Keywords: rain-wind induced vibrations, self-excited systems, nonlinear dynamic, stability, aeroelastic instabilities.

Rain-wind induced vibrations are a fluid-structure interaction phenomena that occurs when rain and wind act simultaneously on cables, hangars and ropes. In 1986, Y. Hikami [1] first detected this kind of vibration behaviour of cables at the Meikonishi Bridge in Japan. The vibration phenomena may induce oscillations with large amplitudes, thus the fatigue of construction elements is possible. This kind of vibration is mainly observed at stay cables of cable stayed bridges and hangars of arch bridges. Oscillations may also occur at overhead lines, ropes of pylons and cables of suspension bridges.

A possible mechanism of excitation of rain-wind induced vibrations is derived in [2]. This mechanism is based on the phenomena of the Prandtl tripwire and considers the rivulets as a movable disturbance. Following this approach the occurrence of the lower and the upper limit of the critical velocity can be explained and all kinds of observable vibrations may be described.

In contrast to the Prandtl tripwire an interaction between the motion of the cable and the rivulets exists in the special case of rain-wind induced vibrations. This interaction is due to the reduction of the drag and stiffness of the cable. For the overall mechanism it is important that rivulets oscillate around the point of transition of the flow with the same frequency as the cable. The result of this motion is a periodical transition of the flow between subcritical and supercritical flow. An energy transfer between the flow around the cable and the elastic structure is induced as a result of the different pressure distributions of the subcritical and supercritical flow. The interaction between cable, rivulets and flow determines the development of a self-excitation mechanism. This leads to vibrations with large amplitudes parallel or perpendicular to the wind direction depending on the location of rivulets.

The equations of motion in two dimensions for cables and rivulets is formulated under consideration of the phenomenology and the assumption that cables are infinitely long cylinders. It is essential to include the influence of the rivulets and its resulting interaction with the structure in the equations of motion of the cable cross-section and in the quasi-stationary theory. Furthermore, the equations of motion of the rivulets are developed. Therefore the effects of the boundary layer based on the Prandtl boundary layer equations and fundamentals of the physics of drops are considered. The equations of motion of the cables are formulated in an inertial system. Neglecting eigenrotations there are six non-linear coupled differential equations. These are required to calculate the unknown translations of cable, the unknown angles of rivulets and the reactive forces for both rivulets.

The numerical analysis of the model equations is very complex, due to the high-grade nonlinearity and the coupled character of the differential equations system. Especially for the complete system of coupled equations the prediction of the stability of the solution appears as very difficult. Mathematical transformations enable the stability of the solution of the nonlinear differential equations system. As a result of this fact the equations of motion for the cable are formulated in Cartesian coordinates, whereby the equations of motion for the rivulets are developed in polar coordinates. The six equations of motion may be reduced to four essential equations. This is not appropriate, since it results in numerical instabilities again. Furthermore the equations become more complex. Therefore, it is advisable to introduce the two reactive forces as pseudo variables. The direct determination of the both reactive forces is also useful under physical aspects because they correspond to the separation condition of the rivulets. These differential equations describe self-excitated, parameter-excitated or mixed forms of oscillations.

The equations are investigated under consideration of different solution methods and numerical integration techniques. Parameter studies and the analysis of sensitivities are carried out in order to compute the upper and the lower critical velocities. Besides the computation of the limit cycle of the rain-wind induced vibrations is demonstrated. The lower limit of the critical velocity range is calculated for various cable inclinations and angles of incidence. A fluid mechanical interpretation is given for the existence of the upper limit. Also the determination of a critical diameters is possible. Furthermore the numerical investigations show that Rain-wind induced oscillations occur also on vertical cables. The analysis of stability of the nonlinear dynamic system leads in many cases to determinate chaotic systems.

1

Hikami, Y., Shiraishi, N., "Rain-Wind Induced Vibrations of Cables in Cable Stayed Bridges", Journal of Wind Engineering and Industrial Aerodynamics, 29, 409-418, 1988. doi:10.1016/0167-6105(88)90179-1

2

Seidel, C., Dinkler, D., "Phänomenologie und Modellierung Regen-Wind induzierter Schwingungen", Bauingenieur, 2004.

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