Keywords: local nonlinearity, piezoelectric control, imposed curvature, oscillations, layered beams.

Important elements of any structural control system are the transducers used for implementation of the control. Sensors and actuators provide the link between the controller and the mechanical system to be controlled and their design and implementation is of prime importance. In general, control transducers come in three main forms [1]: point transducers, arrays of point transducers or continuously distributed transducers. In this paper, the interest is focused on the control of structural vibrations of slender beam structures by means of integrated distributed actuators realized by embedding layers of piezoelectric material. Activation renders piezoelectrically induced strains that are imposed to generate the distributed input of the control system. Special emphasis is given to the identification of the piezoelectric actuation as a source of selfstress [2].

To analyse linear oscillations of the structures under consideration, classical modal analysis can be applied only if the corresponding damping matrix is proportional to both mass and stiffness matrices. Otherwise, for example in case of structures with a single external damping device, an alternative or approximate solution procedure for determining the dynamic response has to be chosen. Vibration problems of linear structures with spatially localized nonlinearities are related to those non-classically damped systems. Such systems are characterized by the fact that their nonlinear behaviour is largely restricted to a limited number of single points in the structure.

Based on an idea of Miller and Iwan [3] the solution of the underlying boundary value problem is found by means of a problem-oriented decomposition in the frequency domain, which has been adopted from methods for structures under multiple support excitations. The (nonlinear) interaction force is expressed as the sum of two separate forces: the first develops due to the imposed piezoelectric curvature at the device's location, and the second contribution arises due to an imposed time-harmonic support excitation with no other external forces acting on the structure.

The main advantageous of this formulation is the fact, that by increasing the degrees of freedom considered for the structure, the numerical effort in solving the coupled system is extended only marginally since the nonlinearity stays "isolated".

Example problems are given for layered beam structures with time-variant imposed piezoelectric curvature, where single supports are connected to nonlinear devices.

1

C.R. Fuller, S.J. Elliott, P.A. Nelson, "Active Control of Vibration", Academic Press, London, United Kingdom, 1996.

2

R. Heuer, H. Irschik, F. Ziegler, "Thermo-piezoelectric vibrations of three-layer elastic plates", in R.B. Hetnarski, N. Noda, (Editors), "Proceedings of the Second International Symposium on Thermal Stresses and Related Topics", 451-454, RIT, Rochester, USA, 1997.

3

R.K. Miller, W.D. Iwan, "The peak harmonic resonance of locally non-linear systems", Earthquake Engineering and Structural Dynamics, 6, 79-87, 1978. doi:10.1002/eqe.4290060109

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