Response of thin-walled cylinders to aerodynamic excitation

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Response of thin-walled
cylinders to aerodynamic
excitation
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• A Doctoral Thesis. Submitted in partial fulfilment of the requirements
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University.
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RESPONSE OF THIN-WALLED CYLINDERS TO AERODYNAMIC EXCITATION
by
RAJASEKHARIAH NATARAJA
B.E., M.E.
·A Doctoral .Thesis
Submitted in partial fulfilment of the requirements
for the award of
Doctor of Philosophy of the
Loughborough University of Technology
·November 1974
Supervisor: Prof. D. J. Johns,
Ph.D., C.Eng., F.R.Ae.S., A.F.A.I.A.A.
Department of Transport Technology
@.
by Rajasekhariah Nataraja, 1974
Louqhbc'cu,h University
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RESPONSE OF THIN-WALLED CYLINDERS TO
AERODYNAMIC EXCITATION
SUMMARY
Non-linear vibrations of thin-walled shells under
aerodynamic excitation are investigated using Flugge's thin
shell theory, modified to include the effects of large deform-
ations. The theory is applicable to any type of boundary
conditions and various types of normal loading. The formula-
tion includes mean or initial deformations of the median surface.
A probabilistic-deterministic analysis of fatigue, representative
of wind effects on earth-borne structures, is proposed based
on the derived stresses and the Palmgren-Miner rule. Extentions
of the non-linear theory to include structural damping and to
analyse single mode. static collapse of thin shells are also
outlined.
In the random vibration analysis; based on energy methods,
the multi-mode random response of thin shells under wind loading
are studied. A fixed-free shell configuration is investigated in
detail, though the formulation is applicable to any type of
boundary conditions.
Oscillatory pressures round· rigid cylinders and flexible
shells are measured in the tests conducted in the 43" x 30" low
speed wind tunnel. Three low pressure transducers-DISA type
51F32-are employed for the purpose. The tests cover the range
5 5
of Reynolds numbers of 0.4 x 10 to 2.85 x 10 •
The effect of shell flexibility 6n the shedding of vortices
is studied by comparison .with the results of the rigid cylinder·
pressure measurements. Also, the 3-D effects on the oscillatory
(U)
pressures are identified.
The r:esponse of two thin shells under wind excitation
are studied experimentally, using the results from strain
The oscillatory pressures and the response
gauge records.
are analysed for the rms values by using analog methods.
Spectral analysis of these signals ar'e carried out on the
Fourier Analyser HP5451-A which employs digital analysis
techniques. Measured response spectra are compared with
those computed from the random vibration theory developed
here.
(Hi)
ACKNOVILEDGEr1ENTS
I would like to express my sincere thanks to
Professor D.J. Johns for his inspiring guidance and advice
throughout this work.
My thanks are also due to Dr. S. Gopalacharyulu for
the discussions we had at various stages of this research.
The help and assistance provided by all the techni
cian staff in fabricating the rig is gratefully acknowledged.
My thanks are due to Mr. J. Britton, Mr. B. Goodall
and Mr. M. Cook for their invaluable assistance in the
experimental work and to Miss V. Johnson for typing the
thesis.
The financial assistance provided by the Department
of Trade and Industry is gratefully acknowledged.
( iv)
CON TEN T S
PART A - THEORY
I
/
Page No.
1. Introduction 1
1.1 Summary 1
1.2 General 1
1.3 Choice of the Thin Shell Theory 2
1.3.1 Basic Thin Shell Theory 2
1.3.2 Large Deformation Considerations 5
1.4 Ground Wind Distribution 6
1.5 Flow Field Round a Circular Cylinder 7
1.6 Transfer Function Evaluation 9
1.7 Statement of the Problem- 10
1.8 Layout of the Dissertation 11
2. Formulation 13
2.1 Introduction 13
2.2 Strain-Displacement and Stress-Strain 14
Relations
2.3 Strain Energy . 15
2.3.1 Linear Strain Energy 16
2.3.2 Non-Linear Strain Energy 16
2.4 Kinetic Energy 17
2.5 Work Potential 18
2.5.1 Concentrated Load at the Tip 18
2.5.2 Line Load - Beam Function Distribution 19
2.5.3 Distributed Load - Harmonic Distribution 19
2.5.4 Wind Load Distribution 20
2.6 Derived Stresses 21
2.7 Random Loading 25
2.7.1 General 25
2.7.2 Work Potential 25
2.8 Fatigue Consideration under Wind Loading 26
2.8.1 General 26
2.8.2 Wind Distribution Function 29
2.8.3 Method of Approach 30
(v)
Page No.
3. Dynamic Analyses 34
3.1 Introduction 34
3.2 Assumed Displacement Functions 34
3.3. Characteristic Equations 35
3.4 Solution Procedure 39
3.4.1 Free Linear Vibrations 39
3.4.2 Free Non-Linear Vibrations 40
3.4.3 Forced Non-Linear Vibrations 42
3.4.4 Comparison with Duffing's Equation 43
4. Static Analysis 45
4.1 Introduction 45
4.2 Characteristic Equations and Their 45
Solution
4.2.1 Assumed Displacements 45
4.2.2 Work Potential under Steady Wind 46
4.2.3 Governing Equations 47
4.3 Stress Resultants 50
4.4 . Hedian Stresses 51
5. Combined Method of Analysis: 52
Static Plus Dynamic
5.1 Introduction 52
5.2 Hodified Displacement Functions 52
5.3 Combined Characteristic Equations 53
5.4 Solution Procedure 55
5 .. 5 Random Response Technique 56
5.6 Response to Distributed Wind Pressure 58
PART B - EXPERIMENTATION
6. Experimental Concept 62
6.1 Introduction 62
6.2 Vibration Characteristics 64
6.3 Flow Properties Around Rigid Cylinder 64
6.4 Flow Properties Around Flexible Cylinder 65
6.5 Hethods of Analysis of Analogue Signals 66