Inextensible String Research Articles

Anomalous curvature evolution and geometric regularization of energy focusing in the snapping dynamics of a flexible body

We examine the focusing of kinetic energy and the amplification of various quantities during the snapping motion of the free end of a flexible structure. This brief but violent event appears to be a regularized finite-time singularity, with remarkably large spikes in velocity, acceleration, and tension easily induced by generic initial and boundary conditions. A numerical scheme for the inextensible string equations is validated against available experimental data for a falling chain and further employed to explore the phenomenon. We determined that the discretization of the equations, equivalent to the physically discrete problem of a chain, does not provide the regularizing length scale, which in the absence of other physical effects must then arise from the geometry of the problem. An analytical solution for a geometrically singular limit, a falling perfectly folded string, accounts surprisingly well for the scalings of several quantities in the numerics, but can only indirectly suggest a behavior for the curvature, one which seems to explain prior experimental data but does not correspond to the evolution of the curvature peak in our system, which instead displays a newly observed anomalously slow scaling. A simple model, incorporating only knowledge of the initial conditions along with the anomalous and singular-limit scalings, provides reasonable estimates for the amplifications of relevant quantities. This is a first step to predict and harness arbitrarily large energy focusing in structures, with a practical limit set only by length scales present in the discrete mechanical system or the initial conditions.

Comprehending Optimality of Finger Flexor Tendon Pulley System Using Computational Analysis.

Existing prosthetic/orthotic designs are rarely based on kinetostatics of a biological finger, especially its tendon-pulley system (TPS). Whether a biological TPS is optimal for use as a reference, say for design purposes, and if so, in what sense, is also relatively unknown. We expect an optimal TPS to yield a high range of flexion while operating with lower tendon tension, bowstringing, and pulley stresses. To gain insight into the TPS designs, we present a parametric study which is then used to determine optimal TPS configurations for the flexor mechanism. A compliant, flexure-based computational model is developed and simulated using the pseudo-rigid body method, with various combinations of pulley/tendon attachment point locations, pulley heights, and widths. Results suggest that three distinct types of TPS configurations corresponding to a single stiff pulley, two stiff pulleys, or one stiff and one flexible-inextensible pulley per phalange can be optimal. For a TPS configuration similar to a biological one, the distal pulleys on the proximal and intermediate phalanges need to be like flexible-inextensible string loops that effectively model the behavior of joint and cruciate pulleys. We reckon that a biological flexor TPS may have evolved to maximize flexion range with minimum possible actuation tension, bowstringing, and pulley stress. Our findings may be useful in not only developing efficient hand devices but also in improving TPS reconstruction surgery procedures.

Light Inextensible Strings (Thread) Under Tension in the Schwarzschild Geometry

Light inextensible string under tension is a stalwart feature of elementary physics. Here I show how considering such a string in the vicinity of a black hole, with the help of computer algebra systems, can generate insight into the Schwarzschild geometry in the context of an undergraduate homework problem. Light taut strings minimize their proper length, given by integrating the spatial component of the Schwarzschild metric along the string. The path itself is given by straightforward numerical solution to the Euler–Lagrange equations. If the string is entirely outside the event horizon, its closest approach to the singularity is tangential. At this point the string is visibly curved, surely a memorable and informative insight. The geometry of the Schwarzschild metric induces some interesting nonlocal phenomena: if the distance of closest approach is less than about [Formula: see text], the string self-intersects, even though it is everywhere under tension. Light taut strings furnish a third interpretation of the concept “straight line”, the other two being null geodesics and free-fall world lines. All the software used is available under the GPL.1

Introducing the trapezoidal pendulum: dynamics of coupled pendula suitable for distance teaching

Two pendula of length l, suspended from points separated by a horizontal distance a, can be joined in a trapezium configuration by linking the bobs with a light inextensible string of length b < a. Here we introduce this ‘trapezoidal pendulum’ as a charming and novel oscillatory system suitable for introductory physics laboratories. In particular, we show that the time period τ is given by a scale-invariant equation of the form (τ/τs) = P(a, b, l), where (with g gravity), and P(a, b, l) is a dimensionless function of a, b, and l. The trapezoidal pendulum exhibits richer behaviour than the canonical simple pendulum, but is nevertheless easy to construct from basic household items, making it an ideal system for home experimentation. Indeed, here we verify the τ dependencies empirically using modest, improvised apparatus. These investigations suggest exciting possibilities for distance teaching activities, such as collaborative studies of the dimensionless scalings, using data obtained by students working independently, and on apparatus of different sizes.

Modeling and Optimal Control of an Octopus Tentacle

We present a control model for an octopus tentacle based on the dynamics of an inextensible string with curvature constraints and curvature controls. We derive the equations of motion together with an appropriate set of boundary conditions, and we characterize the corresponding equilibria. The model results in a system of fourth-order evolutive nonlinear controlled PDEs, generalizing the classic Euler's dynamic elastica equation, that we approximate and solve numerically by introducing a finite difference scheme. We proceed by investigating a reachability optimal control problem associated to our tentacle model. We first focus on the stationary case, establishing a relation with the celebrated Dubins car problem. Moreover, we propose an augmented Lagrangian method for its numerical solution. Finally, we address the evolutive case obtaining first order optimality conditions, then we numerically solve the optimality system by means of an adjoint-based gradient descent method.

The gradient flow of the potential energy on the space of arcs

We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of generalized solutions to the corresponding nonlinear evolutionary PDE and their exponential decay to the equilibrium. We also observe that the system admits solutions backwards in time, which leads to non-uniqueness of trajectories.

Conics and a generalised conical pendulum

There is a long tradition of using geometry to solve problems from mechanics. Unfortunately this tradition is not practised much in schools and university any more.With this exposition we would like to demonstrate how elementary properties of ellipses can be used to solve a problem related to the conical pendulum. The problem of the conical pendulum is to consider a mass attached to one end of a light inextensible string of length with the other end attached at the top of a vertical rod. The mass is moving about the rod in uniform circular motion in a horizontal plane. Given the angular velocity of the mass, the question is to determine the angle the string makes with the rod.

Instability of catenary-type flexible risers conveying fluid in subsea environments

As water depths for oil and gas exploration and extraction increase, structures such as flexible risers, mooring lines, and umbilical cables are increasingly being used for subsea environments. Compared to conventional fixed-type structures and vertical risers, the dynamics of flexible risers is significantly more complex. In particular, the flexible structures may be prone to dynamic instabilities. The goal of the present paper is to provide a comprehensive study of the dynamics, stability, and vibration of flexible risers. We use Kirchhoff's theory of an extensible, flexible rod that resists torsion to develop a set of nonlinear equations for the dynamics of risers. The resulting model incorporates drag and the effects of the fluid being transported internally. Using a nonlinear stability criterion, our analyses show the nonlinear stability of a simple catenary-type riser modeled either as an inextensible or extensible string. For the more advanced rod models, we use a linear stability analysis to show how the internal fluid being conveyed can destabilize certain static configurations.

Initial-boundary value problem for the degenerate hyperbolic equation of a hanging string

We consider an initial-boundary value problem for the degenerate linear hyperbolic equation as a model of the motion of an inextensible string fixed at one end in the gravity field. We shall show the existence and the uniqueness of the solution and study the regularity of the solution.

Generalized solutions for inextensible string equations

We study the system of equations of motion for inextensible strings. This system can be recast into a discontinuous system of conservation laws as well as into the total variation wave equation. We prove existence of generalized Young measure solutions with non-negative tension after transforming the problem into a system of conservation laws and approximating it with a regularized system for which we obtain uniform estimates of the energy and the tension. We also discuss sufficient conditions for non-negativity of the tension for strong solutions.

Numerical Study of the Elastic Pendulum on the Rotating Earth

The elastic pendulum is a simple physical system represented by nonlinear differential equations. Analytical solutions for the bob trajectories on the rotating earth may be obtained in two limiting cases: for the ideally elastic pendulum with zero unstressed string length and for the Foucault pendulum with an inextensible string. The precession period of the oscillation plane, as seen by the local observer on the rotating earth, is 24 hours in the first case and has a well-known latitude dependence in the second case. In the present work, we have obtained numerical solutions of the nonlinear equations for different string elasticities in order to study the transition from one precession period to the other. It is found that the transition is abrupt and that it occurs for a quite small perturbation of the ideally elastic pendulum, that is, for the unstressed string length equal to about 10−4 of the equilibrium length due to the weight of the bob.

Modeling and Numerical Simulation of the Nonlinear Dynamics of the Parametrically Forced String Pendulum

The string pendulum consists of a mass attached to the end of an inextensible string which is fastened to a support. Applying an external forcing to the pendulum’s support is motivated by understanding the behavior of suspension bridges or of tethered structures during earthquakes. The forced string pendulum can go from taut to slack states and vice versa, and is capable of exhibiting interesting periodic and chaotic dynamics. The inextensibility of the string and its capacity to go slack make simulation and analysis of the system complicated. The string pendulum system is thus formulated here as a piecewise-smooth dynamical system using the method of Lagrange multipliers to obtain a system of differential algebraic equations (DAE) for the taut state. In order to develop a formulation for the forced string pendulum system, we first turn to similar but simpler pendulum systems, such as the classic rigid pendulum, the elastic spring pendulum and the elastic spring pendulum with piecewise constant stiffness. We perform a perturbation analysis for both the unforced and forced cases of the spring pendulum approximation, which shows that, for large stiffness, this is a reasonable model of the system. We also show that the spring pendulum with piecewise constant stiffness can be a good approximation of the string pendulum, in the limit of a large extension constant and a low compression constant. We indicate the behavior and stability of this simplified model by using numerical computations of the system’s Lyapunov exponents. We then provide a comparison of the spring pendulum with piecewise constant stiffness with the formulation of the taut-slack pendulum using the DAE for the taut states and derived switching conditions to the slack state.

Small oscillations of a rotating inextensible string with end masses

The characteristic value problem for free vibrations of a steadily rotating variable area inextensible string with end masses is solved numerically for strings with constant area, constant stress, and a type of exponential taper. Several sets of numerical results are presented (in both tabular and graphical formats), which, it is hoped, will be helpful in evaluating numerical methods used in space tether applications. In addition, approximate formulas are developed for quick natural frequency estimation in two limiting cases.

The motion of whips and chains

We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations η t t = ∂ s ( σ η s ) , σ s s − | η s s | 2 σ = − | η s t | 2 , | η s | 2 ≡ 1 with boundary conditions η ( t , 1 ) = 0 and σ ( t , 0 ) = 0 . We prove local existence and uniqueness in the space defined by the weighted Sobolev energy ∑ ℓ = 0 m ∫ 0 1 s ℓ | ∂ s ℓ η t | 2 d s + ∫ 0 1 s ℓ + 1 | ∂ s ℓ + 1 η | 2 d s , when m ⩾ 3 . In addition we show persistence of smooth solutions as long as the energy for m = 3 remains bounded. We do this via the method of lines, approximating with a discrete system of coupled pendula (a chain) for which the same estimates hold.

Uniform rotations of a variable-length pendulum

Let a pendulum represent the material point P of a weight mg attached to the end of a weightless inex tensible string. The motion occurs in a uniform gravity field in the fixed vertical plane Oxy passing through a motionless point O (Fig. 1). The string length OP is the set function of time t: = . It is assumed that the string remains stretched during the pendulum motion; i.e., the coordinates x and y of the point P satisfy the equation of the supporting geometrical constraint

Asymptotic solution of the mathematical model of nonlinear oscillations of absolutely elastic inextensible weightless string

The mathematical model of nonlinear oscillations of weightless string is analyzed. To find an asymptotic solution of the problem, uniformly valid in a long interval of time, an averaged system of integral differential equations has been constructed. A method for constructing special approximations of its solutions is proposed.

The simulation of in-plane tyre modal behaviour: a broad modal range comparison between analytical and discretised modelling approaches

In-plane tyre modal behaviour determines the response of tyres to ride excitations and braking/traction manoeuvres. In many studies, the interest is limited to relatively low frequencies and a detailed investigation into the ability of models to accurately simulate higher-order responses is unnecessary. In cases where an in-plane model is to be used for the generation of the contact deformation and stresses, or where modal reduction methods are implemented, a detailed knowledge of the modal response is desirable. The present work forms a study on the ability of a number of frequently used modelling approaches to generate realistic modal data throughout a wide frequency range. The analytical ring on elastic foundation model is used as a benchmark throughout the paper. Its predictions are compared with those of two discretised models, namely a truss- and a beam-based model. The sensitivity of the ring’s response to a number of physical parameters is discussed. The results are used to inform the comparison between the analytical ring and the discretised models, providing explanations for any discrepancies observed. The limited applicability of the truss model is pointed out, while the accuracy of the beam-based model is enhanced by a circumferential inextensible string element. Both the ring and the enhanced beam models are further improved with the addition of a nonlinear string-based sidewall that accounts for the change in sidewall stiffness with inflation pressure. The findings may offer a reference when setting up in-plane models, including the stage of planning modal tests for parameter identification.

Tensile radial stress in the spinal cord related to arachnoiditis or tethering: a numerical model

Spinal arachnoiditis comprises fibrous scarring of the subarachnoid space, following spinal trauma or inflammation, and is often associated with syringomyelia. We hypothesised that cord-to-dura attachments could cause transient tensile cord radial stress, as pressure waves propagate. This was tested in a fluid-structure interaction model, simulating three types of cord tethering, with 'arachnoiditis' confined to a short mid-section of the cord. The annular system was excited abdominally with a short transient, and the resulting Young and Lamb waves and reflections were analysed. Radial mid-section tethering was less significant than axial tethering, which gave rise to tensile radial stress locally when the cord was not fixed cranially. Simulated as inextensible string connections to the dura, arachnoiditis caused both localised tensile radial stress and localised low pressure in the cord as the transient passed. The extent of these effects was sensitive to the relative stiffness of the dura and cord. Tensile radial stress may create a syrinx in previously normal cord tissue, and transiently lowered pressure may draw in interstitial fluid, causing the syrinx to enlarge if fluid exit is inhibited. The suggested mechanism could also explain the juxtaposition of syrinxes to regions of arachnoiditis.

The skipping rope curve

The equilibrium of a flexible inextensible string, or chain, in the centrifugal force field of a rotating reference frame is investigated. It is assumed that the end points are fixed on the rotation axis. The shape of the curve, the skipping rope curve or troposkien, is given by the Jacobi elliptic function sn.

Toy models: The jumping pendulum

We consider a simple pendulum consisting of a mass attached to an inextensible string of negligible mass. For small or large initial velocities, the motion of the pendulum is along a circle. When given sufficient but not too large an initial velocity, the mass will reach a certain height and leave the circle. After such a jump, it will follow a parabolic path until the string is again fully extended and the motion is again constrained by the string. We assume that the radial component (along the string) of the velocity of the mass instantaneously vanishes when the string becomes taut and that the mass loses some of its energy in the shock and resumes its circular motion. What is the dynamics of such a pendulum? Can it jump more than once? How many times can it jump?