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张永顺, 郑罡, 张晓东, 曾广榕, 杨钰峰, 王保權. 4分点集中阻尼弦系统阻尼特性的可叠加性问题. 力学学报, 待出版. DOI: 10.6052/0459-1879-23-594
引用本文: 张永顺, 郑罡, 张晓东, 曾广榕, 杨钰峰, 王保權. 4分点集中阻尼弦系统阻尼特性的可叠加性问题. 力学学报, 待出版. DOI: 10.6052/0459-1879-23-594
Zhang Yongshun, Zheng Gang, Zhang Xiaodong, Zeng Guangrong, Yang Yufeng, Wang Baoquan. The superposition of damping characteristics of a taut string with concentrated viscous damping systems of a quartile. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-23-594
Citation: Zhang Yongshun, Zheng Gang, Zhang Xiaodong, Zeng Guangrong, Yang Yufeng, Wang Baoquan. The superposition of damping characteristics of a taut string with concentrated viscous damping systems of a quartile. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-23-594

4分点集中阻尼弦系统阻尼特性的可叠加性问题

THE SUPERPOSITION OF DAMPING CHARACTERISTICS OF A TAUT STRING WITH CONCENTRATED VISCOUS DAMPING SYSTEMS OF A QUARTILE

  • 摘要: 带有集中阻尼的张紧弦系统在力学模型上属于混杂动力学系统, 为了解系统的阻尼特性以满足工程应用需求, 通常采用近似方法求解其本征问题. 为更深入的理解系统的动力特性, 文章以含有3项集中黏性阻尼的张紧弦系统作为研究对象, 从解析角度分析了系统的阻尼特性变化规律, 并重点探讨了阻尼特性的可叠加性问题. 推导阻尼布置于4等分点时系统超越函数形式的复频率方程, 给出该方程经换元后的通用代数形式. 在此基础上, 将代数形式的复频率方程依次简化为3类退化系统的特定方程, 即单阻尼系统序列、双阻尼系统序列和3阻尼系统, 在代数层面解析求解3类系统的复本征值, 将其表达为以阻尼系数为参数的显式解析式. 分析阻尼系数对各型系统衰减特性的影响, 利用对称多项式讨论各型系统衰减特性的可叠加性问题, 导出考虑有限阶振动时各型系统间阻尼特性的比例关系. 结果表明, 相同集中阻尼个数的各系统之间复本征值实部之和相等, 且不随集中阻尼的位置坐标而改变; 不同集中阻尼个数的各系统之间复本征值实部之和存在比例关系, 且不随阻尼系数而改变. 最后, 以20分点阻尼弦系统为例, 说明可叠加性为系统本身的固有特性, 并不依赖于复本征值的求解方法.

     

    Abstract: The taut string system with concentrated damping belongs to hybrid dynamic system in mechanical model. In order to understand the damping characteristics of the system to meet the needs of engineering application, the approximate method is usually used to solve its eigenvalue problem. In order to understand the dynamic characteristics of the system more deeply, this paper takes the taut string system with three concentrated viscous damping as the research object, analyzes the variation law of the damping characteristics of the system from an analytical point of view, and focuses on the superposition of the damping characteristics. The complex frequency equation in the form of transcendental function is derived when the damping is arranged at the quartering point, and the general algebraic form of the equation after substitution is given. On this basis, the algebraic complex frequency equation is simplified to the specific equations of three kinds of degenerate systems, namely, single damping system sequence, double damping system sequence and triple damping system. The complex eigenvalues of the three kinds of systems are solved analytically at the algebraic level and expressed as explicit analytical expressions with damping coefficients as parameters. The influence of damping coefficient on attenuation characteristics of various systems is analyzed, and the superposition of attenuation characteristics of various systems is discussed by using symmetric polynomials, and the proportional relationship of damping characteristics among various systems considering finite-order vibration is derived. The results show that the sum of the real parts of complex eigenvalues between systems with the same number of concentrated damping is equal and does not change with the position coordinates of concentrated damping. The sum of the real parts of complex eigenvalues between systems with different numbers of concentrated damping is proportional and does not change with damping coefficient. Finally, taking the twenty equal points damping string system as an example, it is shown that superposition is an inherent characteristic of the system itself and does not depend on the solution method of complex eigenvalues.

     

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