Dynamic analysis of suspended cables carrying moving oscillators

Abstract

An efficient procedure for analyzing in-plane vibrations of flat-sag suspended cables carrying an array of moving oscillators with arbitrarily varying velocities is presented. The cable is modelled as a mono-dimensional elastic continuum, fully accounting for geometrical nonlinearities. By eliminating the horizontal displacement component through a standard condensation procedure, the nonlinear integro-differential equation governing vertical cable vibrations is derived. Due to the dynamic interaction at the contact points with the moving oscillators, such equation is coupled to the set of ordinary differential equations ruling the response of the travelling sub-systems. An improved series representation of vertical cable displacement is proposed, which allows to overcome the inability of the traditional Galerkin method to reproduce the kinks and abrupt changes of cable configuration at the interface with the moving sub-systems. Following the philosophy of the well-known ''mode-acceleration'' method, the convergence of the series expansion of cable response in terms of appropriate basis functions is improved through the introduction of the so-called ''quasi-static'' solution. Numerical results demonstrate that, despite the basis functions are continuous, the improved series enables to capture with very few terms the abrupt changes of cable profile at the contact points between the cable and the moving oscillators.

Modelling and transient planar dynamics of suspended cables with moving mass

“…This results in a very slow convergence of the sine series, as also observed in Al-Qassab et al (2003), Sofi and Muscolino (2007), where the improvements obtainable with more sophisticated expansions were highlighted. In particular, the latter authors proposed to account for the pseudo-static contribution of the truncated high-frequency modes in the series expansion of the transverse displacement v(x,t) (Eq.…”

“…In this study, the dimensional parameters and material properties of the suspended cable are as follows: the area of the cross-section A = 1.175 Â 10 À3 m 2 , the mass per unit length of the cable m = 9.7 Kg/m, the Young modulus of the suspended cable E = 147 GPa, the cable span l = 500 m, the sag of the cable b = 15 m. Except for the sag b, the material properties and dimensions of the suspended cable are chosen to be identical to those of Sofi and Muscolino (2007). Table 1 shows the rate of convergence of the first eight natural frequencies of the suspended bare cable.…”

“…The approximate dynamic strain exhibits two contributions due to the relative acceleration of the moving mass, which arise within the variational formulation of the equations of motion and ensue from retaining the inertia term of the moving mass and the friction in the condensation procedure. Both effects are neglected in the classical condensation, where the dynamic strain of the suspended cable turns out to be uniform along the cable span and is only a function of time t (Wang and Zhao, 2006;Srinil and Rega, 2007;Sofi and Muscolino, 2007). In contrast, in the present improved condensation, the strain (Eq.…”

“…Compared with the classical condensed planar model used by Sofi and Muscolino (2007), some main differences occur in the expression (21) and in the condensed integro-differential equation of motion (22). The approximate dynamic strain exhibits two contributions due to the relative acceleration of the moving mass, which arise within the variational formulation of the equations of motion and ensue from retaining the inertia term of the moving mass and the friction in the condensation procedure.…”

“…They also applied the classical Galerkin method and the wavelet-Galerkin method to discuss the dynamic behavior of the cable due to the moving mass along its length. Sofi and Muscolino (2007) derived the equations governing the planar motion of the coupled cable-moving oscillators system, and investigated the vibration of a shallow suspended cable carrying an array of moving masses with arbitrarily varying velocities. They treated the interaction force between the oscillators and the cable as a vertical external load entering the integro-differential equation of motion of the classical condensed model of the bare cable (Luongo et al, 1984).…”