Responses of SDOF Structures With SPIS-Ⅱ Dampers Under Random Seismic Excitation
-
摘要:
提出了基于Clough-Penzien谱的混联Ⅱ型惯容结构地震动响应封闭解的计算方法，并基于所提方法研究了单自由度混联Ⅱ型惯容结构的减震性能及影响因素。首先，建立混联Ⅱ型惯容结构的运动方程，获得了结构位移与惯容出力等结构响应频域解的统一表达式。其次，基于有理式分解与留数定理分别获得了频率响应特征值函数与Clough-Penzien谱的二次正交式，进而获得了结构响应功率谱的二次正交式。最后，得到了结构与惯容随机地震动响应0 ~ 2阶谱矩的简明封闭解。利用所提方法和虚拟激励法分别对一算例进行分析，验证了所提方法的正确性；同时，利用所提方法分析了惯容各参数对结构抗震性能的影响。研究表明：该文方法获得的响应解为封闭解，在计算效率与计算准确性上都优于虚拟激励法。当μω不变时，随着μm与μξ增大，减震性能增强；当μω=1时，对减震性能的影响最大。
-
关键词:
- 惯容 /
- Clough-Penzien谱 /
- 二次正交化 /
- 谱矩 /
- 封闭解
Abstract:A closed-form solution of responses of SDOF structures with SPIS-Ⅱ dampers under seismic excitation modeled with the Clough-Pezien spectrum was proposed, and the shock absorption performance and influential factors of this system were studied based on the proposed method. Firstly, the motion equation for the SPIS-Ⅱ damper was established, and the unified expressions of frequency domain solutions of structural responses, such as the structural displacement and the inerter force, were obtained. Secondly, based on the rational expression decomposition and the residue theorem, the quadratic orthogonal equations of the frequency response eigenvalue function and the Clough-Pezien spectrum were obtained respectively, and in turn the quadratic orthogonal equation of the structural response power spectrum was deduced. Thirdly, the concise closed-form solutions of the 0~2nd-order spectral moments of the structural responses were acquired. The proposed method and the virtual excitation method were used to analyze a case respectively, which verifies the correctness of the proposed method. Finally, the proposed method was used to analyze the effects of the inerter parameters on the seismic performances of the structure. The research shows that, the proposed method gives closed-form solutions better than those given by the virtual excitation method in terms of computation efficiency and accuracy. The damping performance will improve with the increase of μm and μξ for a constant μω and the damping performance will reach the optimum for μω=1.
-
Key words:
- inerter /
- Clough-Penzien spectrum /
- quadratic orthogonalization /
- spectral moment /
- closed-form solution
-
图 4 惯容质量
${m_{{\text{in}}}}$ 对结构位移的影响Figure 4. Effects of inerter coefficient
${m_{{\text{in}}}}$ on structural displacements
图 6 惯容阻尼
${c_{\text{d}}}$ 对位移的影响Figure 6. Effects of inerter damping
${c_{\text{d}}}$ on displacements
图 5 惯容刚度
${k_{\text{s}}}$ 对结构位移的影响Figure 5. Effects of inerter stiffness
${k_{\text{s}}}$ on structural displacements表 1 谱矩对比
Table 1. Comparison of spectral moments between the presented method and the virtual excitation method
calculation method integration step $\varDelta/( { {\text{rad/s} } } )$ spectral moment ${\alpha _{x,0} }/( {10^{ - 5} }\;{ {\text{m} }^2})$ ${\alpha _{x,1} }/( {10^{ - 4} }\;{ {\text{m} }^{\text{2} } }/{\text{s} })$ ${\alpha _{x,2}}$ $/( { { {\text{m} }^{\text{2} } }/{ {\text{s} }^2} } )$ present method – 7.369214319 1.721475154 0.040331515 PEM 0.01 7.369214317 1.721475145 0.040331509 0.1 7.247097750 1.692932655 0.039664397 0.5 1.465220153 3.434894652 0.080641421 
下载: 导出CSV
表 2 惯容出力对比
Table 2. Comparison of inerter forces between the virtual excitation method and this method
calculation method integration interval integration step inerter force error present method – – 2.466×102 kN – PEM [0, 500] 0.01 rad/s 2.466×102 kN 0 0.05 rad/s 2.466×102 kN 0 0.5 rad/s 2.314×102 kN 4.6%
下载: 导出CSV
-
[1] HOUSNER G W, BERGMAN L A, CAUGHEY T K, et al. Structural control: past, present, and future[J]. Journal of Engineering Mechanics, 1997, 123(9): 897-971. doi: 10.1061/(ASCE)0733-9399(1997)123:9(897) [2] 方同, 薛璞. 振动理论及应用[M]. 西安: 西北工业大学出版社, 1998.FANG Tong, XUE Pu. Vibration Theory and Application[M]. Xi’an: Northwestern Polytechnical University Press, 1998. (in Chinese) [3] 贺辉, 谭平, 刘彦辉, 等. 圆形高耸结构两级变阻尼TMD风振控制[J]. 振动工程学报, 2020, 33(3): 6 doi: 10.16385/j.cnki.issn.1004-4523.2020.03.008HE Hui, TAN Ping, LIU Yanhui, et al. Wind-induced vibration control of circular section high-rise structures employing TMD with two-stage damping level[J]. Journal of Vibration Engineering, 2020, 33(3): 6.(in Chinese) doi: 10.16385/j.cnki.issn.1004-4523.2020.03.008 [4] 钟文坤, 吴玖荣, 孙连杨. 考虑挡板间水动力相互作用影响的矩形TLD水箱阻尼比分析[J]. 应用数学和力学, 2021, 42(1): 71-81ZHONG Wenkun, WU Jiurong, SUN Lianyang. Damping ratio analysis of rectangular TLD tank with hydrodynamic interaction effects between baffles[J]. Applied Mathematics and Mechanics, 2021, 42(1): 71-81.(in Chinese) [5] 石晟, 杜东升, 徐敬海, 等. 基础隔震结构基于实时监测数据的多级预警及其在南京博物院老大殿中的应用[J]. 建筑结构学报, 2022, 43(4): 47-57 doi: 10.14006/j.jzjgxb.2021.0056SHI Sheng, DU Dongsheng, XU Jinghai, et al. Multi-level early-time monitoring data and its application on the old hall of Nanjing Museum[J]. Journal of Building Structures, 2022, 43(4): 47-57.(in Chinese) doi: 10.14006/j.jzjgxb.2021.0056 [6] 韩庆华, 曹馨元, 刘铭劼. 平面张弦结构粘弹性阻尼器振动控制研究[J]. 工程力学, 2021, 38(12): 57-72 doi: 10.6052/j.issn.1000-4750.2020.11.0849HAN Qinghua, CAO Xinyuan, LIU Mingjie. Research on the vibration control of plane string structures using viscoelastic dampers[J]. Engineering Mechanics, 2021, 38(12): 57-72.(in Chinese) doi: 10.6052/j.issn.1000-4750.2020.11.0849 [7] 张瑞甫, 曹嫣如, 潘超. 惯容减震(振)系统及其研究进展[J]. 工程力学, 2019, 36(10): 8-27ZHANG Ruifu, CAO Yanru, PAN Chao. Inerter system and its state-of-the-art[J]. Engineering Mechanics, 2019, 36(10): 8-27.(in Chinese) [8] 黄绪宏, 许维炳, 王瑾, 等. 考虑惯容的多颗粒阻尼器等效力学模型及试验验证[J]. 振动与冲击, 2021, 40(18): 102-111 doi: 10.13465/j.cnki.jvs.2021.18.014HUANG Xuhong, XU Weibing, WANG Jin, et al. Equivalent mechanical model and experimental validation of multi-particle dampers considering inertia capacity[J]. Journal of Vibration and Shock, 2021, 40(18): 102-111.(in Chinese) doi: 10.13465/j.cnki.jvs.2021.18.014 [9] 郭梓龙, 王琳, 倪樵, 等. 接地惯容式减振器对悬臂输流管稳定性和动态响应的影响研究[J]. 力学学报, 2021, 53(6): 1769-1780 doi: 10.6052/0459-1879-21-105GUO Zilong, WANG Lin, NI Qiao, et al. Study of the effect of grounded inertial capacitance damper on the stability and dynamic response of cantilevered flow transport tubes[J]. Journal of Mechanics, 2021, 53(6): 1769-1780.(in Chinese) doi: 10.6052/0459-1879-21-105 [10] 王勇, 李昊轩, 姜文安, 等. 高速列车车下惯容悬吊设备动态特性研究[J]. 振动与冲击, 2022, 41(2): 246-254 doi: 10.13465/j.cnki.jvs.2022.02.030WANG Yong, LI Haoxuan, JIANG Wenan, et al. Research on dynamic characteristics of inertial capacity suspension equipment under high-speed train cars[J]. Journal of Vibration and Shock, 2022, 41(2): 246-254.(in Chinese) doi: 10.13465/j.cnki.jvs.2022.02.030 [11] NAKAMURA Y, WATANABE H, KAWAMATA S. Seismic response control of structures by accelerated liquid mass damper[C]//Proceedings of the 9th World Conference on Earthquake Engineering. Tokyo, Japan, 1988: 785-790. [12] ARAKAKI T, KURODA H, ARIMA F, et al. Development of seismic devices applied to ball screw, part 2: performance test and evaluation of RD-series[J]. AIJ Journal of Technology and Design, 1999, 5(9): 265-270. doi: 10.3130/aijt.5.265 [13] SAITO K, INOUE N. A study on optimum response control of passive control systems using viscous damper with inertial mass: substituting equivalent nonlinear viscous elements for linear viscous elements in optimum control systems[J]. AIJ Journal of Technology and Design, 2007, 13(26): 457-462. doi: 10.3130/aijt.13.457 [14] ARAKAKI T, KURODA H, ARIMA F, et al. Development of seismic devices applied to ball screw, part 1: basic performance test of RD-series[J]. Journal of Architecture and Building Science, 1999, 5(8): 239-244. (in Japanese) [15] SAITO K, KURITA S, INOUE N. Optimum response control of 1-DOF system using linear viscous damper with inertial mass and its Kelvin-type modeling[J]. Journal of Structural Engineering, 2007, 53: 53-66. [16] PAN C, ZHANG R. Design of structure with inerter system based on stochastic response mitigation ratio[J]. Structural Control and Health Monitoring, 2018, 25(6): e2169. doi: 10.1002/stc.2169 [17] 吴应雄, 郑祥浴, 翁锦华, 等. 长周期地震动作用下惯容-层间隔震结构地震响应分析[J]. 振动工程学报, 2022, 35(5): 1222-1232 doi: 10.16385/j.cnki.issn.1004-4523.2022.05.020WU Yingxiong, ZHENG Xiangyu, WENG Jinhua, et al. Seismic response analysis of inertia-layer interval seismic structures under long-period ground shaking[J]. Journal of Vibration Engineering, 2022, 35(5): 1222-1232.(in Chinese) doi: 10.16385/j.cnki.issn.1004-4523.2022.05.020 [18] HOUSNER G W. Characteristics of strong motion earthquakes[J]. Bulletin of the Sesimological Society of America, 1947, 37: 19-31. doi: 10.1785/BSSA0370010019 [19] KANAI K. An empirical formula for the spectrum of strong earthquake motions[J]. Bulletin of Earthquake Research Institute, University of Tokyo, 1961, 39: 86-95. [20] TAJIMI H A. A statistical method of determining the maximum response of a building structure during an earthquake[C]//Proceedings of the 2nd World Conference on Earthquake Engineering. Kyoto, Japan, 1960: 781-797. [21] CLOUGH R W, PENZIEN J. Dynamics of Structures[M]. New York: McGraw Hill, 1993. [22] ZHANG R F, ZHAO Z, PAN C, et al. Damping enhancement principle of inerter system[J]. Structural Control and Health Monitoring, 2020, 27(5): e2523. [23] 赵志鹏, 张瑞甫, 陈清军, 等. 基于减震比设计方法的惯容减震结构分析[J]. 工程力学, 2019, 36(S1): 125-130ZHAO Zhipeng, ZHANG Ruifu, CHEN Qingjun, et al. Analysis of structures with inerter systems based on the response mitigation ratio design method[J]. Engineering Mechanics, 2019, 36(S1): 125-130.(in Chinese) [24] 潘超, 张瑞甫, 王超, 等. 单自由度混联Ⅱ型惯容减震体系的随机地震响应与参数设计[J]. 工程力学, 2019, 36(1): 129-137PAN Chao, ZHANG Ruifu, WANG Chao, et al. Stochastic seismic response and parametric design of single-degree-of-freedom mixed-link type Ⅱ inertial capacity damping system[J]. Engineering Mechanics, 2019, 36(1): 129-137.(in Chinese) [25] GE X G, GONG J H, ZHAO C J, et al. Structural dynamic responses of building structures with non-viscous dampers under Kanai-Tajimi spectrum excitation[J]. Journal of Sound and Vibration, 2022, 517: 116556. doi: 10.1016/j.jsv.2021.116556 [26] 葛新广, 张梦丹, 龚景海, 等. 频响函数二次正交法在Davenport风速谱下结构系列响应简明封闭解的应用研究[J]. 振动与冲击, 2021, 40(21): 207-214 doi: 10.13465/j.cnki.jvs.2021.21.028GE Xinguang, ZHANG Mengdan, GONG Jinghai, et al. Application of quadratic quadrature method of frequency response function to the concise closed solution of structural series response under Davenport wind speed spectrum[J]. Journal of Vibration and Shock, 2021, 40(21): 207-214.(in Chinese) doi: 10.13465/j.cnki.jvs.2021.21.028 [27] 方同. 工程随机振动[M]. 北京: 国防工业出版社, 1995.FANG Tong. Engineering Random Vibration[M]. Beijing: National Defense Industry Press, 1995. (in Chinese) [28] GE X G, AZIM I, GONG J H, et al. Structural dynamic responses of linear structures subjected to Kanai-Tajimi excitation[J]. Structures, 2021, 34: 3958-3967. doi: 10.1016/j.istruc.2021.08.092 [29] 李创第, 陈明杰, 葛新广. 基于Clough-Penzien谱激励的指数型非黏滞阻尼结构随机地震动响应简明封闭解[J]. 应用数学和力学, 2021, 42(3): 282-291LI Chuangdi, CHEN Mingjie, GE Xinguang. A simple closed response solution to random ground motion for exponential non-viscous-damping structures based on the Clough-Penzien spectrum excitation[J]. Applied Mathematics and Mechanics, 2021, 42(3): 282-291.(in Chinese) [30] 林家浩. 随机地震响应的确定性算法[J]. 地震工程与工程振动, 1985, 1: 89-94 doi: 10.13197/j.eeev.1985.01.009LIN Jiahao. Deterministic algorithm for stochastic seismic response[J]. Earthquake Engineering and Engineering Dynamics, 1985, 1: 89-94.(in Chinese) doi: 10.13197/j.eeev.1985.01.009 [31] 林家浩, 张亚辉. 随机振动的虚拟激励法[M]. 北京: 科学出版社, 2004.LIN Jiahao, ZHANG Yahui. Virtual Excitation Method for Random Vibration[M]. Beijing: Science Press, 2004. (in Chinese) [32] 林家浩, 张亚辉, 赵岩. 虚拟激励法在国内外工程界的应用回顾与展望[J]. 应用数学和力学, 2017, 38(1): 1-32.LIN Jiahao, ZHANG Yahui, ZHAN Yan. The pseudo-excitation method and its industrial applications in China and abroad[J]. Applied Mathematics and Mechanics, 2017, 38(1): 1-32. (in Chinese) [33] 孙妍. 复变函数与积分变换[M]. 北京: 机械工业出版社, 2016.SUN Yan. Complex Function and Integral Transformation[M]. Beijing: Machinery Industry Press, 2016. (in Chinese) -
计量
- 文章访问数: 497
- HTML全文浏览量: 177
- PDF下载量: 47
- 被引次数: 0
渝公网安备50010802005915号