Stability analysis of planar equilibrium configurations of elastic rods subjected to end loads

2009 ◽  
Vol 211 (1-2) ◽  
pp. 73-87 ◽  
Author(s):  
S. V. Levyakov ◽  
V. V. Kuznetsov
Keyword(s):  
Stability Analysis ◽  
Elastic Rods ◽  
Equilibrium Configurations ◽  
Planar Equilibrium

scholarly journals GLOBAL SOLUTIONS OF THE EQUATION OF THE KIRCHHOFF ELASTIC ROD IN SPACE FORMS

2012 ◽  
Vol 88 (1) ◽  
pp. 70-80 ◽  
Author(s):  
SATOSHI KAWAKUBO
Keyword(s):  
Space Form ◽  
Global Solutions ◽  
Infinite Length ◽  
Elastic Rods ◽  
Elastic Rod ◽  
Lagrange Equations ◽  
Initial Value ◽  
Space Forms ◽  
Equilibrium Configurations ◽  
Rod Of Finite Length

AbstractThe Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

On the Static Stability Analysis of Viscoelastic Perfect Columns

1985 ◽  
Vol 9 (3) ◽  
pp. 157-164
Author(s):  
W. Szyszkowski ◽  
P.G. Glockner
Keyword(s):  
Stability Analysis ◽  
Static Analysis ◽  
Element Model ◽  
Static Stability ◽  
Model Material ◽  
Zero Eigenvalue ◽  
Equilibrium Configurations ◽  
Static Approach ◽  
Elastic Column ◽  
New Interpretation

In this article the direct static equilibrium approach for stability analysis is used to study the behaviour of a perfect column made of a linear three-element model material and subjected to a concentric load. The study confirms that such a traditional static analysis admits only one non-zero eigenvalue, namely the load a the instant of application, referred to as the Euler load, PE, for the corresponding elastic column. A new interpretation of adjacent equilibrium configurations for viscoelastic structures is introduced which permits an ‘exact’ static analysis of the problem. The results from this analysis agree, in part, with those obtained from a general dynamic stability analysts. They help to clear up some misinterpretations resulting from the application of the static approach and show that time, being inherently an asymmetric parameter, generates effects typical of asymmetric influences and decreases the critical load of the structure.

Stability of Doubly Periodic Deformed Configurations of Plates and Shallow Shells

1970 ◽  
Vol 37 (3) ◽  
pp. 641-650 ◽  
Author(s):  
C. S. Hsu ◽  
S. S. Lee
Keyword(s):  
Stability Analysis ◽  
Nonlinear Analysis ◽  
Large Deflection ◽  
Multiple Equilibrium ◽  
Jump Phenomenon ◽  
Periodic Surface ◽  
Shallow Shells ◽  
Equilibrium Configurations

Presented here is a nonlinear analysis of infinite plates and shallow shells, subjected to doubly periodic surface loadings. The drastically different behaviors predicted by the linear and the nonlinear theories are analyzed and discussed. It turns out that the transition from the small to the large deflection behavior involves nonlinear bifurcation and the existence of multiple equilibrium configurations, and it entails the question of stability. Seen in this light, it is easy to explain various features special to problems in this class, including the jump phenomenon. From the viewpoint of stability analysis, this class of problems is distinct and interesting in that the perturbations which can lead to instability have actually a higher degree of symmetry than the unperturbed configurations.

5.—Qualitative Aspects of the Spatial Deformation of Non-linearly Elastic Rods.

1975 ◽  
Vol 73 ◽  
pp. 85-105 ◽  
Author(s):  
Stuart S. Antman ◽  
Kathleen B. Jordan
Keyword(s):  
Boundary Value Problem ◽  
Geometric Structure ◽  
Constitutive Relations ◽  
Equilibrium States ◽  
Linear Functions ◽  
Order System ◽  
Elastic Rods ◽  
Qualitative Behaviour ◽  
Spatial Deformation ◽  
Planar Equilibrium

In this article we examine the qualitative behaviour of non-planar equilibrium states ofnon-linearly elastic rods subject to terminal loads. In our geometrically exact theory, a rod is endowed with enough geometric structure for it to undergo flexure, torsion, axial extension, and shear. The constitutive equations give appropriate stress resultants and couples as non-linear functions of appropriate strains. These constitutive relations must meet minimal conditions ensuring that they be physically reasonable. It turns out that the equilibrium states of such a rod are governed by a boundary value problem for a quasilinear fifteenth-order system of ordinary differential equations.

Determination and Stability Analysis of Equilibrium Configurations of Objects Suspended From Multiple Aerial Robots

2012 ◽  
Vol 4 (2) ◽  
Author(s):  
Qimi Jiang ◽  
Vijay Kumar
Keyword(s):  
Stability Analysis ◽  
Hessian Matrix ◽  
Polynomial Equation ◽  
Search Algorithms ◽  
Cable System ◽  
Aerial Robots ◽  
Equilibrium Configurations ◽  
Position And Orientation ◽  
The Stability ◽  
Direct Kinematics

This work addresses the problem for determining the position and orientation of objects suspended with n cables from n aerial robots. This is actually the direct kinematics problem of the 3D cable system. First, the problem is formulated based on the static equilibrium condition. Then, an analytic algorithm based on resultant elimination is proposed to determine all possible equilibrium configurations of the planar 4-bar linkage. As the nonlinear system can be reduced to a polynomial equation in one unknown with a degree 8, this algorithm is more efficient than numerical search algorithms. Considering that the motion of a 3D cable system in its vertical planes of symmetry can be regarded as the motion of an equivalent planar 4-bar linkage, the proposed algorithm is used to solve the direct kinematics problem of objects suspended from multiple aerial robots. Case studies with three to six robots are conducted for demonstration. Then, approaches for stability analysis based on Hessian matrix are developed, and the stability of obtained equilibrium configurations is analyzed. Finally, experiments are conducted for validation.

scholarly journals A Theoretical MHD Model for Extragalactic Jets and its Comparison with the Observations

1990 ◽  
Vol 140 ◽  
pp. 441-442
Author(s):  
P. Pietrini
Keyword(s):  
Stability Analysis ◽  
Equilibrium Model ◽  
Stable Equilibrium ◽  
Point Of View ◽  
Observational Constraints ◽  
Radio Sources ◽  
Equilibrium Configurations ◽  
Extragalactic Jets ◽  
Thermal Confinement ◽  
The Stability

Two aspects of the MHD stationary equilibrium model developed by Chiuderi et al.(1989) to describe extragalactic jets are analyzed and compared with the observational constraints: the global energy flux convected by the cylindrical jet and the ranges of the equilibrium parameters allowed by the stability analysis. In particular, the results obtained from the temporal stability analysis are converted into a spatial point of view. In this context, it is easier to find essentially “stable” equilibrium configurations for shorter jets. In conclusion, the fundamental hypotheses of this model (like thermal confinement and substantial equipartition among the various forms of energy considered) are such that the model turns out to be suitable for the description of class I jets, associated with rather low-power radio sources.

Rotating massive stars in general relativity

1967 ◽  
Vol 296 (1445) ◽  
pp. 189-200 ◽  
Keyword(s):  
General Relativity ◽  
Stability Analysis ◽  
Simple Formula ◽  
Massive Star ◽  
Massive Stars ◽  
Rotation Velocity ◽  
Weak Field ◽  
Slow Rotation ◽  
Equilibrium Configurations ◽  
Schwarzschild Radius

Equilibrium models of uniformly rotating massive stars are investigated, using a weak field, slow rotation approximation, which is shown to be adequate for all cases of interest. The fate of radial perturbations about these equilibrium configurations is investigated using a linear­ized stability analysis to determine the oscillation frequency σ in a peturbation ∝ e iσ t . An eigenvalue equation for σ 2 is obtained which can be made self adjoint with respect to the spatial metric, and a variational principle to determine σ 2 is derived. Numerical determinations of σ 2 have been carried out for a variety of masses, radii and rotational velocities, and these results are incorporated in a simple formula that gives the dependence of σ 2 on these quantities. The condition for instability, σ 2 negative, is determined, and it is found that for large masses and maximum rotation velocity, so that when centrifugal force balances gravity at the surface, a massive star becomes unstable when its radius is 208 times the Schwarzschild radius 2 GM/c 2 .

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