Geometric and Local Material Characteristics

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Electromechanical Analogies

Electric and mechanical discrete systems with n degrees of freedom have the same mathematical model: a system of n second order ordinary di¤erential equations. So that, once a mathematical model for a mechanical and for an electric system with the same number of degrees of freedom is given, it is possible to associate at each physical mechanical quantity the electrical one that plays the same role in the model. In this fashion electromechanical analogies for discrete systems are developed. The same procedure can be applied to continuous systems, as studied in ([23], [26]). Example 8 Let us consider the one degree of freedommechanical system in …gure 3-2. If we choose as the state variable the displacement u of the mass m; the following power balance must hold for each virtual velocity u_ ¤ Pint = Pext +Pa.

Periodic Systems and Homogenized Models

Let us consider a periodic one-dimensional electric lattice whose basic cell is composed by a parallel RLC element to ground G and a line element L as represented in …gure 3-5. Let be d the constant space interval between two cells, such that the n¡ th cell is the position x = nd and the (n+1) ¡ th cell is the position x = (n+1)d: The resultant system is represented in …gure 3-6 and it is a generalized lumped electric transmission line.

Homogenized Model

An homogenized model of the one-dimensional electric lattice presented in the previous section is a
continuous electric transmission line. Once the virtual velocity …elds for the continuous transmission line are chosen, the balance equations can be found by the power balance. On the other hand, the constitutive equations will be deduced giving a mapping between the kinematics of the homogenized and the lumped models and prescribing that the virtual powers spent in corresponding virtual velocities must be the same.

Continuous Layered Composite PEM Beam

An axially homogeneous three layered beam with a material and geometrical symmetry with respect to the beam central axis2 , such as that in …gure 4-1, will be considered. Assuming the kinematics, the constitutive equations and the power balance of the system as a 3D piezoelastic continuum and a kinematical mapping between the 3D and the 1D representations, the power balance and the constitutive relations for the 1D model will be derived de…ning sectional sti¤ness, capacitance and coupling coe¢cients. By the power balance a weak formulation of the balance equations will be directly deduced. A strong form of them will be obtained with the boundary conditions after integrations by parts. In this framework it will be shown3 that for an axially homogeneous beam: ² the piezoelectric e¤ect on the mechanical system reduces to a pair of equal and opposite forces applied on the ends of the PZT layers, with a module proportional to the applied voltage; ² the PZT transducer is electrically equivalent to a capacitance in parallel with a current generator with an imposed current proportional to the time derivative of the change in length of the PZT sheet.
The cases in which the upper and lower PZT layers are connected one to each other to couple the applied potential di¤erence with the beam bending mode (out of phase connection, …gure 4-4) and with the beam extensional mode (in phase connection, …gure 4-3) will be treated separately, getting the respective coupling coe¢cients.

Weak Formulation

The weak formulation for the dynamical problem of a continuous piezoelectromechanical layered beam can be derived directly by the expression of the power balance (4.52) substituting in it the constitutive equations.
Since we are treating continuous systems the virtual velocities must be elements of an opportune functional space. In particular, as underlined in the statement of the virtual power principle, they are required to.
1. be smooth enough to evaluate the integrals involved in the virtual power principle.
2. satisfy the homogeneous version of the prescribed essential boundary conditions.
The spatial distribution of the electric variable Á in the continuous body was prescribed in function of its value at the boundaries such that it satis…es the requirement 1., 2. for each value of _Á ¤ uu; _Á ¤ ul; _Á ¤ lu; _Á ¤ velocities, a distinction must be made between the vertical and the horizontal components u_¤ 3;u_¤ 1 because in the problem statement the spatial derivative up to the second order for u_ ¤ 3 and up to the …rst for u_¤ 1 are present. Let us denote by ² H1 0 the space of functions having square integrable9 derivatives up to the …rst order and satisfying the homogeneous version of the boundary conditions prescribed directly on them (essential boundary conditions for u1). ² H2 0 the space of functions having square integrable derivatives up to the second order and satisfying the homogeneous version of the boundary conditions prescribed on them and on their spatial derivatives up to …rst order (essential boundary conditions for u3).

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Table of contents :

1 Introduction 
1.1 Background and Motivations
1.2 Literature Review
1.3 Ideas and Research Objectives
1.4 Outline
2 Preliminaries 
2.1 Continuum Kinematics
2.1.1 Body, References, Coordinates
2.1.2 Deformations of Continuum Bodies
2.1.3 Beam Kinematics
2.2 Virtual Power Principle
2.2.1 Introduction
2.2.2 De…nitions
2.2.3 Statement
2.2.4 Considerations
2.3 Piezoelectric Materials
2.3.1 Linear Constitutive Relations
2.3.2 Voigt Notation
2.3.3 Uniaxial States
2.3.4 PZT Transducers
3 Electrical Systems 
3.1 Discrete Systems
3.2 Electromechanical Analogies
3.3 Periodic Systems and Homogenized Models
3.3.1 Lumped Transmission Line
3.3.2 Homogenized Model
4 Layered Composite PEM Beam 
4.1 Continuous Layered Composite PEM Beam
4.1.1 System Description
4.1.2 Kinematics
4.1.3 Constitutive Relations
4.1.4 Power Balance
4.1.5 Balance Equations
4.1.6 Weak Formulation
4.2 Elastic Beam
4.2.1 Equilibrium Equations
4.2.2 Weak Formulation
4.2.3 Dimensional Analysis and Approximations
4.3 Beam with PZT Transducers
4.3.1 Internal Powers
4.3.2 External Powers
4.3.3 Power Balance
4.4 Results Review
5 Periodic PEM Structures and Homogenized Models 
5.1 Bending Coupling
5.1.1 Re…ned Model of the Basic Cell
5.1.2 Homogenized Continuous Model
5.2 Extensional Coupling
5.2.1 Re…ned Model of the Basic Cell
5.2.2 Homogenized Continuous Model
6 Comparison of Optimal Network Con…gurations 
6.1 Wave Form Solutions
6.2 Waves in PiezoElectroMechanical Beams
6.2.1 Transversal-Electric Coupling
6.2.2 Longitudinal-Electric Coupling
6.3 Optimization for Vibrations Suppression
6.3.1 Performance Index
6.3.2 Optimization Method
6.4 Transversal-Electric Waves
6.4.1 Isolated Resonant Shunts (IRS)
6.4.2 Transmission Line with Line Resistance and Inductance(TL-Rl-Ll)
6.4.3 Transmission Line with Line Inductance and Ground Resistance(TL-Rg-Ll) .
6.4.4 Comparison of Network Con…gurations
6.5 Longitudinal-Electric Waves
6.5.1 Isolated Resonant Shunts (IRS)
6.5.2 Transmission Line with Line Resistance and Inductance(TL-Rl-Ll)
6.5.3 Transmission Line with Line Inductance and Ground Resistance(TL-Rg-Ll) .
6.5.4 Comparison of Network Con…gurations
7 Experiments 
7.1 Goals
7.2 System Design and Realization
7.2.1 Beam with PZT Transducers
7.2.2 Electric Networks
7.3 Experimental Modal Analysis
7.3.1 Instrumentation
7.3.2 Experimental Setup for Mechanical and Electrical FRF Measures
7.3.3 Identi…cation Procedure
7.4 Results
7.4.1 Beam Modal Parameters
7.4.2 Synthetic Inductors Characterization
7.4.3 Beam with Resonant Shunted PZT
7.5 Next Steps
7.6 Conclusions
Appendix A – Physical Dimensions 
Appendix B – Ma terial Properties and External Actions 
B.1 Materi al Properties
B.2 Extern al Actions
B.3 Physical Dimensions
Appendix C – Numerical Values 
C.1 Geometric and Local Material Characteristics
C.2 Sectional Material Characteristics
C.3 Homogenized Material Characteristics
C.4 Dimensionless Parameters
C.4.1 Bending Coupling
C.4.2 Longitudinal Coupling

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