techniques to assess acoustic-structure interaction in liquid rocket engines

163 Pages·2008·1.71 MB·English

TECHNIQUES TO ASSESS ACOUSTIC-STRUCTURE
INTERACTION IN LIQUID ROCKET ENGINES
by
R. Benjamin Davis
Department of Mechanical Engineering and Materials Science
Duke University
Date:
Approved:
Dr. Lawrence N. Virgin, Supervisor
Dr. Andrew M. Brown
Dr. Earl H. Dowell
Dr. Linda P. Franzoni
Dr. Tod A. Laursen
Dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in the Department of Mechanical Engineering and Materials Science
in the Graduate School of
Duke University
2008
ABSTRACT
TECHNIQUES TO ASSESS ACOUSTIC-STRUCTURE
INTERACTION IN LIQUID ROCKET ENGINES
by
R. Benjamin Davis
Department of Mechanical Engineering and Materials Science
Duke University
Date:
Approved:
Dr. Lawrence N. Virgin, Supervisor
Dr. Andrew M. Brown
Dr. Earl H. Dowell
Dr. Linda P. Franzoni
Dr. Tod A. Laursen
An abstract of a dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in the Department of Mechanical Engineering and Materials Science
in the Graduate School of
Duke University
2008
Copyright (cid:176)c 2008 by R. Benjamin Davis
All rights reserved
Abstract
Acoustoelasticity is the study of the dynamic interaction between elastic structures
and acoustic enclosures. In this dissertation, acoustoelasticity is considered in the
context of liquid rocket engine design. The techniques presented here can be used to
determine which forcing frequencies are important in acoustoelastic systems. With
a knowledge of these frequencies, an analyst can either find ways to attenuate the
excitation at these frequencies or alter the system in such a way that the prescribed
excitations do not result in a resonant condition. The end result is a structural
component that is less susceptible to failure.
The research scope is divided into three parts. In the first part, the dynamics of
cylindrical shells submerged in liquid hydrogen (LH ) and liquid oxygen (LOX) are
2
considered. The shells are bounded by rigid outer cylinders with closed rigid ends.
This configuration gives rise to two fluid-filled cavities—an inner cylindrical cavity
and an outer annular cavity. Such geometries are common in rocket engine design.
The natural frequencies and modes of the fluid-structure system are computed by
combiningtherigidwallacousticcavitymodesandthein vacuo structuralmodesinto
a system of coupled ordinary differential equations. Eigenvalue veering is observed
near the intersections of the curves representing natural frequencies of the rigid wall
acoustic modes and the in vacuo structural modes. In the case of a shell submerged
in LH , system frequencies near these intersections are as muchas 30% lowerthan the
2
corresponding in vacuo structural frequencies. Due to its high density, the frequency
reductions in the presence of LOX are even more dramatic. The forced responses of
a shell submerged in LH and LOX while subject to a harmonic point excitation are
2
also presented. The responses in the presence of fluid are found to be quite distinct
from those of the structure in vacuo.
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In the second part, coupled mode theory is used to explore the fundamental fea-
tures of acoustoelastic systems. The result is the development of relatively simple
techniques that allow analysts to make informed decisions concerning the importance
of acoustic-structure coupling without resorting to more time-consuming and com-
plex methods. In this part, a new nondimensional parameter is derived to quantify
the fundamental strength of a particular acoustic-structure interaction irrespective of
material and fluid properties or cavity size. It is shown that, in some cases, reason-
able approximations of the coupled acoustic-structure frequencies can be calculated
without explicit knowledge of the uncoupled component mode shapes. Monte Carlo
simulations are performed to determine the parameter values over which approxi-
mate coupled frequency expressions are accurate. General observations concerning
the forced response of acoustoelastic systems are then made by investigating the
response of a simplified two mode system.
The third part of this research discusses the implementation of a component
mode synthesis (CMS) technique for use with geometrically complex acoustoelastic
systems. The feasibility of conceptually similar techniques was first demonstrated
over 30 years ago. Since that time there have been remarkable advancements in
computational methods. It is therefore reasonable to question the extent to which
CMS remains a computationally advantageous approach for acoustoelastic systems
of practical interest. This work demonstrates that relative to the most recent release
of the popular finite element software package, ANSYS, CMS techniques have a
significant computational advantage when the forced response of an acoustoelastic
system is of interest. However, recent improvements to the unsymmetric eigensolver
availableinANSYShaverenderedCMSalessefficientoptionwhencalculatingsystem
frequencies and modes. The CMS technique is also used to generate new results
relating to geometrically complex acoustoelastic systems.
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Contents
Abstract iv
List of Tables ix
List of Figures x
Acknowledgements xix
1 Introduction 1
1.1 Acoustic-Structure Interaction in Context . . . . . . . . . . . . . . . 1
1.2 Acoustic-Structure Interaction Methods . . . . . . . . . . . . . . . . . 2
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Contributions and Outline . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Theoretical Formulation 13
2.1 The Acoustic-Structure Equations of Motion . . . . . . . . . . . . . . 13
2.1.1 Acoustic Cavity Equations of Motion . . . . . . . . . . . . . . 14
2.1.2 Structural Equations of Motion . . . . . . . . . . . . . . . . . 17
2.1.3 Complete System of Equations . . . . . . . . . . . . . . . . . . 19
2.2 Solving the Acoustic-Structure Equations . . . . . . . . . . . . . . . . 20
2.2.1 The Nonsymmetric Eigenvalue Problem . . . . . . . . . . . . . 23
2.2.2 Transformation to Second-Order Form . . . . . . . . . . . . . 24
2.2.3 Determination of Modal Force . . . . . . . . . . . . . . . . . . 27
2.2.4 System Free Response to Initial Conditions . . . . . . . . . . . 28
2.2.5 Forced Response to Harmonic Excitation . . . . . . . . . . . . 29
3 Cylindrical Shells Submerged in Liquid Rocket Fuel or Oxidizer 31
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3.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Acoustic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Cavities with Rigid Ends . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Other End Boundary Conditions . . . . . . . . . . . . . . . . 39
3.3 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Simply Supported Boundary Conditions . . . . . . . . . . . . 44
3.3.2 Other Structural Boundary Conditions . . . . . . . . . . . . . 46
3.4 Coupled Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 49
3.5 System Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Convergence of System Frequencies . . . . . . . . . . . . . . . . . . . 60
3.7 Coupled Mode Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8 Structural Forced Response . . . . . . . . . . . . . . . . . . . . . . . 65
3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Approximate Techniques for Acoustoelastic Systems 70
4.1 Approximate Techniques in Context . . . . . . . . . . . . . . . . . . . 71
4.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 Approximate Expressions of System Natural Frequency . . . . 74
4.3.2 Dimensionless Coupling Parameter . . . . . . . . . . . . . . . 77
4.3.3 Dimensionless System Frequency Expression . . . . . . . . . . 78
4.3.4 Uncertainty Propagation . . . . . . . . . . . . . . . . . . . . . 83
4.4 Coupling Behavior of Systems with Common Geometries . . . . . . . 84
4.4.1 Rectangular Enclosure . . . . . . . . . . . . . . . . . . . . . . 84
4.4.2 Cylindrical Enclosure . . . . . . . . . . . . . . . . . . . . . . . 86
vii
4.4.3 Annular Enclosure . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5 Practical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.7 Forced Response of a Two Mode System . . . . . . . . . . . . . . . . 99
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 Techniques for Acoustoelastic Systems with Complex Geometry 107
5.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Component Mode Synthesis of Acoustoelastic Systems . . . . . . . . 110
5.2.1 Potential Advantages of CMS . . . . . . . . . . . . . . . . . . 111
5.2.2 Implementation of CMS Method . . . . . . . . . . . . . . . . . 115
5.3 Benchmark Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1 Part I: Calculation of System Frequencies . . . . . . . . . . . 117
5.3.2 Part II: Structural Forced Response to Harmonic Excitation . 120
5.4 Acoustoelastic Analysis of the Frusta of Cones . . . . . . . . . . . . . 122
5.5 Acoustoelastic Analysis of a Shell with a Ring Stiffener . . . . . . . . 126
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Conclusions and Outlook 131
6.1 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Appendix 133
Bibliography 136
Biography 143
viii
List of Tables
3.1 Coupling rules and coefficients for different fluid boundary conditions 50
4.1 The highest Ψ values (and their associated wavenumbers) for a plate
that is simply supported on all edges and backed by a rectangular
fluid-filled cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1 Performance of ANSYS 10 and 11 as well as the CMS method in Part
I of the benchmark test . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2 Performance of ANSYS 11 and the CMS method in Part II of the
benchmark test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
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List of Figures
1.1 Acoustic-structure interaction methods tree . . . . . . . . . . . . . . 3
1.2 Different views of the gimbal joint flowliner system . . . . . . . . . . 8
2.1 Schematic of a fluid-filled enclosure . . . . . . . . . . . . . . . . . . . 14
3.1 Schematic of a cylindrical shell in contact with both cylindrical and
annular fluid cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Behavior of the characteristic equation corresponding to a rigid cylin-
drical duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Behaviorofthecharacteristicequationcorrespondingtoarigidannular
enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 In vacuo modeshapesandnodalpatternsforasimplysupportedcylin-
drical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Fundamental frequency behavior of a cylindrical shell vibrating in vacuo 47
3.6 System frequency behavior and veering diagrams for a shell in contact
with LH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2
3.7 System frequency behavior and veering diagram for a shell in contact
with LOX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.8 Acoustoelastic system frequencies compared to those calculated by an
added mass model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.9 Parameter space for a cylindrical shell filled with LH . . . . . . . . . 58
2
3.10 Percentage frequency convergence ratios for a cylindrical shell in con-
tact with cylindrical and annular cavities filled with LH and LOX . . 61
2
3.11 Comparison of coupled and uncoupled mode shapes for a simply sup-
ported cylindrical shell filled with LH . . . . . . . . . . . . . . . . . 63
2
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