Reliability Based Optimum Design of Gear Trains

1984 ◽  
Vol 106 (1) ◽  
pp. 17-22 ◽  
Author(s):  
S. S. Rao ◽  
G. Das
Keyword(s):  
Optimum Design ◽  
Bending Strength ◽  
Random Variables ◽  
Kinematic Chain ◽  
Gear Train ◽  
Design Parameters ◽  
Minimum Mass ◽  
Mean Values ◽  
Gear Trains ◽  
The Mean

A reliability based approach is presented for the minimum mass design of gear trains. The gear train is idealized as a weakest link kinematic chain and the optimum design is sought for a specified value of the reliability of the gear train with respect to bending strength and surface wear resistance. The design parameters such as the power transmitted, the geometric dimensions, and the material properties are treated as normally distributed random variables. A linear combination of the mean value and standard deviations of the mass of the gear train is considered as the objective function while treating the mean values of the face widths of the gears as design variables. The reliability based optimization results are compared with those obtained by the deterministic procedure. The effects of variation of parameters like the reliability of the gear train and the coefficients of variation of the random variables are also studied. The minimum mass of the gear trains is found to increase as the specified value of the reliability is increased.

Genetic Algorithm Based Design for Gear Trains

2000 ◽  
Author(s):  
Tae Hyong Chong ◽  
Joung Sang Lee
Keyword(s):  
Genetic Algorithm ◽  
Optimum Design ◽  
Initial Guess ◽  
Gear Train ◽  
Design System ◽  
Contact Ratio ◽  
Design Factor ◽  
Integer Variables ◽  
Gear Trains ◽  
Volume Size

Abstract The design of gear train is a kind of mixed problems which have to determine various types of design variables; i.e., continuous, discrete, and integer variables. Therefore, the most common practice of optimum design using the derivative of objective function has difficulty in solving those kinds of problems and the optimum solution also depends on initial guess because there are many sophisticated constraints. In this study, the Genetic Algorithm is introduced for the optimum design of gear trains to solve such problems and we propose a genetic algorithm based gear design system. This system is applied for the geometrical volume (size) minimization problem of the two-stage gear train and the simple planetary gear train to show that genetic algorithm is better than the conventional algorithms for solving the problems that have continuous, discrete, and integer variables. In this system, each design factor such as strength, durability, interference, contact ratio, etc. is considered on the basis of AGMA standards to satisfy the required design specification and the performance with minimizing the geometrical volume (size) of gear trains.

scholarly journals On deterministic approximation of Markov processes by ordinary differential equations

1998 ◽  
Vol 4 (2) ◽  
pp. 99-114 ◽  
Author(s):  
L. I. Rozonoer
Keyword(s):  
Differential Equations ◽  
Markov Model ◽  
Ordinary Differential Equations ◽  
Chemical Reaction ◽  
Markov Processes ◽  
Random Variables ◽  
Mean Values ◽  
Multidimensional Lattice ◽  
The Mean

For a class of Markov processes on the integer multidimensional lattice, it is shown that the evolution of the mean values of some random variables can be approximated by ordinary differential equations. To illustrate the approach, a Markov model of a chemical reaction is considered

scholarly journals Multi-Objective Optimization of Two-Stage Helical Gear Train Using NSGA-II

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
R. C. Sanghvi ◽  
A. S. Vashi ◽  
H. P. Patolia ◽  
R. G. Jivani
Keyword(s):  
Bending Strength ◽  
Gear Train ◽  
Design Parameters ◽  
Uniform Motion ◽  
Nsga Ii ◽  
Surface Fatigue ◽  
Face Width ◽  
Design Variables ◽  
Load Carrying ◽  
Do So

Gears not only transmit the motion and power satisfactorily but also can do so with uniform motion. The design of gears requires an iterative approach to optimize the design parameters that take care of kinematics aspects as well as strength aspects. Moreover, the choice of materials available for gears is limited. Owing to the complex combinations of the above facts, manual design of gears is complicated and time consuming. In this paper, the volume and load carrying capacity are optimized. Three different methodologies (i) MATLAB optimization toolbox, (ii) genetic algorithm (GA), and (iii) multiobjective optimization (NSGA-II) technique are used to solve the problem. In the first two methods, volume is minimized in the first step and then the load carrying capacities of both shafts are calculated. In the third method, the problem is treated as a multiobjective problem. For the optimization purpose, face width, module, and number of teeth are taken as design variables. Constraints are imposed on bending strength, surface fatigue strength, and interference. It is apparent from the comparison of results that the result obtained by NSGA-II is more superior than the results obtained by other methods in terms of both objectives.

Use of Prediction Methods to Assess Laboratory Bias and Mean Values Associated with an Interlaboratory Study for Method Validation and/or Proficiency Testing

2014 ◽  
Vol 97 (2) ◽  
pp. 624-629 ◽  
Author(s):  
Foster D McClure ◽  
Jung K Lee
Keyword(s):  
Random Variables ◽  
Least Square ◽  
Best Linear Unbiased Predictor ◽  
Least Square Estimator ◽  
Mean Values ◽  
Interlaboratory Studies ◽  
Repeatability And Reproducibility ◽  
Best Linear Unbiased ◽  
The Mean ◽  
The Individual

Abstract Two methods of prediction of random variables, best predictor (BP) and best linear unbiased predictor (BLUP), are discussed as potential statistical methods to predict laboratory true mean and bias values using the sample laboratory mean (yi) from interlaboratory studies. The predictions developed here require that the interlaboratory and/or proficiency study be designed and conducted in a manner consistent with the assumptions of a one-way completely randomized model (CRM). Under the CRM the individual laboratory true mean and bias are not parameters but are defined to be random variables that are unobservable and considered as realized values that cannot be estimated but can be predicted using methods of “prediction.” The BP method is applicable when all salient parameters are known, e.g., the consensus true overall mean (μ) and repeatability and reproducibility components (σr2 and σR2), while the BLUP method is useful when σ2r and σR2 are known, but μ is estimated by the generalized least square estimator. Although the derivations of predictors are obtained by minimizing the mean-square error under the CRM assumptions, the predictors are the expected laboratory true mean and bias given the sample laboratory mean, i.e., conditional expectation.

Random Vibration Analysis of Planetary Gear Trains

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
Jianming Yang ◽  
Ping Yang
Keyword(s):  
Sample Path ◽  
Planetary Gear ◽  
Gear Train ◽  
Lumped Parameter Model ◽  
State Variables ◽  
Random Loads ◽  
Planetary Gear Trains ◽  
Lumped Parameter ◽  
Gear Trains ◽  
The Mean

This article investigates the vibration response of a planetary gear train under excitations of both deterministic and random loads. A lumped parameter model has been used in this investigation and the random excitations are represented by white noise. One version of the stochastic Newmark algorithms is employed to solve for both sample path response and the statistics of the response. The mean and the variance for all state variables are obtained through the same algorithm. The effects of three different levels of noise on the statistics are compared against each other.

Kinematic Analysis on Running Clearance in a Trochoidal-Type Machine

1995 ◽  
Author(s):  
John B. Shung ◽  
Yi Zhang
Keyword(s):  
Mathematical Model ◽  
Optimal Design ◽  
Robust Design ◽  
Upper Bound ◽  
Kinematic Analysis ◽  
Design Parameters ◽  
Mean Values ◽  
Type Machine ◽  
The Mean ◽  
Design Optimum

Abstract A methodology to design tight running clearance between rotor and chamber in a trochoidal-type machine is presented. A mathematical model to describe the running clearance is developed. Only kinematic design parameters are considered. The effect of the mean values and tolerances of the design parameters on the running clearance is studied by applying robust design. Mean values of design parameters which provide running clearance to be less sensitive to the tolerances are obtained. The effect of the upper bound of the running clearance on the tolerance is also studied by applying the probabilistic optimal design. Optimum tolerances which minimize a cost function are obtained. Therefore, one can apply this methodology to design running clearance by choosing appropriate mean values and tolerances of the design parameters.

Reliability Analysis and Design of Epicyclic Gear Trains

1979 ◽  
Vol 101 (4) ◽  
pp. 625-632 ◽  
Author(s):  
S. S. Rao
Keyword(s):  
Reliability Analysis ◽  
Structural Reliability ◽  
Design Criterion ◽  
Gear Train ◽  
Design Parameters ◽  
Analysis And Design ◽  
Input Shaft ◽  
Reliability Based Design ◽  
Epicyclic Gear ◽  
Gear Trains

The concepts of system reliability are applied for the structural reliability analysis and design of epicyclic gear trains. The reliability analysis is based on the representation of an epicyclic gear train as a series-parallel network. The power transmitted, the speed of the input shaft, the center distance between the gear pairs and the permissible stresses are assumed to be random variables following normal distribution. The layout of the gears and the speed ratios are assumed to be known. The face widths of the gears are taken as random design parameters. The design criterion is that the reliability of the gear train either in bending or surface wear failure mode at any of the output speeds must be equal to a specified value. The design of an epicyclic transmission system which gives four forward speeds and one reverse speed is considered for illustration. The results of the reliability-based design of the gear train are compared with those of the conventional deterministic design.

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