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.

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.

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.

An Improved Method of Jiles-Atherton Hysteresis Model Based on Variable Time-Step Simulation

2021 ◽  
Author(s):  
Yinguo Yang ◽  
Liling Xiang ◽  
Yitan Guo ◽  
Zhendong Tan ◽  
Yankan Song
Keyword(s):  
Hysteresis Model ◽  
Improved Method ◽  
Time Step ◽  
Variable Time ◽  
Model Based ◽  
Variable Time Step

Dynamic Analysis of Sliding Friction in a Gear Pair

2003 ◽  
Author(s):  
Rajendra Gunda ◽  
Rajendra Singh
Keyword(s):  
Load Distribution ◽  
Sliding Friction ◽  
Spur Gear ◽  
Time Instant ◽  
Friction Torque ◽  
Speed Ratio ◽  
Gear Pair ◽  
Mesh Stiffness ◽  
Static Mode

Chief objective of this article is to evaluate the role of sliding friction in gear dynamics, and more specifically the effect of the periodic variations in mesh stiffness, load distribution and friction torque during a mesh cycle. A non-unity speed ratio spur gear is considered. Only the torsional degree of freedom of the gear pair, with ideal Coulomb friction law, is analyzed. Previous studies by Vaishya and Singh [1–3] make idealized assumptions about temporal (or spatial) variation of mesh stiffness and load sharing in order to obtain more tractable analytical solutions. In our formulation, an accurate Finite Element/Contact Mechanics analysis code [4] is run in the static mode to compute the mesh stiffness and load distribution at every time instant of the mesh. The computed parametric variation of stiffness is then incorporated into our dynamic formulation that includes frictional torques. Next, we use appropriate numerical techniques to solve for the dynamic response in time domain. This study, though preliminary in nature, examines the effects of pinion speed, coefficient of friction and mean input torque. This, along with work in progress, should yield further insights into the role of friction sources in gear vibro-acoustics.

Intelligent Time-Stepping for Practical Numerical Simulation

2021 ◽  
Author(s):  
Soham Sheth ◽  
Francois McKee ◽  
Kieran Neylon ◽  
Ghazala Fazil
Keyword(s):  
Heuristic Method ◽  
Simulation Models ◽  
Model Complexity ◽  
Large Field ◽  
Optimal Time ◽  
Time Step ◽  
Non Linear ◽  
Set Up ◽  
Reservoir Simulator ◽  
Step Selection

Abstract We present a novel reservoir simulator time-step selection approach which uses machine-learning (ML) techniques to analyze the mathematical and physical state of the system and predict time-step sizes which are large while still being efficient to solve, thus making the simulation faster. An optimal time-step choice avoids wasted non-linear and linear equation set-up work when the time-step is too small and avoids highly non-linear systems that take many iterations to solve. Typical time-step selectors use a limited set of features to heuristically predict the size of the next time-step. While they have been effective for simple simulation models, as model complexity increases, there is an increasing need for robust data-driven time-step selection algorithms. We propose two workflows – static and dynamic – that use a diverse set of physical (e.g., well data) and mathematical (e.g., CFL) features to build a predictive ML model. This can be pre-trained or dynamically trained to generate an inference model. The trained model can also be reinforced as new data becomes available and efficiently used for transfer learning. We present the application of these workflows in a commercial reservoir simulator using distinct types of simulation model including black oil, compositional and thermal steam-assisted gravity drainage (SAGD). We have found that history-match and uncertainty/optimization studies benefit most from the static approach while the dynamic approach produces optimum step-sizes for prediction studies. We use a confidence monitor to manage the ML time-step selector at runtime. If the confidence level falls below a threshold, we switch to traditional heuristic method for that time-step. This avoids any degradation in the performance when the model features are outside the training space. Application to several complex cases, including a large field study, shows a significant speedup for single simulations and even better results for multiple simulations. We demonstrate that any simulation can take advantage of the stored state of the trained model and even augment it when new situations are encountered, so the system becomes more effective as it is exposed to more data.

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