This paper considers the problem of natural vibrations of a deformable structure containing elements made of piezomaterials. The piezoelectric elements are connected through electrodes to an external electric circuit, which consists of resistive, inductive and capacitive elements. Based on the solution of this problem, the parameters of external electric circuits are searched for to allow optimal passive control of the structural vibrations. The solution to the problem is complex natural vibration frequencies, the real part of which corresponds to the circular eigenfrequency of vibrations and the imaginary part corresponds to its damping rate (damping ratio). The analysis of behaviour of the imaginary parts of complex eigenfrequencies in the space of external circuit parameters allows one to damp given modes of structure vibrations. The effectiveness of the proposed approach is demonstrated using a cantilever-clamped plate and a shell structure in the form of a semi-cylinder connected to series resonant RL circuits.

An algorithm for numerical realisation of a mathematical statement of the natural vibrations problem for electro-viscoelastic bodies with passive external electric circuits (i.e. shunting circuits) with an arbitrary configuration using the finite element method is proposed in the present paper. The proposed algorithm allows considering the viscoelastic properties of materials using the model of linear hereditary viscoelasticity with complex dynamic moduli and is used to solve 3D solid structure problems that are compatible for ANSYS package element types. This technique implies the usage of the global assembled matrices of stiffness and mass, formed in the ANSYS package. The basis of the algorithm is a novel approach that allows performing decomposition of the global assembled stiffness matrix formed in the ANSYS software package into constituents that are needed for calculation of the natural vibration frequencies of the objects under study. These matrix components are used in the program that was written in FORTRAN (Formula Translation) language. This problem could be efficiently applied for analysis of the dynamic processes in smart systems based on piezoelectric materials and could also form a basis for the development of numerical finite element algorithms for optimization of the dissipative characteristics of electromechanical systems with shunted piezoelectric elements.

The dissipative properties of electromechanical systems based on structure with elements made of piezomaterial can be controlled by attaching external electric circuits to the piezoelements. One can change electric circuit parameters in such a way as to ensure the greatest possible energy dissipation on an external electric circuit and, thereby, the best damping of the system’s specified oscillation frequency. Since the external electric circuits are a collection of elements with lumped parameters attached to a system with distributed parameters, the solution for such a system of electro-viscoelasticity problems in the complete formulation by the finite element method leads to a large solving system of algebraic equations. The solution of this system requires considerable time and computational resources. There are known approaches in mechanics that make it possible to represent mechanical systems with distributed parameters in the form of discrete systems with lumped parameters, such as a spring–mass–damper. In this article, it is proposed to model electromechanical systems with external electric circuits based on their electrical analogue in the form of equivalent electric substitution circuits, which are discrete electrical systems with lumped parameters. These discrete systems are analogues of the initial electromechanical systems in terms of frequency characteristics and the electrical processes that take place in them. The equivalent substitution circuit is based on the Van Dyke model and allows one to obtain the required number of complex eigenfrequencies of the electromechanical system under consideration.

Algorithm for the layout of a piezoelectric element in an elastic medium providing the maximal piezoelectric effect within a specified frequency range,

The article is devoted to the study of the mechanical response of an electro-viscoelastic structure if an electric voltage with given characteristics supplied to a piezoelectric element attached to the surface of the structure. A cantilever plate made of a viscoelastic material is considered as a structure under study. The amplitude of displacements of the free end of the plate in resonance modes occurring at forced steady-state vibrations is considered as the mechanical response. Excitation of vibrations is realized by a harmonic movement applied to the clamped end of the plate. The control of the mechanical response within the framework of this work is supposed to be performed by supplying an electric signal to the piezoelectric element. This signal changes harmonically and has the form of a potential difference which changes according to a harmonic law. The magnitude of the vibration amplitude of the structure is determined numerically by the finite element method implemented in the ANSYS software package. The viscoelastic properties of the plate material are described according to the relations of hereditary theory of viscoelasticity in terms of complex dynamic moduli. Within the framework of this work, the relations between the mechanical response of the structure and the magnitude and polarity of the electric voltage applied to the piezoelectric element were obtained. It is shown that by a proper selection of the characteristics of the electric voltage, it is possible to control the magnitude of the amplitude of forced steady-state vibrations.

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