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The nonlinear dynamic behavior of rain-wind-induced vibration of inclined cable is investigated with the consideration of the equilibrium position of the moving rivulet. The partial differential governing equations of three-degree-of-freedom on the model of rain-wind-induced cable vibration are established, which are proposed for describing the nonlinear interactions among the in-plane, out-of-plane vibration of the cable and the oscillation of the moving rivulet. The Galerkin method is applied to discretize the partial differential governing equations. The approximately analytic solution is obtained by using the method of averaging. The unique correspondence between the wind and the equilibrium position of the rivulet is ascertained. The presence of rivulet at certain positions on the surface of cable is then proved to be one of the trigger for wind-rain-induced cable vibration. The nonlinear dynamic phenomena of the inclined cable subjected to wind and rain turbulence are then studied by varying the parameters including mean wind velocity, Coulomb damping force, damping ratio, the span length, and the initial tension of the inclined cable on the model. The jump phenomenon is also observed which occurs when there are multiple solutions in the system.

Dynamic behavior of corresponding differential systems has been of extensive concern and investigated by many scholars [

On the basis of the measured aerodynamic coefficients, the kinetic equations of rain-wind-induced vibration have been established. However, before the measurement of the aerodynamic coefficients on the moving rivulet, the motion of rivulet was assumed to be harmonic [

The influences of equilibrium position of rivulet on cable have been studied, in which the equilibrium position of rivulet was always assumed to be independent of wind. The equilibrium position of rivulet was altered factitiously to inspect its effects on the vibration while the mean wind velocity was fixed [

In comparison with the literature above, the equilibrium position of the moving rivulet is considered and the nonlinear dynamic characteristics of rain-wind-induced vibration are investigated analytically by using the approximate method of averaging. The present paper established the continuum model of the inclined cable subjected to wind and rain turbulence with the consideration of the equilibrium position of the rivulet. The reliance of the equilibrium position of the rivulet on wind is figured out and the effects of the equilibrium position of the rivulet on the cable vibration are observed and proved. Furthermore, the amplitude response curves of parameters including mean wind velocity, Coulomb damping force, damping ratio, the span length, and the initial tension of the inclined cable present abundant dynamic behaviors. The jump phenomenon is also observed when multivalued solutions exit in the system.

Assumptions of inclined cable are made in present paper: (1) The flexural rigidity, torsional stiffness and shear stiffness are ignored; (2) The constitutive relation of cable deformation submits to Hooke's law and the points bear the stress evenly; (3) The axial motion of the cable is ignored; (4) The influences of bridge and tower are disregarded.

The analytic model of rain-wind-induced vibration of inclined cable is shown in Figures

Mechanical model of inclined cable.

Spatial model of cable

Planar model of cable

Force analysis of the cable element

Force analysis of rivulet

Figure

Balance equations of the system are obtained from the force analysis of the cable element and rivulet (see Figures

In (

Nondimensionalize system (

Given that the rivulet vibrates mainly with the frequency of cable,

Make

Parameters are selected as follows [

Figure

Displacements of in-plane and out-of-plane vibration.

Time-history

Time-history of rivulet vibration

Vibration frequencies.

In-plane vibration

Rivulet vibration

Figure

Comparison diagram of analytic and numerical solutions.

In-plane amplitude of cable

Out-of-plane amplitude of cable

Rivulet amplitude

The correspondences among the equilibrium position of rivulet, mean wind velocity, and rivulet mass are calculated from (

The correspondences between the equilibrium position of rivulet and mean wind velocity with a certain rivulet mass. Arrows indicate the growing direction of

Equilibrium position of rivulet versus mean wind velocity

Amplitude of cable versus mean wind velocity

Amplitude of cable versus equilibrium position of rivulet

The nonlinear dynamic behavior of cable reflected by parameters is given in Figure

The nonlinear response curve with different parameters.

Amplitude of cable versus mean wind velocity

Amplitude of cable versus Coulomb damping force

Amplitude of cable versus damping ratio

Amplitude of cable versus the span length

Amplitude of cable versus the initial tension

The relation curves of amplitude and the span length of cable

The continuum model of inclined cable subjected to wind with moving rivulet on its surface is established considering the equilibrium position of the rivulet. The Galerkin and average methods are adopted to analyze the cable system analytically. The correspondence between mean wind velocity and the equilibrium position of the rivulet is ascertained considered the different rivulet mass. The equilibrium position of the rivulet is proved to be one of the major roles in determining the maximum amplitude of the oscillating cable.

The nonlinear dynamic behavior of the rain-wind-induced cable vibration is then investigated considering the continuous change of parameters including mean wind velocity, Coulomb damping force, damping ratio, the span length, and the initial tension of the inclined cable. Mean wind velocity is observed to be a significant factor in jump phenomena, which is discovered in the range of multiple solutions.

The authors wish to express their special gratitude to the reviewers and the editor for their valuable comments and suggestions that led to a truly significant improvement of the paper. This work was supported by the National Natural Science Foundation of China (no. 51009107), Tianjin Natural Science Foundation (no. 13JCQNJC0420 -0), and Key Project of Tianjin Natural Science Foundation (no. 13JCZDJC27100).