Continuous transition between traveling mass and traveling oscillator using mixed variables

• Absolute or relative displacements of traveling object are replaced by coupling force.
• Continuous exchange between traveling mass and traveling oscillator.
• Very stiff ODEs are avoided by mixed variables.
• Dynamic stability is treated in time-domain and spectral Floquet domain.
• Consistent kinematics for identifying normal part of total acceleration for traveling mass.
• Common kinematics for strains and traveling mass moving along circle-like curved beam.

The interaction between cars or trains and bridges has been often described by means of a simplified model consisting of a beam loaded by a traveling mass, or by a traveling oscillator.Among others, two aspects are essential when dealing with masses traveling along flexible vibrating supports: (i) a complete relative kinematics; and (ii) a continuous transition between a traveling mass, rigidly coupled, and a traveling oscillator, elastically coupled with the support.The kinematics is governed by normal and tangential components—with respect to the curved trajectory—of the acceleration. However in literature these parts are oriented with reference to the undeformed beam configuration. This model is improved here by a non-linear second-order enriched contribution.The transition between a traveling oscillator and a traveling mass is governed by the stiffness k of the elastic or viscoelastic coupling which, in the latter case (i.e. rigid coupling), has to tend towards infinity.However, very large stiffness values cause high frequencies and significant problems are mentioned in the literature in order to establish numerically stable and reliable results and in order to realize a continuous evolution between absolute and relative formulations.By using mixed state variables, generalized displacements and coupling forces, the contribution from the stiffness changes from k   to its inverse 1/k1/k, the coupling force itself becomes a member of the solution-space and the problems, which have been mentioned in the literature, disappear. As a matter of fact, the coupling force can also take into account a viscoelastic contribution; moreover, a larger number of traveling oscillators can be considered, too.Finally, for a periodic sequence of moving oscillators the dynamic stability is treated in the time-domain along several periods, as well as in the spectral domain, by using Floquet׳s theorem.