Combination parametric resonance of an oscillator that moves uniformly along a beam on a periodically inhomogeneous foundation

Abstract

A new form of high-speed transportation, which involves a magnetically levitated vehicle that travels through a vacuum tube, is currently under development. Those parts of the trajectory along which the tube is supported by columns---which possess both axial and bending stiffness---requires the use of a model that allows to study a form of instability that is inherent to this type of support. Specifically, the stability is studied of a point mass with a lateral and vertical degree-of-freedom that moves with constant velocity along an infinite Euler-Bernoulli beam on coupled lateral-vertical periodically inhomogeneous foundation. The beam model is able to deflect in both the lateral and vertical direction and the concentrated mass is attached to this beam by a lateral and vertical contact spring. With the help of a perturbation method it is shown that the system's vibrations can become unstable. As for the model that only exhibits degrees-of-freedom in the vertical direction, the underlying physical phenomenon is parametric resonance, which occurs because of the periodic variation of the foundation stiffnesses. For the lateral-vertical model, this form of instability is referred to as combination parametric resonance, which yields four instability domains in the velocity-mass parameter space as opposed to the one domain for the vertical-only model. The center lines of these four domains depend strongly on the period of inhomogeneity; the larger the period, the higher the velocity at which instability occurs. Vehicle-structure interaction (i.e. the maglev system) also affects the center lines considerably; including contact springs between the beam and the point mass reduces the velocity at which instability occurs. To obtain the complete instability domains and to be able to investigate the mitigating effect of foundation damping, the method from another paper is recommended to be used for follow-up research.