Herrmann Dynamic Stability of Structures

Proceedings of an International Conference Held at Northwestern University, Evanston, Illinois, October 18-20, 1965
1. Auflage 2014
ISBN: 978-1-4832-2324-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 324 Seiten

Dynamic Stability of Structures covers the proceedings of an International Conference on Dynamic Stability of Structures, held in Northwestern University, Evanston, Illinois on October 18-20, 1965, jointly sponsored by the Air Force of Scientific Research and Northwestern University. The conference aims to delineate the various categories of dynamic stability phenomena. This book is organized into six sections encompassing 20 chapters that tackle general topics such as mathematical methods of analysis, physical phenomena, design applications in engineering, and reports of field research. The first two sections deal with the fundamentals, principles, and concept of dynamic stability, as well as an introduction to the use of computing machines as an aid in studying the motions of complicated dynamical systems. The succeeding two sections highlight the statistical aspects in the structural stability theory and certain problems of structural dynamic. These sections also look into the dynamic buckling of elastic structures and the buckling of long slender ships due to wave-induced whipping. The last two sections explore the stability and vibration problems of mechanical systems under harmonic excitation and the dynamic buckling under step loading. These sections also include discussions on the nonlinear dynamic response of shell-type structures and of a column under random loading, as well as Italian research in the field. Structural and mechanical engineers will find this book invaluable.

Autoren/Hrsg.

Herrmann, George

Inhaltsverzeichnis

1;Front Cover;1
2;Dynamic Stability of Structures;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;SESSION I;12
6.1;CHAPTER 1. WELCOMING ADDRESS;14
6.2;CHAPTER 2. OPENING REMARKS;16
6.3;CHAPTER 3. DYNAMIC STABILITY OF
STRUCTURES;18
6.3.1;INTRODUCTION;18
6.3.2;DEFINITION AND CLASSIFICATION OF PROBLEMS OF DYNAMIC STABILITY;20
6.3.3;BUCKLING CRITERIA;30
6.3.4;THE TANGENT MODULUS LOAD;38
6.3.5;CONCLUSIONS;50
6.3.6;REFERENCES;50
7;SESSION II;54
7.1;CHAPTER 4. STABILITY OF CONTINUOUS
SYSTEMS;56
7.1.1;1. INTRODUCTION;56
7.1.2;2. DEFINITIONS AND CRITERIA FOR STABILITY;56
7.1.3;3. THE DYNAMICS OF THE MOTION OF COLD AND WARM FRONT IN THE ATMOSPHERE;59
7.1.4;4. MOTIONS OF ELASTIC SURFACES OF REVOLUTION;61
7.1.5;REFERENCES;63
7.2;CHAPTER 5. STABILITY THEORY
AND THE ASYMPTOTIC BEHAVIOR OF DYNAMICAL SYSTEMS;64
7.2.1;1. INTRODUCTION;64
7.2.2;2. AUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS;65
7.2.3;3. NON AUTONOMOUS SYSTEMS;70
7.2.4;4. AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION;71
7.2.5;REFERENCES;73
8;SESSION III;76
8.1;CHAPTER 6. STATISTICAL ASPECTS IN THE THEORY OF STRUCTURAL STABILITY;78
8.1.1;INTRODUCTION;78
8.1.2;1. FOKKER–PLANCK–KOLMOGOROV EQUATION FOR AN ELASTIC SYSTEM SUBJECTED TO RANDOM FORCES;79
8.1.3;2. SOME EXACT SOLUTIONS OF THE FOKKER–PLANCK–KOLMOGOROV EQUATION AND ITS MECHANICAL MEANING;81
8.1.4;3. THE EXPANDED STABILITY PROBLEM AS A RELIABILITY PROBLEM. PONTRIAGIN'S EQUATION;83
8.1.5;4. AN APPROXIMATE METHOD OF SOLUTION;84
8.1.6;5. BEHAVIOR AND RELIABILITY OF A THIN CURVED ELASTIC PANEL SUBJECTED BY RANDOM FORCES;89
8.1.7;REFERENCES;91
8.2;CHAPTER 7. DYNAMIC BUCKLING OF ELASTIC STRUCTURES: CRITERIA
AND ESTIMATES;94
8.2.1;INTRODUCTION;94
8.2.2;STATIC AND DYNAMIC LOADS;94
8.2.3;STATIC BUCKLING;95
8.2.4;SINGLE-MODE ANALYSES OF DYNAMIC BUCKLING;99
8.2.5;GENERALIZED CRITERIA FOR DYNAMIC BUCKLING;105
8.2.6;SOME MULTI-MODE STUDIES;109
8.2.7;A SPECIAL MANY-MODE STUDY;113
8.2.8;CONCLUDING REMARKS;117
8.2.9;REFERENCES;117
9;SESSION IV;118
9.1;CHAPTER 8. SURVEY OF PROBLEMS OF
STRUCTURAL DYNAMIC STABILITY IN VEHICLE DESIGN;120
9.2;CHAPTER 9. STABILITY PROBLEMS IN THE CONTROL OF SATURN LAUNCH VEHICLES;124
9.2.1;INTRODUCTION;124
9.2.2;THE BASIC CONTROL SYSTEM PROBLEM;124
9.2.3;ANALYSIS OF STABILITY;126
9.2.4;PROBLEM AREAS;135
9.2.5;CONCLUSION;138
9.2.6;REFERENCES;138
9.3;CHAPTER 10. BUCKLING
OF LONG SLENDER SHIPS DUE TO WAVE-INDUCED WHIPPING;140
9.3.1;INTRODUCTION;140
9.3.2;FULL SCALE TRIALS;144
9.3.3;DIFFERENTIAL EQUATIONS OF WHIPPING;153
9.3.4;COMPARISON OF THEORY AND EXPERIMENT;162
9.3.5;SUMMARY;167
9.3.6;REFERENCES;168
9.4;CHAPTER 11. ELIMINATION OF THE POGO INSTABILITY FROM THE GEMINI LAUNCH VEHICLE;170
9.4.1;INTRODUCTION;170
9.4.2;SYSTEM DESCRIPTION;171
9.4.3;FEEDLINE DYNAMIC CHARACTERISTICS;176
9.4.4;SUPPRESSION DEVICES;178
9.4.5;CONCLUDING REMARKS;185
9.4.6;REFERENCES;185
10;SESSION V;186
10.1;CHAPTER 12. STABILITY AND VIBRATION PROBLEMS OF MECHANICAL SYSTEMS UNDER HARMONIC EXCITATION;188
10.1.1;1. THE MECHANICAL SYSTEMS;188
10.1.2;2. THE METHOD OF AVERAGING;190
10.1.3;3. GENERAL RESULTS;194
10.1.4;4. THREE CHARACTERISTIC EXAMPLES;198
10.1.5;5. ADDITIONAL TOPICS;204
10.1.6;REFERENCES;206
10.2;CHAPTER 13. DYNAMIC PLASTIC BUCKLING;208
10.2.1;I. INTRODUCTION;208
10.2.2;II. THE MODERATELY THICK
CYLINDRICAL SHELL. UNIFORM RADIAL IMPULSE ;209
10.2.3;III. THE ROD IN AXIAL PLASTIC FLOW BUCKLING WITHIN A COMPRESSION WAVE;219
10.2.4;IV. THE THIN
CYLINDRICAL SHELL. UNIFORM RADIAL IMPULSE ;227
10.2.5;V. THE THIN STRIP BUCKLING
ELASTICALLY WITHIN AN AXIAL COMPRESSION WAVE;229
10.2.6;VI. THE THIN CYLINDRICAL SHELL BUCKLING
ELASTICALLY WITHIN AN AXIAL COMPRESSION WAVE;234
10.2.7;REFERENCES;237
11;SESSION VI;240
11.1;CHAPTER 14. DYNAMIC BUCKLING UNDER STEP LOADING;242
11.1.1;1. INTRODUCTION;242
11.1.2;2. GENERAL THEORY FOR THE PERFECT SYSTEM;246
11.1.3;3. GENERAL THEORY FOR IMPERFECT SYSTEMS;249
11.1.4;4. INTERPRETATION OF THE GENERAL THEORY;257
11.1.5;5. EXPERIMENTAL VERIFICATION;259
11.1.6;REFERENCES;262
11.2;CHAPTER 15. SOME STUDIES ON THE NONLINEAR DYNAMIC RESPONSE OF SHELL-TYPE STRUCTURES;264
11.2.1;INTRODUCTION;264
11.2.2;SYMBOLS;264
11.2.3;NONLINEAR FLEXURAL VIBRATIONS OF THIN-WALLED CIRCULAR CYLINDERS;265
11.2.4;DISCUSSION OF THE CYLINDER RESULTS;269
11.2.5;DYNAMIC SNAP-THROUGH BUCKLING OF ELASTIC SHALLOW CONICAL AND SPHERICAL CAPS;274
11.2.6;CONCLUDING REMARKS;280
11.2.7;REFERENCES;281
11.3;CHAPTER 16. DYNAMIC STABILITY OF A COLUMN UNDER RANDOM LOADING;282
11.3.1;INTRODUCTION;282
11.3.2;FORMULATION;282
11.3.3;GAUSSIAN WHITE NOISE LOAD VARIATION;283
11.3.4;NON-WHITE LOAD VARIATION;285
11.3.5;SMALL RANDOM VARIATION IN AXIAL LOAD;288
11.3.6;REFERENCES;292
11.4;CHAPTER 17. INTERACTION OF MECHANICAL AND AEROELASTIC
INSTABILITIES OF A CIRCULAR CYLINDRICAL SHELL;294
11.4.1;NOTATION;294
11.4.2;INTRODUCTION;295
11.4.3;EXPERIMENTAL OBSERVATONS;296
11.4.4;FORMULATION OF A MATHEMATICAL PROBLEM;302
11.4.5;SOLUTION OF THE DIFFERENTIAL EQUATIONS;305
11.4.6;THE EIGENVALUE PROBLEM;307
11.4.7;CONCLUSION;312
11.4.8;REFERENCES;312
11.5;CHAPTER 18. DYNAMIC BUCKLING OF A CIRCULAR RING CONSTRAINED IN A RIGID CIRCULAR SURFACE;314
11.5.1;INTRODUCTION;314
11.5.2;THEORETICAL FORMULATION;314
11.5.3;RESULTS;317
11.5.4;CONCLUSIONS;325
11.5.5;REFERENCES;326
11.6;CHAPTER 19. ENERGY CONSIDERATIONS IN THE ANALYSIS OF STABILITY
OF NONCONSERVATIVE STRUCTURAL SYSTEMS;328
11.6.1;1. INTRODUCTION;328
11.6.2;2. CONSERVATIVE SYSTEMS;329
11.6.3;3. NONCONSERVATIVE SYSTEMS;330
11.6.4;4. STABILITY CRITERION;331
11.6.5;5. EXAMPLE;333
11.6.6;REFERENCES;336
11.7;CHAPTER 20. RECENT ITALIAN CONTRIBUTIONS IN THE FIELD OF DYNAMIC STABILITY OF
STRUCTURES;338
11.7.1;REFERENCES;341

Leseproben

DYNAMIC STABILITY OF STRUCTURES†
Nicholas J. Hoff,     Stanford University, Stanford, California Abstract
Dynamic stability is defined and classified, and examples are given for the various classes of problems. Criteria are developed for practical stability and it is shown that in a practical elastic column tested in a conventional testing machine stress reversal always precedes the attainment of the maximum load. The two coincide, however, in the limit when the initial deviations of the column axis from straightness and the loading speed tend to zero. INTRODUCTION
THIRTY-FIVE years ago, when he was working in the airplane industry, the author became aware of the importance of dynamic stability. As a beginning engineer, he had to carry out the complete dynamic, aerodynamic, and structural analysis of a new training plane which was then being designed by four of his equally inexperienced colleagues. It may be of interest to add that in one year’s time the design and analysis were completed and the prototype manufactured and successfully test flown; the story of the difficulties encountered is not part of the present paper. In accordance with governmental airworthiness requirements, the static stability of the straight-line flight of the plane had to be checked under climbing, horizontal flying and gliding conditions. This was done by plotting first against the angle of attack the aerodynamic wing moments with respect to an axis through the center of gravity of the airplane perpendicular to the plane of symmetry of the airplane. The curve obtained always indicates instability for the conventional airplane: If the angle of attack a of the wing is increased, the aerodynamic wing moment M also increases (see Fig. 1)‡
FIG. 1 Change in moment equilibrium of airplane in consequence of change in angle of attack.‡ The airplane is stabilized with the aid of the horizontal tail surfaces. The change in the moment of the tail surface with respect to its own centroidal axis is insignificant compared to the change in the moment of the vertical upward force acting on the tail surface with respect to the center of gravity of the airplane. An increase in the angle of attack a of the wing causes an increase in the angle of attack of the tail surface which increases the lifting force PH of the horizontal tail surface. This increment ?PH multiplied with the distance h between the center of gravity of the airplane and the line of action of the tail force is a negative moment capable of counterbalancing the positive increment ?M in the wing moment if h and the area of the tail surface are sufficiently large. The job of the designer–analyst was therefore to insure, under all normal flight conditions, that h(dPH/da) da had a greater absolute value than (dM/da) da. It was quite a revelation for the author to find out from the literature [1] that the condition mentioned was only a necessary but not a sufficient condition of an automatically stable flight of the airplane. If the dynamic equations of the motion of a rigid airplane in its own plane of symmetry are written for small oscillations about the steady-state conditions, and if the aerodynamic coefficients are inserted with their steady-state values, assumption of an exponential solution results in a quartic. Solution of the quartic can lead to any of the four different behavior patterns shown in Fig. 2. Curve a indicatesan asymptotic return to the initial state after a disturbance, which obviously means stability of the initial state. A second stable pattern is shown in curve b where the disturbance is followed by damped oscillations and eventually a return to the initial state. Curve c corresponds to static and dynamic instability. Finally, the airplane characterized by curve d is statically stable but dynamically unstable; the static restoring force or moment acts in the sense necessary for stability but the ensuing oscillations increase in amplitude rather than damp out.
FIG. 2 Response of airplane to disturbance. In the literature of structural stability one can also find systems that appear to be fully stable when investigated by static methods and whose displacements from the state of initial equilibrium nevertheless increase with time following a disturbance, just as it is indicated in Fig. 2d. Such systems evidently must be analyzed with the aid of the dynamic, or kinetic, method in which the motion of the system following a disturbance is studied. The dynamic equations of motion are also needed when the loads applied to the structure vary significantly with time. In a recent paper Herrmann and Bungay [2] followed usage established in the theory of aeroelasticity when they proposed that structural instability of type (c) be designated as divergence, and that of type (d) as flutter. DEFINITION AND CLASSIFICATION OF PROBLEMS OF DYNAMIC STABILITY
Definition
In the introduction to the first English edition of their monumental textbook entitled Engineering Dynamics, Biezeno and Grammel [3] explain that, following Kirchhoff’s definition, dynamics is the science of motion and forces, and thus includes statics, which is the study of equilibrium, and kinetics, which treats of the relationship between forces and motion. According to this interpretation of the meaning of the words, the dynamic test of equilibrium mentioned in the Introduction of this paper should be called a kinetic test, and this is indeed the terminology adopted by Ziegler [4] in his studies of the stability of non-conservative systems. But dynamics is generally accepted as the antonym of statics in everyday usage, and this is the sense in which it is used in the title of the International Conference on Dynamic Stability of Structures. A number of significantly different concepts can be included in the meaning of the term dynamic stability of structures. One of them is the stability of motion of an elastic system subjected to forces that are functions of time. Another is the study of the stability of a system subjected to constant forces as long as the study is carried out with the aid of the dynamic equations of motion; such an investigation is designated by Ziegler as a stability analysis with the aid of the kinetic criterion. In this paper any stability problem analyzed with the aid of Newton’s equations of motion, or by any equivalent method, will be considered a dynamic stability problem. The classification of the problems that follow is not fundamental in any sense of the word. Its purpose is simply to group together problems that are usually treated by similar mathematical methods, or analyzed by the same group of research men. A more fundamental classification could be based on the principles proposed by Ziegler [5] in his article in Advances in Applied Mechanics. Parametric resonance
Among the problems of the dynamic stability of structures probably the best known subclass is constituted by the problems of parametric excitation, or parametric resonance. A typical example is the initially straight prismatic column whose two ends are simply supported and upon which a periodic axial compressive load is acting (Fig. 3). Such a column is known to develop lateral oscillations if its straight-line equilibrium is disturbed. Depending upon the magnitude and the frequency of the pulsating axial load, the linear Hill or Mathieu equation defining the lateral displacements of the column may yield bounded or unbounded values for these displacements. The structural analyst can be useful to the design engineer if he points out the regions in the frequency-amplitude plane that must be avoided if the column should never deviate noticeably from its initial straight-line equilibrium configuration.
FIG. 3 Parametric resonance. According to Bolotin [6], the first solution of this problem was given by Beliaev [7] in 1924; this was followed by an analysis by Krylov and Bogoliubov [8] in 1935. In the United States, Lubkin [9], a student of Stoker, solved the problem in a doctoral dissertation submitted to New York University in 1939; the results are more easily available in an article by Lubkin and Stoker [10] printed in 1943. The results of a theoretical and experimental investigation of the subject were published by Utida and Sezawa [11] in 1940. Another early solution of the parametric excitation problem of the column is due to Mettler [12] (1940); as a matter of fact, this problem is called Mettler’s problem in Ziegler’s comprehensive work on elastic stability. The parametric excitation of thin flat plates was first discussed by Einaudi [13] (1936). A rather complete treatment of known solutions of the parametric resonance problem can be found in a book by Bolotin [6] in which the effects of friction and nonlinearities are also discussed. It is perhaps unfortunate that the phenomena...