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,通常采用近似方法求解本征值以满足工程应用需求,为进一步明确该类系统的振动特性,有必要从解析角度探讨其本征问题。针对四分点集中阻尼弦系统,推导并化简其频率方程的代数形式,求得代数方程的解,经过换元逆过程,获得原频率方程的所有闭合解。发现闭合解共存在3组,其中两组互为共轭,根据代数基本定理讨论闭合解的结构。结果表明:系统存在两种相同的运动特征,其单位时间对数衰减率相同,频率互为相反数;系统中3个解支对应的单位时间对数衰减率和频率,总是随着阶次的增加而呈周期循环的;在同一解支下的各单值分支也呈周期性循环,即随着阶次的增加,其对应频率增加4π的整数倍,而单位时间对数衰减率保持不变。综上可知,集中阻尼弦系统的运动特性总是随着阶次的增加而重复变化,且变化周期与阻尼安装位置相关。
The tensioned string system with concentrated damping belongs to the hybrid dynamical system in the mechanical model,and the approximation method is usually used to solve the eigen values to meet the needs of engineering applications.In order to further clarify the vibration characteristics of this type of system,it is necessary to explore its eigen problems from the analytical point of view.For a four-point centrally damped string system,the algebraic form of its frequency equation was derived and simplified,the solution of the algebraic equation was obtained,and all the closed solutions of the original frequency equation were obtained after the commutative inverse process.The structure of the closed solutions of the system was discussed according to the fundamental theorem of algebra.It is found that there exist three sets of closed solutions in total,two of which are conjugate to each other.The structure of the closed solutions was discussed according to the fundamental theorem of algebra.The results show that the system has two identical motion characteristics,the logarithmic attenuation rate per unit time is the same,and the frequencies are opposite to each other.The logarithmic attenuation rate and frequency per unit time corresponding to the three de expenditures in the system always cycle periodically with the increase of order.Each single-valued branch under the same debranching also shows a periodic cycle,that is,with the increase of order,its corresponding frequency increases by an integer multiple of 4π,while the logarithmic attenuation rate per unit time remains unchanged.Based on the above,it can be concluded that the motion characteristics of the centralized damping chord system always change repeatedly with the increase of order,and the change period is related to the damping installation position.
作者
郑罡
王保權
王梦丽
张永顺
曾广榕
ZHENG Gang;WANG Bao-quan;WANG Meng-li;ZHANG Yong-shun;ZENG Guang-rong(State Key Laboratory of Mountain Bridge and Tunnel Engineering,Chongqing Jiaotong University,Chongqing 400074,China)
出处
《科学技术与工程》
北大核心
2024年第20期8403-8408,共6页
Science Technology and Engineering
基金
国家自然科学基金(51978112,52178272)。
关键词
张紧弦
本征值
集中阻尼
本征函数
非经典线性系统
taut string
eigenvalue
concentrated damping
eigenfunction
nonclassical linear system
作者简介
第一作者:郑罡(1972-),男,汉族,四川宜宾人,博士,研究员,博士研究生导师。研究方向:斜拉索振动控制、桥梁抗震设计及结构健康监测。E-mail:zhenggang@cqjtu.edu.cn。
