MS14.6 - Moving Loads

In this talk we discuss the aspects of non-material modelling of a flexible structure moving between two qualitatively different domains with a configurational force, acting at the boundary. In particular, we consider the dynamics of a flexible rod, partially inserted into an inclined rigid channel with little or no friction. Initially, the rod will slide downwards because of the gravitational force. The free segment vibrates with growing frequency and decreasing amplitude as the rod is injected further. The falling will be decelerated by the longitudinal contact force, which is acting at the interface between the free segment and the part inside the channel. This configurational force is expressed in terms of the work needed to straighten the curved rod to pull it into the channel. Eventually it will outweigh gravity such, that injection will change to ejection. Under circumstances the rod will fully eject out of the opening. The formulation of this dancing rod problem was first suggested by the authors of [1], who also performed physical experiments and provided a fascinating video of the dynamic process as supplementary material. Being restricted to the case of vanishing inertia of the rod with a concentrated mass at its tip, their mathematical model is based on the equations of Newtonian mechanics and reproduces the experimental observations. The numerical approach of the present contribution features a finite element discretization of the deformation of the free segment with respect to a normalized coordinate. The material length of the free segment is considered as an additional degree of freedom. This results into a mixed Eulerian-Lagrangian kinematic description of a sliding beam in the spirit of [2,3]: while the material particles move across the boundaries of the elements, we still manage to obtain expressions for the potential and the kinetic energy of the entire rod. Numerical experiments using both, classical Kirchhoff rod model and shear deformable Simo-Reissner theory demonstrate the existence of the critical initial length of the free segment, which, in dependence on other parameters, determines, whether the motion is quasi-periodic or the rod will completely eject out of the channel. Further investigations on the configurational force feature analytical solution of the contact problem for a flexible beam, confined in a channel with small width, which allows to consistently account for frictional interaction between the rod and the channel in the general setting. References [1] Armanini, C., Dal Corso, F., Misseroni, D., & Bigoni, D. (2019). Configurational forces and nonlinear structural dynamics. Journal of the Mechanics and Physics of Solids, 130, 82-100. [2] Humer, A., Steinbrecher, I., & Vu-Quoc, L. (2020). General sliding-beam formulation: A non-material description for analysis of sliding structures and axially moving beams. Journal of Sound and Vibration, 480, 115341. [3] Vetyukov, Y. (2018). Non-material finite element modelling of large vibrations of axially moving strings and beams. Journal of Sound and Vibration, 414, 299-317.

Moving load problems have attracted the scientific community for many years since the first railway lines were build. Numerical assessment of the dynamic behaviour of structures subject to moving loads are under huge development, as are other approaches, to mention e.g. semianalytical methods and dynamic Green’s function. This contribution is focused on an infinite beam supported by three viscoelastic layers, which, due to its computational efficiency and relatively good approximation of reality, is a quite common model of a railway line. New developments that are presented concern the instability of a moving mass. The critical velocity in this context will be used for the lowest velocity that separates stable and unstable behaviour. The two above-mentioned methods are compared in terms of computational efficiency and accuracy of the obtained results. All results are presented in dimensionless form to cover a wide range of possible scenarios. There is no emphasis on adapting the data to any real part of the railway track. First, the model is presented along with its basic assumptions and simplifications. A set of dimensionless parameters necessary to describe the model is identified, along with a range of possible numerical values. The governing equations are stated, and solutions are presented by these two methods. Several cases are selected to demonstrate the results. It turns out that there are several situations with quite unpredictable behaviour, especially at low damping. In the semianalytical approach, the so-called lines of instability are tracked as a function of the ratio of velocity and moving mass. It is shown that there are several such lines. They can have quite a strange evolution, since they can form a closed curve, but they usually end up with asymptotes tending to infinite mass ratio at a fixed velocity ratio, and always one of them is tending to zero mass ratio at infinite velocity. It is also shown that the expected relation to the critical velocity of the moving force is not confirmed. First, three values of this critical velocity do not always exist. Sometimes there is only one, then the others are compensated by pseudocritical velocities, which may or may not be dominant, affecting their influence on instability lines. Second, especially for low damping levels, there are more than three asymptotes at fixed velocity ratios, so they cannot correspond to three possible values.

We study the dynamics of one-dimensional slender structures such as beams and arches carrying a moving load. Nonlinear, geometrically-exact rod theory is used to model the structure, which is allowed to undergo arbitrary three-dimensional flexural and twisting deformations. The equations of motion are solved by using the generalised-alpha method for both spatial and temporal discretisation. We find that large deformations (geometrical nonlinearities) have a detuning effect on the resonance and cancellation phenomena of the structure under moving loads. For these results we have obtained new exact expressions for the natural frequencies of circular beams and arches. We find that the collapse scenario (bifurcation) of a circular arch depends strongly on the opening angle of the arch. The speed of the moving load has a very strong effect on vibrations of the structures, including on the free vibrations after passage of the load. High speed tends to stabilise an arch or even to prevent in-plane collapse or out-of-plane instability where a static load of the same magnitude would induce it. We comment on the wider phenomenon of bifurcation delay due to finite rates of application of loads.