Damping the vibrations of a rectangular plate with piezoelectric actuators

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 __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 78–84, February 2008.     

Damping the resonant flexural vibration of a hinged plate with actuators

The paper addresses the active damping of the resonant flexural vibration of a hinged viscoelastic rectangular plate with distributed piezoelectric actuators. The thermomechanical behavior of passive and active materials is described using the concept of complex characteristics. The interaction of the mechanical and temperature fields is taken into account. To solve the problem, the variational and Bubnov–Galerkin methods are used. The effect of the temperature of dissipative heating on the effectiveness of the active damping of resonant vibration is studied     

Resonant vibrations of a clamped viscoelastic rectangular plate

The paper examines the effect of dissipative heating on the performance of a sensor in a viscoelastic rectangular plate undergoing resonant vibrations. The thermoviscoelastic behavior of materials is described using the concept of complex characteristics. The coupling of the electromechanical and thermal fields is taken into account. The nonlinear problem is solved by the Bubnov–Galerkin method. The effect of the mechanical boundary conditions and dissipative-heating temperature on the performance of the sensors is analyzed     

Resonant vibrations of a clamped thermoviscoelastic rectangular plate with sensors and actuators

The paper discusses the active damping of the resonant flexural vibrations of a clamped thermoviscoelastic rectangular plate with distributed piezoelectric sensors and actuators. The thermoviscoelastic behavior of the passive and active materials is described using the concept of complex characteristics. The interaction of the mechanical and thermal fields is taken into account. The Bubnov–Galerkin method is used. The effect of self-heating, the dimensions of the piezoelectric inclusions, and the feedback factor on the effectiveness of active damping of the resonance vibrations of the plate is studied     

Dynamic modelling and active vibration control of a submerged rectangular plate equipped with piezoelectric sensors and actuators

The active vibration control of a rectangular plate either partially or fully submerged in a fluid was investigated. Piezoelectric sensors and actuators were bonded to the plate, and the assumed mode method was used to derive a dynamic model for the submerged plate. The properties of the piezoelectric actuators and sensors, as well as their coupling to the structure, were used to derive the corresponding equations of their behaviour. The fluid effect was modelled according to the added virtual mass obtained by solving the Laplace equation. The natural vibration characteristics of the plate both in air and in water were obtained theoretically and were found to be consistent with the experimental results, and the changes in the natural frequencies resulting from submersion in fluid can be accurately predicted. A multi-input, multi-output positive position feedback controller was designed by taking the natural vibration characteristics into account and was then implemented by using a digital controller. The experimental results show that piezoelectric sensors and actuators along with the control algorithm can effectively suppress the vibration of a rectangular plate both in air and submerged in a fluid.     

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

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     

四边固支矩形薄板自由振动的哈密顿解析解

在哈密顿体系中利用辛几何方法求解了四边固支矩形薄板自由振动问题的解析解。首先,从基本方程出发,将问题表示成Hamilton正则方程,然后利用辛几何方法导出本征值问题,从而得到本征函数解,使之满足边界条件;再由方程组有非零解的条件,最终推导出四边固支矩形薄板的自振频率方程,得到频率的解析解。计算了不同长宽比情况下四边固支矩形薄板的频率,结果与已有文献完全一致。该解法有望推广至更多尚未得到解析解的矩形板的振动问题。     

Stability of viscoelastic rectangular plate with a piezoelectric layer subjected to follower force

The effects of a piezoelectric layer on the stability of viscoelastic plates subjected to the follower forces are evaluated. The differential equation of motion of the viscoelastic plate with the piezoelectric layer is formulated using the two-dimensional viscoelastic differential constitutive relation and the thin plate theory. The weak integral form of the differential equations and the force boundary conditions are obtained. Using the element-free Galerkin method, the governing equation of the viscoelastic rectangular plate with elastic dilatation and Kelvin–Voigt distortion is derived, subjected to the follower forces coupled with the piezoelectric effect. A generalized complex eigenvalue problem is solved, and the force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force; this force is used to improve the stability of the non-conservative viscoelastic plate. For the viscoelastic plate with various boundary conditions, the results for the instability type and the critical loads are presented to show the variations in these factors with respect to the location of the piezoelectric layers and the applied voltages. The stability of the viscoelastic plates can be effectively improved by the determination of the optimal location for the piezoelectric layers and the most favorable voltage assignment.     

Bending of a partially clamped rectangular plate

Summary The present study deals with the bending of a rectangular plate with two opposite simply-supported edges and other two free edges having any portion, clamped in such a way that the slope is zero. The solution is obtained by means of a superposition proposed by Kurata and Hatano [3]. The case of a square plate under a uniform load or a point load has been studied in detail.

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Übersicht Es wird die Biegung einer rechteckigen Platte untersucht, von der zwei Ränder einfach unterstützt sind, während die anderen Ränder zum Teil frei sind, zum Teil so eingespannt, daß die Neigung verschwindet. Die Lösung wird mit Hilfe einer Superposition erhalten, die von Kurata und Hatano [3] vorgeschlagen wurde. Für den Fall einer quadratischen Platte unter einer gleichförmigen Last oder einer Einzellast werden die Berechnungen weiter ausgeführt.

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This paper is a part of author's Ph. D. thesis submitted to the Indian Institute of Technology, Kharagpur in 1965.     

A clamped rectangular plate containing a crack

In this paper the general problem of a rectangular plate clamped along two parallel sides and containing a crack parallel to the clamps is considered. The problem is formulated in terms of a system of singular integral equations and the asymptotic behaviour of the stress state near the corners is investigated. Numerical examples are considered for a clamped plate without a crack and with a centrally located crack, and the stress intensity factors and the stresses along the clamps are calculated.     

Forced axisymmetric vibrations and self-heating of a circular viscoelastic plate,controlled with piezoelectric sensors and actuators

The forced monoharmonic vibrations and self-heating of a circular thermoviscoelastic plate with piezoelectric sensor and actuator are studied. The viscoelastic behavior of the passive (without piezoeffect) and piezoactive materials is described using the concept of complex moduli. The problems of electroelasticity and heat conduction are solved numerically, assuming that the mechanical load is unknown. The effect of self-heating on the active damping of the vibrations of the plate is analyzed     

Tension of a clamped rectangular plate containing a central crack

Using the fundamental solution of a single crack and the Fourier transform solution of an infinite strip, the tension problem of a clamped rectangular plate containing a central crack is reduced to solve a system of singular integral equations. Then, the normal stress on clamped side and the stress intensity factors of the central crack are carried out by means of Gauss-Jacobi integration formulas. The comparison of numerical results is shown in the “table of stress intensity factors”. This work was supported by the Science Fund of the Academy of Sciences of China.     

Resonant vibrations of a hinged viscoelastic rectangular plate

The paper examines the effect of dissipative heating on the performance of a sensor in a hinged thermoviscoelastic rectangular plate undergoing resonant flexural vibrations. The thermoviscoelastic behavior of materials is described using the concept of complex characteristics. The coupling of the electromechanical and thermal fields is taken into account. The nonlinear problem is solved by the variational and Bubnov–Galerkin methods. The effect of the dissipative-heating temperature and the dimensions of the sensor on its performance during resonant vibrations of the plate is analyzed.     

Resonant vibrations of a hinged thermoviscoelastic rectangular plate with sensors and actuators

The paper discusses the active damping of the resonant flexural vibrations of a hinged thermoviscoelastic rectangular plate with distributed piezoelectric sensors and actuators. The thermoviscoelastic behavior of the passive and active materials is described using the concept of complex characteristics. The interaction of mechanical and thermal fields is taken into account. The Bubnov–Galerkin method is used. The effect of dissipative heating, the dimensions of the piezoelectric inclusions, and the feedback factor on the effectiveness of active damping of resonance vibrations of the plate is studied     

Flexural vibration of a square plate clamped along its contour

Analytical expressions are constructed for calculating the natural frequencies and mode shapes of flexural vibrations of a square homogeneous plate clamped along its contour. An error estimate is given by comparing predicted results with those of known high-precision calculations. Also the results of analytical calculations are compared with experimental data obtained by the author using the resonance method. The analytical and corresponding numerical results coincide with the experimental data to within less than 1%.     

A convergent method of solution for clamped rectangular plate

The conventional method of solution for clamped rectangular plate lies in releasing the constraints for rotation at the clamped sides and substituting them by equivalent distributed bending moments. This paper gives a general formulation of this method for arbitrary load with proof of the convergence, and discusses the rate of convergence for displacements and bending moments. The method and its convergence can be extended to the case of arbitrary boundary conditions and to plate on elastic foundation.The author expresses his sincere gratitude to Prof. Hu Hai-chang for guidance in this work.     

Active control of structural acoustic pressure in a rectangular cavity using piezoelectric actuators

Active control of structural acoustic pressure in a rectangular cavity with a flexible beam is simulated numerically. The wave equation of the acoustic pressure and the equation of motion of the beam are approximated via the series expansions, and is then expressed in state space form. The control of structural acoustic pressure and vibration of the beam was implemented by applying the optimal voltage on piezoelectric actuators through an LQR controller. Two cases of different external forces acting on the piezoelectric laminated beam are illustrated. Results demonstrate that such a control system can efficiently reduce the structural acoustic pressure.     

The bending of a thick rectangular plate with three clamped edges and one free edge

The exact solution of the bending of a thick rectangular plate with three clamped edges and one free edge under a uniform transverse load is obtained by means of the concept of generalized simply-supported boundary^[1] in Reissner’s theory of thick plates. The effect of the thickness h of a plate on the bending is studied and the applicable range of Kirchhoffs theory for bending of thin plates is considered.