Characteristics of an Eigensystem Describing the Elastic Stability of a Thin Cantilever Beam

1962 ◽  
Vol 66 (623) ◽  
pp. 722-724 ◽  
Author(s):  
B. Saravanos
Keyword(s):  
Equilibrium Problem ◽  
Cantilever Beam ◽  
Elastic Stability ◽  
Additional Component ◽  
Connecting Rod ◽  
The Stability

The stability of a uniform cantilever beam subjected to an articulated tip load has been analysed as an equilibrium problem of static elastic stability. It was found there that the articulation of the connecting rod applying the tip load introduced additional component forces during the process of beam deformation.

scholarly journals The elastic stability of an annular plate

1924 ◽  
Vol 106 (737) ◽  
pp. 268-284 ◽  
Keyword(s):  
Annular Plate ◽  
Practical Interest ◽  
Middle Surface ◽  
Elastic Stability ◽  
Thin Plates ◽  
Thin Shells ◽  
Stability Equation ◽  
Inner Radius ◽  
Plane Plate ◽  
The Stability

1. Introduction and Summary. —This paper deals with the elastic stability of a circular annular plate under uniform shearing forces applied at its edges. Investigations of the stability of plane plates are altogether simpler than those necessary in the case of curved plates or shells. In the first place, as shown by Mr. R. V. Southwell, two of the three equations of stability relate to a mode of instability that is not of practical interest, and are entirely independent of the third equation which gives the ordinary mode of instability resulting in the familiar bending of the middle surface of the plate. Consequently with a plane plate there is only one equation of stability to be solved, as contrasted with the case of a shell where the three equations are dependent, and must all be solved. In the second place the theory of thin shells can be used with confidence in a plane plate problem, though a more laborious procedure is necessary to deal adequately with a shell. The only stability equation required for the annular plate is therefore deduced without trouble from the theory of thin shells, and its solution presents no difficulty in the case of uniform shearing forces. A numerical discussion is given of the stability of the plate under such forces, the “favourite type of distortion” and the stess that will produce it being obtained for plates with clamped edges in wich the ratio of the outer to the inner radius exceeds 3·2. To some extent to results have been checked by experiment, in which part of the work the viter is indebted to Prof. G. I. Taylor for his valuable help and advice. Distrtion of the type predicted by the theory took place in the two thin plates of rober different ratio of radii, which were used. The disposition of the loci of points which undergo maximum normal displace nt gives some idea of the appearance of the plate after distortion has taken pce. The points have been calculated for a plate in which the ratio of radii 4·18, and the loci are shown on a diagram, which may be compared with a potograph of a distorted plate in which this ratio is 4·3. The ratio of normal dplacements of points of the plate can be seen from contours drawn on the ne diagram. (See pp. 280, 281.)

Vibrations of Elastic Connecting Rod of a High-Speed Slider-Crank Mechanism

1971 ◽  
Vol 93 (2) ◽  
pp. 636-644 ◽  
Author(s):  
Peter W. Jasinski ◽  
Ho Chong Lee ◽  
George N. Sandor
Keyword(s):  
Small Parameter ◽  
High Speed ◽  
Elastic Stability ◽  
Method Of Averaging ◽  
Connecting Rod ◽  
Transverse Vibrations ◽  
Elastic Bar ◽  
The Moment ◽  
Steady State Solutions ◽  
Crank Mechanism

The research involved in this paper falls into the area of analytical vibrations applied to planar mechanical linkages. Specifically, a study of the vibrations, associated with an elastic connecting-bar for a high-speed slider-crank mechanism, is made. To simplify the mathematical analysis, the vibrations of an externally viscously damped uniform elastic connecting bar is taken to be hinged at each end (i.e., the moment and displacement are assumed to vanish at each end). The equations governing the vibrations of the elastic bar are derived, a small parameter is found, and the solution is developed as an asymptotic expansion in terms of this small parameter with the aid of the Krylov-Bogoliubov method of averaging. The elastic stability is studied and the steady-state solutions for both the longitudinal and transverse vibrations are found.

On A Formulation of the Elastic Stability Equations of Circular Cylindrical Shells Under Variable Internal Pressure

1985 ◽  
Vol 9 (2) ◽  
pp. 64-69
Author(s):  
J.L. Urrutia-Galicia ◽  
A.N. Sherbourne
Keyword(s):  
Mathematical Model ◽  
Internal Pressure ◽  
Cylindrical Shells ◽  
Direct Analysis ◽  
Elastic Stability ◽  
Equilibrium Path ◽  
Circular Cylindrical Shells ◽  
The Stability ◽  
Variable Internal Pressure ◽  
The Mathematical Model

The mathematical model of the stability analysis of circular cylindrical shells under arbitrary internal pressure is presented. The paper consists of a direct analysis of the equilibrium modes in the neighbourhood of the unperturbed principal equilibrium path. The final stability condition results in a completely symmetric differential operator which is then compared with current theories found in the literature.

EFFECT OF ADDED TIP MASS ON THE NONLINEAR FLAPWISE AND CHORDWISE VIBRATION OF CANTILEVER COMPOSITE BEAM UNDER BASE EXCITATION

2012 ◽  
Vol 12 (02) ◽  
pp. 285-310 ◽  
Author(s):  
M. EFTEKHARI ◽  
M. MAHZOON ◽  
S. ZIAEI-RAD
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Laminated Composite ◽  
Base Excitation ◽  
System A ◽  
Autoparametric Resonance ◽  
The Stability ◽  
Tip Mass ◽  
Made In

In this paper, a comparative study is performed for a symmetrically laminated composite cantilever beam with and without a tip mass under harmonic base excitation. The base is subjected to both flapwise and chordwise excitations tuned to the primary resonances of the two directions and conditions of 2:1 autoparametric resonance. In the literature, the governing nonlinear equations of the same problem without tip mass have been derived using the extended Hamilton's principle. Extension is made in this study to include the effect of a tip mass on the response of the beam. The natural frequencies are obtained numerically using the diversity guided evolutionary algorithm (DGEA). Next, the multiple scales method is applied to determine the nonlinear response and stability of the system. A set of four first-order differential equations describing the modulation of the amplitudes and phases of interacting modes are derived for the perturbation analysis. For verification, the above equations are reduced to the special case of the cantilever beam without tip mass for comparison with existing results. Finally, the effect of the tip mass on the stability of the fixed points and on the amplitude of oscillation about the equilibrium points in both the frequency and force modulation responses is examined.

Adhesion of lipid tubules in an assembly

1998 ◽  
Vol 9 (5) ◽  
pp. 485-506 ◽  
Author(s):  
RICCARDO ROSSO ◽  
EPIFIANO G. VIRGA
Keyword(s):  
Surface Tension ◽  
External Field ◽  
Equilibrium Problem ◽  
Biological Systems ◽  
Critical Value ◽  
Stability Diagram ◽  
Energy Functional ◽  
Highly Nonlinear ◽  
The Stability ◽  
Lipid Tubules

We study a unilateral equilibrium problem for the energy functional of a lipid tubule subject to an external field. These tubules, which constitute many biological systems, may form assemblies when they are brought in contact, and so made to adhere to one another along at interstices. The contact energy is taken to be proportional to the area of contact through a constant, which is called the adhesion potential. This competes against the external field in determining the stability of patterns with flat interstices. Though the equilibrium problem is highly nonlinear, we determine explicitly the stability diagram for the adhesion between tubules. We conclude that the higher the field, the lower the adhesion potential needed to make at interstices energetically favourable, though its critical value depends also on the surface tension of the interface between the tubules and the isotropic fluid around them.

Determination of the Fluid-Elastic Stability Threshold in the Presence of Turbulence: A Theoretical Study

1989 ◽  
Vol 111 (4) ◽  
pp. 407-419 ◽  
Author(s):  
J. H. Lever ◽  
G. Rzentkowski
Keyword(s):  
Theoretical Study ◽  
Cross Flow ◽  
Elastic Stability ◽  
Mass Ratio ◽  
Response Curves ◽  
Stability Threshold ◽  
Pitch Ratio ◽  
The Stability ◽  
Experimental Findings

A model has been developed to examine the effect of the superposition of turbulent buffeting and fluid-elastic excitation on the response of a single flexible tube in an array exposed to cross-flow. The modeled response curves for a 1.375-pitch ratio parallel triangular array are compared with corresponding experimental data for the same array; reasonably good qualitative agreement is seen. Turbulence is shown to have a significant effect on the determination of the stability threshold for the array, with increasing turbulent buffeting causing a reduction in the apparent critical velocity. The dependence of turbulence response on mass ratio is also found to yield a slight independence between mass and damping parameters on stability threshold estimates, which may account for similar experimental findings. Different stability criteria are compared, and an attempt is made to provide some guidance in the interpretation of response curves from actual tests.

Elastic Stability of Rubber Products

1987 ◽  
Vol 60 (5) ◽  
pp. 957-965 ◽  
Author(s):  
Farhad Tabaddor
Keyword(s):  
Finite Element ◽  
Symmetric Solution ◽  
Biaxial Loading ◽  
Elastic Stability ◽  
Equilibrium Solutions ◽  
Unstable Solutions ◽  
Symmetrical Loading ◽  
Symmetric Loading ◽  
The Stability

Abstract Due to severe nonlinearities, inherent in the finite-element elasticity, uniquely defined boundary-value problems of rubber elasticity may have multiple stable and unstable solutions. An early example was given by Rivlin, who considered the problem of a Neo-Hookean cube, in a state of pure homogeneous deformations, and subjected to three pairs of equal and opposite forces acting normally on the faces of the cube and distributed uniformly over them. He found that, for forces below a certain value, the only possible solution is the symmetric solution, as might be expected. Beyond that certain value, however, there are seven possible equilibrium solutions. One of these seven solutions is the symmetric solution. It is interesting to notice that the symmetric solution, which is initially stable, becomes unstable when loads have reached a certain threshold. The stability problems of homogeneous deformations of Mooney-Rivlin type of materials, under symmetric loading, for triaxial loading and for the plane stress and plane strain cases, are dealt with in Reference 3. It was shown that a finite-element method can be applied for such analyses. The stability of a sheet of Mooney-Rivlin type of material has been studied for a symmetrical loading condition. Such instability phenomenon was first observed by Treloar. In this work, the problem of a sheet of Mooney-Rivlin type of material, subject to general biaxial loading, is studied both analytically and by finite element. An energy approach to the problem is first presented. This problem represents the biaxial loading of rubber sheets or combined extension and inflation of rubber tubes, which are often used in experimental work for characterization of rubber materials. It is shown that the problem has multiple solutions for a certain domain of loading. The equilibrium state, actually attained, is dependent on the manner of quasistatic loading. Various stable solutions are obtained by finite element.

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.

Buckling of a Thin Annular Plate Under Uniform Compression

1958 ◽  
Vol 25 (2) ◽  
pp. 267-273
Author(s):  
N. Yamaki
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Critical Loads ◽  
Equilibrium Equation ◽  
Annular Plate ◽  
Elastic Stability ◽  
Infinite Strip ◽  
Uniform Compression ◽  
The Stability

Abstract This paper deals with the elastic stability of a circular annular plate under uniform compressive forces applied at its edges. By integrating the equilibrium equation of the buckled plate, the problem is solved in its most general form for twelve different combinations of the boundary conditions of the edges. For each case cited the lowest critical loads are calculated with the ratio of its radii as the parameter. It is clarified that the assumption of symmetrical buckling, which has been made by several researchers, often leads to the overestimate for the stability of the plate. Discussions for the limiting cases of the circular plate and infinite strip also are included.

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