Several methods are presented for the modeling and analysis of uncertain beams and other structural elements/systems. By representing each uncertain parameter as an interval number, the vibration problem associated with any uncertain system can be expressed in the form of a system of nonlinear interval equations. The resulting equations can be solved using the exact or a truncation-based interval analysis method. An universal grey number-based approach and an interval-discretization method are proposed to obtain more efficient and/or more accurate solutions. Specifically, the problem of the coupled bending-torsional vibration of a beam involving uncertainties is considered. It is found that the range of the solution (response) increases with increasing levels of uncertainty in all the methods. Numerical examples are presented to illustrate the computational aspects of the methods presented and also to indicate the high accuracy of the interval-discretization approach in finding the solution of practical uncertain systems. The results given by the different interval analysis methods (including the universal grey number-based analysis) are compared with those given by the Monte Carlo method (probabilistic approach) and the results are found to be in good agreement with those given by the interval analysis-based methods for similar data.

References

References
1.
Timoshenko
,
S.
,
Young
,
D. H.
, and
Weaver
,
W.
, Jr.,
1974
,
Vibration Problems in Engineering
,
Wiley
,
New York
.
2.
Rao
,
J. S.
, and
Carnegie
,
W.
,
1970
, “
Solution of the Equations of Motion of Coupled-Bending Bending Torsion Vibrations of Turbine Blades by the Method of Ritz-Galerkin
,”
Int. J. Mech. Sci.
,
12
(
10
), pp.
875
882
.10.1016/0020-7403(70)90024-X
3.
Tanaka
,
M.
, and
Bercin
,
A. N.
,
1999
, “
Free Vibration Solution for Uniform Beams of Nonsymmetrical Cross Section Using Mathematica
,”
Comput. Struct.
,
71
(
1
), pp.
1
8
.10.1016/S0045-7949(98)00236-3
4.
Hashemi
,
S. M.
, and
Richard
,
M. J.
,
2000
, “
Free Vibrational Analysis of Axially Loaded Bending-Torsion Coupled Beams: A Dynamic Finite Element
,”
Comput. Struct.
,
77
(
6
), pp.
711
724
.10.1016/S0045-7949(00)00012-2
5.
Banerjee
,
J. R.
,
Guo
,
S.
, and
Howson
,
W. P.
,
1996
, “
Exact Dynamic Stiffness Matrix of a Bending-Torsion Coupled Beam Including Warping
,”
Comput. Struct.
,
59
(
4
), pp.
613
621
.10.1016/0045-7949(95)00307-X
6.
Yamamura
,
K.
,
2000
, “
Finding all Solutions of Nonlinear Equations Using Linear Combinations of Functions
,”
Reliab. Comput.
,
6
(
2
), pp.
105
113
.10.1023/A:1009956920204
7.
Yamamura
,
K.
, and
Tanaka
,
S.
,
2002
, “
Finding all Solutions of Systems of Nonlinear Equations Using the Dual Simplex Method
,”
BIT Numer. Math.
,
42
(
1
), pp.
214
230
.10.1023/A:1021938622923
8.
Yamamura
,
K.
, and
Suda
,
K.
,
2007
, “
An Efficient Algorithm for Finding All Solutions of Separable Systems of Nonlinear Equations
,”
BIT Numer. Math.
,
47
(
3
), pp.
681
691
.10.1007/s10543-007-0132-1
9.
Rao
,
S. S.
,
2007
,
Vibration of Continuous Systems
,
Wiley
,
Hoboken, NJ
.
10.
Rao
,
S. S.
, and
Berke
,
L.
,
1997
, “
Analysis of Uncertain Structural Systems Using Interval Analysis
,”
AIAA J.
,
35
(
4
), pp.
727
735
.10.2514/2.164
11.
Deng
,
J. L.
,
1982
, “
Control Problems of Grey Systems
,”
Syst. Control Lett.
,
1
(
5
), pp.
288
294
.10.1016/S0167-6911(82)80025-X
12.
Wang
,
Q. Y.
,
1996
,
Foundation of Grey Mathematics
,
Huazhong University of Science and Technology
,
Wuhan, China (in Chinese)
.
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