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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (4): 623-641.doi: 10.3969/j.issn.1005-3085.2024.04.003

• • 上一篇    下一篇

基于平均法的余弦激励分数阶振子系统动力学分析

师  玮1,2,3,  郭  蓉4,  解加全1,2,3,  张彦杰1,2,3,  王  涛1,2,3,  黄庆学1,2,3   

  1. 1. 太原理工大学机械与运载工程学院，太原 030024
    2. 金属成形技术与重型装备全国重点实验室，太原   030024
    3. 先进金属复合材料成形技术与装备教育部工程研究中心，太原 030024
    4. 中北大学理学院，太原 030051
  • 收稿日期:2022-07-23 接受日期:2023-06-07 出版日期:2024-08-15
  • 通讯作者: 解加全 E-mail: xjq371195982@163.com
  • 基金资助:
    国家重点研发计划 (2018YFB1308702); 国家自然科学基金 (51905372; 52005360; 52105557)；中央引导地方科技发展专项资金 (YDZX20191400002149)；山西省研究生教育创新项目 (2020BY142).

Dynamic Analysis of Fractional Oscillator System with Cosine Excitation Based on Average Method

SHI Wei1,2,3,  GUO Rong4,  XIE Jiaquan1,2,3,  ZHANG Yanjie1,2,3,  WANG Tao1,2,3,  HUANG Qingxue1,2,3   

  1. 1. College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024
    2. National Key Laboratory of Metal Forming Technology and Heavy Equipment, Taiyuan 030024
    3. Engineering Research Center of Advanced Metal Composites Forming Technology and Equipment, Ministry of Education, Taiyuan 030024
    4. School of Science, North University of China, Taiyuan 030051
  • Received:2022-07-23 Accepted:2023-06-07 Online:2024-08-15
  • Contact: J. Xie. E-mail address: xjq371195982@163.com
  • Supported by:
    The National Key Research and Development Program (2018YFB1308702); the National Natural Science Foundation of China (51905372; 52005360; 52105557); the Special Funding for Guiding Local Scientific and Technological Development of the Central (YDZX20191400002149); the Graduate Education Innovation Program of Shanxi Province (2020BY142).

摘要：

提出了一种求解余弦激励下分数阶振子系统位移响应的解析和数值算法。其中解析方法是用平均法来求出系统的稳态响应解和瞬态响应解，总位移响应解即为稳态解和瞬态解的总和。提出一种数值方法，利用Gr$\ddot{\rm u}$nwald-Letnikov分数阶导数的定义将系统中的分数阶微分项进行离散化处理，降低了原系统阶数。在一般的周期激励下，系统的近似响应解可以采用傅里叶级数展开法和线性系统叠加原理得到。最后，利用数值模拟对所提方法的有效性和可行性进行了验证，并对分数阶阶次、线性阻尼系数和分数阶导数系数对系统稳态响应振幅和总位移响应的影响进行了分析。

关键词: 平均法, 分数阶振子系统, Gr$\ddot{\rm u}$nwald-Letnikov分数阶导数, 位移响应

Abstract:

An analytical and numerical algorithm for solving the displacement response of fractional oscillator system under cosine excitation is presented. The analytical method means that steady-state response and transient response solutions of the system can be obtained by the average method. The total displacement response solution is the sum of the steady-state solution and transient solutions. In the numerical method, the Grunwald-Letnikov definition of fractional derivative is used to discretize the fractional differential term in the system, so as to reduce the order of the original system. Considering the general periodic excitation, the approximate response solution of the system can be obtained by using the Fourier series expansion method and the linear system superposition principle. Finally, the effectiveness and feasibility of the proposed method are verified by numerical simulation. The effects of fractional order, linear damping coefficient and fractional derivative coefficient on steady-state response amplitude and total displacement response of the system are analyzed.

Key words: average method, fractional oscillator system, Gr$\ddot{\rm u}$nwald-Letnikov fractional derivative, displacement response

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