Reflection and Transmission of Oblique Acoustic Waves by a Sub-Critical Elastic Barrier with

109 Pages·2017·3.77 MB·English

by  Mauricio Villa

Reflection and Transmission of Oblique Acoustic Waves by a
Sub-Critical Elastic Barrier with Discontinuities Using
Analytical Numerical Matching
by
Mauricio Villa
Department of Mechanical Engineering and Materials
Duke University
Date:
Approved:
Donald B. Bliss, Supervisor
Linda Franzoni
Earl Dowell
Thomas Witelski
Thesis submitted in partial fulfillment of the requirements for the degree of
Master of Science in the Department of Mechanical Engineering and Materials
Science in the Graduate School of Duke University
2017
Abstract
Reflection and Transmission of Oblique Acoustic Waves by a
Sub-Critical Elastic Barrier with Discontinuities Using
Analytical Numerical Matching
by
Mauricio Villa
Department of Mechanical Engineering and Materials
Duke University
Date:
Approved:
Donald B. Bliss, Supervisor
Linda Franzoni
Earl Dowell
Thomas Witelski
An abstract of a thesis submitted in partial fulfillment of the requirements for
the degree of Master of Science in the Department of Mechanical Engineering and
Materials Science in the Graduate School of Duke University
2017
Copyright (cid:13)c 2017 by Mauricio Villa
All rights reserved except the rights granted by the
Creative Commons Attribution-Noncommercial License
Abstract
Thisworkdevelopsmodelsforthecoupledstructural-acousticvibrationofboundaries
that reflect and transmit sound. First, the case of an infinitely long, fluid-loaded,
sub-critical membrane that is periodically fixed and forced by oblique incident acous-
tic waves is considered. The method of Analytical Numerical Matching (ANM) is
extended to deal with the resulting spatially-periodic and discrete phase-shifted forc-
ing. The high resolution content of the solution near the constraints is analytically
treated with a polynomial known as the Local Solution. The remaining, rapidly con-
verging, part of the solution is treated modally and is known as the Global Solution.
The Composite ANM Solution is then determined for the motion of the structure,
and the far-field acoustic fields can be efficiently described. It is shown that the
use of ANM effectively addresses the sensitivity of the acoustic fields and structure
motion to the accuracy of which the local region near the structural discontinuities
is resolved. The use of ANM is extended to demonstrate a method to deal with the
mathematical difficulty of acoustic coincidence. The second module of this thesis
presents ongoing work on the development of a model for a fluid-loaded finite mem-
brane in an infinite baffle. Corrections to the in-vacuo structural wavenumber are
developed to model the additional inertance and dissipative effects of the surround-
ing fluid media. The resulting dissipated energy as a function of frequency of the
modified finite membrane is compared to energy radiated by the infinite, periodically
driven, fluid loaded membrane to motivate further refinements of the finite model.
iv
Contents
Abstract iv
List of Figures ix
List of Symbols xii
Acknowledgements xv
1 Introduction 1
1.1 The Periodically Fixed Infinite Membrane . . . . . . . . . . . . . . . 4
1.1.1 Decomposition into Unconstrained and Driven Membrane . . . 4
1.2 The Baffled Finite Structure . . . . . . . . . . . . . . . . . . . . . . . 5
2 Unconstrained Infinite Membrane 7
2.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Acoustic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Method 1: Complete Fluid Loaded Problem . . . . . . . . . . . . . . 11
2.3 Method 2: Hard Wall Reflection and Membrane Radiation . . . . . . 12
2.3.1 Hard Wall Reflection . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Membrane Radiation . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.3 Assembled Pressure Fields . . . . . . . . . . . . . . . . . . . . 15
2.4 Analysis of Unconstrained Membrane . . . . . . . . . . . . . . . . . . 15
2.4.1 Membrane Behavior . . . . . . . . . . . . . . . . . . . . . . . 15
v
2.4.2 Acoustic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Periodically Driven Membrane 18
3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 Phase-Shifted Periodic Line Forcing . . . . . . . . . . . . . . . 19
3.1.2 Fluid Back-Loading . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.3 Resultant Membrane Governing Equation . . . . . . . . . . . 21
3.2 Analytical Numerical Matching . . . . . . . . . . . . . . . . . . . . . 21
3.3 Local Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Transferring the Fluid Pressures from the Local Solution . . . 22
3.3.2 Modified Linear Operator for Oblique Incidence . . . . . . . . 23
3.3.3 Form of the Modified Local Solution . . . . . . . . . . . . . . 24
3.3.4 Effect of Discontinuous Loading . . . . . . . . . . . . . . . . . 25
3.3.5 Development of Local Solution Constraints . . . . . . . . . . . 25
3.3.6 Solving for the Local Solution . . . . . . . . . . . . . . . . . . 29
3.4 Global Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.1 Modal Description of the Smoothed Forcing . . . . . . . . . . 31
3.4.2 Modal Description of Fluid Pressures from Local Solution . . 34
3.4.3 Solving for Global Solution . . . . . . . . . . . . . . . . . . . . 36
3.5 Driven Membrane: Assembly of ANM Solution . . . . . . . . . . . . . 37
3.6 Comparison of ANM to Classical Modal Approach . . . . . . . . . . . 38
3.6.1 Classical Modal Decomposition . . . . . . . . . . . . . . . . . 38
3.6.2 Convergence Comparison of Methods . . . . . . . . . . . . . . 38
3.7 Resultant Acoustic Fields . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7.1 The Discrete Spectrum . . . . . . . . . . . . . . . . . . . . . . 43
3.7.2 Cut-off Phenomenon . . . . . . . . . . . . . . . . . . . . . . . 44
vi
3.7.3 ANM Treatment of Acoustic Coincidence . . . . . . . . . . . . 45
3.7.4 Radiating Pressure Waves from Driven Membrane . . . . . . . 46
4 Periodically Fixed Membrane and Acoustic Fields 47
4.1 Scaling and Superposition: Membrane Motion . . . . . . . . . . . . . 47
4.2 Scaling and Superposition: Acoustic Fields . . . . . . . . . . . . . . . 49
4.3 Radiated Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.1 Interpreting Intensity Expressions . . . . . . . . . . . . . . . . 51
4.4 Sample Intensity Distributions . . . . . . . . . . . . . . . . . . . . . . 52
5 Finite Baffled Structure 58
5.1 In-Vacuo Membrane Model . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Membrane on a Viscous Suspension . . . . . . . . . . . . . . . . . . . 61
5.3 Membrane Inertance Correction Model . . . . . . . . . . . . . . . . . 64
5.4 Power Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4.1 Radiated Power using Rayleigh Integral . . . . . . . . . . . . . 69
5.4.2 High Frequency Approximation . . . . . . . . . . . . . . . . . 71
5.4.3 Power Matching to Determine Damper Value . . . . . . . . . 72
5.4.4 Damper Functional Dependence . . . . . . . . . . . . . . . . . 74
5.5 Frequency Expansion Analysis . . . . . . . . . . . . . . . . . . . . . . 76
5.5.1 Effect of Inertance Correction on Damper Value and Power . . 82
6 Comparison of Finite and Infinite Membrane Models 84
6.1 Power Comparison: Damper Suspension vs. Fluid Loading . . . . . . 85
6.1.1 Power Comparison: Normal Incidence . . . . . . . . . . . . . . 87
6.1.2 Periodic Extension of Finite Membrane on a Viscous Suspension 88
7 Conclusions 90
7.1 Analytical Numerical Matching and Ongoing Efforts . . . . . . . . . . 90
vii
7.2 Finite Structure Model and Ongoing Efforts . . . . . . . . . . . . . . 91
References 93
viii
List of Figures
1.1 Summary of ANM Components [1] . . . . . . . . . . . . . . . . . . . 2
1.2 TheLocalSolution(Top)isforcedbyadiscreteloadp thatisreplaced
a
by a smoothed forcing. The calculated smoothed forcing is applied to
asmallregionandservesasthecouplingtotheGlobalSolution(Bottom) 3
1.3 The Periodically Fixed Membrane is forced by an acoustic wave, and
will result in a reflected and transmitted wave, as well as radiated (or
scattered) waves due to structural reverberation . . . . . . . . . . . . 4
1.4 The Finite Membrane is forced by an acoustic wave and will result in
radiated (or scattered) waves into both fluid media, with significant
energy content in the primary angles of reflection and transmission . 6
2.1 Unconstrained Membrane problem statement . . . . . . . . . . . . . . 8
2.2 Hard wall reflection of incident acoustic wave problem statement . . . 13
2.3 Corresponding membrane radiation problem statement . . . . . . . . 14
3.1 Periodically driven, fluid-loaded infinite membrane. . . . . . . . . . . 19
3.2 LocalSolutionandappliedloads. Thedisplacementinsidethesmooth-
ing length is prescribed using a polynomial description, and outside
this region it is set to zero. . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Real part of Local Solution displacement within the smoothing region 30
3.4 Phase of Local Solution displacement within the smoothing region . . 31
3.5 Absolute value of resultant smooth forcing within the smoothing region 32
3.6 Phase of resultant smooth forcing within the smoothing region . . . . 32
3.7 Magnitude of Fourier Coefficients for Smoothed and Discrete Forces . 33
ix
3.8 Magnitude of Fourier Coefficients for Smoothed and fluid surface pres-
sure from Local Solution . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.9 Magnitudes of structure displacement from the (Top) Local Solution
and (Bottom) global and composite solutions of the driven membrane. 37
3.10 Magnitude of Fourier Coefficients ofstructuraldisplacement; Classical
and Composite ANM Solutions . . . . . . . . . . . . . . . . . . . . . 39
3.11 Magnitude of Fourier Coefficients ofstructuraldisplacement; Classical
Solution and ANM Components . . . . . . . . . . . . . . . . . . . . . 40
3.12 Convergence of structural displacement spatial average for Driven
Membrane; Classical and ANM Solutions . . . . . . . . . . . . . . . 41
3.13 ConvergenceofstructuraldisplacementatdrivepointforDrivenMem-
brane; Classical and ANM Solutions . . . . . . . . . . . . . . . . . . 42
4.1 Periodically Fixed Membrane with an Oblique Acoustic Wave . . . . 47
4.2 Real part of Periodically Fixed Membrane motion. Unconstrained
Membrane shown for comparison . . . . . . . . . . . . . . . . . . . . 48
4.3 IntensityDistributionforsampleproblem. Distributionsfor(Left)top
fluid and (Right) bottom fluid vs radiating angles . . . . . . . . . . . 52
4.4 Intensity Distribution for sample problem offand near resonance. Dis-
tributions for (Left) top fluid and (Right) bottom fluid vs radiating
angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 IntensityDistributionforsampleproblemathigherresonantfrequency.
Distributions for (Left) top fluid and (Right) bottom fluid vs radiating
angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.6 Percent of energy redistributed to radiation angles other than the
primary angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.7 ConvergenceofstructuraldisplacementatdrivepointforDrivenMem-
brane at k 78.09; Classical and ANM Solutions . . . . . . . . . . 56
1
(cid:16)
5.1 Fluid-loadedfinitemembraneinaninfinitebaffledrivenwithanobliquely
incident acoustic wave . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Left: the red box represents the segment of the Unconstrained Mem-
brane used in modeling the finite membrane. Right: the configura-
tion of the in-vacuo Forced-Forced Membrane, and its components
described by η and η . . . . . . . . . . . . . . . . . . . . . . . . 59
ff1 ff2
x