粘弹性结构振动的非线性动力学研究

发布时间：2018-11-04 21:08

【摘要】：粘弹性结构发生非线性振动时,由于材料的粘弹性特性和各种非线性因素的影响,会表现出复杂的非线性动力学行为。正是这些复杂动力学行为的存在使粘弹性结构非线性振动现象成为重点关注的前沿课题。对这一系统的研究不仅从非线性动力学角度分析了粘弹性结构振动的动力学特性,而且也促进了现代非线性动力学理论和方法的发展和完善。因此,针对实际工程中最基本的两种构件——粘弹性梁和粘弹性柱,在外部激励作用下发生非线性振动时,采用理论分析和数值模拟的手段来研究其非线性动力学行为。本文的主要工作和研究成果如下:(1)在查阅和总结大量国内外相关文献资料的基础上,论述了非线性动力学的研究方法和意义,以及粘弹性结构振动问题研究的工程背景和意义;从不同的方面对粘弹性梁和粘弹性柱非线性振动的国内外研究现状进行了全面的综述。(2)考虑微分型的粘弹性本构关系、两端简支的边界条件和外部激励的作用,并采用微分求积法,分别建立了粘弹性梁和柱振动的非线性动力学模型。利用Matlab程序编写四阶龙格——库塔算法,分别对其方程进行数值模拟,绘制相平面图、时程曲线图、功率谱图、庞加莱截面图。固定一组初始条件和参数,粘弹性梁随着粘弹性阻尼系数的增加,逐渐呈现出单倍周期运动、混沌运动、到混沌运动、三倍周期运动、再回归到单倍周期运动;体现了通往混沌的途径——阵发性混沌道路。随着外部激励振幅值的增大,粘弹性梁又会由单倍周期运动、准周期运动、到完全进入混沌运动状态;也体现了准周期环面破裂道路。(3)同样固定一组初始条件和参数,粘弹性柱随着粘性系数的变化,逐渐呈现出单倍周期运动、二倍周期运动、六倍周期运动、再回到六倍周期运动、二倍周期运动、单倍周期运动。随着外部激励振幅值的增大,粘弹性柱会由单倍周期运动、六倍周期运动、准周期运动,到完全进入混沌运动状态;也体现了准周期环面破裂道路。(4)结果表明,在一定参数范围内,两种粘弹性结构都会交替性地出现周期运动、倍周期运动和混沌运动,也体现了混沌运动和确定性运动的区别。基于相同的理论和数值方法,由于结构的不同,各自呈现出的运动形态却不相同。
[Abstract]:When nonlinear vibration occurs in viscoelastic structures, the viscoelastic properties of materials and the influence of various nonlinear factors will lead to complex nonlinear dynamic behavior. It is the existence of these complex dynamic behaviors that makes nonlinear vibration of viscoelastic structures a hot topic. The study of this system not only analyzes the dynamic characteristics of viscoelastic structure vibration from the point of view of nonlinear dynamics, but also promotes the development and perfection of modern nonlinear dynamic theory and method. Therefore, the nonlinear dynamic behaviors of viscoelastic beams and viscoelastic columns are studied by means of theoretical analysis and numerical simulation when nonlinear vibration occurs under external excitation. The main work and results of this paper are as follows: (1) on the basis of consulting and summarizing a large number of domestic and foreign literature, the research methods and significance of nonlinear dynamics are discussed. And the engineering background and significance of the research on the vibration of viscoelastic structures; The research status of nonlinear vibration of viscoelastic beams and viscoelastic columns at home and abroad is summarized from different aspects. (2) considering the differential viscoelastic constitutive relation, the boundary conditions of simple support at both ends and the effect of external excitation. The nonlinear dynamic models of viscoelastic beam and column vibration are established by differential quadrature method. The fourth order Runge-Kutta algorithm is programmed by Matlab program. The equations are numerically simulated, and the phase plane diagram, time history curve, power spectrum diagram and Poincare section diagram are drawn. When a group of initial conditions and parameters are fixed, the viscoelastic beam with the increase of viscoelastic damping coefficient gradually presents the motion of haploperiod, chaos, tripling period, and then returning to the motion of haptic period. It embodies the path to chaos-paroxysmal chaos. With the increase of the amplitude of external excitation, the viscoelastic beam will move from the monoperiodic motion, quasi-periodic motion to the state of chaotic motion. It also reflects the quasi-periodic torus rupture road. (3) with the change of viscosity coefficient, the viscoelastic column gradually presents the motion of haploperiod, double period, six-fold period, and fixed a set of initial conditions and parameters. And then back to the six-fold periodic motion, the double periodic motion, the haploperiodic motion. With the increase of the amplitude of external excitation, the viscoelastic column will move from the motion of haploperiod, the motion of six times period, the motion of quasi period, to the state of chaotic motion completely. (4) the results show that in a certain range of parameters, the periodic motion, periodic motion and chaotic motion occur alternately between the two viscoelastic structures. It also reflects the difference between chaotic motion and deterministic motion. Based on the same theory and numerical method, the motion patterns are different from each other because of the different structures.
【学位授予单位】：广西科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TU311.3

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