Vertex method for computing functions of fuzzy variables

Dong, Weimin; Shah, Haresh C.

doi:10.1016/0165-0114(87)90114-x

Cited by 513 publications

(206 citation statements)

References 4 publications

Supporting

Mentioning

202

Contrasting

Unclassified

Order By: Relevance

“…7(a). However, this approach is unsuitable for 10 problems involving 'extreme points' [27] (i.e. local maxima or minima) within the fuzzy values of the response.…”

Section: Transformation Methodsmentioning

confidence: 99%

Fuzzy stability analysis of regenerative chatter in milling

Sims

Manson

Mann

2010

Journal of Sound and Vibration

View full text Add to dashboard Cite

During machining, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece.Consequently it is desirable to predict, and hence avoid the onset of this instability. Regenerative chatter is a function of empirical cutting coefficients, and the structural dynamics of the machine-tool system. There can be significant uncertainties in the underlying parameters, so the predicted stability limits do not necessarily agree with those found in practice.In the present study, fuzzy arithmetic techniques are applied to the chatter stability problem. It is first shown that techniques based upon interval arithmetic are not suitable for this problem due to the issue of recursiveness. An implementation of fuzzy arithmetic is then developed based upon the work of Hanss and Klimke. The arithmetic is then applied to two techniques for predicting milling chatter stability: the classical approach of Altintas, and the time-finite element method of Mann. It is shown that for some cases careful programming can reduce the computational effort to acceptable levels. The problem of milling chatter uncertainty is then considered within the framework of Ben-Haim's information-gap theory.It is shown that the presented approach can be used to solve process design problems with robustness to the uncertain parameters. The fuzzy stability bounds are then compared to previously published data, to investigate how uncertainty propagation techniques can offer more insight into the accuracy of chatter predictions.

show abstract

“…7(a). However, this approach is unsuitable for 10 problems involving 'extreme points' [27] (i.e. local maxima or minima) within the fuzzy values of the response.…”

Section: Transformation Methodsmentioning

confidence: 99%

Fuzzy stability analysis of regenerative chatter in milling

Sims

Manson

Mann

2010

Journal of Sound and Vibration

View full text Add to dashboard Cite

show abstract

“…To avoid, or at least ameliorate, such overestimation, one can use the same techniques used in the context of interval analysis, such as centered forms, interval splitting, and Taylor models (Section 3.4). However, in the context of fuzzy arithmetic, various other approaches have arisen, such as the "vertex method" (Dong & Shah, 1987) and its various enhancements (e.g., Wood et al, 1992;Otto et al, 1993;Yang et al, 1993;Anile et al, 1995;Chang & Hung, 2006), and the "transformation method" (Hanss, 2002(Hanss, , 2005, itself an extension and generalization of the vertex method. These methods, some of which have been compared by Seng et al (2007) on numerical test problems, attempt to exploit special properties, such as monotonicity and known extrema, of the function to be bounded over the intervals of interest.…”

Section: Fuzzy Arithmeticmentioning

confidence: 99%

Computing fuzzy trajectories for nonlinear dynamic systems

Măceş

Stadtherr

2013

Computers & Chemical Engineering

View full text Add to dashboard Cite

One approach for representing uncertainty is the use of fuzzy sets or fuzzy numbers. A new approach is described for the solution of nonlinear dynamic systems with parameters and/or initial states that are uncertain and represented by fuzzy sets or fuzzy numbers. Unlike current methods, which address this problem through the use of sampling techniques and do not account rigorously for the effect of the uncertain quantities, the new approach is not based on sampling and provides mathematically and computationally rigorous results. This is achieved through the use of explicit analytic representations (Taylor models) of state variable bounds in terms of the uncertain quantities. Examples are given that demonstrate the use of this new approach and its computational performance.

show abstract

“…The extension principle, though well defined, is more difficult to compute for continuous fuzzy subsets than for discrete fuzzy subsets. Various approaches include non-linear programming (Baas and Kwakernaak, 1977), L-R representation (Dubois and Prade, 1980), standard fuzzy arithmetic (Buckley, 1985;Kaufmann and Gupta, 1985), the vertex method involving alphacut representation (Dong et al, 1985;Dong and Wong, 1987;Dong and Shah, 1987;Liou and Wang, 1992a), parametric representation of the membership grades of fuzzy numbers and of the fuzzy numbers resulting from algebraic operations Project Appraisal September 1994…”

Section: Extension Principlementioning

confidence: 99%

Fuzzy applications in project appraisal: Part 2 fuzzy numbers

Smith

1994

Project Appraisal

View full text Add to dashboard Cite

Vertex method for computing functions of fuzzy variables

Cited by 513 publications

References 4 publications

Fuzzy stability analysis of regenerative chatter in milling

Fuzzy stability analysis of regenerative chatter in milling

Computing fuzzy trajectories for nonlinear dynamic systems

Fuzzy applications in project appraisal: Part 2 fuzzy numbers

Contact Info

Product

Resources

About