scholarly journals Optimum Selection of Discrete Tolerances

1989 ◽  
Vol 111 (2) ◽  
pp. 243-251 ◽  
Author(s):  
W.-J. Lee ◽  
T. C. Woo
Keyword(s):  
Reliability Index ◽  
Probabilistic Approach ◽  
Minimum Cost ◽  
Random Variable ◽  
Manufacturing Cost ◽  
Discrete Optimization Problem ◽  
Integer Programming Problem ◽  
Optimum Selection ◽  
Manufacturing Assembly ◽  
Selection Of

Tolerancing involves considerations from all phases of the life cycle of a product including design, manufacturing, assembly, and inspection. Along with minimum cost and maximum functionality and interchangeability, the practice of tolerancing urges a designer to choose an appropriate manufacturing (or inspection) process as well. This situation is formalized as a discrete optimization problem. For an optimum selection of tolerances from a given discrete model involving various manufacturing processes, minimization of manufacturing cost is achieved under the constraint of tolerance stack-up. A random variable and its standard deviation are associated with a dimension and its tolerance. This probabilistic approach enables a trade-off between performance and tolerance (cost). But it also suggests probabilistic optimization. With the aid of a notion called the reliability index [8], tolerance selection is formulated as an integer programming problem. A branch and bound algorithm for ensuring optimum selection is developed by exploiting the special structure of the constraints. To make the enumeration tree small, monotonic relations among the reliability index, cost, and tolerance are examined. The algorithm is tested with examples.

Tolerances: Their Analysis and Synthesis

1990 ◽  
Vol 112 (2) ◽  
pp. 113-121 ◽  
Author(s):  
Woo-Jong Lee ◽  
T. C. Woo
Keyword(s):  
Reliability Index ◽  
Cost Minimization ◽  
Probabilistic Approach ◽  
Convex Polytope ◽  
Random Variable ◽  
Structural Safety ◽  
Worst Case Analysis ◽  
Manufacturing Cost ◽  
Worst Case ◽  
Tolerance Synthesis

Tolerance, representing a permissible variation of a dimension in an engineering drawing, is synthesized by considering assembly stack-up conditions based on manufacturing cost minimization. A random variable and its standard deviation are associated with a dimension and its tolerance. This probabilistic approach makes it possible to perform trade-off between performance and tolerance rather than the worst case analysis as it is commonly practiced. Tolerance (stack-up) analysis, as an inner loop in the overall algorithm for tolerance synthesis, is performed by approximating the volume under the multivariate probability density function constrained by nonlinear stack-up conditions with a convex polytope. This approximation makes use of the notion of reliability index [10] in structural safety. Consequently, the probabilistic optimization problem for tolerance synthesis is simplified into a deterministic nonlinear programming problem. An algorithm is then developed and is proven to converge to the global optimum through an investigation of the monotonic relations among tolerance, the reliability index, and cost. Examples from the implementation of the algorithm are given.

The Comparison of Methods of Assessing the Reliability of the Steel Frame

2015 ◽  
Vol 797 ◽  
pp. 11-18
Author(s):  
Agnieszka Dudzik ◽  
Urszula Radoń
Keyword(s):  
Reliability Index ◽  
Limit State ◽  
Probabilistic Approach ◽  
Random Variables ◽  
Random Variable ◽  
Design Parameters ◽  
Variable Sensitivity ◽  
Index Sensitivity ◽  
The Impact ◽  
The Mathematical Model

The study presents a probabilistic approach to the problems of static analysis of a steel building. Structural design parameters were defined as deterministic values and random variables. The latter were not correlated. The criterion of structural failure is expressed by limit functions related to the ultimate and serviceability limit state. The description of limit functions by the Mathematica program was generated. The Hasofer-Lind index was used as a reliability measure. In the description of random variables were used the normal distribution and, for comparison, different types of probability distribution appropriate to the nature of the variable. Sensitivity of reliability index to the random variables was defined. If the reliability index sensitivity due to the random variable Xi is low when compared with other variables, it can be stated that the impact of this variable on failure probability is small. Therefore, in successive computations it can be treated as a deterministic parameter. Sensitivity analysis leads to simplify the description of the mathematical model, determine the new limit functions and values of the Hasofer-Lind reliability index. The primary research method is the FORM method. In order to verify the correctness of the calculation SORM, Monte Carlo and Importance Sampling methods were used. In the examples of reliability analysis the STAND program was used.

scholarly journals A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

2020 ◽  
Vol 20 (4) ◽  
pp. 799-813
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Asymptotic Relation ◽  
Probabilistic Approach ◽  
Random Variable ◽  
High Order ◽  
Relative Accuracy ◽  
The Difference ◽  
New Perspective

AbstractThe aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble–Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements {P_{k}} and {P_{m}} ({k<m}). Then we analyze the asymptotic relation between these two probabilistic laws when the difference {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

scholarly journals Optimum Selection of Variable Pitch for Chatter Suppression in Face Milling Operations

Materials ◽  
2018 ◽  
Vol 12 (1) ◽  
pp. 112 ◽  
Author(s):  
Alex Iglesias ◽  
Zoltan Dombovari ◽  
German Gonzalez ◽  
Jokin Munoa ◽  
Gabor Stepan
Keyword(s):  
Removal Rate ◽  
Face Milling ◽  
Heavy Duty ◽  
Variable Pitch ◽  
Thin Walls ◽  
Chatter Suppression ◽  
Milling Operations ◽  
Optimum Selection ◽  
Cutting Capacity ◽  
Selection Of

Cutting capacity can be seriously limited in heavy duty face milling processes due to self-excited structural vibrations. Special geometry tools and, specifically, variable pitch milling tools have been extensively used in aeronautic applications with the purpose of removing these detrimental chatter vibrations, where high frequency chatter related to slender tools or thin walls limits productivity. However, the application of this technique in heavy duty face milling operations has not been thoroughly explored. In this paper, a method for the definition of the optimum angles between inserts is presented, based on the optimum pitch angle and the stabilizability diagrams. These diagrams are obtained through the brute force (BF) iterative method, which basically consists of an iterative maximization of the stability by using the semidiscretization method. From the observed results, hints for the selection of the optimum pitch pattern and the optimum values of the angles between inserts are presented. A practical application is implemented and the cutting performance when using an optimized variable pitch tool is assessed. It is concluded that with an optimum selection of the pitch, the material removal rate can be improved up to three times. Finally, the existence of two more different stability lobe families related to the saddle-node and flip type stability losses is demonstrated.

Design of LQR based PI controller with optimum selection of weights for Time Delay Process

2021 ◽  
Author(s):  
K Harshavardhana Reddy ◽  
Sachin Sharma ◽  
B. Madhuri ◽  
K Shivarama Krishna
Keyword(s):  
Time Delay ◽  
Pi Controller ◽  
Optimum Selection ◽  
Selection Of

scholarly journals A Method for the Statistical Evaluation of Crosstalk Effect Between Three Parallel Conductors

1999 ◽  
Vol 22 (2) ◽  
pp. 111-119
Author(s):  
P. T. Trakadas ◽  
C. N. Capsalis
Keyword(s):  
Closed Form ◽  
Uniform Distribution ◽  
Transmission Lines ◽  
Statistical Evaluation ◽  
Unit Length ◽  
Surrounding Medium ◽  
Probabilistic Approach ◽  
Random Variable ◽  
Mutual Inductance ◽  
Theoretical Results

There are several cases at which, in order to evaluate the crosstalk effect among transmission lines carrying useful signals, there is a need for probabilistic approach. This paper considers the problem of crosstalk estimation between transmission lines consisting of three conductors in a homogeneous surrounding medium, where the distance between the conductors is a random variable described by uniform distribution. The transmission lines are considered as electrically short. A closed-form equation is developed for the statistical distribution of the per-unit-length mutual inductance(lm)and an analytical one is described for the evaluation of the per-unit-length capacitance(cm). Theoretical results are compared with simulated ones for validation purposes.

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