Wind-driven nonlinear oscillations of cables

Abstract

Abstract. The objective of this papcr is the study of the d\'namics of damped cable systems, which are suspended in spacc, and their resonance characteristics. Of interest is tile study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a lirst-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables tire acted t.pon by their own weight in thc perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes tire used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations.In order to investigate the behavior of the cablc motion in detail, the linear and the nonlinear analyses arc discussed separately. Tile lirst part of this paper deals with tile solution to the self adjoint boundary-value problenl for small-amplitude vihrations and the determination of mode shapes and natural frequencies. The second problem dealt with in this paper is tile determination of the phenomena produced by tile primary resonance of the systenl. The method of multiple time scales is used to develop solutions for the resuhing multi-dimensiomd dynamical systcnl with quadratic nonlinearity.Numerical results for the steady state response amplitude, and their variation with external cxcitation and external detuning for various values of intcrnal detuning parameters are obtained. Saturation and jump phenomena tire also obscrved The jump phenomcnon occurs when there arc multi-valued solutions and thcrc exists a variation of kinetic cncrg.v ~lll]ong solutions.

Three-mode interactions in harmonically excited systems with quadratic nonlinearities

“…He did not identify any Hopf bifurcations and consequently did not study any nonperiodic motions. Tadjbakhsh and Wang [41] investigated the response of wind-driven cables. They modeled the cable as a three-degree-of freedom system with quadratic nonlinearities with the internal resonant conditions w 3 = 20) 2 and 0)2 ~" 20)1.…”

Non-linear dynamics of wind turbine wings

“…Forced response with coupled vertical and horizontal responses has been analysed in a number of studies based on Galerkin based reduced two-degrees-of-freedom models. Tadjbakhsh and Wang (1990) investigated the response of wind induced cables modelled as a three-degrees-offreedom system including quadratic nonlinearities with the internal resonant conditions ω 3 2ω 2 and ω 2 2ω 1 with the second mode excited (Ω ω 2 ). They found that the system exhibits both saturation and jump phenomena.…”

The saturation bifurcation in coupled oscillators

“…The saturation effect may prove useful in the design of materials or structures that absorb [7,8], localize [9][10][11], or divert [12] vibrational or wave energy. The effect can also appear in weakly nonlinear approximations of driven hanging cables [13,14] or spring-pendulum systems [15], in which the relevant couplings contain time derivatives.…”