scholarly journals Flutter Instability and Ziegler Destabilization Paradox for Elastic Rods Subject to Non-Holonomic Constraints

2020 ◽  
Vol 88 (3) ◽  
Author(s):  
Alessandro Cazzolli ◽  
Francesco Dal Corso ◽  
Davide Bigoni
Keyword(s):  
Equilibrium Configuration ◽  
The Other ◽  
Structural Systems ◽  
Elastic Rods ◽  
Elastic Rod ◽  
Holonomic Constraints ◽  
Flutter Instability ◽  
Divergence Instability ◽  
Trivial Equilibrium ◽  
Destabilization Paradox

Abstract Two types of non-holonomic constraints (imposing a prescription on velocity) are analyzed, connected to an end of a (visco)elastic rod, straight in its undeformed configuration. The equations governing the nonlinear dynamics are obtained and then linearized near the trivial equilibrium configuration. The two constraints are shown to lead to the same equations governing the linearized dynamics of the Beck (or Pflüger) column in one case and of the Reut column in the other. Although the structural systems are fully conservative (when viscosity is set to zero), they exhibit flutter and divergence instability. In addition, the Ziegler's destabilization paradox is found when dissipation sources are introduced. It follows that these features are proven to be not only a consequence of “unrealistic non-conservative loads” (as often stated in the literature); rather, the models proposed by Beck, Reut, and Ziegler can exactly describe the linearized dynamics of structures subject to non-holonomic constraints, which are made now fully accessible to experiments.

Statics Analysis of the Mechanical Model of Nucleosome

2011 ◽  
Vol 343-344 ◽  
pp. 661-667 ◽  
Author(s):  
Yun Xue ◽  
De Wei Weng ◽  
Gang Ming Gong
Keyword(s):  
Mechanical Model ◽  
The Other ◽  
Elastic Rod ◽  
Equilibrium Equations ◽  
Constraint Force ◽  
Precession Angle ◽  
Nutation Angle ◽  
Analytic Expression ◽  
Set Up ◽  
Calculated Analysis

Mechanical model of nucleoside and its equilibrium equations are set up, and the mechanical properties on the equilibrium position are analyzed. In the case constraint force and electrostatic attraction between cylinder OH and elastic rod are balanced, the analytic expression of nutation angle of the section and its conditions of existence are given. It is show that the cylinder OH can maintain equilibrium at any range of the precession angle. In the other case when unbanced, there is phenomenon of separation of elastic rod from cylinder OH in the spiral wound 2 circles, and numerical solution of the precession angle at separation points are calculated. Analysis of equilibrium of cylinder H1 illustrates that the generatrix of cylinder H1 and OH are not parallel, and the angle between them is obtained

The shape of a Möbius band

1993 ◽  
Vol 440 (1908) ◽  
pp. 149-162 ◽  
Keyword(s):  
Aspect Ratio ◽  
Asymptotic Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Rods ◽  
Large Deflections ◽  
Mechanical Equilibrium ◽  
Elastic Rod ◽  
Flexible Material ◽  
Möbius Band

The shape of a Möbius band made of a flexible material, such as paper, is determined. The band is represented as a bent, twisted elastic rod with a rectangular cross-section. Its mechanical equilibrium is governed by the Kirchhoff–Love equations for the large deflections of elastic rods. These are solved numerically for various values of the aspect ratio of the cross-section, and an asymptotic solution is found for large values of this ratio. The resulting shape is shown to agree well with that of a band made from a strip of plastic.

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.

Axially Symmetric Waves in Elastic Rods

1960 ◽  
Vol 27 (1) ◽  
pp. 145-151 ◽  
Author(s):  
R. D. Mindlin ◽  
H. D. McNiven
Keyword(s):  
Cross Section ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elastic Rods ◽  
Axially Symmetric ◽  
Elastic Rod ◽  
One Dimensional ◽  
Axial Shear ◽  
Wave Numbers ◽  
Complex Wave

A system of approximate, one-dimensional equations is derived for axially symmetric motions of an elastic rod of circular cross section. The equations take into account the coupling between longitudinal, axial shear, and radial modes. The spectrum of frequencies for real, imaginary, and complex wave numbers in an infinite rod is explored in detail and compared with the analogous solution of the three-dimensional equations.

scholarly journals Snapping of a Planar Elastica With Fixed End Slopes

2008 ◽  
Vol 75 (4) ◽  
Author(s):  
Jen-San Chen ◽  
Yong-Zhi Lin
Keyword(s):  
Theoretical Prediction ◽  
Static Load ◽  
Natural Frequencies ◽  
Equilibrium Configuration ◽  
The Other ◽  
Deflection Curve ◽  
Slope Difference ◽  
The Stability ◽  
Fixed End ◽  
Load Deflection Curve

In this paper, we study the deformation and stability of a planar elastica. One end of the elastica is clamped and fixed in space. The other end of the elastica is also clamped, but the clamp itself is allowed to slide along a linear track with a slope different from that of the fixed clamp. The elastica deforms after it is subjected to an external pushing force on the moving clamp. It is observed that when the pushing force reaches a critical value, snapping may occur as the elastica jumps from one configuration to another remotely away from the original one. In the theoretical investigation, we calculate the static load-deflection curve for a specified slope difference between the fixed clamp and the moving clamp. To study the stability of the equilibrium configuration, we superpose the equilibrium configuration with a small perturbation and calculate the natural frequencies of the deformed elastica. An experimental setup is designed to measure the load-deflection curve and the natural frequencies of the elastica. The measured load-deflection relation agrees with the theoretical prediction very well. On the other hand, the measured natural frequencies do not agree very well with the theoretical prediction, unless the mass of the moving clamp is taken into account.

scholarly journals A reduced order model of the spine to study pediatric scoliosis

2020 ◽  
Author(s):  
Sunder Neelakantan ◽  
Prashant K. Purohit ◽  
Saba Pasha
Keyword(s):  
Idiopathic Scoliosis ◽  
Inflection Point ◽  
Reduced Order Model ◽  
The Other ◽  
Sagittal Profile ◽  
Deformation Pattern ◽  
Elastic Rod ◽  
Order Model ◽  
Reduced Order ◽  
Wide Range

AbstractThe S-shaped curvature of the spine has been hypothesized as the underlying mechanical cause of adolescent idiopathic scoliosis. In earlier work we proposed a reduced order model in which the spine was viewed as an S-shaped elastic rod under torsion and bending. Here, we simulate the deformation of S-shaped rods of a wide range of curvatures and inflection points under a fixed mechanical loading. Our analysis determines three distinct axial projection patterns of these S-shaped rods: two loop (in opposite directions) patterns and one lemniscate pattern. We further identify the curve characteristics associated with each deformation pattern showing that for rods deforming in a loop 1 shape the position of the inflection point is the highest and the curvature of the rod is smaller compared to the other two types. For rods deforming in the loop 2 shape the position of the inflection point is the lowest (closer to the fixed base) and the curvatures are higher than the other two types. These patterns matched the common clinically observed scoliotic curves - Lenke 1 and Lenke 5. Our elastic rod model predicts deformations that are similar to those of a pediatric spine and it can differentiate between the clinically observed deformation patterns. This provides validation to the hypothesis that changes in the sagittal profile of the spine can be a mechanical factor in parthenogenesis of pediatric idiopathic scoliosis.

Critical Speeds, Divergence, and Flutter Instability in High-Speed Planetary Gears

2012 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Critical Speed ◽  
Numerical Results ◽  
Planetary Gears ◽  
Critical Speeds ◽  
Flutter Instability ◽  
Divergence Instability ◽  
Translational Mode

The structured properties of the critical speeds and associated critical speed eigenvectors of high-speed planetary gears are given. Planetary gears have only planet, rotational, and translational mode critical speeds. Divergence instability is possible at speeds adjacent to critical speeds. Numerical results verify the critical speed locations. Divergence and flutter instabilities are investigated numerically for each mode type.

scholarly journals Flutter and divergence instability in the Pflüger column: Experimental evidence of the Ziegler destabilization paradox

2018 ◽  
Vol 116 ◽  
pp. 99-116 ◽  
Author(s):  
Davide Bigoni ◽  
Oleg N. Kirillov ◽  
Diego Misseroni ◽  
Giovanni Noselli ◽  
Mirko Tommasini
Keyword(s):  
Experimental Evidence ◽  
Divergence Instability ◽  
Destabilization Paradox

The simulation of structural and thermal systems via flow graphs

SIMULATION ◽  
1967 ◽  
Vol 9 (3) ◽  
pp. 149-156
Author(s):  
Larry J. Feeser ◽  
Chuan C. Feng
Keyword(s):  
Heat Flow ◽  
Oriented Graph ◽  
Initial Boundary ◽  
The Other ◽  
Mesh Network ◽  
Graph Representation ◽  
Structural Systems ◽  
Central Difference ◽  
Physical Systems ◽  
Flow Graphs

This paper presents a study of simulation of structural systems and heat flow problems by use of oriented flow graphs. Linear elastic structural systems are transformed into graph representation using two kinds of parametric analyses. In one case, flexibility parameters of structural components are used as branch transmittances of the or iented graphs, and geometric displacements are the ex ternal sources or inputs to the graph. In the other ap proach, stiffness coefficients as graph transmittances and unbalanced residual loads as source inputs lead to relax ation operations which can be represented by feedback analysis. Each of these approaches is the dual of the other in a topological sense. Two-dimensional problems governed by the Laplacian partial differential equation are solved by using first-order central difference analysis to transform the governing equation into an oriented graph. The graph networks of physical systems of this type can be constructed by in spection of the mesh network without recourse to the difference equations. Initial boundary values are treated as independent source inputs to the graph. A solution of the steady-state heat flow in a thin plate is presented. The simulation of physical systems by oriented flow graphs treats the system as an entity in which the physical characteristics are transformed topologically into graph representations. Oriented graphs are essentially analog setups from which numerical results can be obtained by analog computers. If use is made of the loop-rule equa tions of graph theory, efficient use of the digital computer is indicated. The nature of design of structural systems is complex because of the range of parameters to be con sidered and because of required accuracy. Flow graph simulation can give information on the effect of variation of parameters by way of sensitivity analysis and can be an aid in the study of design.

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