Senthil V. Gopinathan

A review and critique of theories for piezoelectric laminates

A review and critique of different laminate theories used for the modeling and analysis of laminated composite beams or plate structures is presented. Many finite-element models use classical laminate theory (CLT), also known as first-order shear deformation theory (FSDT), for the numerical simulation of active structures. The basic assumptions of this model have evolved from those proposed for composite laminate models and are based on thin-plate theory with resulting approximations for the elastic displacement, stress and strain components. In the case of piezoelectric laminates, the approximations spill over into the electric potential and electric field components. No studies and simulations have been documented for the dynamical electromechanical field variations through the thickness of the laminate structure at the resonant frequencies of the structure. This is essential to the understanding of the validity and range of applicability of thin-plate assumptions for active vibration control of structures. On the one hand, thin-plate models result in a computationally tractable model for smart structures, but they should not compromise on the electromechanical coupling effect, which is at the basis of active control. This paper first presents a three-dimensional (3D) complete field solution for active laminates based on a modal, Fourier series solution approach that is used to compute all the through-thickness electromechanical fields near the dominant resonance frequency of a beam plate with two piezoelectric (sensor and actuator) and one structural layers. Then a detailed review of the extant laminate models used for piezoelectric laminates, emphasizing the underlying assumptions in each case, is presented. The non-zero, through-thickness field components are computed under these assumptions. The results of the 3D model and FSDT model are compared for two aspect ratios ((ARs) - thickness-to-width of the layers). An AR of 20 is at the limit of the FSDT and an AR of 50 well within the assumptions of the FSDT. It is concluded that for moderate ARs, several of the approximations of the FSDT are questionable at resonance frequencies. A detailed set of pertinent and general references to papers dealing with piezoelectric laminates is also included. It is hoped that this study will be a reference source for those who want to use FSDT and for those want to understand the dynamical behavior of the internal fields in a smart laminate.

Finite element simulation of smart structures using an optimal output feedback controller for vibration and noise control

This numerical study presents a detailed optimal control design based on the general finite element approach for the integrated design of a structure and its control system. Linear quadratic (LQ) theory with output feedback is considered on the basis of the state space model of the system. Three-dimensional finite elements are used to model the smart structure containing discrete piezoelectric sensors and actuators by the use of combination of solid, transition and shell elements. Since several discrete piezoelectric patches are spatially distributed in the structure to effectively observe and control the vibration of a structure, the system model is thus utilized to design a multi-input-multi-output (MIMO) controller. A modal analysis is performed to transform the coupled finite element equations of motion into the state space model of the system in the modal coordinates. The output feedback controller is then employed to emulate the optimal controller by solving the Riccati equations from the modal space model. An optimal controller design for the vibration suppression of a clamped plate is presented for both the steady state and the transient case. Numerical simulation is also used to predict the reduction in the sound pressure level inside an enclosure radiated from this optimally controlled plate.

Radiated noise control via structural vibration control

Interior noise control in a cabin enclosure using active vibration control of the walls of the enclosure with discrete piezoelectric actuators and sensors is addressed. A hybrid approach using finite element formulation for the radiating walls of the enclosure. We use an exact 3D formulation without making the usual approximations for the electric field in the piezoelectric devices. The electrical boundary conditions and the charge on the electrodes are treated correctly. Computational time is optimized by using plate elements for the structure and 3D element for the devices with transition elements to connect them. A PD-controlled is used to relate the voltage output of the sensor in an open circuit conditions to the charge input to the actuator via appropriate gains to control vibrations. The acoustic part of the problem is modeled via a modal approach. The modal representation of the pressure is used as a mechanical force term on the structure which can be written in terms of a virtual mass. The driving team for the acoustic field is in turn the displacements on the surface of the radiating walls which is computed from the structural equations. This accounts for the acoustic field-structure interaction and the equations are solved simultaneously. By adjusting the feedback gain, significant noise reduction is achieved globally within the cavity for the dominant vibrational modes of the radiating panel.

Finite element/boundary element simulation of interior noise control using active-passive control technique

This paper presents a finite element/boundary element (FE/BE) formulation for modeling and analysis of active-passive noise control system. Finite element method is proposed to model the smart plate with surface bonded piezoelectric patches and the enclosing walls and the dual reciprocity boundary element method is proposed for modeling the acoustic cavity. The use of FE/BE method facilitates us imposing the impedance boundary conditions at the fluid/passive absorber/structure interface. An output feedback optimal controller design procedure is given for the smart plate system with active patches for the low frequency regime.

Active Noise Control Studies Using the Rayleigh-Ritz Method

The use of piezoelectric materials in controlling the vibration of continuous structures has grown significantly in recent years. A number of studies using Finite Element (FE) method [1, 2, 3, 4] have been made on noise transmission studies for rectangular enclosures through flexible smart panels. In these models, the host plate, actuators and sensors are modeled using 2D and 3D elements, which are later coupled, to the cavity in which the pressure is expressed in terms of rigid cavity modes. Although these earlier FE models predict the behavior of the structural panel and the fluid-structure interaction accurately at low frequencies, at high frequencies the size of the model increases resulting in very long computational time. Further, optimal sensor/actuator placement studies, involve repeated FE remeshing during the iterations, hence a simple model like the RR approach is preferred. The potential and kinetic energies of the panel with surface bonded discrete piezoelectric patches are estimated and the equations of motion for the smart panel are derived using Hamilton’s principle. The electric potential inside the piezoelectric patches are assumed to be a quadratic function of thickness coordinate. Classical laminated plate theory is used for modeling the host plate and the electroelastic theory is used to model the surface bonded patches. In electroelastic theory, reduced charge equation is satisfied inside both sensor and actuator patches. For the acoustic enclosure, the cavity pressure is expressed in terms of rigid cavity modes [5]. For the numerical study and to validate the RR approach, the frequencies obtained using RR approach are compared with the FE results for a smart aluminum plate backed cubic cavity.

Rayleigh-Ritz formulation for active control of the acoustics in cabin enclosures

This numerical study presents a detailed optimal control design based on the Rayleigh-Ritz approach for the smart plate-cavity system. Linear quadratic (LQ) theory with output feedback is considered on the basis of the state space model of the system. A vibroacoustic model, which includes a rectangular shaped cavity, enclosed with a five rigid walls and a flexible smart plate with discrete piezoelectric sensor/actuator pairs bonded to its surface. Classical laminated plate theory is used to model the composite plate and electroelastic theory is used model the discrete piezoelectric patches. Eigenfunctions of a clamped-clamped beam are used as the Ritz functions for the panel and the rigid walled cavity modes are used the model the acoustic cavity. The dynamic equations of motion for the coupled smart panel-cavity system are derived using Hamiltons principle. The forcing term due to the cavity acoustic pressure is determined by using virtual work considerations. For the present study, five collocated pairs of sensor/actuator pairs are attached to the plate at a predetermined placement scheme. The performance index considered for the design of the optimal controller includes both the displacement of the panel and the pressure inside the cavity. Numerical simulation is used to predict the reduction in the sound pressure level inside an enclosure radiated from this optimally controlled plate. The Rayleigh-Ritz approach is found to be faster and a more efficient method for designing control system for simple plate-cavity systems when compared to other numerical methods such as the finite element method.

Closed-loop finite element modeling of smart structures: an overview

Smart structures incorporate sensors, actuators and control electronics that permit the structures to tailor their response to changes in the environment in an optimal fashion. The sensors and actuators are constructed from functional materials such as piezoelectric, electrostrictive, shape memory alloys and magnetostrictive materials and more recently using MEMS (Micro Electro Mechanical Systems) devices. All functional materials and devices therefrom involve coupled fields involving elastodynamic, viscoelastic, electric, magnetic and thermal fields. The materials are anisotropic and often nonlinear. Finite element modeling has been successfully used to model these complex structures. More recently, closed ioop numerical simulation of the tailored response of a smart structure has become possible by combining the finite element equations of the sensor response to applied dynamical and/or thermal loads to the input voltage or current to the actuators via a control algorithm. This hybrid approach permits us to simulate the response of the structure with feedback control. Simple feedback controllers have now been replaced by robust controllers that provide stability under a range of uncertainties and do not require a very accurate system model. The talk will present an overview of the approaches of various researchers and consider numerical applications and comparison with experiments for active vibration damping, noise control and shape modification.

Rayleigh-Ritz/boundary element modeling approach for active/passive control

In this paper, a novel hybrid Rayleigh-Ritz/Boundary Element (RR/BE) solution method is proposed to model acoustic domains with flexible walls with piezoelectric patches. The RR approach is a simple, computationally inexpensive approach when compared to the finite element method for flexible walls with surface mounted piezoelectric patches. The RR method is then combined with the boundary element model of the interior acoustic domain and the coupled fluid-structure model is used for designing an active noise control system. This model also allows a designer to incorporate a passive absorber at the fluid-structure interface. The predicted sound pressure attenuation for three different thicknesses of passive absorber in the frequency range of 200 to 1200 Hz is calculated and an optimal thickness value of for the absorber for the smart panel is calculated. The attenuation in sound pressure levels due to an active control system in the presence of passive absorber is also computed. The system matrices resulting from this method are very smaller in size when compared to the FE models, which makes this approach most suitable for optimization studies. This new approach can be further extended to model the more complicated acoustic enclosures with complex interface.

Electro‐elastic laminate theory for discrete piezoelectric patches on laminated plates

Classical laminated plate theory (CLT) has been applied successfully in the past to laminates with discrete piezoelectric patches bonded to the surface or embedded within the layers (1–7). The basic assumptions made in the earlier models were that the strains inside the patches are assumed to be constant and hence the presence of the sensor and actuator patches were neglected while modeling the dynamic properties of the laminate. The validity of these assumptions, the effect of the size of the patches and these assumptions on the solutions obtained, has not been studied. In this paper, the CLT is applied to a laminate with surface‐bonded piezoelectric patches without the above‐mentioned assumptions. A detailed modeling of the patches is developed by expressing the electric potential inside the patch as a quadratic function of thickness coordinate. The equations of motion are derived for a generally orthotropic laminate and solution method for these equations. Analytical solutions are obtained for a plate bonded with one and five collocated piezoelectric actuator/sensor patches. The effect of the passive and active stiffness of the surface bonded actuator and sensor patches on the dynamic characteristics of host plate structure is studied.

Closed‐loop finite‐element modeling of smart structures—An overview

Smart structures incorporate sensors, actuators, and control electronics that permit the structure to tailor its response to changes in the environment in an optimal fashion. The sensors and actuators are constructed using functional materials such as piezoelectric materials. Finite‐element modeling has been successfully used to model these complex structures. More recently, closed‐loop numerical simulation of the tailored response of a smart structure has become possible by combining the finite‐element equations of the sensor response to applied dynamical loads to the input voltage or current to the actuators via a control algorithm. Optimization of sensor/actuator placement as well as optimization of the control gains will be discussed. Applications to control of cabin noise are studied. Constrained layer damping, incorporating active passive damping, can also be simulated. The talk will present an overview of the approach in relation to others in the extant literature.