基于势能判据的约束模态综合法截断准则
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摘要
为提高大型复杂结构体系的计算效率,在深入分析约束模态综合法原理的基础上,定义了由不同主模态截断阶数产生的位移向量组成的线性空间,建立该空间上的一个函数,通过数学定理详细证明了该函数是线性空间上的范数,并以该范数的最大值定义势能判据,推导出势能判据与子结构主模态截断数量之间的关系,据此提出了一种基于势能判据的子结构主模态截断准则。此外对于约束模态综合法,还推导了以机械能为判据的情况,证明了约束模态法中子结构的机械能随截断主模态数量的变化曲线并不是单调的,故不适用于约束模态法。最后将势能判据截断准则应用到地基土-高层建筑相互作用体系的地震响应分析问题中,进一步对所提出的模态截断准则的计算精度进行对比研究。研究结果表明,所提出的势能判据截断准则是十分有效的,而应用于振型叠加法的机械能判据不宜应用于约束模态综合法中。
To improve computational efficiency,the constrained mode synthesis method was discussed and a linear space consisting of displacement vectors calculated with different mode cut-off numbers was defined.A function was also defined and verified to be a norm of this linear space mathematically.Then,adopting the maximum value of this norm to define a potential energy criterion.The relationship between the potential energy criterion and the mode cut-off number was derived.Consequently,a mode cut-off criterion based on the potential energy criterion was proposed.In addition,the criterion adopting the mechanical energy for the constrained mode synthesis method was considered.It was proved that the mechanical energy is not a monotonic function versus the mode cut-off number,so it is not suitable for the constrained mode synthesis method.Finally,the proposed potential energy criterion was applied to seismic analysis of a soil and multi-story building interaction system,the results were compared with those not using the mode synthesis method.It was shown that the proposed mode cut-off criterion based on the potential energy criterion is effective while based on the mechanical energy criterion generally used in the mode superposition method is not suitable for the constrained mode synthesis method.
To improve computational efficiency,the constrained mode synthesis method was discussed and a linear space consisting of displacement vectors calculated with different mode cut-off numbers was defined.A function was also defined and verified to be a norm of this linear space mathematically.Then,adopting the maximum value of this norm to define a potential energy criterion.The relationship between the potential energy criterion and the mode cut-off number was derived.Consequently,a mode cut-off criterion based on the potential energy criterion was proposed.In addition,the criterion adopting the mechanical energy for the constrained mode synthesis method was considered.It was proved that the mechanical energy is not a monotonic function versus the mode cut-off number,so it is not suitable for the constrained mode synthesis method.Finally,the proposed potential energy criterion was applied to seismic analysis of a soil and multi-story building interaction system,the results were compared with those not using the mode synthesis method.It was shown that the proposed mode cut-off criterion based on the potential energy criterion is effective while based on the mechanical energy criterion generally used in the mode superposition method is not suitable for the constrained mode synthesis method.
引文
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[16]田育民.复杂结构模态分析中的截断准则[J].强度与环境,1994,2:18-21.
[17]Craig R R.Substructure Method in Vibration[J].Transaction of the ASME,1995,117:207-213.
[18]熊洪允,曾绍标,毛云英.应用数学基础[M].天津:天津大学出版社,2004.
[19]路观平,王晓泉,王雨苗.模态综合的交叉子结构方法[J].振动工程学报,2002,15(3):300-304.
[20]Craig R R,Chang C J.A Review of Substructure CouplingMethods for Dynamic Analysis[J].Advances in EngineeringScience,1976,2:393-408.
[21]Pan L J,Zhang B M.ANewMethod for the Determination ofDamping in Cocured Composite Laminates with EmbeddedViscoelastic Layer[J].Journal of Sound and Vibration,2009,319:822-831.
[22]宋修广,钟原,刘西利.基于能量范数的模态密集型系统的模态截断[J].山东大学学报,2002,32(4):321-324.
[23]谢能刚,郭兴文,王德信.结构振动模态截断的能量判据[J].振动工程学报,2003,16(3):302-305.
[24]谢能刚,赵雷,方浩.基于能量法的强夯频域分析[J].岩石力学与工程学报,2006,25(1):3224-3228.
[25]谢能刚,王德信.高拱坝地震响应分析中的能量法[J].土木工程学报,2003,36(6):85-87.
[26]谢能刚,王德信,孙林松.能量函数在高拱坝动力优化设计中的应用[J].应用力学学报,2002,19(2):107-109.
[2]楼梦麟.结构动力分析的子结构方法[M].上海:同济大学出版社,1997.107-120.
[3]Hurty W C.Vibrations of Structural Systems by ComponentMode Synthesis[J].Journal of the Engineering MechanicsDivion,ASCE,1960,86:51-59.
[4]Gladwell G M L.Branch Mode Anslysis of Vibrating Systems[J].Sound Vibration,1964,1:41-59.
[5]Hurty W C.Dynamic Analysis of Structural Systems UsingComponent Mode[J].AAIA J,1964,3(4):678-685.
[6]Craig R R,Bampton M C C.Coupling of Structures forDynamic Analyses[J].AAIA J,1968,6(7):1313-1319.
[7]Hou S N.Review of Modal Synthesis Techniques and A NewApproach[J].Shock and Vibration Bulletin,1969,40:25-30.
[8]白建方.复杂场地土层地震反应分析的并行有限元方法[D].上海:同济大学,2007.
[9]Yin B S,Wang W L,Jin Y Q.The Application ofComponent Mode Synthesis for the Dynamic Analysis ofComplex Structures Using ADINA[J].Computers&Structures,1997,64(5/6):931-938.
[10]Kyung R G,Shin HC.Dynamic Analysis of Structures UsingConstrained Component Mode Synthesis[J].AIAA Modelingand Simulation Technologies Conference and Exhibit.2002Aug 5-8,California,Monterey:AIAA-2002-4797.
[11]Wamsler M.On the Selection of the Mode Cut-off Number inComponent Mode Reduction[J].Engineering withComputers.2009,25:139-146.
[12]丁平,顾彦.子结构模态综合在汽车振动噪声分析的应用[J].上海汽车,2007,11:22-24.
[13]张美艳,韩平畴.结构动力重分析的子结构有理逼近[J].计算力学学报,2008,25(6):878-881.
[14]Wang X Y,Mills J K.Dynamic Modeling of a Flexible-linkPlanar Parallel Platform Using a Substructuring Approach[J].Mechanism and Machine Theory,2006,4:671-687.
[15]管根红,陈常顺,高树滋.基于系统响应的模态截断方法的研究[J].南京理工大学学报,2000,24(6):532-535.
[16]田育民.复杂结构模态分析中的截断准则[J].强度与环境,1994,2:18-21.
[17]Craig R R.Substructure Method in Vibration[J].Transaction of the ASME,1995,117:207-213.
[18]熊洪允,曾绍标,毛云英.应用数学基础[M].天津:天津大学出版社,2004.
[19]路观平,王晓泉,王雨苗.模态综合的交叉子结构方法[J].振动工程学报,2002,15(3):300-304.
[20]Craig R R,Chang C J.A Review of Substructure CouplingMethods for Dynamic Analysis[J].Advances in EngineeringScience,1976,2:393-408.
[21]Pan L J,Zhang B M.ANewMethod for the Determination ofDamping in Cocured Composite Laminates with EmbeddedViscoelastic Layer[J].Journal of Sound and Vibration,2009,319:822-831.
[22]宋修广,钟原,刘西利.基于能量范数的模态密集型系统的模态截断[J].山东大学学报,2002,32(4):321-324.
[23]谢能刚,郭兴文,王德信.结构振动模态截断的能量判据[J].振动工程学报,2003,16(3):302-305.
[24]谢能刚,赵雷,方浩.基于能量法的强夯频域分析[J].岩石力学与工程学报,2006,25(1):3224-3228.
[25]谢能刚,王德信.高拱坝地震响应分析中的能量法[J].土木工程学报,2003,36(6):85-87.
[26]谢能刚,王德信,孙林松.能量函数在高拱坝动力优化设计中的应用[J].应用力学学报,2002,19(2):107-109.
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