Analysis and research on collision restitution coefficient of rocking rigid body based on kinetic energy conversion of discrete elements

doi:10.16265/j.cnki.issn1003-3033.2026.03.1829

Abstract

Abstract:

In order to solve the problem that the classical model of rocking rigid body has a large error and cannot solve the collision restitution coefficient of heterogeneous and irregular rigid bodies, realize more accurate dynamic response prediction, design, safety evaluation and vibration control of rocking structure, based on the energy conversion analysis of mass element, the heterogeneous and irregular rigid bodies were discretized and the kinetic energy conversion analysis of discrete elements was carried out. The collision process of rigid body was divided into three stages. In the first stage, the residual kinetic energy of each discrete unit of rigid body was calculated after the vertical kinetic energy was dissipated. In the second stage, the residual kinetic energy of each discrete element of the rigid body was calculated after the kinetic energy had been dissipated in the direction of the rotation corner after the collision. In the third stage, the residual kinetic energy of each discrete element was converted and calculated under the action of internal force, and the collision recovery coefficient of the whole process was solved. With the help of the Digital Image Correlation (DIC) measurement system, the swing response test was carried out and the method was verified. The results show that the relative error between the collision recovery coefficient of a homogeneous rectangular rigid body obtained by this method and the experimental value is less than 3%, far less than the relative error between the classical model of a rocking rigid body and the experimental value. The relative error between the calculated collision recovery coefficients of heterogeneous and irregular rigid bodies and the experimental values is less than 5%. By this method, the dynamic response of a rocking structure after impact can be predicted more accurately, and a more reasonable structural design and safety evaluation can be provided.

Key words: discrete elements, kinetic energy conversion, rocking rigid body, collision restitution coefficient, kinetic energy distribution

CLC Number:

X915.5

DENG Tongfa, ZHOU Tong, SHEN Botan, MAO Qiuyu. Analysis and research on collision restitution coefficient of rocking rigid body based on kinetic energy conversion of discrete elements[J]. China Safety Science Journal, 2026, 36(3): 74-80.

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References 15

[1]	HOUSNER G W. The behavior of inverted pendulum structures during earthquakes[J]. Bulletin of the Seismological Society of America, 1963, 53(2): 403-417. doi: 10.1785/BSSA0530020403
[2]	贾传果, 潘家富, 李建广, 等. 考虑侧壁影响的偏心单刚体地震响应分析[J]. 振动与冲击, 2022, 41(8): 116-123.
	JIA Chuanguo, PAN Jiafu, LI Jianguang, et al. Seismic analysis of eccentric single-rigid-body considering the influence of colliding-to-wall[J]. Journal of Vibration and Shock, 2022, 41(8): 116-123.
[3]	BECK J L, SKINNER R I. The seismic response of a reinforced concrete bridge pier designed to step[J]. Earthquake Engineering & Structural Dynamics, 1973, 2(4): 343-358.
[4]	杜修力, 周雨龙, 韩强, 等. 摇摆桥墩的研究综述[J]. 地震工程与工程振动, 2018, 38(5):1-11.
	DU Xiuli, ZHOU Yulong, HAN Qiang, et al. State-of-the-arton rocking piers[J]. Earthquake Engineeringand Engineering Dynamics, 2018, 38(5):1-11.
[5]	YIM C, CHOPRA A K, JOSEPH P. Rocking response of rigid blocks to earthquake[J]. Earthquake Engineering and Structural Dynamics, 1980, 8(6): 565-587. doi: 10.1002/eqe.v8:6
[6]	ASLAM M, SCALISE D T, GODDEN W G. Earthquake rocking response of rigid bodies[J]. Journal of the Structural Division, 1980, 106(2): 377-392. doi: 10.1061/JSDEAG.0005363
[7]	CORMACK L G. The design and construction of the major bridges on the mangaweka rail deviation[J]. Transactions of the Institute of Professional Engineers New Zealand: Civil Engineering Section, 1988, 15(1): 17-23.
[8]	周颖, 吴浩, 顾安琪. 地震工程: 从抗震、减隔震到可恢复性[J]. 工程力学, 2019, 36(6): 1-12.
	ZHOU Ying, WU Hao, GU Anqi. Earthquake engineering: From earthquake resistance, energy dissipation and isolation to resilience[J]. Engineering Mechanics, 2019, 36(6): 1-12.
[9]	孙魁, 郭昌锋, 刘洪宇. 内嵌式框架摇摆墙结构抗震性能研究[J]. 中国安全科学学报, 2022, 32(增2): 153-159.
	SUN Kui, GUO Changfeng, LIU Hongyu. Research on seismic performance of embedded frame structure with in-filled rocking wall[J]. China Safety Science Journal, 2022, 32(S2): 153-159. doi: 10.16265/j.cnki.issn1003-3033.2022.S2.0044
[10]	肖水晶, 冯鹏. 新型装配式自复位RC剪力墙设计与性能研究[J]. 工程力学, 2024, 41(11):116-124.
	XIAO Shuijing, FENG Peng. Design and performance investigation on novel prefabricated self-centering RC shear wall[J]. Engineering Mechanics, 2024, 41(11):116-124.
[11]	宋英华, 马建, 张远进. 非平稳随机地震激励下多层框架-摇摆墙结构的响应分析[J]. 中国安全科学学报, 2024, 34(6): 48-56. doi: 10.16265/j.cnki.issn1003-3033.2024.06.0987
	SONG Yinghua, MA Jian, ZHANG Yuanjin. Response determination of multi-story frame-rocking wall structure under non-stationary random seismic excitation[J]. China Safety Science Journal, 2024, 34(6): 48-56. doi: 10.16265/j.cnki.issn1003-3033.2024.06.0987
[12]	徐龙河, 杨智博, 谢行思. 自复位复合阻尼耗能支撑滞回性能研究[J]. 振动与冲击, 2024, 43(14):29-36.
	XU Longhe, YANG Zhibo, XIE Xingsi. Study on the hysteretic performance of a self-centering composite damping energy dissipation brace[J]. Journal of Vibration and Shock, 2024, 43(14):29-36.
[13]	KALLIONTZIS D, SRITHARAN S, SCHULTZ A. Improved coefficient of restitution estimation for free rocking members[J]. Journal of Structural Engineering, 2016, 142(12): DOI: 10.1061/(ASCE)ST.1943-541X.0001598.
[14]	ČEH N, JELENIČ G, BIČANIČ N. Analysis of restitution in rocking of single rigid blocks[J]. Acta Mechanica, 2018, 229: 4623-4642. doi: 10.1007/s00707-018-2246-8
[15]	MAO Qiuyu, DENG Tongfa, SHEN Botan, et al. Research on collision restitution coefficient based on the kinetic energy distribution model of the rocking rigid body within the system of mass points[J]. Buildings, 2024, 14(7): DOI: 10.3390/buildings14072119.

序号	试块名称	质量m/kg	H/m	高宽比
M₁	实心木块	0.424 9	0.2	2
M₂	实心木块	0.644 3	0.3	3
M₃	实心木块	0.869 5	0.4	4
M₄	实心木块	1.097 4	0.5	5

序号	r_e	r_H	r_r	r_d
M₁	0.839 7	0.7	0.830 3	0.836 7
M₂	0.931 7	0.85	0.912 6	0.916 2
M₃	0.954 3	0.911 8	0.947 4	0.949 8
M₄	0.985 0	0.942 3	0.965 0	0.966 7

序号	试块名称	宽/m	高/m	质量/g
G₁	实心钢块	0.05	0.01	190
G₂	实心钢块	0.1	0.01	380.08

编号	构件	理论值	试验值
1	M₁+G₂	0.900 239	0.919 052
2	M₂+G₂	0.946 057	0.942 844
3	M₃+G₂	0.965 72	0.971 819
4	M₄+G₂	0.976 098	0.968 127
5	M₁+G₁(O侧)	0.857 682	0.887 779
6	M₁+G₁(O'侧)	0.913 204	0.946 139
7	M₂+G₁(O侧)	0.925 714	0.946 184
8	M₂+G₁(O'侧)	0.948 613	0.947 244
9	M₃+G₁(O侧)	0.954 417	0.948 216
10	M₃+G₁(O'侧)	0.965 848	0.969 557
11	M₄+G₁(O侧)	0.969 153	0.970 975
12	M₄+G₁(O'侧)	0.975 633	0.982 756