Nonlinear Dynamics Research Of Rain-wind-induced Vibration Of Stayed Cable

The rain-wind-induced vibration nonlinear dynamical system contains the complicated dynamical behaviors. It is these complicated behaviors make it become hot concern. The research on the system not only analyses the characteristics of the special rain-wind-induced vibration phenomenon, but also becomes the source power to promote the development and ever-improving of modern nonlinear dynamicics theory. The research contents and obtained major results of this dissertation are as follows.(1). In-plane nonlinear dynamical characteristics of the rain-wind-induced vibration of stayed cable. Taking the geometric nonlinearity and small sag as the base to eatablish the model, a model of wind-rain-induced vibrations of continuous stayed cable was presented by the elastic mechanics and aeroelasticity theory. The differential equation was diverted into differential equation by the Galerkin method. The averaged equation and the steady state solution were obtained by using the method of multiple scales, and bifurcation analysis was performed by the singularity theory. Theoretical analyses show that the frequency of the rivulet has the corresponding range, when the rain-wind-induced vibration of stay cable happens. At the same time, the influence of the system parameters on the system vibration is also analyzed. It is indicated that the nonlinear equation can reflect more complete dynamic characteristics of the in-plane motion of rain-wind-induced vibration of stay cable, and various bifurcation behaviors exist in the system.(2). The homoclinic bifurcation and chaos of the rain-wind-induced vibration of stayed cable. The averaged equations are obtained by using the normal form method and introducing the undermined fundamental frequency to replace the natural frequency. The expression of homoclinic orbit is derived through hyperbolic perturbation method combined with the coordinate transformation. The threshold value of the chaos is got by substituting the homoclinic orbit into the Melnikov function.(3). The dymamic characterastics of rain-wind-induced vibration. A model of rain-wind-induced vibration of continuous stayed cable was presented. A set of partial differential equations were diverted into differential equations by Galerkin method. The averaging equation was achieved by normal form theory, which showed the existence of stationary solutions. Then the stability of the stationary solutions was proved. The original system was studied by numerical simulation with Mathematica. The phase diagrams of the cable and rivulet were achieved. It was shown that their movements converged to limit circles. The validity of the theoretical research was investigated via numerical simulation. The influence of the damping of stayed cable, and the per unit length quality of rivulet etc. was discussed.(4). The bifurcation of two-degree-of-freedom model of rain-wind-induced vibration of stayed cable. By the analysis of the static bifurcation behaviors according to the singularity theory, the bifurcation problem is described as a 1-codimensional unfolding with Z 2symmetry. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied using the eigenvalues in various parameter regions. The corresponding relationship between the unfolding parameters and physical parameters will convenience people further analyzing the bifurcation behaviors of the stay cable, and it also provides theoretical basis to the design of cable stayed bridge. Meanwhile, it is helpful to the development and application of the engineering projects.(5). The chaos of two-degree-of-freedom model of rain-wind-induced vibration of stayed cable. The simple approach to improve the computational precision of Melnikov method is presented by using the undetermined fundamental frequency and normal form method. We construct the improved Melnikov expression for a triple well nonlinear oscillator subject to principal parametric resonance and external excitation. For the occurrence of chaos, the threshold value approximations of chaotic motion are obtained in the Homoclinicity and Heteroclinicity points of view. It depends on the introduction of undetermined fundamental frequency, and adopting new time transformation for fulfilling the Homoclinic and Heteroclinic orbits, so that the effect of disturbing parameter can be easily detected and embodied in the Melnikov operation.(6). The strategy of predicting the Shilnikov sense horseshoe and Homoclinic orbit of the new PID controller is presented by using the Shilnikov theorem and the invariant manifolds of the saddle-focus equilibrium point. The Shilnikov type Homoclinic orbit is concerned not only about its existence but also the analytical series expression, which distinguishes the present work from the general description of the Shilnikov sense chaos. Based on the Silnikov's criterion, a chaotic system of cubic polynomial ordinary differential equations in three dimensions is studied. Using the cardano formula, the exact range of parameter's value corresponding to chaos is obtained by means of the center manifold theory and the method of multiple scales combining with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory, Lyapunov exponent and Lyapunov dimension are investigated via the numerical simulation, which shows that there is a route to chaos though period-doubling bifurcation and the chaos attractors exist in the system. These results obtained here mean that the chaos occurred in the exact range of parameters' value we got.