Keywords: reliability, design, optimization, structural reliability, dynamic contact.

In the field of deterministic structural optimization, the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. So, the resulting optimal design may represent a lower level of reliability and thus a higher risk of failure. Since structural problems are non-deterministic, it is clear that the introduction of the reliability concept plays an important role in the structural optimization field. The integration of reliability analysis into design optimization problem represents the reliability-based design optimization (RBDO) model. The objective of this model is to design structures which should be both economic and reliable. In the RBDO model for robust system design, the mean values of uncertain system variables are usually used as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a mathematical nonlinear programming problem. Therefore, a RBDO solution that reduces the structural weight in uncritical regions does not only provide an improved design but also a higher level of confidence in the design. This approach can be carried out in two separate spaces: the physical space and the normalized space. Since many repeated searches are required in the above two spaces, the computational time for such an optimization is a big problem. The solution of the above nested problems leads to very high computational cost, especially for large-scale structures. The major difficulty lies in the structural reliability evaluation, which is carried out by a special optimization procedure.

Fortunately, an efficient method called the hybrid method has been elaborated. This method, which is based on simultaneous solution of the reliability and the optimization problem, has successfully reduced the computational time problem and has been shown to verify the optimality conditions. The solution is achieved in a hybrid design space (HDS) containing the deterministic and the random variables, and a specified reliability level is satisfied. Structures with static linear and, or perfectly plastic responses were generally studied using the RBDO model.

The objective of this work is to consider the phenomenon of dynamic contact inside a piezoelectric engine using the RBDO method.

The modelling of piezoelectric engines with a travelling wave implies a large variety of physical and mechanical phenomena. This variety leads to approaches and models many and varied, which rest mainly on phenomenological and numerical analyses.

The specifications, increasingly more demanding imposed on the actuators; confine sometimes the conventional electromagnetic engines with their extreme limits of operation. In this context, the recent development of a new type of resonant vibromotors, resting on the principle of the conversion of a mechanical vibration (often ultrasonic) of the stator, in a rigid movement of body of the rotor, is of unquestionable interest for many industrial applications.

Among the many currently studied devices, the engines with travelling wave represent certainly the most promising way to guarantee the manufacture of mass of rotary engines "extremely couple-low speed" without auxiliary mechanical reducer speed.

The generation of a progressive wave of volume imposes the geometrical constraints and mechanics relating to the structure's periodicity of the engine. Under normal conditions of operation, the engine is subjected to:

• An axial static loading of pre-stressing the actuating axial and radial deformations stator and rotor,
• A dynamic excitation of the stator, involving deformations of the inflection out-of-plan which, while being propagated in the volume of the stator, create by the drive, a rigid displacement of body of the rotor,
• Efforts due to the contact and friction in the interface between the stator and the rotor.

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