Discussion of "dynamic stability of cables subjected to an axial periodic load"

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29942/1/0000300.pd

Asymptotic analysis of a translating cable arch

The stability of a translating cable which stands in the shape of an arch between two supports is investigated. Asymptotic solutions for free in-plane and out-of-plane cable vibrations are derived under the assumption of small arch height. Without translation speed, the cable arch collapses under compressional loading. As translation speed is increased, however, the cable arch becomes tensioned and stable. A previous experiment confirms this prediction. The asymptotic solutions derived here show that the in-plane and out-of-plane vibration modes are stabilized in fundamentally different manners.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27630/1/0000006.pd

Modal interactions in the non-linear response of elastic cables under parametric/external excitation

A theoretical model is derived which describes the non-linear response of a suspended elastic cable to small tangential oscillations of one support. The support oscillations, in general, result in parametric excitation of out-of-plane motion and simultaneous parametric and external excitation of in-plane motion. Cubic non-linearities due to cable stretching and quadratic nonlinearities due to equilibrium cable curvature couple these motion components in producing full, three-dimensional cable response. In this study, a two-degree-of-freedom approximation of the model is employed to examine a class of in-plane/out-of-plane motions that are coupled through the quadratic non-linearities. A first-order perturbation analysis is utilized to determine the existence and stability of the planar and non-planar periodic motions that result from simultaneous parametric and external resonances. The analysis leads to a bifurcation condition governing planar stability and results highlight how planar stability is reduced and non-planar response is enhanced whenever a "two-to-one" internal resonance condition exists between a pair of in-plane and out-of-plane cable modes. This two-to-one resonant behavior is clearly observed in experimental measurements of cable response which are also in good qualitative agreement with theoretical predictions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30182/1/0000567.pd

Three-dimensional oscillations of suspended cables involving simultaneous internal resonances

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43332/1/11071_2004_Article_BF00045006.pd

Hydrodynamic and Geometric Stiffening Effects on the Out-of-Plane Waves of Submerged Cables

This study focuses on the relative importance of two sources of nonlinearities affecting submerged cable response. The first of these is the added fluid damping offered by the surrounding medium while the second is the geometric stiffening offered by the cable through finite extensions of its centerline. The contribution of each nonlinear effect, taken separately and in tandem, is evaluated herein through the study of structural waves that form in the (out-of-plane) direction normal to the cable equilibrium plane.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43320/1/11071_2004_Article_135260.pd

Nonlinear oscillations of suspended cables containing a two-to-one internal resonance

The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43323/1/11071_2004_Article_BF00045648.pd