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The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
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In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical,...
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Straightening: existence, uniqueness and stability.

M Destrade1, R W Ogden2, I Sgura3

  • 1School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Republic of Ireland ; School of Mechanical and Materials Engineering, University College Dublin, Belfield, Dublin 4, Republic of Ireland.

Proceedings. Mathematical, Physical, and Engineering Sciences
|April 9, 2014
PubMed
Summary

This study revisits the large static deformation of a cylinder sector into a rectangular block, analyzing its existence and stability. The impedance matrix method proved most precise for solving incremental stability problems, offering insights into wrinkling phenomena.

Keywords:
nonlinear elasticitystiffeningstraightening instability

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Area of Science:

  • Solid Mechanics
  • Nonlinear Elasticity
  • Continuum Mechanics

Background:

  • The straightening of a cylinder sector into a rectangular block is a poorly understood deformation in incompressible nonlinear elasticity.
  • Understanding the forces and stability of such large static deformations is crucial for material science and engineering applications.

Purpose of the Study:

  • To revisit and analyze the existence and stability issues of a sector of a circular cylinder straightening into a rectangular block.
  • To investigate the forces required for large static deformation, including end couples.
  • To study the influence of geometric parameters and constitutive models on wrinkling.

Main Methods:

  • Analysis of existence and stability of the deformation.
  • Investigation of forces, including end couples, for static deformation.
  • Numerical comparison of methods for solving incremental stability problems.

Main Results:

  • The study addresses the existence and stability of this specific large static deformation.
  • The influence of geometric parameters and constitutive models on wrinkling was investigated.
  • The impedance matrix method, utilizing a matrix Riccati differential equation, was found to be the most precise numerical method for incremental stability analysis.

Conclusions:

  • The impedance matrix method offers superior precision for incremental stability analysis in this context.
  • This research contributes to understanding the complex behavior of materials under large deformations, particularly concerning wrinkling.
  • Further investigation into the mechanics of cylinder sector straightening can inform material design and structural analysis.