Keywords: composite materials, passive vibration control, non-linear visco-elastic behaviour, optimization, simulated annealing, cost functional.

The latest developments in computational mechanics have lead to integrated methodologies that permit not only the structural design and shape optimization of mechanical components but also the tailoring of material properties and consequently the design of new materials. This is particularly evident in the area of composite materials where the unit cell geometry (characterizing the composite material) is a key factor in its effective mechanical properties and thus can significantly improve the structural response of the mechanical component.

The aim of this contribution is to extend the techniques of composite materials design to nonlinear material behaviour and apply it to the design of new materials for passive vibration control. The objectives of passive vibration control typically cover attenuation of the steady-state regime of the structure dynamic response. Then, at relatively low excitation frequencies is it important to reduce the steady-state displacement amplitudes, which are generally obtained by an isolator of a strong material that may exhibit a low damping. On the other hand, at high excitation frequencies, a good isolator performance is considered when the transmissibility is low. Then the isolator must possess high damping properties, which usually implies a soft material. Materials exhibiting high stiffness as well as high damping are not common. Moreover, when the isolator is assumed to operate on a large range of frequencies, it would be ideal to have a material that softens at high frequencies. Unfortunately, it can be proven that within one phase solid materials the tendency is completely opposite, in other words, the real materials strengthen at high frequencies. Therefore it is necessary to design a new composite material with specific dynamic properties, namely, a material that softens at high frequencies.

As a first step a computational tool allowing determination of macroscopic optimized one-dimensional isolator behaviour was developed. Voigt, Maxwell, standard and more complex material models can be implemented. The objective function considers the minimization of the initial reaction and/or displacement peak as well as the minimization of the steady-state amplitude of the reaction and/or displacement. The complex stiffness approach is used to formulate the governing equations in an efficient way. Material stiffness parameters are assumed as non-linear functions of the displacement. Numerical solution is performed in the complex space. Special attention is paid to the initial conditions, because in the complex space the transient solution exhibits an unrealistic component originated by the fact that the excitation frequency is already a part of the complex modulus. The steady-state solution is obtained by an iterative process based on the shooting method which imposes the conditions of periodicity with respect to the known value of the period. Extension of the shooting method to the complex space is presented and verified. Non-linear dependence of the material parameters is then optimized using a generic probabilistic meta-algorithm, simulated annealing. Dependence of the global optimum on several combinations of leading parameters of the simulated annealing procedure, like neighbourhood definition and annealing schedule, is also studied and analyzed. The procedures are programmed in the MATLAB environment.

Further development is directed to the design of composite viscoelastic materials with improved behaviour in terms of dynamic stiffness for passive vibration control identified by the procedure presented in this contribution. This application will have a direct and immediate impact on product design and development, especially in the design of new mechanical components such as engine mounts and new suspension systems.

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