Vibration analysis of shallow cable with horizontal spring and dashpot at one end. (15th May 2020) - Zhang, Peng; Zhu, Huanian; Lu, Wensheng; Lu, Xilin; MacRae, Gregory A.
Highlights: The dynamics equations are proposed for a cable with horizontal spring and dashpot. Nondimensional parameters describing the response are identified. Ranges of parameters useful for cable design/construction are proposed. A methodology to obtain the peak deformation and axial force is developed. Abstract: In this study, a general closed-form equation is derived for the steady-state response of a cable with both ends at the same horizontal level, and a horizontal protective device composed of a horizontal spring and dashpot at one end. The cable is subject to vertical constant-amplitude harmonic loading. Nondimensional parameters describing the response are identified. A comparison is performed with the steady-state response from numerical methods, obtained after the transient solution has become negligible. The effects of the nondimensional parameters on the modal damping ratio, modal frequency, maximum displacement, and axial force are quantified. Using the response information obtained, as well as practical considerations, ranges of parameters useful for cable design/construction are proposed. Finally, a step-by-step methodology to obtain the peak deformation and axial force is developed, and this is illustrated with an example. It was shown that the multiple-scale method solved the equations of motion, thus allowing the force and vertical displacement at any point on the cable to be obtained. Three nondimensional parameters representing spring stiffness γ k, dashpot damping ν, and cable characteristic αb 2, and a load parameter were also obtained from the general solution. Not all quantities were obtained explicitly, but some needed to be solved numerically. The solution was consistent with the published solution when the spring was infinitely rigid. Numerical-method solutions matched the peak closed-form solutions with a maximum error of less than 10%. The modal damping ratio increased with decreasing spring stiffness and increasing cable parameter. The modal frequency tended to increase with increasing spring stiffness and dashpot damping. The values of the peak displacement and peak axial force associated with resonance were reduced by an increased cable parameter and decreased spring stiffness. When considering the likely cost, shallow cable definition, total and spring amplitude, and predictivity of behaviour, the recommended ranges of shallow cable design values were given for the nondimensional dashpot damping, spring stiffness, and cable parameter. A design example following design steps indicated that for a 20 m cable with γ k = 0.38, ν = 12.22, α b 2 = 0.26, and amplitude of external harmonic load 0.5 kN/m, the peak displacement and axial force were 2.74 m and 1.08 × 10 3 kN, respectively.