# Vibration Analysis of Fluid-Filled Piping Systems with Epistemic Uncertainties

Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 27)

## Abstract

Non-determinism in numerical models of real-world systems may arise as a consequence of different sources: natural variability or scatter, which is often referred to as aleatory uncertainties, or so-called epistemic uncertainties, which arise from an absence of information, vagueness in parameter definition, subjectivity in numerical implementation, or simplification and idealization processes employed in the modeling procedure. Fuzzy arithmetic based on the transformation method can be applied to numerically represent epistemic uncertainties and to track the propagation of the uncertainties towards the output quantities of interest. In the current study, the fuzzy arithmetical approach is applied to the vibration analysis of a fluid-filled piping system with a structure attached. The investigation of this system is motivated by an automotive application, namely the brake pipes coupled to the floor panel of a car. The piping system is excited by a pressure pulsation in the fluid. Through fluid-structure interaction, this leads to a vibration of the pipes and thus of the structure attached. The uncertainties inherent to the system are of epistemic type and arise, among other things, from a lack of knowledge about the coupling elements between the pipes and the structure. Finite element simulations are performed to compute the vibration response of the system. These simulations are carried out multiple times in the framework of the fuzzy arithmetical algorithm to compute the uncertainty in the vibration response. Since a large number of simulations are needed, computational time is an important issue. In order to minimize the computational effort, substructuring in terms of the component mode synthesis (CMS) and model reduction techniques based on the Craig-Bampton method are used.

## Keywords

Fuzzy Number Vibration Analysis Frequency Response Function Epistemic Uncertainty Piping System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Contreras H (1980) The stochastic finite-element method. Comp.&Struc. 12:341–348
2. 2.
Craig RR, Bampton MCC (1968) Coupling of substructures for dynamic analysis. AIAA Journal 6:1313–1319
3. 3.
Craig RR, Chang CJ (1977) Substructure coupling for dynamic analysis and testing. NASA CR 2781 Google Scholar
4. 4.
Elishakoff I, Ren YJ (1998) The bird’s eye view on finite element method for structures with large stochastic variations. Comp. Meth. in Appl. Mech. and Eng. 168:51–61
5. 5.
Everstine GC (1981) A symmetric potential formulation for fluid-structure interaction. Journal of Sound and Vibration 79:157–160
6. 6.
Ewins DJ (2003) Modal Testing: Theory, Practice, and Application. Research Studies Press Google Scholar
7. 7.
Gauger U, Turrin S, Hanss M, Gaul L (2007) A new uncertainty analysis for the transformation method. Fuzzy Sets and Systems 159:1273–1291
8. 8.
Ghanem RG, Spanos PD (1991) Stochastic Finite Elements: A Spectral Approach. Springer, New York
9. 9.
Haag T, Reuß P, Turrin S, Hanss M (2009) An inverse model updating procedure for systems with epistemic uncertainties. In: Proc. of the 2nd International Conference on Uncertainty in Structural Dynamics, Sheffield, UK Google Scholar
10. 10.
Handa K, Anderson K (1981) Application of finite element methods in the statistical analysis of structures. In: Moan T, Shinozuka M (eds) Proc. of the 3rd International Conference on Structural Safety and Reliability, Elsevier, Amsterdam, pp 409–417 Google Scholar
11. 11.
Hanss M (2002) The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets and Systems 130(3):277–289
12. 12.
Hanss M (2003) The extended transformation method for the simulation and analysis of fuzzy-parameterized models. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11(6):711–727
13. 13.
Hanss M (2005) Applied Fuzzy Arithmetic – An Introduction with Engineering Applications. Springer, Berlin
14. 14.
Herrmann J, Haag T, Engelke S, Gaul L (2008a) Experimental and numerical investigation of the dynamics in spatial fluid-filled piping systems. In: Proc. of Acoustics, Paris Google Scholar
15. 15.
Herrmann J, Haag T, Gaul L, Bendel K, Horst HG (2008b) Experimentelle Untersuchung der Hydroakustik in Kfz-Leitungssystemen. In: Proc. of DAGA, Dresden Google Scholar
16. 16.
Herrmann J, Spitznagel M, Gaul L (2009) Fast FE-analysis and measurement of the hydraulic transfer function of pipes with non-uniform cross section. In: Proc. of NAG/DAGA, Netherlands Google Scholar
17. 17.
Herrmann J, Maess M, Gaul L (accepted 2009) Substructuring including interface reduction for the efficient vibro-acoustic simulation of fluid-filled piping systems. Mechanical Systems and Signal Processing Google Scholar
18. 18.
Kaufmann A, Gupta MM (1991) Introduction to Fuzzy Arithmetic. Van Nostrand Reinhold, New York
19. 19.
Kleiber M, Hien TD (1993) Stochastic Finite Element Method. John Wiley & Sons, New York Google Scholar
20. 20.
Maess M (2006) Methods for efficient acoustic-structure simulation of piping systems. Ph.D. thesis, Institute of Applied and Experimental Mechanics, University of Stuttgart Google Scholar
21. 21.
Moens D, Vandepitte D (2002) Fuzzy finite element method for frequency response function analysis of uncertain structures. AIAA Journal 40:126–136
22. 22.
Möller B, Beer M (2004) Fuzzy Randomness Uncertainty in Civil Engineering and Computational Mechanics. Springer, Berlin
23. 23.
Oberkampf WL (2007) Model validation under both aleatory and epistemic uncertainty. In: Proc. of NATO AVT-147 Symposium on Computational Uncertainty in Military Vehicle Design, Athens, Greece Google Scholar
24. 24.
Rao SS, Sawyer JP (1995) Fuzzy finite element approach for the analysis of imprecisely defined systems. AIAA Journal 33:2364–2370
25. 25.
Schuëller GI (2007) On the treatment of uncertainties in structural mechanics and analysis. Computers and Structures 85(5–6):235–243
26. 26.
Theissen H (1983) Die Berücksichtigung instationärer Rohrströmung bei der Simulation hydraulischer Anlagen. Ph.D. thesis, RWTH Aachen Google Scholar
27. 27.
Zienkiewicz O, Taylor R (2000) The Finite Element Method. Butterworth-Heinemann, Oxford