Active control of geometrically nonlinear transient vibration of composite plates with piezoelectric actuators

In this paper, the incremental finite element equations for geometric non-linear analysis of piezoelectric smart structures are developed using a total Lagrange approach by using virtual velocity incremental variational principles. A four-node first order shear plate element model with reduced and selective integration is also developed. Geometrically non-linear transient vibration response and control of plates with piezoelectric patches subjected to pulse loads are investigated. Active damping is introduced on the plates by coupling a self-sensing and negative velocity feedback algorithm in a closed control loop. The numerical results show that piezoelectric actuators can introduce significant damping and suppress transient vibration effectively. The effects of the number and locations of the piezoelectric actuators on the control system are also discussed.     

FEM dynamic model for active vibration control of flexible linkages and its application to a planar parallel manipulator

Development of dynamic models of flexible linkages, with flexible motion caused by rigid body motion and electromechanical coupling of transduction devices and the host linkage, is very important for the design of active vibration control laws for flexible-link mechanisms. In the first part of this paper, the Lagrange finite element (FE) formulation is used to derive such a dynamic model for a flexible planar linkage with two translational and one rotational degrees of freedom. Linear electromechanical coupling of surface-bonded lead zirconate titanate (PZT) patches with the host linkage is incorporated into the model. In the second part of this paper, this dynamic model is applied to a flexible-link planar parallel manipulator. Based on standard kineto-elastodynamic assumptions, the linkage dynamic model is simplified and simulation of strain rate feedback control using PZT sensors and actuators is performed. Numerical results show that PZT actuators effectively damp vibration of the flexible linkages.     

Modeling and analysis of laminate composite plates with embedded active-passive piezoelectric networks

The objective of this work is to present the finite element modeling of laminate composite plates with embedded piezoelectric patches or layers that are then connected to active-passive resonant shunt circuits, composed of resistance, inductance and voltage source. Applications to passive vibration control and active control authority enhancement are also presented and discussed. The finite element model is based on an equivalent single layer theory combined with a third-order shear deformation theory. A stress-voltage electromechanical model is considered for the piezoelectric materials fully coupled to the electrical circuits. To this end, the electrical circuit equations are also included in the variational formulation. Hence, conservation of charge and full electromechanical coupling are guaranteed. The formulation results in a coupled finite element model with mechanical (displacements) and electrical (charges at electrodes) degrees of freedom. For a Graphite-Epoxy (Carbon-Fibre Reinforced) laminate composite plate, a parametric analysis is performed to evaluate optimal locations along the plate plane (xy) and thickness (z) that maximize the effective modal electromechanical coupling coefficient. Then, the passive vibration control performance is evaluated for a network of optimally located shunted piezoelectric patches embedded in the plate, through the design of resistance and inductance values of each circuit, to reduce the vibration amplitude of the first four vibration modes. A vibration amplitude reduction of at least 10 dB for all vibration modes was observed. Then, an analysis of the control authority enhancement due to the resonant shunt circuit, when the piezoelectric patches are used as actuators, is performed. It is shown that the control authority can indeed be improved near a selected resonance even with multiple pairs of piezoelectric patches and active-passive circuits acting simultaneously.     

Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate

The nonlinear natural frequency of a rectangular box, which consists of one flexible plate and five rigid plates, is studied in this paper. The flexible plate is assumed to vibrate like a simple piston. The behavior of the structural-acoustic coupling between the flexible plate and the air cavity is analyzed by using the proposed finite element modal method. The system finite element equation is reduced and expressed in terms of the modal coordinates with small degrees of freedom by using the proposed reduction method. The system nonlinear stiffness matrix representing the large amplitude vibration can be transformed to be a constant modal matrix. The natural frequencies are determined by using the harmonic balance method to solve the eigenvalue equations of the structural-acoustic system. The effect of the cavity depth on the natural frequencies and convergence studies are discussed in detail.     

An analytical and experimental study of the vibration response of a clamped ribbed plate

An analytical solution is presented in this paper for the vibration response of a ribbed plate clamped on all its boundary edges by employing a traveling wave solution. A clamped ribbed plate test rig is also assembled in this study for the experimental investigation of the ribbed plate response and to provide verification results to the analytical solution. The dynamic characteristics and mode shapes of the ribbed plate are measured and compared to those obtained from the analytical solution and from finite element analysis (FEA). General good agreements are found between the results. Discrepancies between the computational and experimental results at low and high frequencies are also discussed. Explanations are offered in the study to disclose the mechanism causing the discrepancies. The dependency of the dynamic response of the ribbed plate on the distance between the excitation force and the rib is also investigated experimentally. It confirms the findings disclosed in a previous analytical study [T.R. Lin, J. Pan, A closed form solution for the dynamic response of finite ribbed plates, Journal of the Acoustical Society of America 119 (2006) 917–925] that the vibration response of a clamped ribbed plate due to a point force excitation is controlled by the plate stiffness when the source is more than a quarter plate bending wavelength away from the rib and from the plate boundary. The response is largely affected by the rib stiffness when the source location is less than a quarter bending wavelength away from the rib.     

A sector finite element for dynamic analysis of thick plates

A 24 degree of freedom sector finite element is developed for the static and dynamic analysis of thick circular plates. The element formulation is based on Reissner's thick plate theory. The convergence characteristic of the elements is first studied in a static example of an unsymmetrically loaded annular plate. The obvious advantageous effect of including the twist derivatives of deflection as degrees of freedom is shown. The elements are then used to analyze the natural frequencies of an annular plate with various ratios of inner to outer radius. The results are in good agreement with an alternative solution in which thick plate theory is used. The versatility of this finite element is finally demonstrated by performing free vibration analysis of an example of clamped sector plates with various thicknesses and different sectorial angles.     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

Modelling and characterization of a piezoceramic inertial actuator

An inertial actuator (also known as a proof mass actuator) applies forces to a structure by reacting them against an “external” mass. This approach to actuation may provide some practical benefits in the active control of vibration and structure-borne noise: system reliability may be improved by removing the actuator from a structural load path; effective discrete point-force actuation permits ready attachment to curved surfaces, and an inherent passive vibration absorber effect can reduce power requirements.This paper describes a class of recently developed inertial actuators that is based on mechanical amplification of displacements of an active piezoceramic element. Important actuator characteristics include resonance frequencies, clamped force, and the drive voltage to output the force frequency response function.The paper addresses one particular approach to motion amplification, the “dual unimorph,” in detail. A model of actuator dynamic behavior is developed using an assumed-modes method, treating the piezoelectrically induced stresses as external forces. Predicted actuator characteristics agree well with experimental data obtained for a prototype actuator. The validated actuator dynamic model provides a tool for design improvement.     

Active control of low-frequency sound radiation by cylindrical shell with piezoelectric stack force actuators

A novel active control method of sound radiation from a cylindrical shell under axial excitations is proposed and theoretically analyzed. This control method is based on a pair of piezoelectric stack force actuators which are installed on the shell and parallel to the axial direction. The actuators are driven in phase and generate the same forces to control the vibration and the sound radiation of the cylindrical shell. The model considered is a fluid-loaded finite stiffened cylindrical shell with rigid end-caps and only low-frequency axial vibration modes are involved. Numerical simulations are performed to explore the required control forces and the optimal mounting positions of actuators under different cost functions. The results show that the proposed force actuators can reduce the radiated sound pressure of low-frequency axial modes in all directions.     

Vibration of moderately thick square orthotropic stepped thickness plates

Numerous studies that address the vibration of stepped thickness plates are reported in the literature. Predominately, classical plate theory has been used to formulate studies for both isotropic and anisotropic stepped plates. Mindlin plate theory has been employed to obtain results for thick isotropic stepped thickness plates. Exact solutions, Rayleigh-Ritz, differential quadrature and finite element methods have been employed to compute results for frequency of vibration. Results for frequency of vibration for thick orthotropic stepped thickness plates are presented here using orthorhombic material properties of aragonite. The finite element method has been used to compute frequencies and determine mode shapes for simply supported and clamped square Mindlin plates.     

Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid

In this study, a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics (natural frequencies and associated mode shapes) of rectangular Kirchhoff plates. For the interaction of the Kirchhoff plate–Pasternak foundation, a mixed-type finite element formulation is employed by using the Gâteaux differential. The plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes, and at each node four degrees of freedom are present (one transverse displacement, two bending moments and one torsional moment). Therefore, a total number of 16 degrees-of-freedom are assigned to each element. A consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis. The plate structure considered is assumed clamped or simply supported along its edges and resting on a Pasternak foundation. Furthermore, the plate is fully or partially in contact with fresh water on its one side. For the calculation of the fluid–structure interaction effects (generalized fluid–structure interaction forces), a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid's free surface. It is assumed that the fluid is ideal, i.e., inviscid, incompressible, and its motion is irrotational. It is also assumed that the plate–elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. At the fluid–structure interface, continuity considerations require that the normal velocity of the fluid is equal to that of the structure. The normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements, obtained from the finite element analysis. By using the boundary integral equation method the fluid pressure is eliminated from the problem, and the fluid–structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients (due to the inertial effect of fluid). To asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure, the natural frequencies and associated mode shapes are presented. Furthermore, the influence of the submerging depth on the dynamic behavior is also investigated.     

Plate-shaped non-contact ultrasonic transporter using flexural vibration

We developed a plate-shaped non-contact transporter based on ultrasonic vibration, exploiting a phenomenon that a plate can be statically levitated at the place where its gravity and the acoustic radiation force are balanced. In the experiment, four piezoelectric zirconate titanate elements were attached to aluminum plates, on which lattice flexural vibration was excited at 22.3 kHz. The vibrating plates were connected to a loading plate via flexible posts that can minimize the influence of the flexure induced by heavy loads. The distribution of the vibration displacement on the plate was predicted through finite-element analysis to find the appropriate positions of the posts. The maximum levitation height of this transporter was 256 μm with no load. When two vibrating plates were connected to a loading plate, the maximum transportable load was 4.0 kgf.     

EXPERIMENTAL MEASUREMENTS BY AN OPTICAL METHOD OF RESONANT FREQUENCES AND MODE SHAPES FOR SQUARE PLATES WITH ROUNDED CORNERS AND CHAMFERS

Most of the work done on vibration of plates published in the literature includes analytical and numerical studies with few experimental results available. In this paper, an optical system called the amplitude-fluctuation electronic speckle pattern interferometry for the out-of-plane displacement measurement is employed to investigate the vibration behavior of plates with rounded corners and with chamfers. The boundary conditions are traction free along the circumference of the plate. Based on the fact that clear fringe patterns will appear only at resonant frequencies, both resonant frequencies and corresponding mode shapes can be obtained experimentally using the present method. Numerical calculations by finite element method are also performed and the results are compared with the experimental measurements. Good agreements are obtained for both results. It is interesting to note that the mode number sequences for some resonant modes are changed. The transition of mode shapes from the square plate to the circular plate is also discussed.     

Design and Optimization of the Electromagnetic Actuator in Active-Passive Hybrid Vibration Isolator

The design and optimization of actuators are difficult and critical for the active-passive hybrid vibration control system. In this paper, an electromagnetic actuator model is established based on Ohm’s Law for magnetic circuit considering the leakage flux. The 600N electromagnetic actuators are designed and optimized based on ANSYS simulation according to the engineering request. Its transient characteristics are studied. The effects of different structural parameters on its output force are analyzed. The experimental results show that the structure parameters and output force characteristics of the designed electromagnetic actuators satisfy the practical requirement.     

Comparison between the discrete and finite element methods for modelling an agricultural spray boom—Part 2: Automatic procedure for transforming the equations of motion from force to displacement input and validation

This paper validates the discrete element method for linear flexible multibody systems, elaborated in Part 1 of the paper, of which the flexible bodies are a composition of flexible beams. An automatic procedure is developed to convert the linear equations of motion of a multibody system from force to displacement input. By this procedure, support motions and displacements of actuators between the bodies can be employed as an input to the system. Furthermore, using this procedure, the methodology explained in Part 1, which was valid for tree structured systems can be extended to systems containing closed kinematic chains. The methodology of Part 1 is applied for the discrete and finite element approximations to model the horizontal behaviour of an agricultural spray boom. As the inputs to the spray boom are known under the form of positions, the equations of motions are converted from force to position inputs. The discrete and finite element approximations are compared based on accuracy and the complexity of the resulting models.     

The forced vibration of one-dimensional multi-coupled periodic structures: An application to finite element analysis

A general theory for the forced vibration of multi-coupled one-dimensional periodic structures is presented as a sequel to a much earlier general theory for free vibration. Starting from the dynamic stiffness matrix of a single multi-coupled periodic element, it derives matrix equations for the magnitudes of the characteristic free waves excited in the whole structure by prescribed harmonic forces and/or displacements acting at a single periodic junction. The semi-infinite periodic system excited at its end is first analysed to provide the basis for analysing doubly infinite and finite periodic systems. In each case, total responses are found by considering just one periodic element. An already-known method of reducing the size of the computational problem is reexamined, expanded and extended in detail, involving reduction of the dynamic stiffness matrix of the periodic element through a wave-coordinate transformation. Use of the theory is illustrated in a combined periodic structure+finite element analysis of the forced harmonic in-plane motion of a uniform flat plate. Excellent agreement between the computed low-frequency responses and those predicted by simple engineering theories validates the detailed formulations of the paper. The primary purpose of the paper is not towards a specific application but to present a systematic and coherent forced vibration theory, carefully linked with the existing free-wave theory.     

Geometrically nonlinear dynamic response of stiffened plates with moving boundary conditions

An approach is presented to investigate the nonlinear vibration of stiffened plates. A stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as Euler beams. Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. Besides, the effect caused by boundary movement is transformed into equivalent excitations. The first approximation solution of the non-resonance is obtained by means of the method of multiple scales. The primary parametric resonance and primary resonance of the stiffened plate are studied by using the same method. The accuracy of the method is validated by comparing the results with those of finite element analysis via ANSYS. Numerical examples for different stiffened plates are presented to discuss the steady response of the non-resonance and the amplitude-frequency relationship of the primary parametric resonance and primary resonance. In addition, the analysis on how the damping coefficients and the transverse excitations influence amplitude-frequency curves is also carried out. Some nonlinear vibration characteristics of stiffened plates are obtained, which are useful for engineering design.     

抗磁悬浮振动能量采集器动力学响应的仿真分析

分析了微型抗磁悬浮振动能量采集器中悬浮磁体的受力特性,发现了能量采集器的单稳态和双稳态现象,研究了能量采集器在不同工作状态下该两种稳态类型时的动力学响应特性.当能量采集器处于非工作的单稳态状态时,其动力学响应是在线性系统的基础上加入非线性扰动、幅频响应曲线向右偏转;热解石墨板间距越大,非线性扰动越强烈,右偏现象则越显著.当能量采集器处于非工作的双稳态状态时,其动力学响应比较复杂,出现倍周期、4倍周期以及混沌等非线性系统特有的现象.当能量采集器处于工作状态的双稳态状态时,其振动频率和外界激励频率保持一致,进行周期振动.该研究对抗磁悬浮振动能量采集器的结构设计具有重要的参考价值,为提高能量采集器的响应特性和输出性能提供了理论指导.     

Vibration analysis of circular segment shaped plates

Circular segment shaped plates are analyzed to determine their natural frequencies and mode shapes of vibration. The analysis is based on the finite element approach. The curved sided triangular plate bending element is used for solving the problem. The effect of variation of the size of the plate on the vibrational characteristics is studied and several important conclusions are made.     

Sound radiation by set of L-jointed plates with four pairs of piezoelectric elements

In the paper the possibilities of reducing the amplitude of structural vibrations and radiated noise in the way of use of piezoelectric actuators are discussed. The tests were done for the L-jointed aluminium plates. On one of its surfaces there were mounted four piezoelectric actuators and on the opposite one there were placed four piezoelectric sensors. The one of the four actuators was used as vibrations generator, while the remaining ones were used for vibration control. The influence of activation and mode shape form on the structure's response were investigated. The finite element method [15] analysis was performed, with the application of ANSYS computer package [1]. The numerical solutions were confirmed experimentally.