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By F. Ebrahimi

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Płaczek M. (2010a). Selection of Parameters of External Electric Circuit for Control of Dynamic Flexibility of a Mechatronic System. Solid State Phenomena, Vol. 164, (2010), pp. , & Płaczek M. (2010b). Development of Mathematical Model of a Mechatronic System. Solid State Phenomena, Vol. 164, (2010), pp. , & Płaczek M. (2010c). The analysis of vibrating systems based on the exact end approximate method. International Journal of Modern Manufacturing Technologies, Vol. II, No. 1, (2010), pp. , & Płaczek M.

Gao J. , & Liao W. H. (2005). Vibration analysis of simply supported beams with enhanced self-sensing active constrained layer damping treatments. Journal of Sound and Vibration, Vol. 280, (2005), pp. 329-357 Hagood N. , & von Flotow A. (1991). Damping of structural vibrations with piezoelectric materials and passive electric networks. Journal of Sound and Vibration, Vol. 146, No. 2, (1991), pp. , & Tylikowski A. (1995). Torsional Vibrations Actively Attenuated by Piezoelectric System, Proceedings of the 4th German-Polish Workshop on Dynamical Problems in Mechanical Systems, Berlin, July 1995 Kurnik W.

3. The solution of the system’s characteristic equation approach to limit described by equation: 32 Advances in Piezoelectric Transducers kn  2n  1 , 2l n  1, 2, 3... (10) This solution is precise for n > 3. For the lower values of n solutions should be readout from the graphic solution (Fig. 3) and they are presented in table 4. Fig. 3. The graphic solution of the characteristic equation of the system (equation 7) Taking into account the system’s boundary and initial conditions, after transformations the sequence of eigenfunctions is described by the equation: Xn  x   An cos knl  cosh knl  cos knx  cosh knx   sin knx  sinh kn x.

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