基于频率相关黏性阻尼模型的复模态叠加法
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摘要
黏性阻尼模型存在每周期耗散能量与外激励频率相关的缺陷,复阻尼模型时域计算结果存在发散现象.为克服上述两种阻尼模型的不足,在复阻尼模型基础上,依据时频域转化原则推导了频率相关黏性阻尼模型.频率相关黏性阻尼模型不仅具有每周期耗散能量与外激励频率无关的优点,还保证了结构位移时程的稳定收敛.混合结构由具有不同阻尼特性的材料组成,其阻尼矩阵为非比例矩阵,无法直接采用实模态叠加法.根据频率相关黏性阻尼模型与复阻尼模型的转换关系,提出了适用于混合结构的基于频率相关黏性阻尼模型的复模态叠加法.算例分析结果表明,与基于黏性阻尼模型的复模态叠加法相比,基于频率相关黏性阻尼模型的复模态叠加法不仅计算结果唯一,且不增加矩阵维度,具有较高的计算效率.小阻尼情况下,两种方法的计算结果近似相等,且与复阻尼模型的频域计算结果一致.当阻尼比较大时,两种方法的计算结果差异增大,但频率相关黏性阻尼模型的复模态叠加法与复阻尼模型的频域计算结果仍保持一致.
According to the viscous damping model, energy dissipation is related to external excitation frequencies in each cycle.There is divergence in time domain calculation results of complex damping model. In order to overcome the two kinds of disadvantages of damping models, frequency dependent damping model is obtained by time-frequency transformation principle on the basis of complex damping model. Energy dissipation is not related to external excitation frequencies in each cycle. In addition, time-history calculation of frequency dependent damping model is convergent. Real mode superposition method is not directly adopted, because the damping matrix of mixed structure of multiple damping characteristic materials is non-proportional. By the transformations of frequency dependent damping model and complex damping model, complex mode superposition method is proposed based on frequencydependent damping model. And it can be directly applied to mixed structure. The analysis results of the cases show that compared with complex mode superposition method based on viscous damping model, the calculation results of the complex mode superposition method based on frequency dependent damping model are unique. In addition, the method does not increase matrix dimensions and has high computational efficiency. The analysis results of small damping cases show that calculation results of the two methods are approximately equal. They are consistent with frequency domain calculation results of complex damping model. When the damping ratios are larger, the difference of calculation results by the two methods will increase. While the calculation results of complex mode superposition method based on frequency dependent damping model will remain consistent with frequency domain calculation results of complex damping model.
According to the viscous damping model, energy dissipation is related to external excitation frequencies in each cycle.There is divergence in time domain calculation results of complex damping model. In order to overcome the two kinds of disadvantages of damping models, frequency dependent damping model is obtained by time-frequency transformation principle on the basis of complex damping model. Energy dissipation is not related to external excitation frequencies in each cycle. In addition, time-history calculation of frequency dependent damping model is convergent. Real mode superposition method is not directly adopted, because the damping matrix of mixed structure of multiple damping characteristic materials is non-proportional. By the transformations of frequency dependent damping model and complex damping model, complex mode superposition method is proposed based on frequencydependent damping model. And it can be directly applied to mixed structure. The analysis results of the cases show that compared with complex mode superposition method based on viscous damping model, the calculation results of the complex mode superposition method based on frequency dependent damping model are unique. In addition, the method does not increase matrix dimensions and has high computational efficiency. The analysis results of small damping cases show that calculation results of the two methods are approximately equal. They are consistent with frequency domain calculation results of complex damping model. When the damping ratios are larger, the difference of calculation results by the two methods will increase. While the calculation results of complex mode superposition method based on frequency dependent damping model will remain consistent with frequency domain calculation results of complex damping model.
引文
1杨慧,闫维明,何浩祥.基于残余应力的复合阻尼模型及在R-C梁损伤分析中的应用.振动工程学报,2017,30(3):432-441(Yang Hui,Yan Weiming,He Haoxiang.The damage analysis of R-C beam structure based on the combined damping model of residual stress.Journal of Vibration Engineering,2017,30(3):432-441(in Chinese))
2 Bert CW.Material damping:An introductory review of mathematical models,measures and experimental techniques.Journal of Sound and Vibration,1973,29(2):129-135
3梁超锋,刘铁军,邹笃建等.材料黏滞系数与损耗因子的频率相关性研究.力学学报,2012,44(5):933-937(Liang Chaofeng,Liu Tiejun,Zou Dujian,et al.The frequency-dependent study on viscosity coefficient and loss tangent of viscoelastic materials.Chinese Journal of Theoretical and Applied Mechanics,2012,44(5):933-937(in Chinese))
4吴泽玉,王东炜,李玉河.复阻尼结构动力方程的增维精细积分法.振动与冲击,2017,36(2):107-110(Wu Zeyu,Wang Dongwei,Li Yuhe.Magnified dimension precise integration method for the dynamic equations of complex damped structures.Journal of Vibration and Shock,2017,36(2):107-110(in Chinese))
5 Crandall SH.The role of damping in vibration theory.Journal of Sound and Vibration,1970,11(1):3-18
6 Bert CW.Material damping:An introductory review of mathematic measures and experimental technique.Journal of Sound and Vibration,1973,29(2):129-153
7朱镜清.关于复阻尼理论的两个基本问题.固体力学学报,1992,13(2):113-118(Zhu Jingqing.On the two basic problems of complex damping theory.Acta Mechanica Solida Sinica,1992,13(2):113-118(in Chinese))
8 Mastroddi F,Eugeni F,Erba F.On the modal diagonalization of viscoelastic mechanical systems.Mechanical Systems and Signal Processing,2017,96:159-175
9朱敏,朱镜清.逐步积分法求解复阻尼结构运动方程的稳定性问题.地震工程与工程振动,2001,21(4):59-62(Zhu Min,Zhu Jingqing.Studies on stability of step-by-step methods under complex damping conditions.Earthquake Engineering and Engineering Vibration,2001,21(4):59-62(in Chinese))
10 Lee JH,Kim J.Identification of damping matrices from measured frequency response functions.Journal of Sound and Vibration,2001,240(3):545-565
11尚守平,甘宜诚,蒋林.结构振动阻尼理论模型探究及实验分析.地震工程与工程振动,2015,35(2):166-171(Shang Shouping,Gan Yicheng,Jiang Lin.Theoretical research and experimental analysis of structural damping.Earthquake Engineering and Engineering Vibration,2015,35(2):166-171(in Chinese))
12何钟怡,廖振鹏,王小华.关于复阻尼理论的几点注记.地震工程与工程振动,2002,22(1):1-6(He Zhongyi,Liao Zhenpeng,Wang Xiaohua.Some notes on theory of complex damping.Earthquake Engineering and Engineering Vibration,2002,22(1):1-6(in Chinese))
13 Wang J.Rayleigh coefficients for series infrastructure systems with multiple damping properties.Journal of Vibration and Control,2015,21(6):1234-1248
14 Clough RW,Penzien J.结构动力学.王光远.译.北京:科学出版社,1983(Clough RW,Penzien J.Dynamics of Structures.Wang Guangyuan transl.Beijing:Science Press,1983(in Chinese))
15 15 Chen LY,Chen JT,Chen CH,et al.Free vibration of a SDOFsystem with hysteretic damping.Mechanics Research Communications,1994,21(6):599-604
16 Pan Y,Wang YF.Frequency-domain analysis of exponentially damped linear systems.Journal of Sound and Vibration,2013,332(7):1754-1765.
17 Reggio A,Angelis MD.Modelling and identification of structures with rate-independent linear damping.Meccanica,2015,50(3):617-632
18朱镜清.频率相关黏性阻尼理论及有关问题的解.振动与冲击,1992(4):1-7(Zhu Jingqing.Frequency dependant viscous damping theory and some related problems.Journal of Vibration and Shock,1992(4):1-7(in Chinese))
19刘庆林,傅学怡.基于复阻尼假定的不同材料阻尼特性混合结构抗震分析反应谱CCQC法.土木工程学报,2011,44(3):61-71(Liu Qinglin,Fu Xueyi.A response spectrum CCQC method for seismic analysis of structures of multiple material damping characteristics based on complex damping assumption.China Civil Engineering Journal,2011,44(3):61-71(in Chinese))
20潘旦光.地震反应分析中Rayleigh阻尼系数的优化解.工程力学,2013,30(11):15-20(Pan Danguang.An optimization solution for Rayleigh damping coefficients in seismic response analysis.Engineering Mechanics,2013,30(11):15-20(in Chinese))
21黄维,钱江,周知.基于Rayleigh阻尼模型的竖向混合结构设计阻尼比研究.工程力学,2015,32(10):60-67(Huang Wei,Qian Jiang,Zhou Zhi.Research on equivalent damping ratio of vertically mixed structures based on Rayleigh damping model.Engineering Mechanics,2015,32(10):60-67(in Chinese))
22 Wei H,Jiang Q,Zhi Z,et al.An approach to equivalent damping ratio of vertically mixed structures based on response error minimization.Soil Dynamics and Earthquake Engineering,2015,72:119-128
23 Luco JE,Lanzi L.Optimal Caughey series representation of classical damping matrices.Soil Dynamics and Earthquake Engineering,2017,92:253-265
24汪梦甫.非比例阻尼线性体系地震反应计算的振型分解反应谱法.地震工程与工程振动,2007,27(1):31-37(Wang Mengfu.On seismic response analysis methods of non-proportional damped linear MDOF systems.Journal of Earthquake Engineering and Engineering Vibration,2007,27(1):31-37(in Chinese))
25 Neugebauer R,Scheffler C,Wabner M,et al.Advanced state space modeling of non-proportional damped machine tool mechanics.Cirp Journal of Manufacturing Science and Technology,2010,3(1):8-13
26 Neugebauer R,Scheffler C,Wabner M,et al.State space modeling of non-proportional passive damping in machine tools.International Journal of Advanced Manufacturing Technology,2011,53(9-12):945-952
27 Qin JQ.Seismic response analysis on non-proportional damped system by using perturbation technique.Advanced Materials Research,2013,639-640(1):917-921
28楼梦麟,范幺清.求解非比例阻尼体系复模态的实模态摄动法.力学学报,2007,39(1):112-118(Lou Menglin,Fan Yaoqing.Modal perturbation method for obtaining complex modal characteristics of non-proportional damping systems.Chinese Journal of Theoretical and Applied Mechanics,2007,39(1):112-118(in Chinese))
29国巍,余志武.一种计算非比例阻尼结构地震响应的新方法.力学学报,2011,43(6):1170-1180(Guo Wei,Yu Zhiwu.Anew method for calculating seismic response of non-proportionally damped structures.Chinese Journal of Theoretical and Applied Mechanics,2011,43(6):1170-1180(in Chinese))
30段绍伟,向湘林,沈蒲生.复杂阻尼结构阻尼模型研究.振动与冲击,2011,30(2):73-76(Duan Shaowei,Xiang Xianglin,Shen Pusheng.Damping model of a complex structure.Journal of Vibration and Shock.2011,30(2):73-76(in Chinese))
31 Mohamed K,Noureddine B,Scott C.Appropriation effects in the estimation of modal damping.Advances in Acoustics and Vibration,2017,5:185-193
32 Zoghaib L,Mattei PO.Time and frequency response of structures with frequency dependent,non-proportional linear damping.Journal of Sound and Vibration,2014,333(3):887-900
33朱镜清,朱敏.复阻尼多自由度系统动力分析的模态叠加法.地震工程与工程振动,2004,24(1):55-58(Zhu Jingqing,Zhu Min.Two mode-superposition methods for dynamic analysis of MDOF systems with complex damping characteristics.Earthquake Engineering and Engineering Vibration,2004,24(1):55-58(in Chinese))
34刘庆林,傅学怡,孙占琦.基于复阻尼假定的不同材料阻尼特性混合结构抗震分析复模态叠加法.建筑结构学报,2011,32(9):27-33(Liu Qinglin,Fu Xueyi,Sun Zhanqi.A complex mode superposition method for seismic analysis of structures of multiple material damping characteristics based on complex damping assumption.Journal of Building Structures,2011,32(9):27-33(in Chinese))
35 Makris N,Zhang J.Time-domain viscoelastic analysis of earth structures.Earthquake Engineering and Structural Dynamics,2015,29(6):745-768
36 Lage Y,Cach?o H,Reis L,et al.A damage parameter for HCF and VHCF based on hysteretic damping.International Journal of Fatigue,2014,62(7):2-9
37大崎顺彦.地震动的谱分析入门.吕敏申,谢礼立.译.北京:地震出版社,2008(Osaki Y.Introduction to Spectrum Analysis of Ground Motion.L¨u Minshen,Xie Lili,trans.Beijing:Seismological Press,2008(in Chinese))
38胡聿贤.地震工程学(第2版).北京:地震出版社,2006(Hu Yuxian.Earthquake Engineering,2nd edn.Beijing:Seismological Press,2006(in Chinese))
39王建平,刘宏昭,原大宁等.含复阻尼系统的对偶原则及数值方法研究.振动工程学报,2004,17(1):62-65(Wang Jianping,Liu Hongzhao,Yuan Daning,et al.Dual principle of oscillation systems with complex damping and its numerical method.Journal of Vibration Engineering,2004,17(1):62-65(in Chinese))
40朱镜清.结构抗震分析原理.北京:地震出版社,2002(Zhu Jingqing.Seismic Analysis of Structures.Beijing:Earthquake Press,2002(in Chinese))
41潘玉华,王元丰.复阻尼结构动力方程的高斯精细时程积分法.工程力学,2012,29(2):16-20(Pan Yuhua,Wang Yuanfeng.Gauss precise time-integration of complex damping vibration systems.Engineering Mechanics,2012,29(2):16-20(in Chinese))
2 Bert CW.Material damping:An introductory review of mathematical models,measures and experimental techniques.Journal of Sound and Vibration,1973,29(2):129-135
3梁超锋,刘铁军,邹笃建等.材料黏滞系数与损耗因子的频率相关性研究.力学学报,2012,44(5):933-937(Liang Chaofeng,Liu Tiejun,Zou Dujian,et al.The frequency-dependent study on viscosity coefficient and loss tangent of viscoelastic materials.Chinese Journal of Theoretical and Applied Mechanics,2012,44(5):933-937(in Chinese))
4吴泽玉,王东炜,李玉河.复阻尼结构动力方程的增维精细积分法.振动与冲击,2017,36(2):107-110(Wu Zeyu,Wang Dongwei,Li Yuhe.Magnified dimension precise integration method for the dynamic equations of complex damped structures.Journal of Vibration and Shock,2017,36(2):107-110(in Chinese))
5 Crandall SH.The role of damping in vibration theory.Journal of Sound and Vibration,1970,11(1):3-18
6 Bert CW.Material damping:An introductory review of mathematic measures and experimental technique.Journal of Sound and Vibration,1973,29(2):129-153
7朱镜清.关于复阻尼理论的两个基本问题.固体力学学报,1992,13(2):113-118(Zhu Jingqing.On the two basic problems of complex damping theory.Acta Mechanica Solida Sinica,1992,13(2):113-118(in Chinese))
8 Mastroddi F,Eugeni F,Erba F.On the modal diagonalization of viscoelastic mechanical systems.Mechanical Systems and Signal Processing,2017,96:159-175
9朱敏,朱镜清.逐步积分法求解复阻尼结构运动方程的稳定性问题.地震工程与工程振动,2001,21(4):59-62(Zhu Min,Zhu Jingqing.Studies on stability of step-by-step methods under complex damping conditions.Earthquake Engineering and Engineering Vibration,2001,21(4):59-62(in Chinese))
10 Lee JH,Kim J.Identification of damping matrices from measured frequency response functions.Journal of Sound and Vibration,2001,240(3):545-565
11尚守平,甘宜诚,蒋林.结构振动阻尼理论模型探究及实验分析.地震工程与工程振动,2015,35(2):166-171(Shang Shouping,Gan Yicheng,Jiang Lin.Theoretical research and experimental analysis of structural damping.Earthquake Engineering and Engineering Vibration,2015,35(2):166-171(in Chinese))
12何钟怡,廖振鹏,王小华.关于复阻尼理论的几点注记.地震工程与工程振动,2002,22(1):1-6(He Zhongyi,Liao Zhenpeng,Wang Xiaohua.Some notes on theory of complex damping.Earthquake Engineering and Engineering Vibration,2002,22(1):1-6(in Chinese))
13 Wang J.Rayleigh coefficients for series infrastructure systems with multiple damping properties.Journal of Vibration and Control,2015,21(6):1234-1248
14 Clough RW,Penzien J.结构动力学.王光远.译.北京:科学出版社,1983(Clough RW,Penzien J.Dynamics of Structures.Wang Guangyuan transl.Beijing:Science Press,1983(in Chinese))
15 15 Chen LY,Chen JT,Chen CH,et al.Free vibration of a SDOFsystem with hysteretic damping.Mechanics Research Communications,1994,21(6):599-604
16 Pan Y,Wang YF.Frequency-domain analysis of exponentially damped linear systems.Journal of Sound and Vibration,2013,332(7):1754-1765.
17 Reggio A,Angelis MD.Modelling and identification of structures with rate-independent linear damping.Meccanica,2015,50(3):617-632
18朱镜清.频率相关黏性阻尼理论及有关问题的解.振动与冲击,1992(4):1-7(Zhu Jingqing.Frequency dependant viscous damping theory and some related problems.Journal of Vibration and Shock,1992(4):1-7(in Chinese))
19刘庆林,傅学怡.基于复阻尼假定的不同材料阻尼特性混合结构抗震分析反应谱CCQC法.土木工程学报,2011,44(3):61-71(Liu Qinglin,Fu Xueyi.A response spectrum CCQC method for seismic analysis of structures of multiple material damping characteristics based on complex damping assumption.China Civil Engineering Journal,2011,44(3):61-71(in Chinese))
20潘旦光.地震反应分析中Rayleigh阻尼系数的优化解.工程力学,2013,30(11):15-20(Pan Danguang.An optimization solution for Rayleigh damping coefficients in seismic response analysis.Engineering Mechanics,2013,30(11):15-20(in Chinese))
21黄维,钱江,周知.基于Rayleigh阻尼模型的竖向混合结构设计阻尼比研究.工程力学,2015,32(10):60-67(Huang Wei,Qian Jiang,Zhou Zhi.Research on equivalent damping ratio of vertically mixed structures based on Rayleigh damping model.Engineering Mechanics,2015,32(10):60-67(in Chinese))
22 Wei H,Jiang Q,Zhi Z,et al.An approach to equivalent damping ratio of vertically mixed structures based on response error minimization.Soil Dynamics and Earthquake Engineering,2015,72:119-128
23 Luco JE,Lanzi L.Optimal Caughey series representation of classical damping matrices.Soil Dynamics and Earthquake Engineering,2017,92:253-265
24汪梦甫.非比例阻尼线性体系地震反应计算的振型分解反应谱法.地震工程与工程振动,2007,27(1):31-37(Wang Mengfu.On seismic response analysis methods of non-proportional damped linear MDOF systems.Journal of Earthquake Engineering and Engineering Vibration,2007,27(1):31-37(in Chinese))
25 Neugebauer R,Scheffler C,Wabner M,et al.Advanced state space modeling of non-proportional damped machine tool mechanics.Cirp Journal of Manufacturing Science and Technology,2010,3(1):8-13
26 Neugebauer R,Scheffler C,Wabner M,et al.State space modeling of non-proportional passive damping in machine tools.International Journal of Advanced Manufacturing Technology,2011,53(9-12):945-952
27 Qin JQ.Seismic response analysis on non-proportional damped system by using perturbation technique.Advanced Materials Research,2013,639-640(1):917-921
28楼梦麟,范幺清.求解非比例阻尼体系复模态的实模态摄动法.力学学报,2007,39(1):112-118(Lou Menglin,Fan Yaoqing.Modal perturbation method for obtaining complex modal characteristics of non-proportional damping systems.Chinese Journal of Theoretical and Applied Mechanics,2007,39(1):112-118(in Chinese))
29国巍,余志武.一种计算非比例阻尼结构地震响应的新方法.力学学报,2011,43(6):1170-1180(Guo Wei,Yu Zhiwu.Anew method for calculating seismic response of non-proportionally damped structures.Chinese Journal of Theoretical and Applied Mechanics,2011,43(6):1170-1180(in Chinese))
30段绍伟,向湘林,沈蒲生.复杂阻尼结构阻尼模型研究.振动与冲击,2011,30(2):73-76(Duan Shaowei,Xiang Xianglin,Shen Pusheng.Damping model of a complex structure.Journal of Vibration and Shock.2011,30(2):73-76(in Chinese))
31 Mohamed K,Noureddine B,Scott C.Appropriation effects in the estimation of modal damping.Advances in Acoustics and Vibration,2017,5:185-193
32 Zoghaib L,Mattei PO.Time and frequency response of structures with frequency dependent,non-proportional linear damping.Journal of Sound and Vibration,2014,333(3):887-900
33朱镜清,朱敏.复阻尼多自由度系统动力分析的模态叠加法.地震工程与工程振动,2004,24(1):55-58(Zhu Jingqing,Zhu Min.Two mode-superposition methods for dynamic analysis of MDOF systems with complex damping characteristics.Earthquake Engineering and Engineering Vibration,2004,24(1):55-58(in Chinese))
34刘庆林,傅学怡,孙占琦.基于复阻尼假定的不同材料阻尼特性混合结构抗震分析复模态叠加法.建筑结构学报,2011,32(9):27-33(Liu Qinglin,Fu Xueyi,Sun Zhanqi.A complex mode superposition method for seismic analysis of structures of multiple material damping characteristics based on complex damping assumption.Journal of Building Structures,2011,32(9):27-33(in Chinese))
35 Makris N,Zhang J.Time-domain viscoelastic analysis of earth structures.Earthquake Engineering and Structural Dynamics,2015,29(6):745-768
36 Lage Y,Cach?o H,Reis L,et al.A damage parameter for HCF and VHCF based on hysteretic damping.International Journal of Fatigue,2014,62(7):2-9
37大崎顺彦.地震动的谱分析入门.吕敏申,谢礼立.译.北京:地震出版社,2008(Osaki Y.Introduction to Spectrum Analysis of Ground Motion.L¨u Minshen,Xie Lili,trans.Beijing:Seismological Press,2008(in Chinese))
38胡聿贤.地震工程学(第2版).北京:地震出版社,2006(Hu Yuxian.Earthquake Engineering,2nd edn.Beijing:Seismological Press,2006(in Chinese))
39王建平,刘宏昭,原大宁等.含复阻尼系统的对偶原则及数值方法研究.振动工程学报,2004,17(1):62-65(Wang Jianping,Liu Hongzhao,Yuan Daning,et al.Dual principle of oscillation systems with complex damping and its numerical method.Journal of Vibration Engineering,2004,17(1):62-65(in Chinese))
40朱镜清.结构抗震分析原理.北京:地震出版社,2002(Zhu Jingqing.Seismic Analysis of Structures.Beijing:Earthquake Press,2002(in Chinese))
41潘玉华,王元丰.复阻尼结构动力方程的高斯精细时程积分法.工程力学,2012,29(2):16-20(Pan Yuhua,Wang Yuanfeng.Gauss precise time-integration of complex damping vibration systems.Engineering Mechanics,2012,29(2):16-20(in Chinese))