This paper presents a modeling study of the dynamics of a helical spring element with variable pitch and radius considering both the static stiffness and dynamic response by using the geometrically exact beam theory. The geometrically exact beam theory based on the Euler–Bernoulli beam hypothesis is described, of which the shear deformations are ignored. Unlike the traditional spliced curved beam element method, the helical spring element is described with curvature vector and axial strain by establishing and spline-interpolating a function of the radius, the height, the polar angle, and the torsion angle of the whole spring. In addition, a model smoothing method is developed and applied in the numerical analysis to filter the high-frequency oscillation component of the flexible multibody systems, so as to correct the system dynamic equations and improve the calculation efficiency when solving the static equilibrium of the spring. This study also carries out five numerical trials to validate the above dynamic procedure of the helical spring element. The example of the spring static stiffness design shows that the proposed helical spring procedure enables one to deal with practical engineering applications.
This paper addresses the energy‐efficient train timetabling problem for MSM systems, where both propulsion and suspension energy consumption are considered. The timetable design problem is modelled as a bi‐level model for a complete two‐way MSM line. The upper level determines the train departure time at the first station, which makes train operations more convenient for passengers. The lower level uses an empirical description of the train energy consumption as a function of segment running times, and an energy‐efficient timetable optimization model is built. In doing so, all the services in both directions along a certain planning horizon are considered while attending to a known passengers’ demand. Moreover, the convenience of considering energy consumption as part of a broad objective function that includes other relevant costs is pointed out. Then, a unified sequential solution algorithm is developed for an efficient and accurate solution of the bi‐level model. Experiments show that the proposed framework can generate a holographic timetable of energy‐efficient MSM containing multi‐dimensional variables such as time, space, velocity, and electrical quantities.
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