Damping the vibrations of a rectangular plate with piezoelectric actuators

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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.

“…, which were obtained using the Kirchhoff-Love hypotheses. Therefore, the results are in good agreement with well-known theories and earlier results [17]. …”

Section: Calculated Results and Theirsupporting

confidence: 92%

“…An analysis of these results shows that the amplitudes of the potential difference Va and deflection W increase as the shear moduli decrease. As the shear moduli increase, V a and W decrease asymptotically tending to V a = 21.68 and W = × -5 305 10 5 . , which were obtained using the Kirchhoff-Love hypotheses.…”

Section: Calculated Results and Theirmentioning

confidence: 95%

Influence of shear strains on damping the vibrations of a rectangular plate with piezoelectric actuators

Карнаухов

Tkachenko

2009

Self Cite

“…The effect of the applied voltage on the electroelastic state of the plate is examined Introduction. The principles of the theory of electroelasticity for piezoelectric bodies [5,7,8] and various methods for the analysis of their electroelastic state are outlined in [2,9,10,13,18,22,23]. Of special interest is the electroelastic state of piezoelectric bodies with holes and cracks.…”

mentioning

confidence: 99%

Electroelastic state of an infinite multiply connected piezoelectric plate with known electric potentials applied to its boundaries

Khoroshev

2010

Generalized complex potentials, their expressions for a domain with elliptic holes, and the discrete least-squares method are used to analyze the generalized plane electroelastic state of a piezoelectric plate having holes and cracks with electric potentials applied to their boundaries. There are no mechanical loads. A numerical analysis is conducted. The effect of the applied voltage on the electroelastic state of the plate is examined Introduction. The principles of the theory of electroelasticity for piezoelectric bodies [5,7,8] and various methods for the analysis of their electroelastic state are outlined in [2,9,10,13,18,22,23]. Of special interest is the electroelastic state of piezoelectric bodies with holes and cracks. There are numerous approaches to problems for simply connected domains with a hole [3,6,16,20] or a crack [3,10,16,19]. Methods of solving two-dimensional problems for multiply connected domains based on complex variable theory [1,3,11] are best developed. In [3], generalized complex potentials for two-dimensional static problems of electroelasticity were studied, their expressions for bodies with holes and cracks were derived, and a wide class of problems of practical importance for plates with various geometric characteristics and unelectroded boundaries was solved using the least-squares method to satisfy the boundary conditions. Green's function and boundary integral equations were used in [11] to solve two-dimensional static and dynamic problems of electroelasticity for multiply connected piezoelectric bimorphs, infinite and semi-infinite bodies with unelectroded surfaces and some dynamic problems for bodies with partially electroded surfaces. The methods proposed in [3] were used in [1] to study the generalized plane electroelastic state of a finite piezoelectric plate having electroded holes and cracks with electric potentials applied to their boundaries.In the present paper, we extend this approach to an infinite piezoelectric plate with holes and cracks. We will present numerical results and analyze the solutions of problems for plates with two identical circular holes or cracks.

“…The cited studies dealt only with one mode of elastic vibrations. Some relevant issues were analyzed in [12][13][14][15].The papers [4, 27] outline a method of designing an optimal slewing program that minimizes the relative accelerations of the flexible appendages of a spacecraft. The method allows for the infinite number of modes of elastic vibrations, not going beyond the finite-dimensional mathematical model.…”

mentioning

confidence: 99%

“…The cited studies dealt only with one mode of elastic vibrations. Some relevant issues were analyzed in [12][13][14][15].…”

mentioning

confidence: 99%

Optimal limited-rate slewing of a flexible spacecraft

Zakrzhevskii

2010

The paper presents an efficient method of finding the optimal program control for the reorientation of a spacecraft with flexible appendages at a limited slewing rate. The appendages are incorporated into the mathematical model based on the quasistatic approximation. The problem is solved analytically by parametrizing the functional of a multipoint boundary-value problem. The optimal solution is illustrated graphically for different parts of the attainability domains Introduction. Optimal rest-to-rest maneuvers of spacecraft with flexible appendages have long attracted the attention of researchers and are still of interest because new spacecraft designs, new effectors, new control systems are developed.A common practice is to combine feedforward and feedback controls for such systems. The deformation of flexible appendages can only be measured with distributed sensors. If a spacecraft is not equipped with such sensors, there can be no acceptable quality of feedback. Therefore, whether maneuvers of such spacecraft are successful is strongly dependent on the quality of feedforward control. In this field, a number of fundamental results have been obtained. They allow optimization based on various goal functions. For long-lived spacecraft with a low power-to-weight ratio, it is important to minimize the consumption of energy [17,20,25,26]. For observation satellites that maneuver much, it is important to minimize the maneuver time [5,6,8,9,16,18,19,21,23,24]. The most typical result in this field is bang-bang control, which may appear extremely sensible to simulation errors. Sometimes these goal functions are combined into one with weighted components [11].In some cases, it is important to minimize not the fuel consumption and maneuver time, but rather the moment perturbations of the spacecraft because of the vibrations of flexible appendages. The slewing of such spacecraft should be controlled in such a manner that no vibrations of the flexible appendages occur and affect the operation of the spacecraft equipment. First attempts to solve this problem using goal functions with various combinations of generalized coordinates of elastic displacements and their derivatives were made in the 1970s [2, 3, 10]. Later, it was proposed that feedforward control did not include spectral components of the spacecraft as a whole [7]. Modifications of this approach were applied to elastic nonlinear systems [22], but the residual vibrations were nevertheless significant [11]. The cited studies dealt only with one mode of elastic vibrations. Some relevant issues were analyzed in [12][13][14][15].The papers [4, 27] outline a method of designing an optimal slewing program that minimizes the relative accelerations of the flexible appendages of a spacecraft. The method allows for the infinite number of modes of elastic vibrations, not going beyond the finite-dimensional mathematical model. This is possible if the first several modes are described in the usual formulation and all the higher modes in the quasistatic formulation. Of separat...