Influence Factors of Tangential Restitution Coefficient of Rolling Stone Based on Friction and Deformation Energy Dissipation
-
摘要:
切向恢复系数是滚石碰撞回弹的重要控制参数，目前的理论公式不能完全反映其作用机制，这是滚石动力学研究的一个难点问题.为此，根据滚石不同的回弹状态，提出基于入射角度变化的切向力模型；进一步，以切向接触理论和动能定理为基础，考虑碰撞过程中切向的摩擦耗能与变形耗能，推导了切向恢复系数的理论公式；最后研究入射速度、入射角、被撞击物体的变形模量对切向恢复系数的影响.结果表明：滚动回弹的切向恢复系数主要受切向变形量的影响；滑动回弹时，入射速度对切向恢复系数的影响参数为
\begin{document}$ {v}^{\frac{1}{20}} $\end{document} ，切向恢复系数随着其增加而缓慢减少；入射角度对切向恢复系数的影响参数为
$ \frac{\mathrm{c}\mathrm{o}{\mathrm{s}}^{\frac{1}{20}}{\beta }_{i}}{\mathrm{t}\mathrm{a}\mathrm{n}{\beta }_{i}} $，切向恢复系数随其增加而增大；被撞击物体的变形模量对切向恢复系数的影响参数为
$ {E}_{2}^{-\frac{5}{8}} $，切向恢复系数随其增加而增加.基于摩擦与变形耗能的切向恢复系数计算公式为滚石的碰撞回弹过程提供了新的计算模型.
Abstract:The tangential restitution coefficient is an important control parameter for the rebound of the rolling stone, and the current theoretical formula can not fully reflect its mechanism. Firstly, according to the different rebound states of the rolling stone, a tangential force model based on the change of incident angle is proposed. Further considering the tangential friction energy dissipation and deformation energy dissipation in the collision process, the theoretical formula of tangential restitution coefficient is derived based on tangential contact theory and kinetic energy theorem. Finally, the influence of various factors on the tangential restitution coefficient is studied. The results show that the tangential restitution coefficient of rolling rebound is mainly affected by tangential deformation. When the rolling stone slips, the influence parameter of incident velocity on the tangential restitution coefficient is
\begin{document}$ {v}^{\frac{1}{20}} $\end{document} , and the tangential restitution coefficient decreases slowly as it increases, while the influence parameter of incident angle on tangential restitution coefficient is
$ \frac{\mathrm{c}\mathrm{o}{\mathrm{s}}^{\frac{1}{20}}{\beta }_{i}}{\mathrm{t}\mathrm{a}\mathrm{n}{\beta }_{i}} $, and the tangential restitution coefficient increases with its increase, the influence parameter of the deformation modulus of the impacted object on the tangential restitution coefficient is
$ {E}_{2}^{-\frac{5}{8}} $, and the tangential restitution coefficient increases with its increase. The tangential recovery coefficient based on friction and deformation energy dissipation provides a new computational model for the collision process of rolling stone.
-
图 6 不同入射速度下的切向恢复系数
Fig. 6. Tangential restitution coefficients at different incident velocities
图 7 不同入射速度下切向恢复系数变化曲线
Fig. 7. Variation of restitution coefficient at different incident velocities
图 8 不同入射角下的切向恢复系数
Fig. 8. Tangential restitution coefficient at different incident angles
图 9 不同入射角下的恢复系数($ v= $20 m/s)
Fig. 9. Variation of coefficient of restitution under different incident angles($ v= $20 m/s)
图 10 不同入射角下的耗能比例系数S
Fig. 10. The energy dissipation proportional coefficient S at different incident angles
图 11 恢复系数随$ {E}_{2} $/$ {E}_{1} $变化曲线
Fig. 11. Variation of coefficient of restitution under different ratios of $ {E}_{2} $/$ {E}_{1} $
表 1 滚石冲击计算参数
Table 1. Calculation parameters of rock fall impact
滚石 防护物体 变形模量$ {E}_{1} $(GPa) 泊松比
$ {\mu }_{1} $半径R(m) 密度
ρ(kg· m－3)变形
模量$ {E}_{2} $(GPa)泊松比
$ {\mu }_{2} $摩擦系数
$ f $40 0.2 0.5 2 500 30 0.2 0.5 
下载: 导出CSV
表 2 $ {\mathit{\beta }}_{\mathit{i}} $=1°时不同入射速度碰撞特征量统计
Table 2. Collision typical parameters quantity statistics under different incidence speeds($ {\beta }_{i} $=1°)
(m· s-1)$ v $ $ {t}_{1} $
(ms)$ T $
(ms)$ {F}_{N\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $$ {F}_{T\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $恢复系数 $ {e}_{n} $ $ {e}_{t} $ 10 1.456 3.058 12.264 0.183 0.622 0.743 15 1.343 2.849 19.95 0.299 0.562 0.675 20 1.267 2.711 28.175 0.423 0.523 0.631 25 1.212 2.609 36.825 0.552 0.495 0.599 30 1.170 2.527 45.832 0.687 0.473 0.573 35 1.133 2.461 55.145 0.826 0.455 0.553 40 1.104 2.405 64.728 0.971 0.439 0.536
下载: 导出CSV
表 3 $ {\mathit{\beta }}_{\mathit{i}} $=60°时不同入射速度碰撞特征量统计
Table 3. Collision typical parameters quantity statistics under different incidence speeds($ {\beta }_{i} $=60°)
(m· s-1)$ v $ $ {t}_{1} $
(ms)$ T $
(ms)$ {F}_{N\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $$ {F}_{T\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $恢复系数 $ {e}_{n} $ $ {e}_{t} $ 10 1.669 3.443 5.403 2.701 0.738 0.509 15 1.539 3.209 8.789 4.394 0.667 0.501 20 1.453 3.052 12.412 6.206 0.621 0.491 25 1.389 2.936 16.224 8.112 0.587 0.487 30 1.340 2.844 20.191 10.095 0.561 0.484 35 1.299 2.769 24.294 12.147 0.540 0.481 40 1.265 2.706 28.516 14.258 0.522 0.476
下载: 导出CSV
表 4 不同入射角度碰撞特征量统计($ \mathit{v}= $20 m/s)
Table 4. Collision typical parameters quantity statistics under different incident angles($ v= $20 m/s)
$ {\beta }_{i} $
(°)$ {t}_{1} $
(ms)$ T $
(ms)$ {F}_{N\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $$ {F}_{T\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $恢复系数 $ {e}_{n} $ $ {e}_{t} $ 1 1.267 2.708 28.345 0.438 0.523 0.631 10 1.269 2.710 28.004 4.340 0.524 0.524 20 1.281 2.735 26.476 8.472 0.530 0.419 30 1.302 2.774 23.994 11.997 0.541 0.382 45 1.356 2.874 18.810 9.405 0.569 0.193 60 1.453 3.052 12.412 6.206 0.620 0.491 75 1.658 3.423 5.623 2.811 0.731 0.745 89 2.358 4.825 0.221 0.111 0.829 0.907
下载: 导出CSV
表 5 不同被撞击物体变形模量碰撞特征量
Table 5. Collision typical parameters quantity statistics under deformation modulus of different objects impacted($ v= $20 m/s, $ {\beta }_{i}= $60°)
E2
(MPa)$ {t}_{1} $
(ms)$ T $
(ms)$ {F}_{N\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $$ {F}_{T\mathrm{m}\mathrm{a}\mathrm{x}}/ $
$ MN $恢复系数 $ {e}_{n} $ $ {e}_{t} $ 10 28.578 69.193 0.631 0.315 0.173 0.352 50 15.018 35.519 1.201 0.601 0.211 0.389 100 11.387 26.663 1.584 0.792 0.230 0.405 200 8.638 20.024 2.088 1.044 0.251 0.421 500 6.005 13.736 3.002 1.501 0.283 0.441 1 000 4.574 10.353 3.943 1.972 0.311 0.455 10 000 1.971 4.278 9.151 4.575 0.457 0.483 40 000 1.453 3.052 12.412 6.206 0.621 0.563
下载: 导出CSV
-
[1] Azzoni, A., de Freitas, M. H., 1995. Experimentally Gained Parameters, Decisive for Rock Fall Analysis. Rock Mechanics and Rock Engineering, 28(2): 111-124. https://doi.org/10.1007/bf01020064 [2] Brizmer, V., Kligerman, Y., Etsion, I., 2006. The Effect of Contact Conditions and Material Properties on the Elasticity Terminus of a Spherical Contact. International Journal of Solids and Structures, 43(18): 5736-5749. [3] Buzzi, O., Giacomini, A., Spadari, M., 2012. Laboratory Investigation on High Values of Restitution Coefficients. Rock Mechanics and Rock Engineering, 45(1): 35-43. https://doi.org/10.1007/s00603-011-0183-0 [4] Cagnoli, B., Manga, M., 2003. Pumice-Pumice Collisions and the Effect of the Impact Angle—Art. No. 1636. Geophysical Research Letters, 30(12): 1636. [5] Chau, K., Wong, R., Wu, J., 2002. Coefficient of Restitution and Rotational Motions of Rockfall Impacts. International Journal of Rock Mechanics and Mining Sciences, 39(1): 69-77. doi: 10.1016/S1365-1609(02)00016-3 [6] Chen, Y. L., 2013. Influence of Key Factors on Trajectories of Rockfalls. Chinese Journal of Geotechnical Engineering, 35(Suppl. 2): 191-196(in Chinese with English abstract). [7] Chen, Y. Q., Wang, Q. C., 2018. Correction Calculation of Impact Force of Rockfall Based on Hertz Contact Theory and Thornton Elastoplasticity Hypothesis. Science Technology and Engineering, 18(13): 37-41(in Chinese with English abstract). doi: 10.3969/j.issn.1671-1815.2018.13.006 [8] Day, R. W., 1997. Case Studies of Rockfall in Soft versus Hard Rock. Environmental and Engineering Geoscience, 3(1): 133-140. [9] Dussauge, C., Grasso, J., Helmstetter, A., 2003. Statistical Analysis of Rockfall Volume Distributions: Implications for Rockfall Dynamics. Journal of Geophysical Research: Solid Earth, 108(B6): ETG2-1-ETG2-11. [10] Giani, G. P., Giacomini, A., Migliazza, M., et al., 2004. Experimental and Theoretical Studies to Improve Rock Fall Analysis and Protection Work Design. Rock Mechanics and Rock Engineering, 37(5): 369-389. https://doi.org/10.1007/s00603-004-0027-2 [11] He, S. M., Wu, Y., Li, X. P., 2009. Research on Restitution Coefficient of Rock Fall. Rock and Soil Mechanics, 30(3): 623-627(in Chinese with English abstract). [12] He, S. M., Zhuang, W. L., Zhang, X., et al., 2013. Research on Rockfall Impact Prevention of Chediguan Bridge Pier, Duwen Road. Chinese Journal of Rock Mechanics and Engineering, 32(Suppl. 2): 3421-3427(in Chinese with English abstract). [13] Hu, X. L., Tang, H. M., Zhu, L. X., 2011. Collapse Mode and Mechanism of High Magmatite Rock Slope in Wenchuan Epicentral Area. Earth Science, 36(6): 1149-1154(in Chinese with English abstract). [14] Kharaz, A., Gorham, D., Salman, A., 2001. An Experimental Study of the Elastic Rebound of Spheres. Powder Technology, 120(3): 281–291. doi: 10.1016/S0032-5910(01)00283-2 [15] Labiouse, V., Heidenreich, B., 2009. Half-Scale Experimental Study of Rockfall Impacts on Sandy Slopes. Natural Hazards and Earth System Science, 9(104): 1981-1993. [16] Liu, X. F., Zhao, Y. Q., Wang, X. R., et al., 2022. Current Status and Prospects of Research on Fatigue Damage and Failure Precursors of Rocks. Earth Science, 47(6): 2190-2198(in Chinese with English abstract). [17] Liu, Y. J., 2002. Study on Fluidifying Theory of Large Highspeed Rockslide (Dissertation). Southwest Jiaotong University, Chengdu(in Chinese with English abstract). [18] Luo, G., Cheng, Q. G., Shen, W. G., et al., 2022. Research Status and Development Trend of the High-Altitude Extremely-Energetic Rockfalls. Earth Science, 47(3): 913-934(in Chinese with English abstract). [19] Lü, Q., Sun, H. Y., Zhai, S. K., et al., 2003. Evaluation Models of Rockfall Trajectory. Journal of Natural Disasters, 12(2): 79-84 (in Chinese with English abstract). doi: 10.3969/j.issn.1004-4574.2003.02.014 [20] Maw, N., Barber, J. R., Fawcett, J. N., 1976. The Oblique Impact of Elastic Spheres. Wear, 38(1): 101-114. doi: 10.1016/0043-1648(76)90201-5 [21] Mangwandi, C., Cheong, Y., Adams, M., et al., 2007. The Coefficient of Restitution of Different Representative Types of Granules. Chemical Engineering Science, 62(1): 437-450. [22] Mei, X. F., Hu, X. W., Luo, G., et al., 2019. A Study on the Coefficient of Restitution and Peak Impact of Rockfall Based on the Elastic-Plastic Theory. Journal of Vibration and Shock, 38(8): 14-20(in Chinese with English abstract). [23] Mindlin, R. D., 1949. Compliance of Elastic Bodies in Contact. Journal of Applied Mechanics, 16(3): 259-268. doi: 10.1115/1.4009973 [24] Qin, Z. Y., Lu, Q. S., 2006. Analysis of Impact Process Model Based on Restitution Coefficient. Journal of Dynamics and Control, (4): 294-298(in Chinese with English abstract). doi: 10.3969/j.issn.1672-6553.2006.04.002 [25] Scheiner, S., Pichler, B., Hellmich, C., et al., 2006. Loading of Soil-Covered Oil and Gas Pipelines Due to Adverse Soil Settlements: Protection against Thermal Dilatation-Induced Wear, Involving Geosynthetics. Computers and Geotechnics, 33(8): 371-380. doi: 10.1016/j.compgeo.2006.08.003 [26] Schwager, T., Becker, V., Pöschel, T., 2008. Coefficient of Tangential Restitution for Viscoelastic Spheres. The European Physical Journal E, Soft Matter, 27(1): 107-114. https://doi.org/10.1140/epje/i2007-10356-3 [27] Seifried, R., Schiehlen, W., Eberhard, P., 2005. Numerical and Experimental Evaluation of the Coefficient of Restitution for Repeated Impacts. International Journal of Impact Engineering, 32(1): 508-524. [28] Vu-Quoc, L., Lesburg, L., Zhang, X., 2004. An Accurate Tangential Force-Displacement Model for Granular-Flow Simulations: Contacting Spheres with Plastic Deformation, Force-Driven Formulation. Journal of Computational Physics, 196(1): 298-326. https://doi.org/10.1016/j.jcp.2003.10.025 [29] Wang, Y. S., Cheng, W. Q., Liu, J. W., 2022. Forming Process and Mechanisms of Geo-Hazards in Luding Section of the Sichuan-Tibet Railway. Earth Science, 47(3): 950-958(in Chinese with English abstract). [30] Wu, C. Y., Li, L. Y., Thornton, C., 2003. Rebound Behavior of Spheres for Plastic Impacts. International Journal of Impact Engineering, 28: 929-946. doi: 10.1016/S0734-743X(03)00014-9 [31] Yang, H. Q., Zhou, X. P., 2009. A New Approach to Calculate Trajectory of Rockfall. Rock and Soil Mechanics, 30(11): 3411-3416(in Chinese with English abstract). doi: 10.3969/j.issn.1000-7598.2009.11.032 [32] Ye, S. Q., Gong, S. Q., 2015. Research on Normal Restitution Coefficient of Rockfall Collision by Model Tests. China Railway Science, 36(4): 13-19(in Chinese with English abstract). [33] Ye, S. Q., Gong, S. Q., Wang, L. F., et al., 2018. Research on Value of Tangential Restitution Coefficient for Rockfall Collision. China Railway Science, 39(1): 8-15(in Chinese with English abstract). [34] Zhang, L. Q., Yang, Z. F., Xu, B., 2004. Rock Falls and Rock Fall Hazards. Journal of Engineering Geology, 12(3): 225-231 (in Chinese with English abstract). doi: 10.3969/j.issn.1004-9665.2004.03.001 [35] Zhang, G. C., Tang, H. M., Xiang, X., 2012. Characteristic Parameters Theoretical Analysis of Rockfall Impact on Ground. Chinese Journal of Rock Mechanics and Engineering, 31(Suppl. 1): 2839-2846(in Chinese with English abstract). [36] Zhang, G. C., Xiang, X., Tang, H. M., 2011. Field Test and Numerical Calculation of Restitution Coefficient of Rockfall Collision. Chinese Journal of Rock Mechanics and Engineering, 30(6): 1266-1273(in Chinese with English abstract). [37] 陈颖骐, 王全才, 2018. 基于Hertz弹性理论和Thornton弹塑性假设的滚石冲击力的修正计算. 科学技术与工程, 18(13): 37-41. https://www.cnki.com.cn/Article/CJFDTOTAL-KXJS201813006.htm [38] 陈宇龙, 2013. 滚石运动过程中关键参数的影响分析. 岩土工程学报, 35(增刊2): 191-196. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC2013S2032.htm [39] 何思明, 吴永, 李新坡, 2009. 滚石冲击碰撞恢复系数研究. 岩土力学, 30(3): 623-627. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200903010.htm [40] 何思明, 庄卫林, 张雄, 等, 2013. 都汶公路彻底关大桥桥墩抗滚石冲击防护研究. 岩石力学与工程学报, 32(增刊2): 3421-3427. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX2013S2056.htm [41] 胡新丽, 唐辉明, 朱丽霞, 2011. 汶川震中岩浆岩高边坡破坏模式与崩塌机理. 地球科学, 36(6): 1149-1154. doi: 10.3799/dqkx.2011.121 [42] 刘新锋, 赵英群, 王晓睿, 等, 2022. 岩石疲劳损伤及破坏前兆研究现状与展望. 地球科学, 47(6): 2190-2198. doi: 10.3799/dqkx.2021.186 [43] 刘涌江, 2002. 大型高速岩质滑坡流体化理论研究(博士论文). 成都: 西南交通大学. [44] 罗刚, 程谦恭, 沈位刚, 等, 2022. 高位高能岩崩研究现状与发展趋势. 地球科学, 47(3): 913-934. doi: 10.3799/dqkx.2021.133 [45] 吕庆, 孙红月, 翟三扣, 等, 2003. 边坡滚石运动的计算模型. 自然灾害学报, 12(2): 79-84. https://www.cnki.com.cn/Article/CJFDTOTAL-ZRZH200302013.htm [46] 梅雪峰, 胡卸文, 罗刚, 等, 2019. 基于弹塑性理论的落石碰撞恢复系数和峰值冲击力研究. 振动与冲击, 38(8): 14-20. https://www.cnki.com.cn/Article/CJFDTOTAL-ZDCJ201908003.htm [47] 秦志英, 陆启韶, 2006. 基于恢复系数的碰撞过程模型分析. 动力学与控制学报, (4): 294-298. https://www.cnki.com.cn/Article/CJFDTOTAL-DLXK200604001.htm [48] 王运生, 程万强, 刘江伟, 2022. 川藏铁路廊道泸定段地质灾害孕育过程及成灾机制. 地球科学, 47(3): 950-958. doi: 10.3799/dqkx.2021.179 [49] 杨海清, 周小平, 2009. 边坡落石运动轨迹计算新方法. 岩土力学, 30(11): 3411-3416. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200911041.htm [50] 叶四桥, 巩尚卿, 2015. 落石碰撞法向恢复系数的模型试验研究. 中国铁道科学, 36(4): 13-19. https://www.cnki.com.cn/Article/CJFDTOTAL-ZGTK201504004.htm [51] 叶四桥, 巩尚卿, 王林峰, 等, 2018. 落石碰撞切向恢复系数的取值研究. 中国铁道科学, 39(1): 8-15. https://www.cnki.com.cn/Article/CJFDTOTAL-ZGTK201801003.htm [52] 张路青, 杨志法, 许兵, 2004. 滚石与滚石灾害. 工程地质学报, 12(3): 225-231. https://www.cnki.com.cn/Article/CJFDTOTAL-GCDZ200403000.htm [53] 章广成, 唐辉明, 向欣, 2012. 冲击地面过程中落石特征参量的理论分析. 岩石力学与工程学报, 31(增刊1): 2839-2846. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX2012S1030.htm [54] 章广成, 向欣, 唐辉明, 2011. 落石碰撞恢复系数的现场试验与数值计算. 岩石力学与工程学报, 30(6): 1266-1273. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201106026.htm -
点击查看大图
计量
- 文章访问数: 72
- HTML全文浏览量: 25
- PDF下载量: 15
- 被引次数: 0
