Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: A piezoelectric finite element approach
TL;DR: In this paper, a new structure (shell or plate) containing an integrated distributed piezoelectric sensor and actuator is proposed, where the distributed sensing layer monitors the structural oscillation due to the direct PDE and the distributed actuator layer suppresses the oscillation via the converse PDE.
About: This article is published in Journal of Sound and Vibration.The article was published on 1990-04-08. It has received 642 citations till now. The article focuses on the topics: Piezoelectric sensor & PMUT.
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TL;DR: In this paper, the advances and trends in the formulations and applications of the finite element modeling of adaptive structural elements are surveyed and discussed in a first attempt to survey and discuss the advances.
639 citations
Cites background from "Distributed piezoelectric sensor/ac..."
...Early elements did not deal with layered structures [3,24,25,90,99,100,108]....
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...They were either tetrahedral [3,32] or brick elements [5,24,25,31,38,55,56,58,65,66,90,99,100, 102,105,108]....
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TL;DR: In this paper, a finite element formulation based on the classical laminated plate theory is presented for the shape and vibration control of the functionally graded material (FGM) plates with integrated piezoelectric sensors and actuators.
407 citations
TL;DR: In this article, a simple negative velocity feed back control algorithm coupling the direct and converse piezoelectric effects is used to actively control the dynamic response of an integrated structure through closed loop control.
353 citations
TL;DR: In this paper, the analysis of laminated composite plate structures with piezoelectric actuators and sensors is presented, where the authors implement layerwise representations of displacements and electric potential, and can model both the global and local electromechanical response of smart composite laminates.
342 citations
TL;DR: In this article, the authors present an overview of the modelling that has been proposed by various workers in the field of smart or intelligent structures, and present the main objective of this article.
Abstract: The main objective of this article is to present an overview of the modelling that has been proposed by various workers in the field of smart or intelligent structures. Before the main discussion o...
309 citations
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Book•
01 Jan 1974
TL;DR: In this article, the authors present a formal notation for one-dimensional elements in structural dynamics and vibrational properties of a structural system, including the following: 1. Isoparametric Elements.
Abstract: Notation. Introduction. One-Dimensional Elements, Computational Procedures. Basic Elements. Formulation Techniques: Variational Methods. Formulation Techniques: Galerkin and Other Weighted Residual Methods. Isoparametric Elements. Isoparametric Triangles and Tetrahedra. Coordinate Transformation and Selected Analysis Options. Error, Error Estimation, and Convergence. Modeling Considerations and Software Use. Finite Elements in Structural Dynamics and Vibrations. Heat Transfer and Selected Fluid Problems. Constaints: Penalty Forms, Locking, and Constraint Counting. Solid of Revolution. Plate Bending. Shells. Nonlinearity: An Introduction. Stress Stiffness and Buckling. Appendix A: Matrices: Selected Definition and Manipulations. Appendix B: Simultaneous Algebraic Equations. Appendix C: Eigenvalues and Eigenvectors. References. Index.
6,126 citations
Book•
03 Mar 1971
TL;DR: In this paper, the authors consider the problem of minimizing the sum of a differentiable and non-differentiable function in the context of a system governed by a Dirichlet problem.
Abstract: Principal Notations.- I Minimization of Functions and Unilateral Boundary Value Problems.- 1. Minimization of Coercive Forms.- 1.1. Notation.- 1.2. The Case when ?: is Coercive.- 1.3. Characterization of the Minimizing Element. Variational Inequalities.- 1.4. Alternative Form of Variational Inequalities.- 1.5. Function J being the Sum of a Differentiable and Non-Differentiable Function.- 1.6. The Convexity Hypothesis on $$ {U_{ad}} $$.- 1.7. Orientation.- 2. A Direct Solution of Certain Variational Inequalities.- 2.1. Problem Statement.- 2.2. An Existence and Uniqueness Theorem.- 3. Examples.- 3.1. Function Spaces on ?.- 3.2. Function Spaces on ?.- 3.3. Subspaces of Hm(?).- 3.4. Examples of Boundary Value Problems.- 3.5. Unilateral Boundary Value Problems (I).- 3.6. Unilateral Boundary Value Problems (II).- 3.7. Unilateral Boundary Value Problems (III).- 3.8. Unilateral Boundary Value Problems Case of Systems.- 3.9. Elliptic Operators of Order Greater than Two.- 3.10. Non-differentiable Functionals.- 4. A Comparison Theorem.- 4.1. General Results.- 4.2. An Application.- 5. Non Coercive Forms.- 5.1. Convexity of the Set of Solutions.- 5.2. Approximation Theorem.- Notes.- II Control of Systems Governed by Elliptic Partial Differential Equations.- 1. Control of Elliptic Variational Problems.- 1.1. Problem Statement.- 1.2. First Remarks on the Control Problem.- 1.3. The Set of Inequalities Defining the Optimal Control.- 2. First Applications.- 2.1. System Governed by the Dirichlet Problem Distributed Control.- 2.2. The Case with No Constraints.- 2.3. System Governed by a Neumann Problem Distributed Control.- 2.4. System Governed by a Neumann Problem Boundary Control.- 2.5. Local and Global Constraints.- 2.6. System Governed by a Differential System.- 2.7. System Governed by a 4th Order Differential Operator.- 2.8. Orientation.- 3. A Family of Examples with N = 0 and $$ {U_{ad}} $$ Arbitrary.- 3.1. General Case.- 3.2. Application (I).- 3.3. Application (II).- 4. Observation on the Boundary.- 4.1. System Governed by a Dirichlet Problem (I).- 4.2. Some Results on Non-homogeneous Dirichlet Problems.- 4.3. System Governed by a Dirichlet Problem (II).- 4.4. System Governed by a Neumann Problem.- 5. Control and Observation on the Boundary. Case of the Dirichlet Problem.- 5.1. Orientation.- 5.2. Boundary Control in L2(?).- 5.3. A "Controllability-Like" Problem.- 5.4. Pointwise Control and Observation.- 6. Constraints on the State.- 6.1. Orientation.- 6.2. Control and Constraints on the Boundary.- 7. Existence Results for Optimal Controls.- 7.1. Orientation.- 7.2. Distributed Control.- 7.3. Singular Perturbation of the System.- 7.4. Boundary Control.- 7.5. Control of Systems Governed by Unilateral Problems.- 8. First Order Necessary Conditions.- 8.1. Statement of the Theorem.- 8.2. Proof of the Theorem.- 8.2.1. "Algebraic" Transformation.- 8.2.2. General Remarks on the Utilization of (8.13.).- 8.2.3. Proof that dj,?0.- Notes.- III Control of Systems Governed by Parabolic Partial Differential Equations.- 1. Equations of Evolution.- 1.1. Data.- 1.2. Evolution Problems.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Some Examples.- 1.6. Semi-groups.- 2. Problems of Control.- 2.1. Notation. Immediate Properties.- 2.2. Set of Inequalities Characterizing the Optimal Control.- 2.3. Case (i). Set of Inequalities.- 2.4. Case (ii). Set of Inequalities.- 2.5. Orientation.- 3. Examples.- 3.1. Mixed Dirichlet Problem for a Second Order Parabolic Equation.- 3.1.1. C = Injection Map of L2(0, T V)?L2(Q).- 3.1.2. C = Identity Map of L2(0, T V) into itself.- 3.1.3. Observation of the Final State.- 3.2. Mixed Neumann Problem for a Parabolic Equation of Second Order.- 3.2.1. Case (i).- 3.2.2. Case (ii).- 3.3. System of Equations and Equations of Higher Order.- 3.3.1. System of Equations.- 3.3.2. Higher Order Equations.- 3.4. Additional Results.- 3.5. Orientation.- 4. Decoupling and Integro-Differential Equation of Riccati Type (I).- 4.1. Notation and Assumptions.- 4.2. Operator P(t), Function r(t).- 4.3. Formal Calculations.- 4.4. The Finite Dimensional Case Approximation.- 4.5. Passage to the Limit.- 4.6. Integro-Differential Equation of Riccati Type.- 4.7. Connections with the Hamilton-Jacobi Theory.- 4.8. The Case where Constraints are Present.- 4.9. Various Remarks.- 4.9.1. Direct Study of the "Riccati Equation".- 4.9.2. Another Approach to the Direct Study of the "Riccati Equation".- 4.9.3. Yet Another Approach to the Direct Study of the "Riccati Equation".- 5. Decoupling and Integro-Differential Equation of Riccati Type (II).- 5.1. Application of the Schwartz-Kernel Theorem.- 5.2. Example of a Mixed Neumann Problem with Boundary Control.- 5.3. Example of a Mixed Neumann Problem with Observation of the Final State.- 5.4. Mixed Neumann Problem, Observation of the Final State and Constraints in a Vector Space.- 5.5. Remarks on Decoupling in the Presence of Constraints.- 6. Behaviour as T ? + ?.- 6.1. Orientation and Hypotheses.- 6.2. The Case T = ?.- 6.3. Passage to the Limit as T ? + ?.- 7. Problems which are not Necessarily Coercive.- 7.1. Distributed Observation.- 7.2. Observation of the Final State.- 7.3. Examples where N = 0 and $$ {U_{ad}} $$ is not Bounded.- 8. Other Observations of the State and other Types of Control.- 8.1. Pointwise Observation of the State.- 8.2. Pointwise Control.- 8.3. Control and Observation on the Boundary.- 9. Boundary Control and Observation on the Boundary or of the Final State for a System Governed by a Mixed Dirichlet Problem.- 9.1. Orientation and Problem Statement.- 9.2. Non Homogeneous Mixed Dirichlet Problem.- 9.3. Definition of $$ \frac{{\partial y}}{{\partial {v_A}}} $$ Observation.- 9.4. Cost Function Equations of Optimal Control.- 9.5. Regular Control.- 9.6. Observation of the Final State.- 9.7. Observation of the Final State, Second Order Parabolic Operator.- 10. Controllability.- 10.1. Problem Statement.- 10.2. Controllability and Uniqueness.- 10.3. Super-Controllability and Super-Uniqueness.- 11. Control via Initial Conditions Estimation.- 11.1. Problem Statement. General Results.- 11.2. Examples.- 11.3. Controllability.- 11.4. An Estimation Problem.- 12. Duality.- 12.1. General Remarks.- 12.2. Example.- 13. Constraints on the Control and the State.- 13.1. A General Result.- 13.2. Applications (I).- 13.3. Applications (II).- 14. Non Quadratic Cost Functions.- 14.1. Orientation.- 14.2. An Example.- 14.3. Remarks on Decoupling.- 15. Existence Results for Optimal Controls.- 15.1. Orientation.- 15.2. Non-linear Problem with Distributed Control (I).- 15.3. Non-linear Problem with Distributed Control. Singular Perturbation.- 15.4. Non-linear Problem. Boundary Control.- 15.5. Utilization of Convexity and the Maximum Principle for Second Order Parabolic Equations.- 15.6. Control of Systems Governed by Evolution Inequalities.- 16. First Order Necessary Conditions.- 16.1. Statement of the Theorem.- 16.2. Proof of Theorem 16.1.- 16.2.1. "Algebraic" Transformation.- 16.2.2. Utilization of (16.11.).- 16.2.3. Proof of (16.12.).- 16.3. Remarks.- 17. Time Optimal Control.- 17.1. Problem Statement.- 17.2. Existence Theorem.- 17.3. Bang-Bang Theorem.- 18. Miscellaneous.- 18.1. Equations with Delay.- 18.1.1. Definition of the State.- 18.1.2. Control Problem.- 18.2. Spaces which are not Normable.- Notes.- IV Control of Systems Governed by Hyperbolic Equations or by Equations which are well Posed in the Petrowsky Sense.- 1. Second Order Evolution Equations.- 1.1. Notation and Hypotheses.- 1.2. Problem Statement. An Existence and Uniqueness Result.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Examples (I).- 1.6. Examples (II).- 1.7. Orientation.- 2. Control Problems.- 2.1. Notation. Immediate Properties.- 2.2. Case (2.5.).- 2.3. Case (2.6.).- 2.4. Case (2.7.).- 2.5. Case (2.8.).- 3. Transposition and Applications to Control.- 3.1. Transposition of Theorem 1.1.- 3.2. Application (I).- 3.3. Application (II).- 3.4. Application (III).- 4. Examples.- 4.1. Examples of Hyperbolic Problems. Distributed Control, Distributed Observation.- 4.2. Examples of Hyperbolic Systems. Distributed Control, Observation of the Final State.- 4.3. Petrowsky Type Equation. Distributed Control. Distributed Observation.- 4.4. Petrowsky Type Equation. Distributed Control. Observation of the Final State.- 4.5. Orientation.- 5. Decoupling.- 5.1. Problem Statement. Rewriting as a System of First Order Equations.- 5.2. Rewriting of the Set of Equations Determining the Optimal Control.- 5.3. Decoupling.- 5.4. Riccati Integro-differential Equation.- 5.5. Another Optimal Control Problem. Decoupling.- 6. Control via Initial Conditions. Estimation.- 6.1. Problem Statement.- 6.2. Coercivity of J(?).- 6.3. System of Equations Determining the Optimal Control.- 7. Boundary Control (I).- 7.1. Problem Statement.- 7.2. Definition of the State of the System.- 7.3. Distributed Observation.- 7.4. Boundary Observation.- 8. Boundary Control (II).- 8.1. Problem Statement.- 8.2. Control ? Regular.- 8.3. Examples.- 9. Parabolic-Hyperbolic Systems.- 9.1. Recapitulation of Some General Results.- 9.2. Complement.- 9.3. Control Problems.- 9.4. Example (I).- 9.5. Example (II).- 9.6. Decoupling.- 10. Existence Theorems.- 10.1. Orientation.- 10.2. Example. Introduction of a "Viscosity" Term.- 10.3. Time Optimal Control.- Notes.- V Regularization, Approximation and Penalization.- 1. Regularization.- 1.1. Parabolic Regularization.- 1.2. Application to Optimal Control.- 1.3. Application to Decoupling.- 1.4. Various Remarks.- 1.5. Regularization of the Control.- 2. Approximation in Terms of Systems of Cauchy-Kowaleska Type.- 2.1. Evolution Equation on a Variety.- 2.2. Approximation by a System of Cauchy-Kowaleska Type.- 2.3. Linearized Navier-Stokes Equation.- 3. Penalization.- Notes.
3,539 citations
TL;DR: In this paper, a scaling analysis is performed to demonstrate that the effectiveness of actuators is independent of the size of the structure and evaluate various piezoelectric materials based on their effectiveness in transmitting strain to the substructure.
Abstract: This work presents the analytic and experimental development of piezoelectric actuators as elements of intelligent structures, i.e., structures with highly distributed actuators, sensors, and processing networks. Static and dynamic analytic models are derived for segmented piezoelectric actuators that are either bonded to an elastic substructure or embedded in a laminated composite. These models lead to the ability to predict, a priori, the response of the structural member to a command voltage applied to the piezoelectric and give guidance as to the optimal location for actuator placement. A scaling analysis is performed to demonstrate that the effectiveness of piezoelectric actuators is independent of the size of the structure and to evaluate various piezoelectric materials based on their effectiveness in transmitting strain to the substructure. Three test specimens of cantilevered beams were constructed: an aluminum beam with surface-bonded actuators, a glass/epoxy beam with embedded actuators, and a graphite/epoxy beam with embedded actuators. The actuators were used to excite steady-state resonant vibrations in the cantilevered beams. The response of the specimens compared well with those predicted by the analytic models. Static tensile tests performed on glass/epoxy laminates indicated that the embedded actuator reduced the ultimate strength of the laminate by 20%, while not significantly affecting the global elastic modulus of the specimen.
2,719 citations
TL;DR: In this article, the authors proposed a method for reducing the size of the stiffness matrix by eliminating coordinates at which no forces are applied, based on the procedure used in Ref. 1 for stiffness matrix reduction.
Abstract: Just as it is often necessary to reduce the size of the stiffness matrix in statical structural analysis, the simultaneous reduction of the nondiagonal mass matrix for natural mode analysis may also be required. The basis for one such reduction technique may follow the procedure used in Ref. 1 for the stiffness matrix, namely, the elimination of coordinates at which no forces are applied.
2,418 citations
1,665 citations