168 L. Carassale et al. Fig. 16.1 Rail line and traveling wheel idealization j j+1 j u 1 j u + jф 1 jф+ x 0 1 ( ) u x lg ag kb, cb kp, cp k, c ms ma m a w 1 j v + j v Fig. 16.2 Subsystems 16.2.1 Rail Line Model The rail line is idealized by a continuous Euler-Bernoulli beam supported by the sleepers having a constant spacing l. The sleepers are connected to the ground and to the rail through visco-elastic devices representing, respectively the behavior of the terrain and the flexibility of the railpads [1] (Fig. 16.1). Let kb and cb be the stiffness and damping coefficient of the ground-sleeper connection, while kp andcp are the stiffness and damping coefficient of the rail-sleeper connection; ms is the mass of the sleepers. The motion of the rail is governed by a partial differential equation, however the system can be discretized according to the Finite Element (FE) approach obtaining the equation of motion in the form: M0¨u+C0˙u+K0u =f +g (16.1) where uis the displacement vector organized as u= ⎡ ⎢⎣ u1 . . . uN ⎤ ⎥⎦; uj = ⎡ ⎣ uj φj vj ⎤ ⎦ (16.2) where N is the number of sleepers included in the model, uj and φj are, respectively, the vertical displacement and rotation of the rail at the support point j (Fig. 16.2), and vj is the vertical displacement of the j th sleeper. The vector f contains the
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