# Mémoire Online: Wave models for the flexural vibrations of thin plates Model of the vibrations of polygonal plates by the image source method Vibration damping using the acoustic black hole effect

Sommaire: Wave models for the flexural vibrations of thin plates Model of the vibrations of polygonal plates by the image source method Vibration damping using the acoustic black hole effect

Introduction
I Model of the vibrations of polygonal plates by the image source method
1 Introduction to the image source method
1.1 General considerations on exural wave motion in thin plates
1.2 Motivation for using an image source approach
1.2.1 Statement of the problem
1.2.2 Integral formulation of the exural vibrations of a polygonal plate
1.2.3 Discussion
1.3 The image source method and its applications
1.3.1 General considerations
1.3.2 Previous developments of the image source method
1.4 Overview of the proposed approach
2 The image source method for simply supported polygonal plates
2.1 Introduction
2.2 Green’s functions of convex polygonal plates
2.2.1 Statement of the problem
2.2.2 Modal expansion of the Green’s function
2.2.3 Image source method (ISM)
2.2.4 Obtaining a modal expansion of the Green’s function from ISM
2.2.5 Examples
2.3 E ect of truncation of the image source cloud
2.3.1 Truncation of the image source cloud
2.3.2 In uence of structural damping on the accuracy of AISM
2.3.3 Application of AISM to an arbitrary polygonal plate
2.3.4 In uence of the truncation radius on the accuracy of AISM
2.4 Conclusion
2.A Construction of the image source cloud
2.B Elementary cells of equilateral and half-equilateral triangular plates
2.B.1 Half-equilateral triangular plate
2.B.2 Equilateral triangular plate
2.C Asymptotic modal overlap factor
3 Model of boundary conditions of plates by a state vector approach
3.1 Model of exural wave motion by the state vector approach
3.1.1 Global and local coordinates
3.1.2 Matrix form of the equations of motion
3.1.3 Specifying the eigenvalue and eigenvector matrices
3.2 Model of the boundary conditions
3.2.1 Reection matrix of a plate edge
3.2.2 Scattering matrix of a junction between two plates
3.3 Some examples
3.3.1 Edges
3.3.2 Junctions
3.4 Conclusion
4 Harmonic Green’s functions of semi-in nite and polygonal plates
4.1 Introduction
4.2 Green’s function of an in nite plate
4.3 Green’s function of a semi-in nite plate
4.3.1 Formulation of the problem
4.3.2 Validation of the formulation for simply supported, roller sup-
ported, clamped and free boundary conditions
4.4 Green’s function of a convex polygonal plate
4.4.1 Formulation of the problem
4.4.2 Geometrical construction of image sources
4.4.3 Image sources of rst order
4.4.4 Image sources of second and higher orders
4.4.5 Domain of applicability of the solution
4.5 Numerical implementation
4.6 Results
4.6.1 Levy-type plate: comparison to the exact solution
4.6.2 Arbitrary polygonal plates: comparison to nite element method
4.7 Conclusion
4.A Green’s function of an in nite plate in rectangular coordinates
4.B Case of re
ection without angular dependence
4.C Discrete Fourier transform
4.D Variable separation in di erent coordinate systems
4.E Arbitrary polygonal plate with free boundaries
5 On the extension of the image source method to plate assemblies
5.1 Statement of the problem
5.2 Image source method for an assembly of two plates
5.2.1 Geometrical construction of image sources
5.2.2 Model of the junction
5.2.3 Displacement eld of the assembly
5.3 Validation
5.3.1 Virtual junction between two rectangular plates
5.3.2 Virtual junction between two arbitrarily polygonal plates
5.4 Conclusion
6 Measurement of complex bending sti ness of a at panel
6.1 Introduction
6.2 Model
6.2.1 Green’s problem of the exural motion of the plate
6.2.2 Solution by the image source method
6.3 Experimental setup
6.3.1 Description of the experimental setup
6.3.2 Calibration of the measurement system
6.4 Estimation of the bending sti ness of the panel
6.4.1 Modulus of the bending sti ness
6.4.2 Young’s modulus and structural damping ratio
6.5 Conclusion
II Vibration damping in beams and plates using the acoustic black hole eff ect
7 Acoustic black hole e ect in thin plates
7.1 Introduction
7.2 Existing models
7.2.1 Model of Mironov
7.2.2 Model of Krylov
7.2.3 Discussion
7.3 Model of the acoustic black hole e ect by the state vector approach
7.3.1 Flexural vibrations of a variable-thickness beam covered with a damping layer
7.3.2 Numerical implementation
7.3.3 Illustration of the black hole e ect
7.3.4 E ect of the di erent parameters of the damping layer
7.3.5 Simulated driving-point mobilities
7.4 Experimental results on elliptical plates
7.5 Vibration damping in polygonal plates using the black hole e ect
7.5.1 Statement of the problem
7.5.2 Manufacturing of the plates
7.5.3 Experimental results
7.5.4 Model by the image source method
7.6 Conclusion
8 Acoustic black hole e ect in shape-memory materials
8.1 Introduction
8.2 Model
8.2.1 Statement of the problem
8.2.2 Determination of the complex Young’s modulus of the material
8.2.3 Numerical simulations
8.3 Experimental observations
8.3.1 Experimental setup
8.3.2 Experimental results
8.4 Conclusion
Conclusion
References

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Extrait du mémoire Wave models for the flexural vibrations of thin plates Model of the vibrations of polygonal plates by the image source method Vibration damping using the acoustic black hole effect

Part I: Model of the vibrations of polygonal plates by the image source method
Chapter 1: Introduction to the image source method
Abstract
The purpose of this chapter is to give a general insight into the image source method.
First, the description of exural motion of thin plates within the Kirchho assumptionsis recalled in order to set the variables, coordinate systems and governing equations used throughout this document. Then, the integral representation of the Green’s function of a polygonal plate is reviewed in order to justify the choice of the image source method for modelling high frequency vibrations. The applications and limitations of the method are then detailed. The two di erent approaches of the image source method that are developed in the following chapters are brie y presented here as an introduction.

Wave models for the flexural
1.1 General considerations on exural wave motion in thin plates
The aim of this section is to introduce the di erent variables needed for describing exural motion in a thin plate, and to derive the propagation equation for the transverse displacement. The motion of the plate is considered in the framework of Kirchho ’s theory of thin plates, which ignores shearing of cross sections and rotational inertia [81].
The plate is assumed to be isotropic, with thickness h, density , Young’s modulus E and Poisson’s ratio . Fig. 1.1 shows the di erent forces and moments acting on a plate element of surface area dxdy without any external force, centred at point (x; y) and at a given time t. The state of such surface element is described by the variables [34]

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Mémoire Online: Wave models for the flexural vibrations of thin plates Model of the vibrations of polygonal plates by the image source method Vibration damping using the acoustic black hole effect (11.48 MB) (Cours PDF)