Asymptotic behavior of eigenfrequencies of a thin elastic rod with non-uniform cross-section

2020 ◽  
Vol 72 (1) ◽  
pp. 119-154
Author(s):  
Shuichi JIMBO ◽  
Albert RODRÍGUEZ MULET
Keyword(s):  
Asymptotic Behavior ◽  
Cross Section ◽  
Elastic Rod ◽  
Uniform Cross Section ◽  
Thin Elastic Rod

scholarly journals Branching and Stability of Elasic Rod Equilibrium Forms

2021 ◽  
Author(s):  
Dmitrii Skubov ◽  
Dmitry Yu. Kopnin
Keyword(s):  
Differential Equation ◽  
Boundary Problem ◽  
Cross Section ◽  
Compressive Load ◽  
Elliptic Functions ◽  
Vertical Position ◽  
Loss Of Stability ◽  
Elastic Rod ◽  
Turn Angle ◽  
Thin Elastic Rod

Abstract In our article from the positions of branching theory the classical Euler task about stability of thin elastic rod under action of vertical compressive load is considered. With using of turn-tensor and accompanying vector the deformation of rod is described. In result the conditions of equilibrium in a case of linear determining equation are reduced to boundary problem relatively turn angle of cross section, described by the differential equation of pendulum. With help elliptic functions the diagrams of branching are constructed and are received the exact formulas of rod deformation at the loss of stability of vertical position. Analogy the task of stability of rod at turned force on end of rod is considered. Also, the oscillating loss of rod stability at tracking load is studied.

scholarly journals Asymptotic Behavior of Stable Structures Made of Beams

2021 ◽  
Author(s):  
Georges Griso ◽  
Larysa Khilkova ◽  
Julia Orlik ◽  
Olena Sivak
Keyword(s):  
Asymptotic Behavior ◽  
Cross Section ◽  
Cross Sections ◽  
Linear Elasticity ◽  
Circular Cross Section ◽  
Linear Elasticity Theory ◽  
Linear Elastic ◽  
The Cross ◽  
Small Rotation ◽  
Stable Structures

AbstractIn this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

Cantilever Rod in Cross Wind

1989 ◽  
Vol 56 (3) ◽  
pp. 639-643 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Numerical Integration ◽  
Large Deformation ◽  
Aerodynamic Drag ◽  
Flexural Rigidity ◽  
Relative Importance ◽  
Elastic Rod ◽  
Nondimensional Parameter ◽  
Cantilever Rod ◽  
Thin Elastic Rod

A thin elastic rod is held at one end in a strong cross wind. The nonlinear large deformation equations are formulated and solved by perturbation and numerical integration. The problem is governed by a nondimensional parameter K representing the relative importance of aerodynamic drag to flexural rigidity. For large K, phenomena such as nonuniqueness, instability, and hysteresis may occur.

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