摘要
目前,非线性动力学理论在自然科学、生命科学、社会科学等领域的应用越来越得到重视。特别是在工程领域,基于非线性动力学分析的设计逐步引起人们的重视。
形状记忆合金(SMA)是一类应用前景广阔的智能材料。通过SMA预变形,可以使形状记忆合金产生不同形状的回复力-位移关系曲线,这些曲线形状分别为矩型,三角旗,蝴蝶结,双旗帜型。双旗帜型(Flag-Shaped)的滞回模型更加典型,并且能反映预应力钢结构节点和磁流变阻尼器的力位移特性。作为非线性动力学设计初步探讨,本文研究双旗帜型(Flag-Shaped)的滞回模型特性对动力学行为的影响。
本文建立了形状记忆合金减振器的力学模型,用平均法导出系统的幅频响应方程,用约束分岔奇异性理论计算了转迁集和分岔图。针对滞后模型分段线性的特点,用统一的绝对值函数求和形式来描述滞后特性的各种可能路径,显著减少了路径切换的种类,降低了逻辑判断的难度,简化了程序编制的复杂性。利用编制的matlab数值程序,计算了系统的多种幅频响应分岔图,通过与理论结果对比发现:在参数刚度k数值较小的情况下,数值计算得到数据与理论分析得到的幅频曲线有较高的吻合程度;在参数刚度k数值比较大的情况下,数值计算得到数据与理论分析得到的幅频曲线吻合较差。
最后对形状记忆合金减振器的减振效果进行分析。形状记忆合金的加入,使系统避开线性系统共振区高峰值,并使得系统的最大幅值得到抑制。上述研究对形状记忆合金减振器设计有一定的指导意义。
Nowadays, nonlinear dynamics theory is more and more important in the natural sciences, life sciences, social sciences and other fields of application. Especially in the engineering field, the design based on nonlinear dynamic analysis gradually attracts people's attention.
Shape memory alloy (SMA) is a kind of intelligent material with promising future in application. With the pre-deformation of shape memory alloy (SMA), it could produce different response force - displacement curves. The shape of these curves are rectangle, triangle, bowknot, and flag-shaped. The Flag-Shaped hysteretic model is more typical, and can also reflect the force-displacement characteristics of the prestressed steel nodes and the magneto-rheological damper. As a preliminary study of the design on nonlinear dynamic analysis, this paper mainly studies the influence of the Flag-Shaped hysteresis on the dynamic behavior.
This paper builds the model of a mass-spring system with the shape memory alloy damper. Obtaining amplitude-frequency equation by using the averaging method, and calculate the transition sets and bifurcation diagrams by the theory of singular bifurcation. In order to simplify the complication of the programming, an unified form of absolute value function is introduced to express all the possible routes of the hysteretic model. By comparison with the theoretical results, it is shown that the amplitude-frequency curves based on numerical calculation agree well with those obtained by theoretical analysis for small stiffness parameters,
Finally, the effects of shape memory alloy damper on the vibration reduction are investigated. The results show that the damper reduces the vibration amplitude of the system.
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