European Journal of Mechanics A/Solids 28 (2009) 141–154 Contents lists available at ScienceDirect European Journal of Mechanics A/Solids www.elsevier.com/locate/ejmsol Robust optimization of the non-linear behaviour of a vibrating system M.-L. Bouazizi a,∗ , S. Ghanmi a , R. Nasri b , N. Bouhaddi c a Nabeul Preparatory Engineering Institute (IPEIN), University of 7 Novembre, 8000 M’rezgua, Nabeul, Tunisia b Laboratory of the Systems and Applied Mechanics, Polytechnic school of Tunis, Tunisia c FEMTO-ST Institute UMR 6174- Laboratory of R. Chaleat Applied Mechanics University of Franche-Comté, 24 Chemin de l’Epitaphe, 25000 Besançon, France article info abstract Article history: Received 28 April 2007 Accepted 10 April 2008 Available online 1 May 2008 Keywords: Dynamic absorber Uncertainties Robustness Non-linearities Muti-objective optimization Self Organizing Map In this paper, one proposes to optimize the vibratory behaviour of an absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the design parameters. The total system is modeled by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. The resolution is carried out in the temporal field according to a traditional diagram. Two cases of non-linearity were considered. In the first case, one is interested in the study of the system comprising a combination of the two generalized non-linearities of quadratic and cubic type of stiffness and damping. The second case relates to a non-linearity of non-whole power (in this paper 1.5), combined with the cubic case. It is a question of seeking the optimal responses envelopes of the deterministic and stochastic case and this for the non-linear displacements, phases and forces. The multi-objective optimization step consists in seeking the first Pareto front of several linear and non- linear objective functions by using a genetic algorithm of type “NSGA” (Non-dominated Sorting Genetic Algorithm). The design parameters are: mass, linear and non-linear stiffness and damping of the absorber. To obtain solutions presenting a good compromise between optimality and the robustness, one introduces uncertainties on these design parameters. The robustness is then defined by the dispersion of the parameters (definite as the ratio: mean value/standard deviation) and it is introduced as additional objective function. The use of the clusters resulting from the Self-Organizing Maps of Kohonen (SOM) is also suggested for a rational management of the design space. A study of sensitivity a posteriori can be exploited in order to eliminate the non-significant design parameters.  2008 Elsevier Masson SAS. All rights reserved. 1. Introduction In order to reduce the vibrations in the revolving machines and the mechanical systems, the dynamic absorbers are often used in various mechanical applications (Crankshaft, rotor of the wings of a helicopter, etc.). Each absorber consists of an oscillator: an aux- iliary that one adds with the vibrating structure, in order to carry out a transfer of energy from the principal system to the auxiliary system. This latter can either be added on the principal structure or directly envisaged at the design step. This solution of absorber is frequently used to cure the problems of vibrations of the revolv- ing machines. In the case of the real structures a linear model will be insuffi- cient to describe the dynamic behaviour correctly. It thus appears natural to introduce non-linear models of structures which are able to predict the dynamic behaviour of the real structures. The solutions of these non-linear problems are obtained by approxi- * Corresponding author. Tel.: (+216) 98 913 472; fax: (+216) 72 220 181. E-mail address: lamjed.bouazizi@ipein.rnu.tn (M.-L. Bouazizi). mate methods which exploit iterative algorithms. In the literature, several methods were proposed to approach the solution of the mechanical systems subjected to dynamic stresses. Concerning the mechanical systems with only one degree of freedom, one can quote various works: Natsiavas (1992) studied the response of an absorber of vibration on a machine with cubic non-linearity of Duffing type. Numerical results are given with sev- eral combinations of parameters of the system in order to improve the absorber. For the discrete systems with several dof comprising the ab- sorbers, several works considered the non-linearity. Vakakis and Paipetis (1986) studied the effect of the viscosity of the absorber on a linear conservative system with several dof. Natsiavas and Tratskas (1996) studied a system with two non-linear dof in trans- lation and rotation by the method of the multiple scales. Verros and Natsiavas (2002) studied an oscillator with two dof with cubic non-linearity of stiffness subjected to a harmonic load. They used the method of the disturbances in order to study the effect of the parameters on the stability of the periodic solution. Zhu et al. (2004) studied a discrete system with two dof with two cubic non-linearities of stiffness and damping subjected to 0997-7538/$ – see front matter  2008 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2008.04.006