Evolution of the planar vibrations of a rectangular piezoceramic plate as its aspect ratio is changed

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...

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confidence: 99%

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confidence: 99%

Damping the vibrations of a rectangular plate with piezoelectric actuators

Карнаухов

Tkachenko

2008

“…Substituting (19) into the first equation in (16) and performing algebraic transformations, we arrive at a system of linear differential equations with respect to time: …”

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confidence: 99%

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

Карнаухов

Tkachenko

2009

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.

“…Using the assumptions for the electromechanical field variables, considering the structural symmetry and loading conditions, and performing simple transformations, we reduce the forced-vibration problem [12] to a normal-form system of ordinary differential equations for the complex amplitude of w M r , , J…”

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confidence: 99%

“…The plate is made of passive polymer [10] and the pads (actuators) are made of TsTStBS-2 piezoceramics [1]. Temperature approximations of the complex electromechanical characteristics of these materials are given in [12]. The input data: r 0 =0.05 m, R = 0.2 m, h =0.01 m, g 0 =0, and g g 2 = = s 0.638.…”

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confidence: 99%

Flexural vibrations and vibrational heating of a ring plate with thin piezoceramic pads under single-frequency electromechanical loading

Киричок

2008

The paper deals with the coupled problem of flexural vibrations and dissipative heating of a viscoelastic ring plate with piezoceramic actuators under monoharmonic electromechanical loading. The temperature dependence of the complex characteristics of passive and piezoactive materials is taken into account. The coupled nonlinear problem of thermoviscoelasticity is solved by an iterative method. At each iteration, orthogonal discretization is used to integrate the equations of elasticity and an explicit finite-difference scheme is used to solve the heat-conduction equation with a nonlinear heat source. The effect of the dissipative heating temperature, boundary conditions, and the thickness and area of the actuator on the active damping of the forced vibrations of the plate under uniform transverse harmonic pressure is examined Introduction. Piezoactive compound thin-walled elements such as ring plates are widely used in various energy converters, piezoelectric transformers, vibration detectors and generators [3, 11, 12, etc.]. Recently, active-damping methods based on piezoelectric elements have been intensively developed and put into practice [7,15]. These methods use a passive (no piezoelectric effect) plate with piezoelectric pads or inclusions (actuators) to which a potential difference of appropriate amplitude and phase is applied to counterbalance the most intensive modes of mechanical vibrations.The performance of actuators is affected by the electromechanical properties of piezomaterials, dimensions, geometry, arrangement, boundary conditions, and temperature. The temperature of the plate may increase due to heat exchange with the environment and hysteresis losses during intensive vibrations. The effect of dissipative heating and nonuniform electroding of the outside surfaces of bimorph piezoplates undergoing monoharmonic vibrations on the electromechanical processes in them was examined in [13-16, etc.]. The achievements in solving coupled problems of electrothermoviscoelasticity for thin-walled elements with inhomogeneities of various types, including those mentioned above, under single-frequency electric loading are discussed in [3,8].The paper [4] was apparently the first to study the effect of dissipative heating on the efficiency of damping a clamped circular plate with an infinitely thin actuator. In [4], an analytic solution was used to derive a formula for the electric potential applied to the actuator to counterbalance the external mechanical load. The complex modulus of the passive material was assumed to be linearly dependent on the temperature and the losses in the piezoactive material and the dependence of the stiffness characteristics of the plate on its thickness were neglected. The effect of temperature on the damping of forced axisymmetric vibrations of circular plates with actuators of arbitrary thickness is numerically analyzed in [6,9] with allowance for the temperature dependence of the electroelastic moduli.The present paper finds a numerical solution to describe the forced flexural...