Taut String Research Articles

Algorithms for exact multi-object muscle wrapping and application to the deltoid muscle wrapping around the humerus

Lines of action of muscle forces imply the function and performance of muscles acting around joints. It is not always possible to determine muscle force lines of action in vivo, and so computational techniques are often used to predict them. It is common to model a muscle as a taut elastic string that follows the shortest geodesic path between attachments over the wrapping geometry. A number of studies have been concerned with wrapping paths over single wrapping objects, and those that have considered more objects have applied the single-object solutions with iterative approaches to the search for a solution. This study presents a more efficient methodology for finding the exact solutions to a certain class of wrapping problems in which the path is constrained by multiple surfaces. It also introduces a more general wrapping technique based on the idea of energy minimization, which has been successfully validated against the exact solution. These methods are applied to the case of an element of the deltoid wrapping around the humerus modelled as a composite sphere-cylinder. Comparison of results with those obtained from approximated single-object solutions demonstrates the need to include correct multi-object wrapping algorithms in biomechanical models.

Residual-based localization and quantification of peaks in X-ray diffractograms

We consider data consisting of photon counts of diffracted x-ray radiation as a function of the angle of diffraction. The problem is to determine the positions, powers and shapes of the relevant peaks. An additional difficulty is that the power of the peaks is to be measured from a baseline which itself must be identified. Most methods of de-noising data of this kind do not explicitly take into account the modality of the final estimate. The residual-based procedure we propose uses the so-called taut string method, which minimizes the number of peaks subject to a tube constraint on the integrated data. The baseline is identified by combining the result of the taut string with an estimate of the first derivative of the baseline obtained using a weighted smoothing spline. Finally, each individual peak is expressed as the finite sum of kernels chosen from a parametric family.

Exact expressions for the amplitudes and damped normal frequencies of a taut vibrating linear string of coupled masses

We determine exact expressions for the amplitudes and vibrational eigenmodes of a taut linear string of identical masses that are subjected to a viscous damping that is proportional to their momentum. We first examine the damped displacements of the string in the more familiar limit of a continuous distribution of mass. We then derive a damped version of the linear, one-dimensional wave equation and describe the dispersion curve. Next, we find exact expressions for the damped normal eigenmodes for a finite system. We show that a fraction of these modes have imaginary frequencies that correspond to overdamped oscillations. Next, we show that the remaining modes exhibit underdamped oscillatory behaviour and represent that portion of the dispersion curve graphically. We use methods that are at an intermediate level of presentation.

Generalizations of the Taut String Method

The taut string method is classically used in statistical applications to obtain a sparse estimation for a density given by point measurements. Mostly, a discrete formulation is employed that interpretes the data and the output as piecewise constant splines. This paper deals with the continuous formulation of this algorithm. We show that it is able to deal with continuous data as well as with discrete data interpreted as Dirac measures. In fact, any one-dimensional finite signed Radon measure is suited as input for the method. Moreover, we study the usage of tubes of nonconstant diameter. Examples indicate that such tubes can be useful in various applications. An existence and uniqueness theorem is given for the continuous formulation of the taut string algorithm with arbitrary tubes of nonnegative diameter.

The Equivalence of the Taut String Algorithm and BV-Regularization

It is known that discrete BV-regularization and the taut string algorithm are equivalent. In this paper we extend this result to the continuous case. First we derive necessary equations for the sol...

Empirical formulas to estimate cable tension by cable fundamental frequency

The cable tension plays an important role in the construction, assessment and long-term health monitoring of cable structures. The cable vibration equation is nonlinear if cable sag and bending stiffness are included. The engineering implementation of a vibration-based cable tension evaluation is mostly carried out by the simple taut string theory. However, the simple theory may cause unacceptable errors in many applications since the cable sag and bending stiffness are ignored. From the practical point of view, it is necessary to have empirical formulas if they are simple and yet accurate. Based on the solutions by means of energy method and fitting the exact solutions of cable vibration equations where the cable sag and bending stiffness are respectively taken into account, the empirical formulas are proposed in the paper to estimate cable tension based on the cable fundamental frequency only. The applicability of the proposed formulas is verified by comparing the results with those reported in the literatures and with the experimental results carried out on the stay cables in the laboratory. The proposed formulas are straightforward and they are convenient for practical engineers to fast estimate the cable tension by the cable fundamental frequency.

Forced response of translating media with variable length and tension: Application to high-speed elevators

The lateral response of vertically translating media with variable length and tension, subjected to general initial conditions and external excitation, are determined. The translating media are modelled as a taut string and tensioned beams with pinned and fixed boundaries. In each model a rigid body is attached to the lower end of the medium and has a prescribed displacement along the horizontal direction. The rate of change in the energy of the translating medium is analysed from the control volume and system viewpoints. The models are used to predict the forced response of a moving cable in a high-speed elevator. Three spatial discretization schemes are used to calculate the response and shown to yield the same results. The convergence of the solution for each model is investigated. The approximate solution for the string model with constant tension is compared with its exact solution from the wave method.

Densities, spectral densities and modality

This paper considers the problem of specifying a simple approximating density function for a given data set (x1,…,xn). Simplicity is measured by the number of modes but several different definitions of approximation are introduced. The taut string method is used to control the numbers of modes and to produce candidate approximating densities. Refinements are introduced that improve the local adaptivity of the procedures and the method is extended to spectral densities.

Parametric resonance and nonlinear string vibrations

Periodic changes in the tension of a taut string parametrically excite transverse motion in the string when the driving frequency is close to twice the natural frequency of any transverse normal mode of the string. The literature on this phenomenon is synthesized and extended to include the effects of damping as well as nonlinearity. It is shown that it is nonlinearity rather than damping that limits the growth of a resonantly excited mode, although damping is needed for steady-state oscillations to occur. The validity of the usual approximation that the string tension depends only on time and not on space is checked by modeling a string as point masses joined by massless linear springs. It is found that although this approximation is likely to be violated in practice, the violation does not have a significant effect on the results. The source of the disagreement in the literature for the speed of longitudinal waves in a stretched string is identified.

Non-linear free vibration of a cable against a straight obstacle

The goal of this study was to show how the response of a cable was altered after it impacted a straight obstacle. The two ends of the cable were held at the same level, and the cable was initially vibrating at its first symmetric mode. The free vibration of a cable depends on the amplitude of vibration, the sag-to-span ratio, and the ratio of the elastic to geometric stiffness as measured by Irvine and Caughey's cable parameter λ 2. How these variables affected the response of a cable interacting with a straight obstacle was examined for three limiting cases: a nearly taut string, an extensible shallow-sag cable, and an inextensible deep-sag cable. For a nearly taut string interacting with a straight obstacle in the absence of gravity, the periodicity that occurs for a linear taut string was preserved even when the amplitude of vibration was large. However, the frequency of vibration was modified due to the non-linear effects of large amplitudes. The periodicity was lost if the cable was not taut both in the presence and in the absence of gravity. For the case of a nearly taut string with gravity, the periodicity could be recovered if the obstacle assumed the shape of the equilibrium configuration of the cable. This was because the obstacle no longer behaved as a straight obstacle but as a convex obstacle whose radius of curvature decreased with decreasing amplitude-to-sag ratio. The behavior approached that of a point obstacle in the limit. Shallow-sag cables were analyzed by keeping the amplitude-to-sag ratio constant so that the obstacle behaved in a consistent manner. Comparison of our findings with the results of Irvine and Caughey for the free vibrations of a cable in air showed that the relative frequency based on the time to the first contact was not affected by the presence of the obstacle for all values of λ 2. However, the relative frequency based on the time between the first two contacts was slightly reduced when compared to the values corresponding to a non-interacting cable. This was attributed to the longer persistent contact that developed between the cable and the obstacle. Similar results were found for an inextensible deep-sag cable with various sag-to-span ratios. In both cases of shallow- and deep-sag cables, the ratio of the initial amplitude to the distance between the cable and the obstacle had a minimal effect on the reduction in the relative frequency.

Initial conditions in frequency-domain analysis: the FEM applied to the scalar wave equation

The present paper describes a procedure to consider initial conditions in the frequency-domain analysis of continuous media discretized by the FEM. The frequency-domain formulation presented here is based on a standard DFT procedure, the FFT algorithm being employed to transform from time to the frequency domain and vice versa. The standard Galerkin finite element method (displacement model) is used to replace the original differential governing equation by an integral equation amenable to numerical solution. The scalar wave equation (one- and two-dimensional) is used to illustrate the proposed approach. At the end of the paper, examples of wave propagation for a taut string, a one-dimensional rod and a membrane are presented to illustrate the robustness of the formulation presented here.

Semiactive Damping of Cables with Sag

Cables, such as those used in cable–stayed bridges, suspension bridges, guy wires, transmission lines, and flexible space structures, are prone to vibration due to their low inherent damping characteristics. The mitigation of cable vibration is necessary to minimize negative impact. Transversely attached passive viscous dampers have been implemented on some cables to dampen vibration. However, it can be shown that only minimal damping can be added if the damper attachment point is close to the end of the cable. For long cables, passive dampers may provide insufficient supplemental damping to eliminate vibration problems. A recent study by the authors demonstrated that “smart” semiactive damping can provide significantly superior supplemental damping for a cable modeled as a taut string. This article extends the previous work by adding sag, inclination, and axial flexibility to the cable model. The equations of motion are given. A new control–oriented model is developed for cables with sag. Passive, active, and smart (semiactive) dampers are incorporated into the model. Cable response is seen to be reduced by semiactive dampers for a wide range of cable sag and damper location.

Computational method for muscle-path representation in musculoskeletal models.

This paper presents a new and efficient method to calculate the line-of-action of a muscle as it wraps over bones and other tissues on its way from origin to insertion. The muscle is assumed to be a one-dimensional, massless, taut string, and the surfaces of bones that the muscle may wrap around are approximated by cross-sectional boundaries obtained by slicing geometrical models of bones. Each cross-sectional boundary is approximated by a series of connected line segments. Thus, the muscle path to be calculated is piecewise linear with vertices being the contact points on the cross-sectional boundaries of the bones. Any level of geometric accuracy can be obtained by increasing the number of cross sections and the number of line segments in each cross section. The algorithm is computationally efficient even for large numbers of cross sections.

Classical Casimir effect for beads on a string

Two small beads are situated at distance a apart on an otherwise uniform taut string. A transverse wave of angular frequency ω is incident from one side, exerting longitudinal forces F1 and F2, respectively, on the beads. The effective “force of attraction” between the beads, FC=(F1−F2)/2, is the simplest classical analog to the Casimir effect. We find that FC can be positive or negative, depending on the values of a and ω. For a broad spectrum of incident “noise,” however, the net “Casimir” force is zero.

Nonlinear Interactions: Analytical, Computational, and Experimental Methods

Nonlinear Interactions: Analytical, Computational, and Experimental Methods

INTERNAL RESONANCES IN WHIRLING STRINGS INVOLVING LONGITUDINAL DYNAMICS AND MATERIAL NON-LINEARITIES

Internal resonance mechanisms between near-commensurate longitudinal and transverse modes of a taut spatial string are identified and studied using an asymptotic method, and the influence of material non-linearities on the resulting solutions is considered. Geometrical non-linearities couple longitudinal motions to in-plane and out-of-plane transverse motions, resulting in resonant and non-resonant interactions between linearly orthogonal string modes. Past studies have included only transverse modes in the description of string motions and have predicted periodic, quasi-periodic, and chaotic whirling motions arising from the geometrical non-linearities. This study considers further the inclusion of longitudinal motions and a non-linear material law, which are both appropriate for the study of rubber-like strings. An asymptotic analysis captures the aforementioned whirling motions, as well as a new class of whirling motions with significant longitudinal content. Periodic, quasi-periodic, and aperiodic (likely chaotic) responses are included among these motions. Their existence, hardening–softening characterization, and stability are found to be highly dependent on the magnitude of the material non-linearities.

Nonlinear Dynamics of a Taut String with Material Nonlinearities

A spatial string model incorporating a nonlinear (and nonconservative) material law is proposed using finite deformation continuum mechanics. The resulting model is shown to reduce to the classical nonlinear string when a linear material law is used. The influence of material nonlinearities on the string’s dynamic response to excitation near a transverse natural frequency is shown to be small due to their appearance at high orders only. Material nonlinearities appear at low order in the equations for excitation near a longitudinal natural frequency, and a solution for this case is developed by applying a second order multiple scales method directly to the partial differential equations. The material nonlinearities are found to influence both the degree of nonlinearity in the response and its softening or hardening nature.

A 'taut string algorithm' for straightening a piecewise linear path in two dimensions

A piecewise linear path in two dimensions is formed by drawing straight lines between adjacent points in the sequence (u i : i = 1, 2,..., n}. Let Δ be a given positive number such that the length of each straight line segment is at least 3 Δ. We straighten the path in the following way. For i = 2, 3,..., n-1, we surround u i by a circular ring of radius A and centre u i . Then a piece of string that begins at u l and ends at u n is threaded through all the rings in sequence. The new path is constructed by pulling the string tight. An iterative algorithm is proposed that generates the new path to prescribed accuracy. Its convergence is proved and its efficiency is demonstrated by some numerical results.

Non-destructive determination of leaf area in tomato plants using image processing

SummaryThe objectives of this study were to validate the use of image processing for the measurement of leaf areas in intact tomato plants and to examine the potential of taut- string bounding area as a parameter for wilting. This was achieved by taking images of 18 tomato plants, viewed from the top, from the side and from an oblique angle. In the case of side and oblique views the images were taken as the plant was rotated through 360° with 45° steps. An average value of the leaf area was then calculated for each view. Regression of true leaf area on image-calculated areas for individual views resulted in a second order polynomial which adequately described the relationship. The regression coefficient, R2, adjusted for degrees of freedom, was 0.98. Multiple regression of true leaf area against areas obtained from two views increased R2 to 0.99. When all three view points were taken into consideration, R2 increased further, albeit modestly, to 0.993. Taut-string boundary provides information about the plant’s compactness as it grows. A boundary was obtained by rotating the image co-ordinate system through 16 different angles. After every rotation the extreme points of the image were determined. These points formed the corners of a bounding polygon that approximated to the taut string boundary. The polygon area was calculated from the co-ordinates of the polygon corners. Regression of true leaf area on taut string image area for individual views resulted in a second-order polynomial which adequately described the relationship. Multiple regression improved the regression coefficient. To test the potential usefulness of taut-string area, two plants were water stressed, then irrigated and monitored while they recovered from wilted to turgid. The change in taut string area detected (50–55%) was sufficient to recommend its use as a meaningful parameter for wilting.

FREELY PROPAGATING WAVES IN ELASTIC CABLES

This paper analyzes structural waves that propagate freely along an elastic and sagging cable. Results highlight the controlling role played by equilibrium cable curvature, a role not captured in previous studies of cable wave propagation that employ a simple taut string model. The investigation begins with a mathematical model governing the three-dimensional non-linear response of a long elastic cable. An asymptotic form of this model is presented for linear wave propagation along cables. An asymptotic form of this model is presented for linear wave propagation along cables having small equilibrium curvature. Upon linearization, structural waves lying within the equilibrium plane decouple from those orthogonal to this plane. The out-of-plane waves are non-dispersive and are well described using the taut string approximation. By contrast, the in-plane waves are dispersive and result in coupled longitudinal/transverse cable deflections. Attention focuses on these coupled in-plane waves for which spectral and dispersion relations are derived. Two qualitatively distinct in-plane wave forms exist. One wave type induces mainly longitudinal deformations and propagates with a speed close to that associated with an elastic rod. The other wave type induces mainly transverse deflections and propagates with a speed close to that associated with a taut string. Free response results demonstrate that significant tension waves may follow from purely transverse initial deflections.