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2017 | OriginalPaper | Chapter

2. Applications of the Mechanics of a String

Author : Oliver M. O’Reilly

Published in: Modeling Nonlinear Problems in the Mechanics of Strings and Rods

Publisher: Springer International Publishing

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Abstract

We now turn to applications of the treatment in the previous chapter and develop mechanical models for the dynamics of a string. Many of the systems we consider in this chapter are classical but the analyses of their equations of motion is heavily influenced by treatments of, and novel insights into, these problems that have appeared during the past twenty years. Several of these problems involve chains which we model as a string in the presence of a gravitational body force. The chain fountain, a chain being dropped into a heap, and a chain with a fold are among the problems considered. The analysis of these systems serves to illuminate the roles played by momentum, energy, and material momentum in generating a closed system of equations to determine the dynamics of the string.

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previous chapter Mechanics of a String

next chapter Link, Writhe, and Twist

Our perspective on this matter is influenced by the works of Troger and his coworkers [315, 331] and statements on the lack of energy conservation in some chain problems in the textbooks authored by Lamb [193] and Love [212].

These conditions were first enunciated in a work by Green and Laws [129] on ideal jets of fluid.

The proof is presented in [250].

For a string modeling a three-dimensional linearly elastic body with a cross-sectional area A and mass density per unit volume ρ ₀ ^∗, we find that \(\rho _{0} =\rho _{ 0}^{{\ast}}A\) and so \(\frac{EA} {\rho _{0}} = \frac{E} {\rho _{0}^{{\ast}}}\). We also recall that \(\sqrt{ \frac{E} {\rho _{0}^{{\ast}}}}\) is the propagation speed for longitudinal waves in an infinitely long bar [346, Section 168].

The functional form of n(μ) with its local minimum and local maximum is motivated by constitutive relations for one-dimensional continua that are used in studies on phase transformations by Abeyaratne and Knowles [2, 5], Ericksen [98], and Purohit and Bhattacharya [296] (among many others).

The corresponding stiffness for the strain energy function \(\rho _{0}\psi = \frac{EA} {2} \left (\mu -1\right )^{2}\) is EA.

For the homogeneous strings of interest here, \(\oint \rho _{0}\mathbf{v} \cdot \frac{\partial \mathbf{r}} {\partial \xi } d\xi =\rho _{0}\oint \mathbf{v} \cdot \mathbf{E}_{t}ds\), and thus, the material momentum P and its integral can be related to the circulation v ⋅ E _t and “flow” that were first defined by Kelvin in 1869 and play a prominent role in fluid mechanics. For further details on this topic, the reader is referred to [52, 194, 349].

This solution was first determined by Cayley [53, Page 511] and is labeled c in Figure 2.8.

See [142, 366, 367] for the relevant citations.

I am most grateful to James Hanna for pointing this out.

Problem 17 on Page 252 of Jeans’ textbook [174] is reproduced in Exercise 2.5 at the end of this chapter.

We use the symbol \(\boldsymbol{\omega }\) to denote the angular velocity of a rigid body in this discussion. It should not be confused with the same symbol for the velocity \(\boldsymbol{\omega }\) that appears in later chapters and is associated with a set of directors.

For additional commentary, including slow-motion videos of chain fountains and falling chains, on these works see [114] and Ruina’s website http://ruina.mae.cornell.edu/.

This prescription is identical to the one we used earlier with Cayley’s problem (cf. Eqn. (2.40)).

For instance, Eqn. (2.65) is equivalent to the governing equation for the same problem established by Virga in [356, Eqn. (5.10)] with \((\frac{1} {2} - 2e,\gamma )\) identified with his (1 − f, y). The same differential equation is also equivalent to Grewal et al.’s work with \((\dot{\gamma },\gamma )\) identified with their \((-v = -\dot{x},L - x)\) and \(e = \frac{1} {6}\) for [142, Eqn. (18)] and e = 0 for [142, Eqn. (19)]. Finally, Eqn. (2.65) is equivalent to [151, Eqn. (5)] with \((\frac{1} {2} - 2e,\gamma )\) identified with their (1 −γ, y).

See, for example, [150, 189, 304].

This prescription is identical to the prescriptions (2.40) and (2.66) for B _γ that were used in Sections 2.4 and 2.6, respectively. As with these prescriptions, observe that the prescription (2.84) accommodates cases where \(\dot{\gamma }< 0\).

The video can be accessed at http://stevemould.com/siphoning-beads/.

It might be helpful for some readers to review the three conditions for a steady axial motion discussed in Section 2.2 and to see how they pertain to the chain fountain shown in Figure 2.19.

It may be helpful to examine Eqn. (1.112) to find the prescription for b that we are using here. Note that because the string is assumed to be homogeneous, the expression for this force simplifies dramatically to the derivative of a potential energy density.

We are closely following the discussion of the inverted axially moving catenary in Perkins and Mote [288] and its extension to the nonlinearly elastic case that is discussed in [259].

For extensions of the classic catenary problem to the case where the string is extensible, the reader is referred to Antman [11, 12] and Dickey [87, 88].

This prescription is identical to the prescriptions (2.40), (2.66), and (2.84) for B _γ that were used in Sections 2.4, 2.6, and 2.7, respectively.

Our presentation and scope here is strongly influenced by the earlier analytical and experimental work of Biggins [23].

In [23], the parameters α and β are used to prescribe \(n\left (\gamma _{0}^{+},t\right ) = T(0) = \left (1-\alpha \right )\rho _{0}c^{2}\) and \(n\left (\gamma _{1}^{-},t\right ) = T(w) =\beta \rho _{0}c^{2}\). Thus, examining Eqns. (2.110) and (2.115), we can conclude that \(\alpha = \frac{1} {2} - 2e_{0}\) and \(\beta = \frac{1} {2} - 2e_{1}\). The values of α = 0. 12 and β = 0. 11 that are used in [23] are equivalent to e ₀ = 0. 190 and e ₁ = 0. 195 that are used to construct the results shown in Figure 2.21.

It is left as an exercise for the interested reader to substitute for H, x _a, and U in terms e ₀, e ₁, θ ₀, ρ ₀, and h in the expressions for w and n −ρ ₀ c ² to convince themselves that knowledge of these parameters suffices to determine w and n −ρ ₀ c ².

For further references on, and discussion of, the energy-conserving formulation, we refer interested readers to [142, 366, 367].

Versions of this classic problem are discussed in several textbooks (cf. [174, Problem 8, Page 238], [236, Problem 526], and [237, Sample Problem 4/10]). We highly recommend reading [142] and [237, Sample Problem 4/10] for additional perspectives on the present problem.

It should not be surprising given the earlier comments on the classic formulations of these problems that the choice e = 0. 25 corresponds to the case considered in [236, Problem 526] and [237, Sample Problem 4/10].

The results shown in Figure 1.20 might be of assistance with these computations.

The relationship between B _γ and the driving force f is discussed in Exercise 1.7 on Page 53.

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Title: Applications of the Mechanics of a String
Author: Oliver M. O’Reilly
Publisher: Springer International Publishing
Book: Modeling Nonlinear Problems in the Mechanics of Strings and Rods
Print ISBN: 978-3-319-50596-1

Electronic ISBN: 978-3-319-50598-5

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-50598-5_2

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