Vibrations of a Multi-Span Beam Structure Carrying Many Moving Oscillators

Abstract

A beam structure carrying multiple moving oscillators is a mathematical model for various engineering applications, including rapid transit systems. With many moving oscillators having different speeds and varying inter-distances, the number of oscillators on the structure is time-varying, which inevitably complicates the beam–oscillator interactions. Consequently, the order of a mathematical model for the coupled beam–oscillator system changes with time, with many possibilities. Because of this, it is extremely difficult, if not impossible, for a conventional method to determine the dynamic response of a beam structure carrying many moving oscillators. In the literature, previous investigations have been limited to a beam structure with only one moving oscillator, which may not totally capture the physical behaviors of a structure with many moving oscillators, as seen in certain engineering applications. Developed in this work is a new semi-analytical method that can systematically handle arbitrarily many moving oscillators in both modeling and solution. In the development, an extended solution domain (ESD) is defined and based on the ESD a generalized assumed-mode method is devised. This modeling method completely resolves the issue of changing order in mathematical modeling. Because the proposed method makes use of the exact eigenfunctions of the beam structure (instead of traditional admissible functions), it renders determination of the dynamic response of a coupled beam–oscillator system highly accurate and efficient. The proposed method is demonstrated in several numerical examples. Furthermore, in a benchmark problem, it is shown that for the same accuracy in computation, the elapsed computation time used by the proposed method is just 3.3% of the time required by the finite element method.

Semi-Analytical Approach and Green’s Function Method: A Comparison in the Analysis of the Interaction of a Moving Mass on an Infinite Beam on a Three-Layer Viscoelastic Foundation at the Stability Limit—The Effect of Damping of Foundation Materials

“…Using the Green's function method to calculate the time series of a moving mass on an infinite beam on a three-layer foundation requires elastic contact between the moving mass and the beam and the presence of damping in the foundation materials to ensure numerical stability. The advantage of this method is the possibility of taking into account the nonlinearity of the contact between the moving mass and the beam, namely, the nonlinear character of the elastic contact and the possibility of contact loss [32,40,42]. On the other hand, the D-decomposition method developed in terms of the Green's function method can be applied to calculate the critical velocity of the moving mass on a beam on a three-layer foundation with high accuracy when the foundation is damped.…”

“…The problem of the stability of a moving object on an elastic structure with applications to the vehicle/track system was treated in the past by considering different mechanical representations of the vehicle (moving object) and the track (elastic structure). A railway vehicle can be represented by simple models, such as a moving mass [28,[31][32][33], two moving masses [34,35], a two-mass oscillator [36][37][38] or a train of two-mass oscillators [39][40][41], a three-mass oscillator [42], or a three-body oscillator [43,44]. The set of moving inertial objects was also dealt with in pioneering works [45,46] and, more recently, in [47].…”

“…Achieving this goal requires taking into account the loss of contact between the moving object and the structure. In this case, the equations of motion become nonlinear, and solving them in the time domain can be performed using a method based on the Green's function [32,40]. In this way, a nonlinear stability analysis based on the Hopf bifurcation is possible [42].…”

On the critical velocity of a mass moving along an infinite beam supported by three viscoelastic layers

“…Several models of the moving object as well as of the supporting elastic structure have already been used. A railway vehicle can be simplified as one [1][2][3] or two moving masses [4,5]; one two-mass [6,7] or three-body [8,9] oscillator or a train of moving objects [10][11][12][13]. A more detailed vehicle model is introduced in [14][15].…”

Instability of Vibrations of Mass(es) Moving Uniformly on a Two-Layer Track Model: Parameters Leading to Irregular Cases and Associated Implications for Railway Design

“…Mathematically, such a situation is detected due to the exponential increase in vibration amplitudes, implying that the steadystate solution is never reached even in the presence of damping. Several moving inertial objects on finite structures have been recently analyzed in [15][16][17]. Other works about infinite structures have also implemented the D-decomposition method together with the dynamic Green's function [18][19][20][21] or integral transforms [22,23].…”