Keywords: structural dynamics, mid-frequency, wave based method, deterministic, probabilistic, Monte Carlo simulation.

At present, several numerical prediction techniques are available for the analysis of the steady-state dynamic deformations of mechanical structures. Most of these techniques can be classified as being based on either deterministic or probabilistic modelling principles. The use of the deterministic methods is limited to the so-called low-frequency range. On the one hand because the computational effort of the deterministic techniques typically increases exponentially when frequency increases, since the spatial variation of the dynamic field variables increases. On the other hand because the response of a system becomes increasingly sensitive to small perturbations of its properties as the frequency increases. When the interest lies in predicting the mean response of nominally identical structures higher frequencies, the probabilistic methods, based on energy principles, give good results. The underlying assumptions for limit the use of probabilistic methods to the high-frequency range.

In-between the low- and the high-frequency range, there is a relatively wide mid-frequency-range in which some of the structural subsystems fulfil the requirements for the probabilistic approach and some do not (yet). In this respect, it is useful to develop a deterministic methodology which can handle both the deterministic and the probabilistic behaviour. It has been shown that when there is enough uncertainty in the dynamic properties of a system, the response statistics become insensitive to the way in which the system is randomised. Therefore, this paper proposes an approach based on the deterministic wave based method (WBM) [1] which is able to efficiently perform Monte Carlo simulations with point masses at random locations. The random point mass distributions introduce the desired variability of the population.

The paper starts with a description of the WBM for the steady-state dynamic analysis of the bending behaviour of flat plates [2]. Next, the methodology for adding discrete point masses to the plate is explained. Since only the point mass distributions change in the Monte Carlo simulations, a significant time gain can be achieved by saving on the building time of the system by reusing parts of the system matrix which remain unchanged by the addition of point masses. In the numerical example, it is shown that this implementation leads to a reduction of the calculation time with a factor 7 compared with a conventional finite element (FE) calculation with the same accuracy.

Additionally, due to the enhanced performance of the WBM[3], the method can be used for higher frequencies than the FE method. In that way, the deterministic WBM is an alternative to tackle problems in the mid-frequency region where some parts of the system show a deterministic and some a probabilistic behaviour.

1

W. Desmet, "A Wave based prediction technique for coupled vibro-acoustic analysis", Ph.D. dissertation, Katholieke Universiteit Leuven, Departement Werktuigkunde, Leuven, 1998.

2

C. Vanmaele, D. Vandepitte, W. Desmet, "An efficient wave based prediction technique for plate bending vibrations", Comput. Methods Appl. Mech. Engrg., 196, 3178-3189, 2007. doi:10.1016/j.cma.2007.03.002

3

K. Vergote, C. Vanmaele, D. Vandepitte, W. Desmet, "On the use of an efficient wave based method for steady-state structural dynamic analysis", in "Proceedings of the First Leuven Symposium on Applied Mechanics in Engineering", Leuven, Belgium, 433-459, 2008.

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