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ALE beam using reference dynamics

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Abstract

We present a small strain beam model based on the Arbitrary Lagrangian Eulerian setting for use in multibody dynamics. The key contribution of the present paper is to provide a formulation with large flexible reference motion and small overlaid deflections. We point out that the reference motion is described by actual degrees of freedom of the model. Therefore, we use a vector of generalized positions and an Eulerian coordinate, which itself is a degree of freedom and in which the flow of the beam material through an arbitrary volume is represented. The additional displacements describe small fluctuations around the reference motion. With this idea it is easy to separate the motion of belt drives, cable and rope ways or strings. In particular, the overlaid deflections are described for efficient numeric computation and may be analyzed in an easy way for vibrational behavior. The guiding reference motion is arbitrary, i.e., the transmission ratios are degrees of freedom and may change dynamically affecting also the fluctuations. Contacts with dry friction are foreseen and represented in the present model. It is validated and proven to be efficient in comparison with classic co-rotational and absolute nodal coordinate formulations in our application. The simulation of pushbelt continuously variable transmissions is taken as a high-dimensional industrial example.

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Notes

  1. The absolute change in the position of a mass particle is derived based on \(r(\bar {x})\) with \(\bar {x}\) being constant (cf. (2)).

  2. The derivative of a matrix \(\mathbb {G} \) w.r.t. to a vector \(d\) applied to a vector \(e\) is defined as

    $$\begin{aligned} \frac{\mathrm {d} \mathbb {G} }{ \mathrm {d}d} e = \sum_{i=1}^{k} \frac{\mathrm {d} \mathbb {G} }{ \mathrm {d}d _{i}} e_{i} \end{aligned}$$

    where \(d\) and \(e\) have the dimension \(k\).

  3. \([t\ n\ b]\) refers to the local deformations and does not coincide with \([t _{\mathit {Ref}}\ n _{\mathit {Ref}}\ b _{\mathit {Ref}}]\).

  4. The parameter \(\varTheta \) is defined in such a way that symmetric properties can be used in the numerical evaluation.

  5. Interpolated by \(N_{1}\) and \(N_{3}\).

  6. Interpolated by \(N_{2}\) and \(N_{4}\).

  7. Here on the left side of the ring.

  8. Cases 1, 4 and 7.

  9. Cases 2, 5 and 8.

  10. Cases 3, 6 and 9.

  11. The pulley sheaves deform due to the clamping forces. Thus, elements change the running radius within a pulley arc, which is called spiral running.

  12. For instance ALE0.

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  1. Technische Universität München, Institute of Applied Mechanics, Garching bei München, Germany

    Kilian Grundl, Thorsten Schindler, Heinz Ulbrich & Daniel J. Rixen

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Grundl, K., Schindler, T., Ulbrich, H. et al. ALE beam using reference dynamics. Multibody Syst Dyn 46, 127–146 (2019). https://doi.org/10.1007/s11044-019-09671-7

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