Solution of stochastic eigenvalue problem by improved stochastic inverse power method (I-SIPM)
- 310 Downloads
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.
KeywordsImproved stochastic inverse power method (I-SIPM) Stochastic eigenvalue problem Polynomial chaos expansion (PCE) Stochastic Wielandt deflation method (SWDM)
This work was supported by the JSPS KAKENHI Grant Number 26630451. The authors are grateful for the support.
- 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
- 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.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
- 19.Japanese Industrial Standard (JIS) G3192 (2008) Hot rolled sections section properties of all JIS hot rolled sectionsGoogle Scholar
- 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.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
- 23.Tamai S, Itoh M, Kanayama T (2001) Development of tuned mass damper (In Japanese). IHI Eng Rev 41(1):27–31Google Scholar
- 24.Korenev BG, Reznikov LM (1993) Dynamic vibration absorbers: theory and technical applications. Wiley, ChichesterGoogle Scholar