# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The transformation method for the simulation and analysis of systems with uncertain parameters. (English) Zbl 1015.65005
Summary: The transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters. Based on the concept of $\alpha$-cuts, the method represents a special implementation of fuzzy arithmetic that avoids the well-known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation. Systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results. As a by-product of the implementation scheme, the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output. By this, a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic. Finally, to show the efficiency of the transformation method, the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection.

##### MSC:
 65C60 Computational problems in statistics 93B35 Sensitivity (robustness) of control systems 26E50 Fuzzy real analysis
Full Text:
##### References:
 [1] Anile, A. M.; Deodato, S.; Privitera, G.: Implementing fuzzy arithmetic. Fuzzy sets and systems 72, 239-250 (1995) [2] De Wit, C. Canudas; Olsson, H.; Ström, K. A. \mathring{}; Lischinsky, P.: A new model for control of systems with friction. IEEE trans. Automat. control 40, 419-425 (1995) · Zbl 0821.93007 [3] Dong, W.; Shah, H. C.: Vertex method for computing functions of fuzzy variables. Fuzzy sets and systems 24, 65-78 (1987) · Zbl 0634.94025 [4] Dong, W. M.; Wong, F. S.: Fuzzy weighted averages and implementation of the extension principle. Fuzzy sets and systems 21, 183-199 (1987) · Zbl 0611.65100 [5] Dubois, D.: H. prade, fuzzy sets and systems: theory and applications. Mathematics in science and engineering 144 (1980) · Zbl 0444.94049 [6] L. Gaul, R. Nitsche, Contact pressure control in bolted joint connections, in: L. Gaul, C. Brebbia (Eds.), Computational Methods in Contact Mechanics, vol. IV, WIT Press, Southampton, Boston, 1999, pp. 369--378. [7] Gaul, L.; Nitsche, R.: Friction control for vibration suppression. Mech. syst. Signal process. 14, 139-150 (2000) [8] M. Hanss, On the implementation of fuzzy arithmetical operations for engineering problems, Proc. 18th Internat. Conf. of the North Amer. Fuzzy Informat. Process. Soc.--NAFIPS ’99, New York, USA, 1999, pp. 462--466. [9] M. Hanss, O. Nehls, Simulation of the human glucose metabolism using fuzzy arithmetic, Proc. 19th Internat. Conf. of the North Amer. Fuzzy Informat. Process. Soc.--NAFIPS 2000, Atlanta, GA, USA, 2000, pp. 201--205. [10] M. Hanss, O. Nehls, Enhanced parameter identification for complex biomedical models on the basis of fuzzy arithmetic, Proc. Joint 9th IFSA and 20th NAFIPS Internat. Conf., Vancouver, BC, Canada, 2001. [11] Hanss, M.; Willner, K.: A fuzzy arithmetical approach to the solution of finite element problems with uncertain parameters. Mech. res. Comm. 27, 257-272 (2000) · Zbl 1058.74636 [12] M. Hanss, S. Hurlebaus, L. Gaul, Fuzzy sensitivity analysis for the identification of material properties of orthotropic plates from natural frequencies, Mech. Syst. Signal Process., in press. [13] M. Hanss, K. Willner, S. Guidati, On applying fuzzy arithmetic to finite element problems, Proc. 17th Internat. Conf. of the North Amer. Fuzzy Informat. Process. Soc.--NAFIPS ’98, Pensacola Beach, FL, USA, 1998, pp. 365--369. [14] Kaufmann, A.; Gupta, M. M.: Introduction to fuzzy arithmetic. (1991) · Zbl 0754.26012 [15] Klir, G. J.: Fuzzy arithmetic with requisite constraints. Fuzzy sets and systems 91, 165-175 (1997) · Zbl 0920.04007 [16] Lorenzen, T. J.; Anderson, V. L.: Design of experiments. (1993) · Zbl 0853.62055 [17] Moore, R. E.: Interval analysis. (1966) · Zbl 0176.13301 [18] R. Nitsche, L. Gaul, Controller design for friction driven systems, Proc. CISM Course: Smart Structures--Theory and Applications, Udine, Italy, 2000. [19] Otto, K. N.; Lewis, A. D.; Antonsson, E. K.: Approximating ${\alpha}$-cuts with the vertex method. Fuzzy sets and systems 55, 43-50 (1993) · Zbl 0931.26010 [20] Wood, K. L.; Otto, K. N.; Antonsson, E. K.: Engineering design calculations with fuzzy parameters. Fuzzy sets and systems 52, 1-20 (1992) [21] Yang, H. Q.; Yao, H.; Jones, J. D.: Calculating functions of fuzzy numbers. Fuzzy sets and systems 55, 273-283 (1993)