PSI - Issue 28

D.A. Oshmarin et al. / Procedia Structural Integrity 28 (2020) 1438–1448 Author name / Structural Integrity Procedia 00 (2019) 000–000

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practice.Modern objects often require to have vibration isolation but the changes to be introduced into them are demanded to be minimal. This requirement is of particular importance in the context of the proliferation of miniature technical products, or in conditions of their inaccessibility for manipulation (for example, in outer space). One of quite widespread options of such structural modifications is to coat the surface of the structure in whole or in part using various viscoelastic materials (Nashif et al., 1985). In this case, it is possible to significantly reduce the amplitude of vibrations in steady-state modes or increase the rate of damping of vibrations in transient modes. However, this approach has its own limitations associated with possible area of coating of a structure surface, allowable thickness of a viscoelastic layer and properties of a material used. Within the context of development of smart-technologies, another approach related to supplementing of an original structure with elements made of piezoelectric elements connected to external electric circuits attracts more and more attention of researchers (Moheimani and Fleming, 2006). The presence of such elements provides additional dissipation of energy thereby increasing the damping properties of structures. Nowadays systems of control of structural vibration damping based on application of elements made of viscoelastic and piezoelectric materials are still applied separately. However, an increasing number of researchers and designers are convinced that they complement each other, providing the same means for controlling structural vibrations in a wide frequency range (Trindade, 2007). Herewith these damping mechanisms can operate as separately as well as simultaneously depending on the mutual arrangement of viscoelastic layers and piezoelectric elements. This method is a combination of the advantages of the capabilities of viscoelastic materials (stability, fail-safety, reliability, simplicity) for dissipating vibrational energy at high frequencies with the advantages of the capabilities of techniques based on the use of elements made of piezoelectric materials (high performance, adaptability, manufacturability) at low frequencies (Trindade and Benjeddou, 2002). In last few decades an approach which is based on capabilities of both viscoelastic materials and piezoelectric materials with external electrical circuits is extensively developed. A high interest to this approach and the modern state-of-the-art was demonstrated in a number of review papers (Stanway, 2003; Benjeddou, 2001). A detailed comparison of application of the approaches under consideration was presented in the following papers (Ghoneim, 1993; Ghoneim, 1996; Ghoneim and Karkoub, 2001). Despite the abundance of articles devoted to this issue, due to its versatility and complexity, there are still fragments that require a solution, for example, a study of the possibilities provided by each of the approaches to ensure reliable and effective damping of given vibration modes. Within the framework of this work, a comparative analysis of the effectiveness of the use of viscoelastic materials and a piezoelectric element is presented, to the electrodes of which a resistive or resonant electric circuit can be connected to damp the first eight vibration modes of the plate. 2. Mathematical statement of the problem The object under study is a piece-wise homogeneous solid body of the volume 1 2 3 V V V V    , where 1 V is the volume of its elastic part, 2 V is the volume of viscoelastic part and 3 V is the volume of piezoelectric elements. In the examined body, the electrode-covered part of the surface 3 el S of volume 3 V can be connected in the general case to the electric circuit of arbitrary configuration, consisting of the resistive, capacitive and inductive elements. The variational equation of the motion for the object under consideration is formulated based on the relations of the linear theory of elasticity and Maxwell's equations for a medium in the quasi-static approximation and it takes the following form (Matveenko et al 2016):       1 2 3 1 2 3 1 1 0 i i ij ij i ij ij i ij ij i i i i V V V L R C u u dV u u dV D E u u dV dtdt dt C L R                                          (1)

Here , i i D E are the components of the vectors of electric flux density and electric field intensity, respectively; ij 