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By Frantisek Trebuna

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WE = 2 σ max . (13) 2⋅ E A calculation of the total strain energy density per half cycle can be realised using a well-known Feltner’s term 36 Applied Mechanics and Mechatronics σ2 2 + max 1 + n' 2 ⋅ E and for the hysteretic energy density per cycle the Morrow’s relationship is often applied, WF = σ max ⋅ ε max p ⋅ (14) 1 − n' . (15) 1 + n' It’s important to note that these relationships are approximate, very simple, what is their facility. Our proposed numerical approach is more complicated but much more accurate.

In [7] the authors defined W (t) taking into account the signs of stresses and strains in order to distinguish the energy under the tension (+) and under the compression (-) as follows 1 sign (σ ) + sign (ε ) . (18) ⋅ σ (t ) ⋅ ε (t ) ⋅ 2 2 For distinguishing between the positive and negative parameter of an strain energy density, the function sign has been introduced into the elastic energy density expression [7]. Another computational approach considering the “positive” and “negative” character of the strain energy density may be defined (from author’s experience) as follows W1 (t ) = 1 (19) ⋅ σ (t ) ⋅ ε (t ) ⋅ sign (ε ) .

611 23 mechanical system. The problem of the optimal placement of actuators and sensors for flexible structures is analysed in works [16]. 2. Modelling of thin-walled structures with piezoelectric active elements The models used for optimal placement of the piezoelectric elements are derived by the application of FE approach to modelling of piezoelectric smart structures and coupled electromechanical behaviour. We can rewrite the second-order structural model obtained from the finite element procedure in the form  + M −1 Dq + M −1 Kq = M −1 B0 u q (1) y = C 0 q q + C 0 v q (2)  are vectors of where M , D, K are mass, damping and stiffness matrices, B0 is input matrix, q, q , q displacements, velocities and accelerations and u is a vector of control parameters.

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