In the engineering sciences the one-dimensional continuum of a string, completely lacking bending and torsional stiffness, is used to describe structures both very thin and long ([1]).Originally, the lecturer's major motivation for dealing with such structures were problems modelling tethered satellite systems. However, in this talk these spacecrafts are regarded merely as a source of inspiration. For the sake of mathematical lucidity and brevity only the string-pendulum in a homogeneous gravitational field is used to illustrate the theoretical results.

We introduce the dynamics of flexible structures by discussing results of experiments conducted with a chain. Surprisingly, the observed phenomena defy a simple explanation. For example, the endpoint of a folded chain-pendulum falls much faster than a body in free falling motion. As we additionally prove that a chain of small rigid bodies represents a discretization of an inextensible string these seemingly paradox results give already a glimpse into the dynamics of the continuum.

According to Newton's second law the motion of a string obeys a system of hyperbolic partial differential equation. Linearization leads to a decoupling of longitudinal waves and motions orthogonal to the configuration curve. Just like these characteristic motions, the possible shock structures essentially fall into two separate classes: longitudinal shock-waves with continuously varying tangents and general kinks with continuous tension ([2]). We show that the latter are already reflected by a simple chain with sufficient accuracy. Since kinks are contact discontinuities they behave like highly curved transversal waves. Although this fact facilitate the numerical treatment considerably we present also a method to calculate the propagation of a kink explicity.

However, the problem of the folded inextensible string mentioned above turns out to be a special case. Due to the degeneration of a one-dimensional problem, the balance of linear momentum admits a one-parameter family of possible motions. They physically reasonable selection from this a-priori-non-unique solutions is found only by means of a two-dimensional well posed problem. Since the ultimate result is in contradiction to solutions given in the literature ([3]) it could motivate further discussions.

[1] S. S. Antman (1995). Nonlinear Problems of Elasticity. Volume 107 of Applied Mathematical Sciences. Springer-Verlag, New York.

[2] M. Shearer (1985). Elementary Wave Solutions of the Equations Describing the Motion of an Elastic String. SIAM J. Math. Anal., 16:447-459.

[3] J. L. Meriam and L.G. Kraige (1998). Dynamics. Volume 2 of Engineering Mechanics. John Wiley & Sons, New York, 4th Edition.

 
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