected to a deformable ring-gear discretized into beam elements. Symmetric tip relief on external and internal gear meshes are introduced as normal deviations along the lines of contact superimposed on a stiffness distribution aimed at simulating position- and time-varying mesh stiffness functions. The equations of motion are solved by the combination of a Newmark’s time-step integration scheme and a contact algorithm to account for possible partial or total contact losses. Symmetric linear profile modifications are then optimized by using a genetic algorithm with the objective of minimizing dynamic tooth loads over a speed range. Finally, the interest of the corresponding optimum profile modifications with regard to speed and torque variations is analyzed.

Analytical and Numerical Dynamic Analysis of Gears in the Presence of Engine Acyclism

2007 ◽  
Author(s):  
G. Sika ◽  
P. Velex
Keyword(s):  
Speed Ratio ◽  
Mesh Stiffness ◽  
Time Step ◽  
Contact Conditions ◽  
Contact Algorithm ◽  
Newmark’S Method ◽  
Tooth Number ◽  
Gear Geometry ◽  
Set Up ◽  
Variable Time Step

A one degree of freedom model is set up which incorporates time-varying mesh stiffness functions and the influence of unsteady input rotations due to engine acyclism. In order to investigate spur and helical gears, a piecewise and a sine function are used to simulate mesh stiffness fluctuations. Contact conditions are considered in order to deal with contact losses and back strikes, i.e., when contacts occur on the back flanks of the teeth. The dynamic response is determined by combining several analytical and numerical techniques. It is shown that acyclism modulations can generate additional response peaks on either side of the main resonance area. This is due to frequency and amplitude modulations between the mesh excitations and the harmonics of the engine rotational speed. The analytical results compare particularly well with those delivered by a time-step numerical integration by Newmark’s method with controlled variable time-step coupled with a contact algorithm. This excellent agreement shows that Newmark’s method can be extended to the dynamic simulation of geared drives with unsteady rotational speeds provided that time-steps are carefully calibrated and readjusted. Finally, the influence of gear geometry (module, tooth number, speed ratio) along with the acyclism parameters (frequencies, amplitudes) is studied and some general trends are presented concerning the resulting dynamic tooth loads.

Influence of Clearances and Thermal Effects on the Dynamic Behavior of Gear-Hydrodynamic Journal Bearing Systems

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
R. Fargère ◽  
P. Velex
Keyword(s):  
Degrees Of Freedom ◽  
Integration Scheme ◽  
Time Step ◽  
Specific Element ◽  
Normal Contact ◽  
Contact Algorithm ◽  
Gear Teeth ◽  
Proposed Model ◽  
Elastic Couplings ◽  
Definition Of

A global model of mechanical transmissions is introduced which deals with most of the possible interactions between gears, shafts, and hydrodynamic journal bearings. A specific element for wide-faced gears with nonlinear time-varying mesh stiffness and tooth shape deviations is combined with shaft finite elements, whereas the bearing contributions are introduced based on the direct solution of Reynolds' equation. Because of the large bearing clearances, particular attention has been paid to the definition of the degrees-of-freedom and their datum. Solutions are derived by combining a time step integration scheme, a Newton–Raphson method, and a normal contact algorithm in such a way that the contact conditions in the bearings and on the gear teeth are simultaneously dealt with. A series of comparisons with the experimental results obtained on a test rig are given which prove that the proposed model is sound. Finally, a number of results are presented which show that parameters often discarded in global models such as the location of the oil inlet area, the oil temperature in the bearings, the clearance/elastic couplings interactions, etc. can be influential on static and dynamic tooth loading.

Optimization of Profile Modifications With Regard to Dynamic Tooth Loads in Single and Double-Helical Planetary Gears With Flexible Ring-Gears

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
M. Chapron ◽  
P. Velex ◽  
J. Bruyère ◽  
S. Becquerelle
Keyword(s):  
Equations Of Motion ◽  
Integration Scheme ◽  
Time Varying ◽  
Time Step ◽  
Planetary Gears ◽  
Contact Algorithm ◽  
Beam Elements ◽  
Internal Gear ◽  
Gear Meshes ◽  
Optimum Profile

This paper is mostly aimed at analyzing optimum profile modifications (PMs) in planetary gears (PGTs) with regard to dynamic mesh forces. To this end, a dynamic model is presented based on 3D two-node gear elements connected to deformable ring-gears discretized into beam elements. Double-helical gears are simulated as two gear elements of opposite hands which are linked by shaft elements. Symmetric tip relief on external and internal gear meshes are introduced as time-varying normal deviations along the lines of contact and time-varying mesh stiffness functions are deduced from Wrinckler foundation models. The equations of motion are solved by coupling a Newmark time-step integration scheme and a contact algorithm to account for possible partial or total contact losses. Symmetric linear PMs for helical and double-helical PGTs are optimized by using a genetic algorithm with the objective of minimizing dynamic tooth loads over a speed range. Finally, the sensitivity of these optimum PMs to speed and load is analyzed.

Simulations of Gear-Rolling Element Bearing Interactions in Geared Transmissions

2003 ◽  
Author(s):  
F. Lahmar ◽  
P. Velex
Keyword(s):  
Equations Of Motion ◽  
Rolling Element Bearing ◽  
Integration Scheme ◽  
Time Step ◽  
Normal Contact ◽  
Contact Algorithm ◽  
Rolling Element ◽  
Non Linear ◽  
External Forces ◽  
Gear Rolling

The modular model of geared systems presented in this paper makes it possible to simultaneously account for the contact conditions in gears and rolling element bearings. Gears are modeled as two rigid cylinders connected by distributed mesh stiffnesses while ball and roller bearings contribute to the equations of motion as time-varying, non-linear external forces. Solutions are obtained by combining a Newmark time-step integration scheme, a Newton-Raphson method for ball bearing non-linearity and a normal contact algorithm that deals with the contact problem between the teeth. It is found that the static gear-bearing couplings are generally more important than the dynamic couplings with a significant influence of the gear on the bearing response. Finally, it is shown that, in certain conditions, bearings can generate non-linear parametric excitations of the same orders of magnitude as those associated with the meshing of helical gears.

Simulations of the dynamic response of planetary gears in the presence of localised tooth faults

2019 ◽  
Vol 233 (21-22) ◽  
pp. 7212-7223 ◽  
Author(s):  
Q Thoret-Bauchet ◽  
P Velex ◽  
M Guingand ◽  
P Casanova
Keyword(s):  
Planetary Gear ◽  
Contact Forces ◽  
Vibration Monitoring ◽  
Integration Scheme ◽  
Ring Gear ◽  
State Equations ◽  
Time Step ◽  
Normal Contact ◽  
Planetary Gears ◽  
Contact Algorithm

This paper is aimed at analysing the influence of local tooth faults such as pitting on the dynamic behaviour of planetary gears. A model of one-stage planetary gear combining lumped parameters and Timoshenko beam elements is presented, which accounts for deformable shafts and ring gears. Local tooth fault are simulated by material removal from tooth flanks, which can be positioned on the sun-gear, the planets and the ring-gear. The corresponding state equations are solved by combining a Newmark time-step integration scheme combined with a unilateral normal contact algorithm, which verifies that all contact forces on gear teeth are compressive and that no contact can occur outside the contact areas. A number of results are presented, which show the influence of tooth fault positions, depths and extents on displacement and acceleration signals. The contribution of a deformable ring-gear is analysed and the possibility to detect such localised tooth faults from vibration monitoring is discussed.

Modeling of Spur and Helical Gear Planetary Drives With Flexible Ring Gears and Planet Carriers

2006 ◽  
Vol 129 (1) ◽  
pp. 95-106 ◽  
Author(s):  
V. Abousleiman ◽  
P. Velex ◽  
S. Becquerelle
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Lumped Parameter Model ◽  
Integration Scheme ◽  
Ring Gear ◽  
Time Step ◽  
Contact Algorithm ◽  
Lumped Parameter ◽  
Epicyclic Gear ◽  
Internal Gear

A model is presented which enables the simulation of the three-dimensional static and dynamic behavior of planetary/epicyclic spur and helical gears with deformable parts. The contributions of the deflections of the ring gear and the carrier are introduced via substructures derived from 3D finite element models. Based on a modal condensation technique, internal gear elements are defined by connecting the ring-gear substructure and a planet lumped parameter model via elastic foundations which account for tooth contacts. Discrete mesh stiffness and equivalent normal deviations are introduced along the contact lines, and their values are recalculated as the mating flank positions vary with time. A constraint mode substructuring technique is used to simulate the planet carrier as a superelement which is connected to the planet center. Planetary/epicyclic gear models are completed by assembling lumped parameter sun gear/planet elements along with shaft elements, lumped stiffness, masses and inertias. The corresponding equations of motion are solved by combining a time-step integration scheme and a contact algorithm for all simultaneous meshes. Several quasistatic and dynamic results are given which illustrate the potential of the proposed hybrid model and the interest of taking into account ring gear and carrier deflections.

scholarly journals Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 129
Author(s):  
Yuan Li ◽  
Ni Zhang ◽  
Yuejiao Gong ◽  
Wentao Mao ◽  
Shiguang Zhang
Keyword(s):  
Side Effect ◽  
Domain Boundary ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Integration Scheme ◽  
Computational Accuracy ◽  
Time Step ◽  
Corner Points ◽  
Discontinuous Elements ◽  
The Time Domain

Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided.

The influence of tooth pitting on the mesh stiffness of a pair of external spur gears

2016 ◽  
Vol 106 ◽  
pp. 1-15 ◽  
Author(s):  
Xihui Liang ◽  
Hongsheng Zhang ◽  
Libin Liu ◽  
Ming J. Zuo
Keyword(s):  
Spur Gears ◽  
Mesh Stiffness
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