MS15.5 - Nonlinear Dynamics and Dynamic Stability

In this paper, we studied a vibration of a string. The string is attached to a fixed point on one end while on the other hand, the position of the contact point can vary along a smooth obstacle. By applying a boundary fixing transformation, we transform the problem from a linear problem with moving boundary, to a nonlinear problem with fixed boundary. It is assumed that the vibrations around the stationary position of the string is small. Explicit approximations of the solution are obtained by using a multiple time-scales perturbation method. Depending on the parameters in the problem, it turns out that three different cases for the obstacle boundary condition have to be considered, that is, Dirichlet, or Neumann, or Robin type of boundary conditions. To avoid infinite-dimensional system of ordinary differential equations that occurs in the modal interactions of the string vibrations, characteristic coordinates are used together with a multiple time-scales approach to analyze the string dynamics in terms of traveling waves in opposite directions. A comparison of a direct numerical integration of the PDE problem and and the results obtained by using afromentioned perturbation approach shows an excellent agreement in the results.

Hyperloop is a new emerging transportation system that is in the development stage. Its design minimises the air resistance by having the vehicle travel inside a de-pressurised tube (near vacuum) and eliminates the wheel-rail contact friction by using an electro-magnetic levitation system (similar to the ones used by Maglev trains). By doing so, it can potentially reach much higher velocities than conventional railways, thus being a climate-friendly competitor to air transportation. It is well known that a vehicle travelling on a supporting structure can become unstable when its velocity exceeds a certain critical velocity. Instability leads to excessive amplification of the response, which can lead to derailment in extreme cases. This is rarely the case in conventional railway tracks, but the Hyperloop vehicle, due its predicted very high velocities, can very well exceed the critical velocity in the system. Consequently, knowing in which velocity regimes the Hyperloop system can be unstable is of high practical importance for its design. Previous studies have investigated this by simplifying the supporting structure (tube, pillars, soil, etc.) to a single degree-of-freedom system. While this approach is a good starting point, it neglects the frequency/wavenumber and vehicle velocity dependent reaction force of the supporting structure. This study aims to investigate the influence of correctly representing the reaction force on the stability of the system. To this end, the Hyperloop is modelled as an infinite beam continuously supported by a visco-elastic foundation (the periodic nature of the supports is neglected at this stage) subject to a moving mass. The interaction between the mass and the supporting structure occurs through an electro-magnetic force governed by a conventional proportional and derivative (PD) control. This study can help engineers designing the Hyperloop system avoid undesired excessive vibrations that can lead to fatigue problems and, in extreme cases, to derailment.

The prediction of the so-called seismic site response (i.e., the response of the top soil layers induced by seismic waves) is important for designing structures in areas prone to earthquakes. For seismic loads that induce large soil strains, accounting for the nonlinear behaviour of the soil can be of importance for accurate predictions. In a previous work, the authors proposed a nonlinear gradient elasticity model for predicting the seismic site response. In the said model, the nonlinear constitutive behaviour of the soil is governed by the hyperbolic soil model, in which the secant shear modulus is dependent on the shear strain through a non-polynomial (hyperbolic) relation. Moreover, the classical wave equation was extended to a nonlinear gradient elasticity model to capture the effects of small-scale heterogeneity/micro-structure. Compared to the classical continuum, higher-order gradient terms are introduced into the equation of motion, which lead to dispersive effects prohibiting the formation of un-physical jumps in the response. The aforementioned model is used in this work too, in which a Gaussian pulse is imposed as an initial condition and the solution is determined using a novel finite difference scheme. This work investigates the behaviour of the proposed model for different levels of initial nonlinearity (i.e., induced by the initial conditions). More specifically, we focus on explaining and studying the appearance of a non-zero plateau trailing behind as the initial shape propagates away. It is shown that the higher the initial nonlinearity, the more pronounced the plateau, indicating that the non-zero plateau is a characteristic of the system’s nonlinearity. The in-depth investigation of the proposed model's characteristics can be helpful when using it to accurately predict the seismic site response.