Keywords: free vibration analysis, piezoelectric adaptive structures, electro-mechanical coupling coefficient, passive shunted damping, active damage detection.

For classical structures, modal properties extraction is a critical step in the performance evaluation of piezoelectric adaptive structures applications, either in open-loop (shunted damping, damage identification) or closed-loop (active vibration control, active noise control). Boundary conditions (BC) play a more crucial role here since modal properties depend not only on the mechanical boundary conditions but also on the electrical boundary conditions. Hence, if piezoelectric devices are electroded, they can either be short circuited (SC), that is, is zero electric potential, or grounded or open-circuited (OC), that is, zero electric displacement or charge free. The distinction between these two electric boundary conditions (BC) has been well known in the electro-acoustics community since the early seventies [1] but not so well known in the mechanical community working on piezoelectric adaptive structures [2].

Thus, depending on the electromechanical coupling effect representation, many electric degrees of freedom (DOF), free finite elements (FE), can only provide either SC or OC natural frequencies. The former case results from consideration of the piezoelectric effect via equivalent electric loads only while the latter is the result of either analytical or numerical static condensation of the electric potential. This shortcoming is also true for several analytical three-dimensional solutions. In fact, only a few studies have provided both SC and OC natural frequencies [3].

Although non-electroded electric BC is physically different from previous cases, it is numerically handled the same as the OC case, except that the absence of an electrode imposes the continuity for both electric potential and transverse displacement at the interface. Nevertheless, the presence of an electrode requires that the nodal electric potentials are identical (for free-vibration) and equal to a constant and that the sum of the nodal charges is constant (nil for free-vibration) for the electroded area (equipotential surface) nodes [1].

It is worth recalling that SC and OC modal frequencies can be extracted from the resonances of voltage-driven (admittance) and charge-driven (impedance) frequency response functions (FRF), respectively [1]. The latter problem can be derived using static condensation of the electric potentials from electromechanically coupled equations of motion. These are often named actuator and sensor equations but must not be handled separately as is sometimes shown in the literature. If this is done then they are no longer coupled. It is also worth mentioning that SC and OC frequencies can be measured using an impedance analyser. These correspond to the resonance and anti-resonance frequencies of the electric admittance, or anti-resonance and resonance of the electric impedance of the piezoelectric device.

Early piezoelectric FE was based on the electric potential as an independent variable [4]. They are well suited for the extraction of the SC modal properties while OC modal properties could be obtained after static condensation of the voltage DOF. Traditionally, no electric BC is imposed for OC eigenvalue FE analysis. Although necessary, the fulfilment of the above charge condition due to the presence of the electrodes is rarely mentioned in the open-literature results. This is probably due to the difficulty in handling a charge-based condition using voltage-based FE. This may not be the case using charge-based FE, such as mixed or hybrid FE since the electric displacements or charge are retained as electric DOF. However, the fulfilment of the voltage-based condition, due to the presence of an electrode, is now not mentioned in the literature although it is necessary. Here also, the static condensation of the electric charge/displacement DOF is generally considered. So, care should be taken in the interpretation of the resulting modal properties.

The main objective this paper is to clarify the evaluation procedure of the modal properties of piezoelectric adaptive structures and their interpretation and use for smart structures applications. It is hoped that this will avoid some confusing comparisons during benchmarking of new formulations. For this, an electromechanical coupling performance criterion is proposed. It consists of the so-called electromechanical coupling coefficient (EMCC) which uses both SC and OC frequencies and measures the conversion rate of the mechanical energy to the electrical energy and vice versa. It is then applied to passive shunted damping and active damage detection for illustration purpose.

1

H. Allik, K.M. Webman, "Vibrational response of sonar transducers using piezoelectric finite elements", The Journal of the Acoustical Society of America, 56, 1782-1791, 1974. doi:10.1121/1.1903513

2

A. Benjeddou, "Modelling and simulation of adaptive structures and composites: current trends and future directions", in "Progress in Computational Structures Technology", B.H.V Topping and C.A. Mota Soares (Editors), Saxe-Coburg Publications, Stirling, Scotland, Chapter 10, 251-280, 2004.

3

J.-F. Deü, A. Benjeddou, "Free-vibration analysis of laminated plates with embedded shear-mode piezoceramic layers", International Journal of Solids and Structures, 42, 2059-2088, 2005. doi:10.1016/j.ijsolstr.2004.09.003

4

A. Benjeddou, "Advances in piezoelectric finite element modelling of adaptive structural elements: a survey", Computers and Structures, 76, 347-363, 2000. doi:10.1016/S0045-7949(99)00151-0

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