The paper addresses the active damping of nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators. The problem is solved by two methods: (i) the classical method of balancing the fundamental vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamic-programming method that reduces the problem to an algebraic Riccati equation. The results produced by both approaches are presented and compared Introduction. Thin plates are widely used in many fields of modern engineering including space technology, aircraft engineering, automotive industry, shipbuilding, mechanical engineering, radio electronics, etc. Such plates are often subjected to nonstationary and harmonic mechanical loads. Especially dangerous are resonant vibrations occurring when the frequency of the harmonic force becomes equal to the natural frequency of the plate. Intensive mechanical nonstationary vibrations are no less dangerous. This brings about the task of damping stationary and nonstationary vibrations of thin plates. For this purpose, passive damping (elements with high hysteresis losses embedded into the plate) is widely used. Numerous Ukrainian and foreign publications on passive damping of vibrations of thin-walled elements are reviewed in [7][8][9]15].Recently, active damping with distributed piezoelectric inclusions (so-called sensors and actuators) has been used. The essence of active-damping methods is in the following. The use of eigenfunction expansion reduces many vibration problems for thin-walled elements to a system of ordinary differential equations and sometimes to even one ordinary differential equation (this is so, for example, for rectangular plates and cylindrical panels with hinged ends), i.e., to a one-degree-of-freedom system. The solution describing the forced vibrations of a one-degree-of-freedom system with viscous friction under a harmonic force consists of two terms.The first term describes accompanying natural vibrations that exponentially decay and depend on the initial conditions. The damping rate depends on the damping factor: the greater this coefficient, the quicker the vibrations decay.The second term describes purely forced vibrations with the frequency of the exciting force. The natural vibrations eventually decay and only the purely forced vibrations remain. The intensity of the latter depends on the amplitude and frequency of the exciting force and the damping factor. The higher the amplitude of the force, the closer its frequency to the natural frequency, the less the damping factor, the higher the amplitude of forced vibrations. There are two damping methods. One employs only actuators to which a potential difference is applied to balance the mechanical load. This decreases the amplitude of the exciting force and, hence, the amplitude of the forced vibrations. The primary task here is to calculate the necessary potential difference to be applied to the actuator(s). Thus, the former method substantially reduces the intensity o...

The paper examines the harmonic vibrations of an infinitely long thin cylindrical shell made of a nonlinear elastic piezoceramic material and subjected to periodic electric loading. Amplitude-frequency characteristics are plotted for different amplitudes of the load. Points of these characteristics are analyzed for stability. The transients occurring while harmonic vibrations attain the steady state are studied Keywords: piezoceramics, cylindrical shell, harmonic vibrations, amplitude-frequency characteristics, nonlinear elastic piezoceramic materialIntroduction. Effects typical of nonlinear systems, such as the amplitude dependence of resonant frequencies, the hysteretic frequency dependence of the amplitude, etc., are observed when piezoelectric bodies are excited at resonant frequencies even in weak electric fields. Nonlinearities of two types are observed in the vibrations of thin-walled elements.One type is due to the heating generated during the dissipation of electromechanical energy. This nonlinearity results from the temperature dependence of the material properties and the nonlinear dependence of the dissipation function on strains and temperature. It is adequately detailed in [6-10]. The vibrations of piezoelectric and viscoelastic plates are studied in [3][4][5].The other type is associated with the dependence of the material characteristics on the amplitudes of independent field quantities. This nonlinearity has been inadequately studied. The papers [11][12][13] deal with some simple problems for piezoceramic bodies made of a material with second-type nonlinearity.Since thin piezoelectric cylindrical shells are widely used in various engineering applications, this paper employs the harmonic balance method to examine harmonic vibrations in such shells made of materials with second-type nonlinearity. The amplitude-frequency characteristics of the shell will be plotted for different levels of electric loading. It will be shown that these characteristics may differ both quantitatively and qualitatively for different amplitudes of the load. Points of the amplitude-frequency characteristics will be analyzed for stability, depending on load amplitudes.1. Problem Statement. Consider a radially polarized, infinitely long, piezoceramic, cylindrical shell with thickness h and mid-radius R. The shell is coated with infinitely thin electrodes to which an electric potentialV t 0 ( )is applied. The momentless equation of motion of the shell is as follows:

The problem of active damping of the nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators is solved using Timoshenko's hypotheses. To this end, two methods are employed: (i) a classical method of balancing the principal vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamical-programming method that reduces the problem to the matrix algebraic Riccati equation. The results obtained by both approaches are compared. The effect of the shear modulus on the amplitude of the damping potential difference and the deflection of the plate is analyzed Keywords: piezoelectric actuator, damping, hinged plate, method of dynamical programming, matrix algebraic Riccati equationIntroduction. Anisotropic plates are widely used in various fields of modern engineering. To assure the operability of plate-like structural members under nonstationary and harmonic (including resonant) dynamic loads, it is necessary to damp the vibrations induced by these loads. To this end, both passive and active damping methods are used [7-9, 12-14, 22-27]. The vibrations of piezoelectric and viscoelastic plates are studied in [10, 13-16, -21]. Recently, piezoelectric inclusions, so-called sensors and actuators, have come to be used for the active damping of the stationary and nonstationary vibrations of thin-walled members. In the case of actuators, the chief task is to establish how the potential difference applied to the actuator should vary to balance the mechanical load. This potential difference is determined using various models of thin-walled members and design methods. If the plate is strongly anisotropic, refined models of thin-walled members, including refined Timoshenko-type hypotheses should be employed. Also of interest is to compare the potential differences calculated by the classical approach (where the most intensive vibration modes are balanced by actuators) and by optimal-control methods.The present paper addresses the problem of active damping of the nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators based on the Timoshenko hypotheses [6]. To solve the problem, we will use (i) the classical approach of balancing the principal vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamic-programming method that reduces the problem to the matrix algebraic Riccati equation. We will compare the results obtained by both approaches and analyze the effect of the shear modulus on the amplitude of the damping potential difference and the deflection of the plate.

The active damping of the resonant vibrations of a flexible cylindrical panel with rectangular planform and clamped edges is considered. The damping is done with distributed piezoelectric sensors and actuators. It is shown that the amplitude of the resonant vibrations can be substantially decreased by choosing the appropriate feedback factors. The combined effect of geometrical nonlinearity and dissipative properties of the material on the effectiveness of damping is analyzed Keywords: piezoelectric sensor, piezoelectric actuator, active damping, flexible cylindrical panel with hinged edgeIntroduction. Cylindrical panels made of passive (no piezoelectric effect) elastic and viscoelastic materials are widely used in various fields of modern engineering [5,9]. Harmonic loads with nearly resonance frequency induce intensive vibrations, which may cause failure of the panel due to fatigue or other factors. To prevent these consequences, such vibrations should be damped. Wide use is made of the passive method to damp stationary and nonstationary vibrations of a structure with elements with high hysteresis losses [6,7,10].Recent trends are toward using active methods to damp the vibrations of thin-walled structures with piezoactive inclusions [11][12][13][14][15][16][17][18][19]. There are two basic active-damping methods. One of them employs only one type of piezoelectric inclusions, so-called actuators to which a voltage is applied to balance the resonant component of the mechanical load. This voltage is calculated with classical methods (use of actuators to balance some most energy-intensive vibration modes) or optimal-control methods [12,13].The second method applies to the actuator a voltage proportional to the first derivative of the sensor voltage with respect to time. Then there is not only energy dissipation in the viscoelastic material (passive damping), but also additional damping determined by the feedback factor. Resonant vibrations, high levels of mechanical loading, and small thickness of the element may cause amplitudes comparable to the panel thickness, which requires taking geometrical nonlinearity into account.It should be noted that the modern literature on active damping pays little attention to the collective effect of geometrical nonlinearity and viscoelasticity on the effectiveness of active damping of panels. The latter is commonly described by using elementary differential models or by adding a term proportional to the derivative of the positional variable with respect to time to the equation of motion.The present paper addresses the forced harmonic vibrations of flexible cylindrical panels with rectangular planform and hinged edges. Geometrical nonlinearity is described using an elementary model that allows for squared angles of rotation. To describe the viscoelastic behavior of the material, use is made of an integral viscoelastic model [3]. To solve the problem, we will use the Bubnov-Galerkin method, which allows reducing the original nonlinear problem to a system of ordinary nonlinear inte...