Modelling a string with varying length using the finite element method

Abstract eng:
This paper presents an investigation into using the finite element method to model a string with a moving constraint at one boundary, causing the string to change in length with time. This simple finite element model using straight elements provides insight into how this may be implemented in more complex models. Two approaches are considered for modelling a constraint which moves along the string. In the first the lengths of all the elements in the finite element model change equally with time, while in the second only the length of the element nearest the end being wound in is varied. When this length becomes small enough, the element is merged with the next element and hence the number of elements in the model decreases with time. This resembles the well understood spaghetti problem, in that essentially one end of a string is being pulled, and this provides a means of validation. Numerical results are compared with those of an analytic model and a perturbation solution and the agreement is very good. The difference in run-time between the two finite element models is also considered. Although run-time is not an issue for the simple string considered here, it may become so when applied to more complex models. Reducing the number of elements with time decreases the run-time, but modifying the system matrices between time steps increases the run-time.

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