Abstract
The description of real-life engineering structural systems is usually associated with some amount of uncertainty in specifying material properties, geometric parameters and boundary conditions. In the context of structural dynamics it is necessary to consider joint probability distribution of the natural frequencies in order to account for these uncertainties. Current methods to deal with such problems are dominated by approximate perturbation methods. In this paper a new approach based on an asymptotic approximation of multidimensional integrals is proposed. A closed-form expression for general order joint moments of arbitrary number of natural frequencies of linear stochastic systems is derived. The proposed method does not employ the small-randomness and Gaussian random variable assumption usually used in the perturbation based methods. Joint distributions of the natural frequencies are investigated using numerical examples and the results are compared with Monte Carlo simulation.
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Abbreviations
- \(\kappa_{jk}^{(r_1,r_2)}\) :
-
(r 1,r 2)th order cumulant of jth and kth natural frequencies
- D (•) (x ):
-
Hessian matrix of (•) at x
- d (•) (x ):
-
gradient vector of (•) at x
- I :
-
identity matrix
- K :
-
stiffness matrix
- M :
-
mass matrix
- μ :
-
mean of parameter vector x
- O :
-
null matrix
- \(\phi_{j}\) :
-
jth mode shape of the system
- Σ:
-
covariance matrix of parameter vectors x
- \({\Sigma}_{\omega_{jk}}\) :
-
covariance matrix of jth and kth natural frequencies
- θ :
-
optimal point
- x :
-
basic random variables
- \(\epsilon_{m}, \epsilon_{k}\) :
-
strength parameters associated with mass and stiffness coefficients
- \(\mu_{jk}^{(r_{1},r_{2})}\) :
-
\((r_{1},r_{2})\)th order joint moment of jth and kth natural frequencies
- ω j :
-
jth natural frequencies of the system
- \(\widetilde{\bf s}\) :
-
vector of complex Laplace parameters s 1 and s 2
- L (x) :
-
negative of the log-likelihood funtion
- m :
-
number of basic random variables
- \(M_{\omega_{j},\omega_{k}} (s_{1},s_{2})\) :
-
joint moment generating function of ω j and ω k
- N :
-
degrees-of-freedom of the system
- p (•) :
-
probability density function of (•)
- \((\bullet)^{\rm T}\) :
-
matrix transpose
- \(\mathbb{C}\) :
-
space of complex numbers
- \(\mathbb{R}\) :
-
space of real numbers
- Cov \(({\bullet,\bullet})\) :
-
covariance of random quantities
- \(\Vert{\bullet}\Vert\) :
-
determinant of a matrix
- exp:
-
exponential function
- \({\bf E}[{\bullet}]\) :
-
expectation operator
- \(\in\) :
-
belongs to
- \(\mapsto\) :
-
maps into
- \(\mid \bullet \mid\) :
-
l 2 norm of (•)
- \(\overline{(\bullet)}\) :
-
deterministic value corresponding to (•)
- dof:
-
degrees-of-freedom
- jpdf:
-
joint probability density function
- pdf:
-
probability density function
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Adhikari, S. Joint statistics of natural frequencies of stochastic dynamic systems. Comput Mech 40, 739–752 (2007). https://doi.org/10.1007/s00466-006-0136-8
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DOI: https://doi.org/10.1007/s00466-006-0136-8