Natural frequencies of structures with interval parameters

Abstract This paper deals with the evaluation of the lower and upper bounds of the natural frequencies of structures with uncertain-but-bounded parameters. The solution of the generalized interval eigenvalue problem is pursued by taking into account the actual variability and dependencies of uncertain structural parameters affecting the mass and stiffness matrices. To this aim, interval uncertainties are handled by applying the improved interval analysis via extra unitary interval (EUI), recently introduced by the first two authors. By associating an EUI to each uncertain-but-bounded parameter, the cases of mass and stiffness matrices affected by fully disjoint, completely or partially coincident uncertainties are considered. Then, based on sensitivity analysis, it is shown that the bounds of the interval eigenvalues can be evaluated as solution of two appropriate deterministic eigenvalue problems without requiring any combinatorial procedure. If the eigenvalues are monotonic functions of the uncertain parameters, then the exact bounds are obtained. The accuracy of the proposed method is demonstrated by numerical results concerning truss and beam structures with material and/or geometrical uncertainties.

[1]  Zhiqing He,et al.  Computation of bounds for eigenvalues of structures with interval parameters , 2010, Appl. Math. Comput..

[2]  Huinan Leng,et al.  Computing bounds to real eigenvalues of real‐interval matrices , 2008 .

[3]  Isaac Elishakoff,et al.  Whys and Hows in Uncertainty Modelling , 1999 .

[4]  A. Sofi,et al.  Bounds for the stationary stochastic response of truss structures with uncertain-but-bounded parameters , 2013 .

[5]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[6]  Suhuan Chen,et al.  The Rayleigh quotient iteration method for computing eigenvalue bounds of structures with bounded uncertain parameters , 1995 .

[7]  J. H. Starnes,et al.  The bound set of possible eigenvalues of structures with uncertain but non-random parameters , 1996 .

[8]  Xiaojun Wang,et al.  Eigenvalue bounds of structures with uncertain-but-bounded parameters , 2005 .

[9]  Quan Yuan,et al.  An evolution strategy method for computing eigenvalue bounds of interval matrices , 2008, Appl. Math. Comput..

[10]  Assem S. Deif,et al.  Advanced matrix theory for scientists and engineers , 1990 .

[11]  Makoto Ohsaki,et al.  Optimization And Anti-Optimization Of Structures Under Uncertainty , 2010 .

[12]  Andrzej Pownuk.  General interval FEM program based on sensitivity analysis , 2007 .

[13]  Wei Gao.  Interval natural frequency and mode shape analysis for truss structures with interval parameters , 2006 .

[14]  G. Meng,et al.  An efficient method for evaluating the natural frequencies of structures with uncertain-but-bounded parameters , 2009 .

[15]  Robert L. Mullen,et al.  Natural Frequencies of a Structure with Bounded Uncertainty , 2006 .

[16]  Xiao Wei Yang,et al.  Interval eigenvalue analysis for structures with interval parameters , 2003 .

[18]  S. H. A. Chen,et al.  Natural frequencies of structures with uncertain but nonrandom parameters , 1995 .

[19]  Huinan Leng,et al.  Real eigenvalue bounds of standard and generalized real interval eigenvalue problems , 2014, Appl. Math. Comput..

[20]  Su-huan Chen,et al.  Interval finite element method based on the element for eigenvalue analysis of structures with interval parameters , 2001 .

[21]  Huinan Leng,et al.  Computing eigenvalue bounds of structures with uncertain-but-non-random parameters by a method based on perturbation theory , 2006 .

[22]  Xiaojun Wang,et al.  Solution theorems for the standard eigenvalue problem of structures with uncertain-but-bounded parameters , 2005 .

[23]  A. Sofi,et al.  Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis , 2012 .