Issue 49

D. Oshmarin et alii, Frattura ed Integrità Strutturale, 49 (2019) 800-813; DOI: 10.3221/IGF-ESIS.49.13 801 thereby generating the mechanism of passive control of vibration. Because peizoelectric elements and components of the external circuits are small in weight, such a system does not lead to serious changes in the spectral pattern of the initial structure and can be used as a complementary device to developed technical solutions. The possibility of applying such devices to damping the vibrations of a mechanical structure was first demonstrated by Forward [1] and theoretically substantiated by Hagood and von Flotow [2]. A key to the problem of vibration damping with shunted piezoelectric elements is to find the parameters of a simplest shunt that is most effective in damping the vibrations of a particular structure at a group of frequencies in the specified frequency range. The shunt circuit should meet the following requirements: it should provide maximum damping of mechanical vibrations and preserve its stability; it should operate without a power source; its design costs and weight should be reduced to a minimum; and, since the circuit is integrated in the structure, the size of its elements should be minimal [3]. Over the last decade, a variety of passive shunt circuits have been designed and rated. According to the classification presented in Moheimani and Fleming [3], passive shunts can be linear or nonlinear. Among the linear passive circuits, one can distinguish between the resistive (including only resistors) and resonant (including resistors and inductors) circuits. The resistive circuits used to shunt piezoelectric elements cause the structure to behave as if it is made of viscoelastic material. The use of resonant RL circuits consisting of a parallel- or series-connected inductance coil and resistor leads to the creation of the electric oscillatory circuit due to the interaction of the external circuit inductance and inherent capacitance of the piezoelectric element. As a result, the frequency spectrum of the system (the structure with a piezoelectric element and external circuit) extends to include an additional resonance frequency (eigenfrequency of the oscillatory circuit). By varying the parameters of the elements of the electric circuit, one can adjust this additional resonance to the natural vibration frequency of the structure. This allows the energy of vibrations to be transferred to the external circuit, which provides an effective suppression of the structure vibrations at a given frequency. In practice, the requirements commonly imposed on structures are the absence of vibrations at the frequencies of a certain group in some excitation frequency range. Therefore, the development of techniques for multimodal damping of structure vibrations is an urgent problem. At present, one can distinguish the following basic approaches to multimodal damping: – the employment of one piezoelectric element and a complex external circuit in which the possibility of damping several modes of vibrations is realized in one way or another [4-15]; – the usage of several piezoelectric elements forming an integrated network and connected by one or another method to a single external circuit [16-22]; – the usage of several piezoelectric elements, each having its own electrical circuit that is not coupled with others to form a joint circuit [23-27]. Each of these approaches has its own advantages and drawbacks. Thus, for example, in the presence of complicated branched circuits the complexity of their tuning increases, because each branch of the shunt circuit cannot be considered as a separate independent shunt [13]. In this case, it is advisable to use blocking or conducting circuits connected in series to shunt branches [9, 12], which allows one to consider them as independent. However, in practice this leads to very cumbersome electrical circuits due to the use of many inductive elements, which certainly affects the mass characteristics of damping devices. This problem can be partially solved by using electronic analogues of inductive elements. However, such a substitution is only an approximation to ideal inductive elements and does not completely eliminate the necessity of tuning the branches and determining the circuit parameters [7]. The difficulty with the circuit tuning can be readily solved with the use of several piezoelectric elements connected to simple RL circuits, since in this case each piezoelectric element with a connected circuit represents an isolated system and parameters of the shunt circuits can be selected independently. However, as shown in [4], in the case of multimodal damping, this approach is not necessarily successful because a great number of elements in the circuit can essentially increase the mass of the structure and have an ill effect on its dynamic characteristics. The variety of approaches and techniques based on the application of piezoelectric elements connected to external electric circuits for the problem of multimodal vibration damping was demonstrated in the paper [28]. But at the same time the authors proposed new technique for solving the stated problem. The essence of their technique is in consideration of an impedance of a shunting circuit as a controller in the state space of the model of electromechanical system. In order to achieve multimodal vibration damping it is required to find the electric circuit that has the desirable impedance. Hence, nowadays, the problem accosiated with the use of piezoelectric elements to damp several vibration modes still requires further investigation and development of new approaches to the implementation of multimodal damping. This paper presents one possible application of the natural vibration problem for electroelastic structures that include piezoelectric elements connected to external circuits. The results of the solution to the problem are complex values of