# Parametric Instability of an Oscillator moving on a Railway Track

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## Abstract

The instability of a moving mass / oscillator due to surpassing the velocity of the minimum group wave velocity (in a continuous homogeneous structure) has been studied extensively and is well understood. In contrary to that, Parametric Instability of a moving mass / oscillator on a continuous periodic structure has been studied less extensive and therefore the mechanism behind the instability is unknown. Literature that is avaiable on this topic mainly focuses on continuous periodic inhomogeneous structures, namely where the foundation is modeled as a continuous periodic inhomogeneous structure. Even less well studied are models where instead of a continuous periodic inhomogeneous foundation discrete periodic supports have been used. So far as known to the author there has also been no studies where the discrete supports are coupled through a medium. In order to solve the transition curves discerning the stable and unstable domains concerning the parametric instability of a moving mass / oscillator the analogy with the Mathieu equation is used, which tells us that the solution on those transition curves will be periodic with once or twice the period of the parametric excitation. In this thesis we have focussed on studying the use of the analogy on continuous structures founded upon periodic supports.

The Mathieu equation describes the motion of a parametric oscillator, for example a pendulum with a length that periodically varies over time. The theory that predicts the solution to the Mathieu equation is called Floquet theory and ascociated with the solution are the Floquet exponents, these exponents dictate whether the solution will be periodic and bounded or unbounded. By solving for the Floquet exponents of the Mathieu one sees that for an increase of the amplitude of the parametric excitation the system will experience a greater exponential growth. For a greater mistuning between the parametric excitation and the natural frequency of the equivalent non-parametrically forced equation we see that the system will experience a smaller exponential growth. Outside the instability domains the solutions will be bounded and periodic and contain a wide variety of frequencies. When damping is introduced, the value of the damping coefficient (if written in the canonical form of a viscously damped single degree of freedom equation) will be subtracted from the value of the undamped Floquet exponents and by that result in an upward shift and narrowing of the transition curves. Furthermore, outside the instability domains we will see two regions: the first lies close the the transition curves and is asymptotically stable with a period equal to that of the transition curve, the second covers the remaining stable region and is asymptotically stable and periodic with a wide variety of frequencies.

Regarding the Parametric Instability of a moving mass / oscillator we have studied three different models: a continuous Euler-Bernoulli beam on periodic spring supports, a continuous Euler-Bernoulli beam on periodic supports that are complex (i.e. modeled as an oscillator between two springs), and a continuous Euler-Bernoulli beam on periodic supports that are founded upon a 2-dimensional lattice. Of these models we have conducted a parametric study as to study what the effects are of the various parameters. We have seen that whenever the ratio between the stiffness of the supports and that of the beam is increased, the instability domains will shift to higher velocities and become wider. For certain combinations also 'islands' of instability may appear, where these 'islands' indicate stable areas between regions of instability. If damping is introduced into the system, this will generally narrow the instability domains and shift them to higher values of the mass and lower values of the velocity. However, for certain parameter combinations adding damping will lead to a widening of the instability domain. In the case of a moving oscillator the instability domains will narrow and be shifted to lower values. If the support is modeled with a mass it will affect the general trend of the transition curves through its own resonance, hence for a complex structure it is advised to model the supports with the correct dynamic equations. Last but not least, if the supports are coupled through a medium (a 2-dimensional lattice in this case) one will generally see a similar effect as adding damping has.

In this thesis we have also studied three real world cases: a regular railway track, a high-speed railway slab-track, and the Hyperloop. In the first case we have seen that Parametric Instability will most likely have no influence. For the slab-track, being much more stiff, Parametric Instability will be important. However, a more extensive study with several cases must confirm this. In the last case, namely the Hyperloop, we have seen that for a moving mass the instability domains are relatively large as compared with the other cases. We have also studied the Parametric Instability of a test-pod, which showed that its instability domains that are negligible. Of course, this was merely a test-pod not capable of transporting people, hence when a larger pod is studied it may be expected that the instability domains may not be neglected.