Computational Issues Associated With Gear Rattle Analysis: Part I — Problem Formulation

1992 ◽  
Author(s):  
C. Padmanabhan ◽  
T. E. Rook ◽  
Rajendra Singh
Keyword(s):  
Numerical Solution ◽  
Mathematical Formulation ◽  
Order Reduction ◽  
Problem Formulation ◽  
Reduction Gear ◽  
Type Problem ◽  
Contact Ratio ◽  
Processing Stage ◽  
Gear Contact ◽  
Gear Rattle

Abstract This paper proposes a new procedure for formulating the gear rattle type problem analytically before attempting a numerical solution. This step is necessary due to the nature of the mathematical formulation with vibro-impacts, which is non-analytical and hence causes numerical “stiffness”. The procedure is essentially an “intelligent” pre-processing stage and is based on our vast experience in simulating such systems. Important concepts such as order reduction, gear contact ratio, appropriate choice of non-dimensionalization parameters are illustrated through several examples.

Computational Issues Associated with Gear Rattle Analysis

1995 ◽  
Vol 117 (1) ◽  
pp. 185-192 ◽  
Author(s):  
C. Padmanabhan ◽  
R. C. Barlow ◽  
T. E. Rook ◽  
R. Singh
Keyword(s):  
Mathematical Formulation ◽  
Evaluation Criteria ◽  
Numerical Algorithms ◽  
Order Reduction ◽  
Simulation Models ◽  
Reduction Gear ◽  
Contact Ratio ◽  
Model Order ◽  
Domain Integration ◽  
Gear Rattle

This paper proposes a new procedure for formulating the gear rattle type problem analytically before attempting a numerical solution. It also outlines appropriate evaluation criteria for direct time domain integration algorithms used to solve such problems. The procedure is necessary due to the non-analytical nature of the mathematical formulation describing vibro-impacts, which can lead to numerical “stiffness” problems. The method is essentially an “intelligent” pre-processing stage and is based on our experience in simulating such systems. Important concepts such as model order reduction, gear or clutch stiffness contact ratio, appropriate choice of non-dimensionalization parameters are illustrated through examples. Several case studies of increasing complexity are solved using various well known numerical algorithms; solutions are compared qualitatively and quantitatively using the proposed evaluation criteria, and specific numerical problems are identified. Some of the simulation models have also been validated by comparing predictions with experimental data.

Effect of Contact Ratio on Spur Gear Dynamic Load With No Tooth Profile Modifications

1996 ◽  
Vol 118 (3) ◽  
pp. 439-443 ◽  
Author(s):  
Chuen-Huei Liou ◽  
Hsiang Hsi Lin ◽  
F. B. Oswald ◽  
D. P. Townsend
Keyword(s):  
Dynamic Load ◽  
Spur Gear ◽  
Tooth Size ◽  
Contact Ratio ◽  
Gear Dynamics ◽  
Wide Range ◽  
Gear Contact ◽  
Center Distance ◽  
Selection Of ◽  
Better Than

This paper presents a computer simulation showing how the gear contact ratio affects the dynamic load on a spur gear transmission. The contact ratio can be affected by the tooth addendum, the pressure angle, the tooth size (diametral pitch), and the center distance. The analysis presented in this paper was performed by using the NASA gear dynamics code DANST. In the analysis, the contact ratio was varied over the range 1.20 to 2.40 by changing the length of the tooth addendum. In order to simplify the analysis, other parameters related to contact ratio were held constant. The contact ratio was found to have a significant influence on gear dynamics. Over a wide range of operating speeds, a contact ratio close to 2.0 minimized dynamic load. For low-contact-ratio gears (contact ratio less than two), increasing the contact ratio reduced gear dynamic load. For high-contact-ratio gears (contact ratio equal to or greater than 2.0), the selection of contact ratio should take into consideration the intended operating speeds. In general, high-contact-ratio gears minimized dynamic load better than low-contact-ratio gears.

A Nonlinear Analysis Formulation for Bend Stiffeners

2007 ◽  
Vol 51 (03) ◽  
pp. 250-258 ◽  
Author(s):  
M. A. Vaz ◽  
C. A. D. de Lemos ◽  
M. Caire
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Nonlinear Analysis ◽  
Cross Section ◽  
Contact Force ◽  
Oil And Gas ◽  
Constitutive Relations ◽  
Mathematical Formulation ◽  
Power Series Expansion ◽  
Gas Production

Bend stiffeners are polymeric structures with a conical shape designed to limit the curvature of flexible risers and umbilical cables at their uppermost connections, protecting them against overbending and from accumulation of fatigue damage. Thus, they are of vital importance to deep water oil and gas production systems. This work develops a mathematical formulation and a numerical solution procedure for the geometrical and material nonlinear analysis of the riser/bend stiffener system considered as a beam bending model. The structures are separately modeled, which allows the numerical calculation of the contact force along the system arc length. The governing differential equations are derived considering geometrical compatibility, equilibrium of forces and moments, and nonlinear asymmetric material constitutive relations, which leads to a shift in the neutral axis position from the cross-section centroid. The eccentricity and the bending moment versus curvature relation for each cross section are numerically calculated and then expressed by a polynomial power series expansion. A set of four first-order nonlinear ordinary differential equations is written and four boundary conditions are specified at both ends. Once the global problem is solved, the contact force may be promptly calculated. A finite difference method is implemented in Fortran code to obtain the numerical solution. A case study is carried out where linear elastic symmetric and nonlinear elastic asymmetric constitutive models are compared and discussed. The results are presented for the riser/bend stiffener deflected configuration, angle, curvature, and contact force distribution. The results demonstrate that an accurate structural analysis of bend stiffeners depends on a precise assessment of the nonlinear asymmetric polyurethane property.

Integrated System for Soft Tissue Dynamic Simulation

2010 ◽  
Author(s):  
Xiaodong Zhao ◽  
Baoxiang Shan ◽  
Assimina A. Pelegri
Keyword(s):  
Finite Element ◽  
Soft Tissue ◽  
State Space ◽  
Soft Tissues ◽  
Mathematical Formulation ◽  
State Space Model ◽  
Order Reduction ◽  
Integrated System ◽  
Space Model ◽  
Tissue Models

An integrated system is built to model and simulate the dynamic response of soft tissues. The mathematical formulation employs finite element and model order reduction approaches to develop a state space model for soft tissues that allows for time-efficient numerical analysis. The stimulus device and signal processing routines are built in Matlab/Simulink and then integrated with the finite element state space model. This integrated system facilitates expeditious numerical evaluation of different soft tissue models subjected to dynamic excitation. It further elucidates the effect of different stimulus sources, as well as relative influences of different sources of uncertainty.

Gear Rattle Modeling Based on Transient Stiffness and Damping Analysis

2012 ◽  
Vol 271-272 ◽  
pp. 936-947
Author(s):  
Ming Qin ◽  
Ning Xie ◽  
Hui Wang ◽  
Kai Zhang ◽  
Xue Ping Liu ◽  
...  
Keyword(s):  
Physical Model ◽  
Traditional Method ◽  
Relative Displacement ◽  
Simulation Program ◽  
Gear Pair ◽  
Modeling And Analysis ◽  
Gear Contact ◽  
Stiffness And Damping ◽  
Gear Rattle

Structural characters of gear contact have a determinative impact on gear rattle. Contrarily traditional method using average stiffness and damp coefficients which weakens the accuracy on modeling and analysis of gear rattle phenomenon, in this paper, a methodology modeling gear rattle process with transient stiffness and damp coefficients is proposed. Gear rattle process is modeled by considering the physical model of gear contact, gear pair movement, and actually geometrical meshing curve. A study of gear rattle is made by the simulation program and an experiment is also done to verify the method. Results show that this method can effectively analyze the frequency and the relative displacement of gear rattle, etc.

The Cross-Sectional Shapes of Longitudinal Hydraulic Fractures

1984 ◽  
Vol 106 (4) ◽  
pp. 554-561 ◽  
Author(s):  
D. Segalman
Keyword(s):  
Mathematical Formulation ◽  
High Stress ◽  
In Situ Stress ◽  
Hydraulic Fractures ◽  
Stress Distributions ◽  
Type Problem ◽  
Cross Sectional ◽  
The Cross ◽  
Sectional Shape

A mathematical formulation has been developed for calculating the cross-sectional shape of hydraulic fractures. This formulation treats the problem as a free-boundary-type problem and is modeled after mathematical formulations developed for contact and lubrication problems. Numerical solution of the resulting equations has been used to address problems involving particularly difficult in-situ stress distributions, including problems in which the fracture breaks through high-stress barriers. The technique is illustrated on two example problems.

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