Journal of Marine Science and Technology

, Volume 23, Issue 4, pp 814–834

# Solution of stochastic eigenvalue problem by improved stochastic inverse power method (I-SIPM)

• Xi Chen
• Yasumi Kawamura
Original article

## Abstract

Eigenvalue analysis is an important problem in a variety of fields. In structural mechanics in the field of naval architecture and ocean engineering, eigenvalue problems commonly appear in the context of, e.g. vibrations and buckling. In eigenvalue analysis, the physical characteristics are often considered as deterministic, such as mass, geometries, stiffness in the structures. However, in many practical cases, they are not deterministic. Such uncertainties may cause serious problems because the influence of the uncertainties is in general unknown. To solve the stochastic eigenvalue problem, in this article, we have proposed two methods. First, the improved stochastic inverse power method (I-SIPM) based on response surface methodology is proposed. The method is different with previous stochastic inverse power method. The minimum eigenvalue and eigenvector of stochastic eigenvalue problems can be evaluated using the proposed method. Second, the stochastic Wielandt deflation method (SWDM) is proposed to evaluate ith (i > 1) eigenvalues and eigenvectors of stochastic eigenvalue problems. This is very important for solving natural mode and buckling mode analysis problem. Next, two example problems are investigated to show the validity of two new methods compared with a Monte-Carlo simulation, i.e. the vibration problem of a discrete 2-DOF system and the buckling problem of a continuous beam. Finally, the uncertainty estimation for the dynamic damper problem is discussed using proposed method. The probability of the natural frequency falling into the range to be avoided is shown when the dynamic damper has a stochastic mass and stiffness.

## Keywords

Improved stochastic inverse power method (I-SIPM) Stochastic eigenvalue problem Polynomial chaos expansion (PCE) Stochastic Wielandt deflation method (SWDM)

## Notes

### Acknowledgements

This work was supported by the JSPS KAKENHI Grant Number 26630451. The authors are grateful for the support.

## References

1. 1.
Stefanou G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Engrg 198(9):1031–1051
2. 2.
Arregui-Mena JD, Margetts L, Mummery PM (2014) Practical application of the stochastic finite element method. Arch Comput Method E 23(1):171–190
3. 3.
Nakagiri S, Hisada T, Itotani Y (1982) Analysis of structure with uncertainty in shape by stochastic finite element method (In Japanese). Trans Jpn Soc Mech Eng Ser A 339–348Google Scholar
4. 4.
Kleiber M, Hien TD (1992) The stochastic finite element method. Wiley, Chichester
5. 5.
Ghanem R, Spsnos P (1991) Stochastic finite elements: a spectral approach. Springer, New York
6. 6.
Ghanem RG, Kruger RM (1996) Numerical solution of spectral stochastic finite element systems. Comput Method Appl Mech Eng 129(3):289–303
7. 7.
Nouy A (2009) Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch Comput Methods Eng 16(3):251–285
8. 8.
Ghanem R (1999) Ingredients for a general purpose stochastic finite elements implementation. Comput Methods Appl Mech Eng 168:19–34
9. 9.
Choi SK, Grandhi RV, Canfield RA (2007) Reliability-based structural design. Springer, Berlin
10. 10.
Honda R (2004) Spectral stochastic boundary element method for elastic problems with geometrical uncertainty. Doboku Gakkai Ronbunshu 2004(759):111–120 (in Japanese)
11. 11.
Chen X, Kawamura Y, Okada T (2016) Stochastic finite element method based on response surface methodology considering uncertainty in shape of structures. 13th international symposium on practical design of ship and other floating structures, CopenhagenGoogle Scholar
12. 12.
Chen X, Kawamura Y, Okada T (2016) Development of structural analysis method with uncertainty in shape to follow non-normal distribution by stochastic finite element method (in Japanese). Trans Jpn Soc Comput Eng Sci 2016:20160019Google Scholar
13. 13.
Verhoosel CV, Gutiérrez MA, Hulshoff SJ (2016) Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems. Int J Numer Meth 68(4):401–424
14. 14.
Sepahvand K, Marburg S. Hardtke HJ (2011) Stochastic structural modal analysis involving uncertain parameters using generalized polynomial chaos expansion. Int J Appl Mech 3(3):587–606
15. 15.
Xiu D, Karniadakis G (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. J Sci Comput 24(2):619–644
16. 16.
Pohlhausen E (1921) Berechnung der Eigenschwingungen statisch-bestimmter Fachwerke. Z Angew Math Mech 1:28–42
17. 17.
Tjalling JY (1995) Historical development of the Newton–Raphson method. Soc Ind Appl Math 37:531–551
18. 18.
Meirovitch L (1980) Computational methods in structural dynamics. Sijthoff and Noordhoff, The Netherlands
19. 19.
Japanese Industrial Standard (JIS) G3192 (2008) Hot rolled sections section properties of all JIS hot rolled sectionsGoogle Scholar
20. 20.
Matsuura Y, Matsumoto K, Mizuuchi M, Arima K, Jouichi H (1985) A study of mechanical dynamic dampers to reduce ship vibration (In Japanese). Kansai Soc N A J 197:117–126Google Scholar
21. 21.
Miyashita K, Kobayashi K, Takaoka Y, Ebira K, Toma Y, Kubo M (2004) The larger LNG carrier (145,000 m3 type) aiming at low operation cost. The 14th international conference and exhibition on liquefied natural gas, DohaGoogle Scholar
22. 22.
Kondo K, Ohta T, Satoh H (1987) On the vibration control of superstructures of ships by means of a dynamic vibration absorber with an adjustable mass (in Japanese). J Soc Nav Archit Jpn 162:356–363
23. 23.
Tamai S, Itoh M, Kanayama T (2001) Development of tuned mass damper (In Japanese). IHI Eng Rev 41(1):27–31Google Scholar
24. 24.
Korenev BG, Reznikov LM (1993) Dynamic vibration absorbers: theory and technical applications. Wiley, ChichesterGoogle Scholar