Modeling and Detection of Localized Tooth Defects in Geared Systems

1999 ◽  
Author(s):  
M. El Badaoui ◽  
V. Cahouet ◽  
F. Guillet ◽  
J. Daniere ◽  
P. Velex
Keyword(s):  
Numerical Simulations ◽  
Gear Tooth ◽  
Equations Of Motion ◽  
Vibration Monitoring ◽  
Detection Methods ◽  
Implicit Time ◽  
Integration Scheme ◽  
Time Step ◽  
Normal Contact ◽  
Detection Techniques

Abstract The early detection of failures in geared systems is an important industrial problem which has still to be addressed from both an experimental and theoretical viewpoint. The proposed paper combines some extensive numerical simulations of a single stage geared unit with localized tooth faults and the use of several detection techniques whose performances are compared and critically assessed. A model aimed at simulating the contributions of local tooth defects such as spalling to the gear dynamic behavior is set up. The pinion and the gear of a pair are modeled as to two rigid cylinders with all six degrees of freedom connected by a series of springs which represent gear body and gear tooth compliances on the base plane. Classical shaft finite elements including torsional, flexural and axial displacements can be superimposed on the gear element together with some lumped stiffnesses, masses, inertias, ... which account for the load machines, bearings and couplings. Tooth defects are modeled by a distribution of normal deviations over a zone which can be located anywhere on the active tooth flanks. Among the numerous available signal processing techniques used in vibration monitoring, cepstrum analysis is sensitive, reliable and it can be adapted to a complex geared system with several meshes. From an analytical analysis of the equations of motion, two complementary detection techniques based upon the acceleration power cesptrum are proposed. The equations of motion and the contact problem between mating flanks are simultaneously solved by coupling an implicit time-step integration scheme and a unilateral normal contact algorithm. The results of the numerical simulations are used as a data base for the proposed detection techniques. The combined influence of the defect location, depth and extent is analyzed for two examples of spur and helical gears with various profile modifications and the effectiveness of the two complementary detection methods is discussed before some conclusions are drawn.

Modeling and Detection of Localized Tooth Defects in Geared Systems

1999 ◽  
Vol 123 (3) ◽  
pp. 422-430 ◽  
Author(s):  
M. El Badaoui ◽  
V. Cahouet ◽  
F. Guillet ◽  
J. Danie`re ◽  
P. Velex
Keyword(s):  
Numerical Simulations ◽  
Gear Tooth ◽  
Equations Of Motion ◽  
Vibration Monitoring ◽  
Detection Methods ◽  
Implicit Time ◽  
Integration Scheme ◽  
Time Step ◽  
Normal Contact ◽  
Detection Techniques

The early detection of failures in geared systems is an important industrial problem which has still to be addressed from both an experimental and theoretical viewpoint. The proposed paper combines some extensive numerical simulations of a single stage geared unit with localized tooth faults and the use of several detection techniques whose performances are compared and critically assessed. A model aimed at simulating the contributions of local tooth defects such as spalling to the gear dynamic behavior is set up. The pinion and the gear of a pair are assimilated to two rigid cylinders with all six degrees of freedom connected by a series of springs which represent gear body and gear tooth compliances on the base plane. Classical shaft finite elements including torsional, flexural and axial displacements can be superimposed to the gear element together with some lumped stiffnesses, masses, inertias, … which account for the load machines, bearings and couplings. Tooth defects are modeled by a distribution of normal deviations over a zone which can be located anywhere on the active tooth flanks. Among the numerous available signal processing techniques used in vibration monitoring, cepstrum analysis is sensitive, reliable and it can be adapted to complex geared system with several meshes. From an analytical analysis of the equations of motion, two complementary detection techniques based upon acceleration power cepstrum are proposed. The equations of motion and the contact problem between mating flanks are simultaneously solved by coupling an implicit time-step integration scheme and a unilateral normal contact algorithm. The results of the numerical simulations are used as a data base for the proposed detection techniques. The combined influence of the defect location, depth and extent is analyzed for two examples of spur and helical gears with various profile modifications and the effectiveness of the two complementary detection methods is discussed before some conclusions are drawn.

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.

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.

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

2012 ◽  
Vol 227 (7) ◽  
pp. 1505-1514 ◽  
Author(s):  
SD Yu ◽  
BC Wen
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Simple Procedure ◽  
Mathematical Problem ◽  
Equations Of Motion ◽  
Inequality Constraints ◽  
Integration Scheme ◽  
Time Step ◽  
Multiple Degrees Of Freedom

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

scholarly journals Investigation of Air Pocket Behavior in Pipelines Using Rigid Column Model and Contributions of Time Integration Schemes

Water ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 785
Author(s):  
Arman Rokhzadi ◽  
Musandji Fuamba
Keyword(s):  
Transient Flow ◽  
Time Integration ◽  
Driving Pressure ◽  
Implicit Time ◽  
Numerical Dissipation ◽  
Integration Scheme ◽  
Time Step ◽  
Column Model ◽  
Integration Schemes ◽  
Backward Euler

This paper studies the air pressurization problem caused by a partially pressurized transient flow in a reservoir-pipe system. The purpose of this study is to analyze the performance of the rigid column model in predicting the attenuation of the air pressure distribution. In this regard, an analytic formula for the amplitude and frequency will be derived, in which the influential parameters, particularly, the driving pressure and the air and water lengths, on the damping can be seen. The direct effect of the driving pressure and inverse effect of the product of the air and water lengths on the damping will be numerically examined. In addition, these numerical observations will be examined by solving different test cases and by comparing to available experimental data to show that the rigid column model is able to predict the damping. However, due to simplified assumptions associated with the rigid column model, the energy dissipation, as well as the damping, is underestimated. In this regard, using the backward Euler implicit time integration scheme, instead of the classical fourth order explicit Runge–Kutta scheme, will be proposed so that the numerical dissipation of the backward Euler implicit scheme represents the physical dissipation. In addition, a formula will be derived to calculate the appropriate time step size, by which the dissipation of the heat transfer can be compensated.

scholarly journals A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

2015 ◽  
Vol 143 (9) ◽  
pp. 3838-3855 ◽  
Author(s):  
Steven Sandbach ◽  
John Thuburn ◽  
Danail Vassilev ◽  
Michael G. Duda
Keyword(s):  
Time Integration ◽  
Standard Test ◽  
Newton Iteration ◽  
Atmospheric Modeling ◽  
Implicit Time ◽  
Integration Scheme ◽  
Baroclinic Wave ◽  
Time Step ◽  
Integration Schemes ◽  
Helmholtz Problem

Abstract An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge–Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank–Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%–20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

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.

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.

Studies on the Accuracy Stability and Efficiency of a New Time Integration Scheme for Ballast Modeling Using the Discrete Element Method

2014 ◽  
Author(s):  
G. F. Mathews ◽  
R. L. Mullen ◽  
D. C. Rizos
Keyword(s):  
Particle Interaction ◽  
Time Integration ◽  
Discrete Element ◽  
Critical Discussion ◽  
Implicit Time ◽  
Integration Scheme ◽  
Computationally Efficient ◽  
Time Step ◽  
Central Difference ◽  
Time Integration Scheme

This paper presents the development of a semi-implicit time integration scheme, originally developed for structural dynamics in the 1970’s, and its implementation for use in Discrete Element Methods (DEM) for rigid particle interaction, and interaction of elastic bodies that are modeled as a cluster of rigid interconnected particles. The method is developed in view of ballast modeling that accounts for the flexibility of aggregates and the arbitrary shape and size of granules. The proposed scheme does not require any matrix inversions and is expressed in an incremental form making it appropriate for non-linear problems. The proposed method focuses on improving the efficiency, stability and accuracy of the solutions, as compared to current practice. A critical discussion of the findings of the studies is presented. Extended verification and assessment studies demonstrate that the proposed algorithm is unconditionally stable and accurate even for large time step sizes. It is demonstrated that the proposed method is at least as computationally efficient as the Central Difference Method. Guidelines for the implementation of the method to ballast modeling are discussed.

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