Damping Model Uncertainty in Structural Dynamics

Adhikari, S.,
International Conference on Noise and Vibration Engineering (ISMA2006), Leuven, Belgium, 2006.

Uncertainties are unavoidable in the description of real-life engineering systems. The quantification of uncertainties plays a crucial role in establishing the credibility of a numerical model. Uncertainties can be broadly divided into two categories. The first type is due to the inherent variability in the system parameters. This type of uncertainty is often referred to as aleatoric uncertainty. If enough samples are present, it is possible to characterize the variability using well established statistical methods and consequently the probably density functions (pdf) of the parameters can be obtained. The second type of uncertainty is due to the lack of knowledge regarding a system, often referred to as epistemic uncertainty or model uncertainty. This kind of uncertainty generally arise in the modelling of complex physical phenomenon such as damping. Quantification of uncertainties associated with the damping forces is difficult because, unlike inertia and stiffness forces, it is not in general clear what are the state variables that govern the damping forces. The most common approach is to use `viscous damping' where the instantaneous generalized velocities are the only relevant state variables. Several studies exist where viscous damping coefficients are assumed to be random variables. Considering randomness in the viscous damping parameters will however not address the fundamental uncertainty that arises due to the use of viscous damping model itself. Viscous damping by no means the only damping model within the scope of linear analysis. Any model which makes the energy dissipation functional non-negative is a possible candidate for a valid damping model. Therefore, to avoid any `model biases', in this study we have used possibly the most general linear damping model[1, 2] given by
F d(t) = ó
õ
t


-¥
G (t,tu(td t     (1)
where u(t) Î RN is the vector of generalized coordinates with t Î R+ denotes time and G (t,t) Î RN× N is the kernel function matrix. Only in the special case when G (t,t) = C   d(t-t), where d(t) is the Dirac-delta function, equation (1) reduces to the case of viscous damping.

This paper is devoted to quantify uncertainty associated with the use of the general damping model (not only the model parameters). Two approaches, based on parametric and non-parametric methods are developed. In the first approach, different equivalent functional forms of G (t,t) are derived and their parameters are selected. The collection of these different equivalent functional forms are then assumed to form the random sample space. Under these settings, the selection of any one model (such as the viscous model) can be regarded as a random event in the sample space of the admissible functions. In the second approach G (t,t) is considered to be a matrix variate random process whose distribution is obtained using the maximum entropy principle[3, 4]. The results obtained from the parametric and non-parametric methods are compared using large-scale engineering dynamic problems.

References

[1]
Adhikari, S., ``Dynamics of non-viscously damped linear systems,'' ASCE Journal of Engineering Mechanics, Vol. 128, No. 3, March 2002, pp. 328--339.

[2]
Wagner, N. and Adhikari, S., ``Symmetric state-space formulation for a class of non-viscously damped systems,'' AIAA Journal, Vol. 41, No. 5, 2003, pp. 951--956.

[3]
Soize, C., ``A nonparametric model of random uncertainties for reduced matrix models in structural dynamics,'' Probabilistic Engineering Mechanics, Vol. 15, 2000, pp. 277--294.

[4]
Soize, C., ``Maximum entropy approach for modeling random uncertainties in transient elastodynamics,'' Journal of the Acoustical Society of America, Vol. 109, No. 5, May 2001, pp. 1979--1996.

BiBTeX Entry
@INPROCEEDINGS{cp25,
    AUTHOR={S. Adhikari},
    TITLE={Damping model uncertainty in structural dynamics},
    BOOKTITLE={International Conference on Noise and Vibration Engineering (ISMA2006)},
    YEAR={2006},
    Address={Leuven, Belgium},
    Month={September},
    Note={}
}

by Sondipon Adhikari