Analyses of Sandwich Beams and Plates with Viscoelastic Cores

216 Pages·2001·1.537 MB·English

by  Gang W.

ABSTRACT
Title of Dissertation: Analyses of Sandwich Beams and Plates
with Viscoelastic Cores
Gang Wang, Doctor of Philosophy, 2001
Dissertation directed by: Associate Professor Norman M. Wereley
Department of Aerospace Engineering
A hybrid damping scheme using passive constrained damping layers (PCLD),
and surface bonded piezoceramic actuators was proposed for interior cabin noise
andvibrationcontrolinhelicopters. Inordertoevaluatetheperformanceofthese
treatments, we need to understand the dynamic behavior of sandwich structures.
The analyses of sandwich structures are complicated by the frequency dependent
stiffness and damping properties of viscoelastic materials. The methods devel-
oped in this thesis specifically deal with finite element methods and assumed
modes methods to this problem.
A spectral finite element method (SFEM) was developed in the frequency
domain for sandwich beam analysis. The results of natural frequencies and fre-
quency responses for two cantilevered beams with different span of PCLD treat-
ments were presented and validated by experimental results and other analyses;
including the assumed modes method (AM), and conventional finite element
method (CFEM). The SFEM method implicitly accounts for frequency depen-
dent stiffness and damping of viscoelastic materials. However, CFEM and AM
method have to use additional internal dissipation coordinates to account for
these properties. The Golla-Hughes-McTavish (GHM) damping method was
used in both analyses. Also SFEM improves accuracy of frequency predictions
compared to the results of CFEM and AM method because of its higher order
interpolation functions.
We expected to extend SFEM method to two-dimensional sandwich plate
structures. But it is extremely difficult to solve the governing equations for a
sandwich plate. An alternative method was developed to update the traditional
AM method by using plate mode shapes. The plate mode shape functions were
solved directly based on the Kantorovich variational method for both transverse
bending and in-plane vibration of isotropic rectangular plates. These plate mode
shapeswereemployed tocalculatesandwich platesinAMmethod. Theresultsof
naturalfrequencies, lossfactorsandfrequency responsefunctionswerecalculated
and validated by experimental data and the results by using beam and rod mode
shapes. The comparable results were achieved for both analyses with less modes
in the case of using plate mode shapes.
Analyses of Sandwich Beams and Plates
with Viscoelastic Cores
by
Gang Wang
Dissertation submitted to the Faculty of the Graduate School
of The University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2001
Thesis Committee:
Associate Professor Norman M. Wereley, Chairman/Advisor
Professor Amr Baz
Professor Inderjit Chopra
Professor Sung W. Lee
Associate Professor Darryll J. Pines
(cid:1)c Copyright by
Gang Wang
2001
DEDICATION
To my parents and my teachers.
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ACKNOWLEDGEMENTS
My sincere gratitude and appreciation goes to my advisor, Dr. Nor-
man Wereley. His kindness, encouragement, and support, helped my
walk through this path. His personal concerns for my family are
especially appreciated.
I would also like to thank my dissertation committee members, Dr.
Baz, Dr. Chopra, Dr. Lee. and Dr. Pines, for their suggestions, and
their enthusiasm in my research.
Overthecourseofmygraduatestudies, Ihavesharedhappymoments
and tough times with my colleagues at the Alfred Gessow Rotorcraft
Center. I am grateful for this.
I wish to thank my wife, Ying, for her patience, help, and love during
those years. Many thanks to Dr. Chang, for his helpful discussions
on mathematics in my research as well as his personal concerns for
my family.
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Above all, I give thanks to God.
Finally, this research was supported by U.S Army Research Office
under the FY96 MURI in Active Control of Rotorcraft Vibration
and Acoustics, with Dr. Gary Anderson and Dr. Tom Doligalski
serving as technical monitors. Lab equipment support was provided
under the FY96 Defense University Research Instrumentation Pro-
gram (DURIP) Contract No. DAAH-0496-10301, and Dr. Gary
Anderson serving as technical monitor.
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TABLE OF CONTENTS
LIST OF TABLES ix
LIST OF FIGURES xiv
1 Introduction 1
1.1 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . 1
1.2 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Viscoelastic Materials . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Sandwich Beams . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Scope of the Present Research . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Sandwich Beam . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Sandwich Plate . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Viscoelastic Materials 19
2.1 Characteristics of Viscoelastic Materials . . . . . . . . . . . . . . 20
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2.2 Classical Damping Models . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Modern Damping Models . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Fractional Derivatives Model . . . . . . . . . . . . . . . . . 26
2.3.2 AFT and ADF Models . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Golla-Hughes-McTavish Model . . . . . . . . . . . . . . . . 28
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Comparison of Analyses of Sandwich Beams 40
3.1 Assumptions and Governing Equations . . . . . . . . . . . . . . . 41
3.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . 43
3.2 Spectral Finite Element Method . . . . . . . . . . . . . . . . . . . 45
3.2.1 Isotropic Rod and Beam . . . . . . . . . . . . . . . . . . . 47
3.2.2 Sandwich Beam . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Conventional Finite Element Method . . . . . . . . . . . . . . . . 57
3.4 Assumed Modes Method . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Solution Type/Methods . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7.1 Modal Frequency Predictions . . . . . . . . . . . . . . . . 64
3.7.2 Number of Elements . . . . . . . . . . . . . . . . . . . . . 65
3.7.3 Frequency Response Functions . . . . . . . . . . . . . . . . 66
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3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Analyses of Sandwich Plates: Part I 80
4.1 Assumptions and Governing Equations . . . . . . . . . . . . . . . 81
4.1.1 Asssumptions . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.2 Sandwich Plate Energies and Governing Equations . . . . 83
4.2 Assumed Modes Method Using Beam and Rod modes . . . . . . . 88
4.2.1 Analytical Validation: Simply Supported . . . . . . . . . . 90
4.2.2 Experimental Validation: All Four Sides Clamped . . . . . 92
4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Plate Mode Shapes 110
5.1 Plate In-plane Mode Shape . . . . . . . . . . . . . . . . . . . . . . 112
5.1.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.2 Validation and Results . . . . . . . . . . . . . . . . . . . . 122
5.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Plate Bending Vibration . . . . . . . . . . . . . . . . . . . . . . . 133
5.3 Results for Plate Bending and In-plane Mode Shape Functions . . 139
6 Analyses of Sandwich Plate: Part II 146
6.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
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