【作者】 陈先龙；

【导师】 何锃；

【作者基本信息】 华中科技大学 ， 工程力学， 2006， 硕士

【摘要】 梁的非线性振动是典型的非线性振动之一,而随着柔性航天器、雷达天线等带有柔性附件的系统在高速、轻质、高精度等方面的发展,要求对这类系统的动力学特性做更深入的分析。但传统梁理论多直接采用结构力学中的小应变假设,因此需要通过合理的假设建立更精确的数学模型来描述这类系统的变形运动。本文从做定轴转动大型刚性结构上的简支梁的平面振动入手,首先建立随体坐标系来描述变形位移场,采用Green应变理论精确分析了梁的耦合变形情况,建立了柔性梁变形运动的几何非线性方程;对于梁的本构关系的描述,采用微分型的Kelvin-Voigt模型;在同时考虑了系统的转动惯量、弯转-拉伸耦合效应和粘性阻尼的影响下,采用微元法,从应力-应变的角度得出了系统的动力学方程;然后在随体坐标系内采用Galerkin方法来离散变形位移场,得出了系统的控制方程。由于梁的变形运动以横向振动为主,故在仍考虑系统的几何非线性的基础上,可先忽略其轴向变形运动,使得系统的控制方程简化为一个强非线性方程,然后采用一种改进的L-P法求出了柔性梁横向振动的一阶近似解。通过与能量积分所得结果的比较表明,所求得的近似解有较好的精度最后采用基于四阶的Runge-Kutta法的数值计算分析了柔性梁的平面变形运动。与传统的模型相比较,表明本文的模型确实能克服传统模型在这一领域的缺陷;分析了在不同阻尼的影响下系统的响应情况,表明粘性阻尼对于这类系统的运动有比较大的影响。更多还原

【Abstract】 The nonlinear vibration of beams is a typical problem of nonlinear one. To meet the demands of high rotating speeds, fast response and accurate positioning for industrial applications such as the design of helicopters, robotic arms, satellite appendages, and turbine blades, thorough investigation for dynamic characteristics of such system is needed by using suitable mathematical models with proper simplifying assumptions concerning the geometric nonlinearity.But the traditional models adopt the small deformation assumption in the structural dynamics. In this paper a refined geometrically nonlinear model for a rotating slender flexible beam is developed. With rotating coordinate system, the dynamics of a rigid-flexible coupled system are analyzed for a central rigid body with rotating motion and a flexible viscoelastic beam. The axial deformation, the transverse deformation and the cross section rotation of the beam are taken into account. The geometric nonlinearity of the coupling system is discussed by Green strain formula and the material of the beam obeys the Kelvin-Voigt constitutive relation. A set of dynamic equations is established by utilizing the differential element method. Then the Galerkin method is used to discretized the equations and the governing equations are obtained.Through neglecting the axial motion, a simplified model, suitable for studying the transverse vibration, is presented. The governing differential equation is strongly nonlinear, so various approximate analytical methods based on perturbation theory for a weakly nonlinear system can’t be used directly. By using the improved L-P method, different from the known, the first order approximate solution which has fair accuracy is obtained.Then the equations of the motion are solved numerically using the fourth order Runge-Kutta method. Graphical results are presented to show the influence of centrifugal stiffening effect, viscoelasticity, geometric nonlinearity. Numerical results indicate that the system is stable in case of high rotating speed and that the viscoelastic damping decreases the amplitude of the axial and the transverse deformation.更多还原