Imperfection-sensitivity of semi-symmetric branching

1977 ◽  
Vol 357 (1689) ◽  
pp. 193-211 ◽  
Keyword(s):  
General Theory ◽  
Potential Function ◽  
Catastrophe Theory ◽  
Bifurcation Theory ◽  
Elastic Stability ◽  
Structural Systems ◽  
Stiffened Plate ◽  
Imperfection Sensitivity ◽  
Branching Points ◽  
Symmetric Points

The general theory of elastic stability is extended to include the imperfection-sensitivity of twofold compound branching points with symmetry of the potential function in one of the critical modes (semi-symmetric points of bifurcation). Three very different forms of imperfection-sensitivity can result, so a subclassification into monoclinal, anticlinal and homeoclinal semi-symmetric branching is introduced. Relating this bifurcation theory to René Thom’s catastrophe theory, it is found that the anticlinal point of bifurcation generates an elliptic umbilic catastrophe, while the monoclinal and homeoclinal points of bifurcation lead to differing forms of the hyperbolic umbilic catastrophe. Practical structural systems which can exhibit this form of branching include an optimum stiffened plate with free edges loaded longitudinally, and an analysis of this problem is presented leading to a complete description of the imperfection-sensitivity. The paper concludes with some general remarks concerning the nature of the optimization process in design as a generator of symmetries, instabilities and possible compound bifurcations.

scholarly journals On Curve Veering and Flutter of Rotating Blades

1994 ◽  
Vol 116 (3) ◽  
pp. 702-708 ◽  
Author(s):  
D. Afolabi ◽  
O. Mehmed
Keyword(s):  
Algebraic Geometry ◽  
Catastrophe Theory ◽  
Algebraic Curves ◽  
Rotation Speed ◽  
Elastic Stability ◽  
Rotating Blades ◽  
Rotating Blade ◽  
Modes Of Vibration ◽  
Curve Veering ◽  
As If

The eigenvalues of rotating blades usually change with rotation speed according to the Stodola-Southwell criterion. Under certain circumstances, the loci of eigenvalues belonging to two distinct modes of vibration approach each other very closely, and it may appear as if the loci cross each other. However, our study indicates that the observable frequency loci of an undamped rotating blade do not cross, but must either repel each other (leading to “curve veering”), or attract each other (leading to “frequency coalescence”). Our results are reached by using standard arguments from algebraic geometry—the theory of algebraic curves and catastrophe theory. We conclude that it is important to resolve an apparent crossing of eigenvalue loci into either a frequency coalescence or a curve veering, because frequency coalescence is dangerous since it leads to flutter, whereas curve veering does not precipitate flutter and is, therefore, harmless with respect to elastic stability.

Initial Postbuckling Behavior of Imperfect, Antisymmetric Cross-Ply Cylindrical Shells Under Torsion

1987 ◽  
Vol 54 (1) ◽  
pp. 174-180 ◽  
Author(s):  
David Hui ◽  
I. H. Y. Du
Keyword(s):  
Cylindrical Shells ◽  
Elastic Stability ◽  
Buckling Load ◽  
Postbuckling Behavior ◽  
Imperfection Sensitivity ◽  
Torsional Buckling ◽  
Buckling Mode ◽  
Young’S Moduli ◽  
Coupling Behavior ◽  
Parameter Variations

This paper deals with the initial postbuckling of antisymmetric cross-ply closed cylindrical shells under torsion. Under the assumptions employed in Koiter’s theory of elastic stability, the structure is imperfection-sensitive in certain intermediate ranges of the reduced-Batdorf parameter (approx. 4 ≤ ZH ≤ 20.0). Due to different material bending-stretching coupling behavior, the (0 deg inside, 90 deg outside) two-layer clamped cylinder is less imperfection sensitive than the (90 deg inside, 0 deg outside) configuration. The increase in torsional buckling load due to a higher value of Young’s moduli ratio is not necessarily accompanied by a higher degree of imperfection-sensitivity. The paper is the first to consider imperfection shape to be identical to the torsional buckling mode and presents concise parameter variations involving the reduced-Batdorf paramter and Young’s moduli ratio.

Modeling of Elastically Coupled Bodies: Part I—General Theory and Geometric Potential Function Method

1998 ◽  
Vol 120 (4) ◽  
pp. 496-500 ◽  
Author(s):  
Ernest D. Fasse ◽  
Peter C. Breedveld
Keyword(s):  
General Theory ◽  
Potential Energy ◽  
Potential Function ◽  
Geometric Modeling ◽  
Function Method ◽  
Rigid Bodies ◽  
Potential Functions ◽  
Energy Functions ◽  
Automatic Computation ◽  
Potential Energy Functions

This paper looks at spatio-geometric modeling of elastically coupled rigid bodies. Desirable properties of compliance families are defined (sufficient diversity, parsimony, frame-indifference, and port-indifference). A novel compliance family with the desired properties is defined using geometric potential energy functions. The configuration-dependent wrenches corresponding to these potential functions are derived in a form suitable for automatic computation.

scholarly journals General theory of elastic stability

1956 ◽  
Vol 14 (2) ◽  
pp. 133-144 ◽  
Author(s):  
Carl E. Pearson
Keyword(s):  
General Theory ◽  
Elastic Stability

Singularities of dispersion relations

1983 ◽  
Vol 95 (3-4) ◽  
pp. 301-316 ◽  
Author(s):  
G. Dangelmayr
Keyword(s):  
Catastrophe Theory ◽  
Bifurcation Theory ◽  
Normal Forms ◽  
Dispersion Relations ◽  
Bifurcation Parameter ◽  
Special Cases ◽  
Wave Numbers ◽  
Imperfect Bifurcation ◽  
Generic Singularities ◽  
Zero Frequency

SynopsisGeneric singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.

scholarly journals V. On the general theory of elastic stability

1914 ◽  
Vol 213 (497-508) ◽  
pp. 187-244 ◽  
Keyword(s):  
General Theory ◽  
Boundary Conditions ◽  
Radial Pressure ◽  
Elastic Stability ◽  
Circular Ring ◽  
Theory Of Elasticity ◽  
Comprehensive Theory ◽  
The Subject ◽  
The Stability ◽  
Straight Rod

Problems which deal with the stability of bodies in equilibrium under stress are so distinct from the ordinary applications of the theory of elasticity that it is legitimate to regard them as forming a special branch of the subject. In every other case we are concerned with the integration of certain differential equations, fundamentally the same for all problems, and the satisfaction of certain boundary conditions; and by a theorem due to Kiechiioff we are entitled to assume that any solution which we may discover is unique. In these problems we are confronted with the possibility of two or more configurations of equilibrium , and we have to determine the conditions which must be satisfied in order that the equilibrium of any given configuration may be stable. The development of both branches has proceeded upon similar lines. That is to say, the earliest discussions were concerned with the solution of isolated examples rather than with the formulation of general ideas. In the case of elastic stability, a comprehensive theory was not propounded until the problem of the straight strut had been investigated by Euler, that of the circular ring under radial pressure by M. Lévy and G. H. Halphen, and A. G. Greenhill had discussed the stability of a straight rod in equilibrium under its own weight, under twisting couples, and when rotating.

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