On the Dynamics of Elastic Strips

Goriely, Alain; Nizette, Michel; Tabor, M.

doi:10.1007/s003320010009

Cited by 66 publications

(85 citation statements)

References 32 publications

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“…From now on we treat static rods only, whose solutions are given by solving (15) and = 0 which correspond to equilibrium solutions as described in [28]. These form a system of first order differential equations with initial value…”

Section: Coupling the Kirchhoff Equations To The Reference Framementioning

confidence: 99%

“…Figure 4 contains filaments with 3000 steps with circular cross-section, (a) and (b) are almost closed solutions. In order to produce them, specially tuned boundary conditions were chosen as described in [28]. In this sense, the method is unable to produce closed solutions without further tuning of boundary values and other parameters.…”

Section: Solving the Initial Value Problem With And Without External mentioning

confidence: 99%

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Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

Xavier¹

2014

Journal of Computational Methods in Physics

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This paper shows how to apply a simple Runge-Kutta algorithm to get solutions of Kirchhoff equations for static filaments subjected to arbitrary external and static forces. This is done by suitably integrating at once Kirchhoff and filament reference system equations under appropriate initial boundary conditions. To show the application of the method, we display several numerical solutions for filaments including cases showing the effect of gravity.

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Section: Coupling the Kirchhoff Equations To The Reference Framementioning

confidence: 99%

Section: Solving the Initial Value Problem With And Without External mentioning

confidence: 99%

Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

Xavier¹

2014

Journal of Computational Methods in Physics

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“…More recently, Champneys and Thompson [3] and van der Heijden and Thompson [27] performed bifurcation analyses of an anisotropic elastic rod but did not focus on the question of stability. Goriely, Nizette, and Tabor [10] considered the dynamic stability of the unbuckled configurations for both the isotropic and anisotropic cases, and van der Heijden et al [26] inferred stability for unbuckled and buckled configurations for the isotropic case from the shape of solution branches in a particular "distinguished" coordinate system. Neukirch and Henderson [21] performed an in-depth classification of buckled solutions for the isotropic problem, including the computation of two-dimensional sheets of equilibria.…”

Section: An Elastic Strut Clamped At Each End With a Relative Twismentioning

confidence: 99%

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

Hoffman¹,

Manning²,

Paffenroth³

2002

SIAM J. Appl. Dyn. Syst.

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Abstract. We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler-Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values σ ≤ 1 at which a second ODE (the Jacobi equation) has a solution vanishing at 0 and σ. Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter λ, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of λ of interest.Here we present an alternative approach that takes advantage of the presence of a parameter λ. Rather than search for conjugate points σ ≤ 1 at a fixed value of λ, we search for a set of special parameter values λm (with corresponding Jacobi solution ζ m ) for which σ = 1 is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any λ equals the number of these ζ m for which ζ m , Sζ m < 0, where S is the Jacobi differential operator at λ. This computation is particularly simple when λ appears linearly in S.We apply this approach to the elastic strut, in which the force appears linearly in S, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE.

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“…Based on Newton's second law, the Kirchhoff rod model provides a theoretical frame describing the static and dynamic behaviors of elastic rods [7,8]. The Kirchhoff model holds for small curvatures of rods, but Kirchhoff rods can undergo large changes of shape [9].…”

Section: Introductionmentioning

confidence: 99%

“…Nizette and Goriely gave a parameterized analytical solution for Kirchhoff rods with circular cross-section and further made a systematic classification of all kinds of equilibrium solutions [7]. Goriely et al studied the dynamical stability of elastic strips by analyzing the amplitude equations governing the dynamics of elastic strips [8].…”

Section: Introductionmentioning

confidence: 99%

A Symbolic-Numeric Method for Solving Boundary Value Problems of Kirchhoff Rods

Liu

Weber

2005

Computer Algebra in Scientific Computing

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Abstract. We study solution methods for boundary value problems associated with the static Kirchhoff rod equations. Using the well known Kirchhoff kinetic analogy between the equations describing the spinning top in a gravity field and spatial rods, the static Kirchhoff rod equations can be fully integrated. We first give an explicit form of a general solution of the static Kirchhoff equations in parametric form that is easy to use. Then by combining the explicit solution with a minimization scheme, we develop a unified method to match the parameters and integration constants needed by the explicit solutions and given boundary conditions. The method presented in the paper can be adapted to a variety of boundary conditions. We detail our method on two commonly used boundary conditions.

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On the Dynamics of Elastic Strips

Cited by 66 publications

References 32 publications

Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

Static Kirchhoff Rods under the Action of External Forces: Integration via Runge-Kutta Method

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

A Symbolic-Numeric Method for Solving Boundary Value Problems of Kirchhoff Rods

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