随机激励下一类含分数阶阻尼的轮胎的振动响应
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摘要
基于随机平均法研究了Kanai-Tajimi噪声激励下含分数阶阻尼的轮胎动力学系统的响应.首先将地震波近似为Kanai-Tajimi噪声,结合点接触模型和分数阶导数模型,建立轮胎的动力学方程,然后运用随机平均法求解振动位移的稳态概率密度函数的解析解,最后通过Monte-Carlo数值模拟验证了该方法的有效性.利用振动位移的概率密度求解聚丁二烯橡胶、丁基B252橡胶轮胎振动位移的均值与方差,并以此为依据考察这两类橡胶的减振性能.研究结果表明,轮胎振动位移的均值和方差随橡胶的储能模量的增大而增大,随耗散模量的增大而减小,这说明减小橡胶的储能模量或增大耗散模量可有效改善轮胎的减振性能.所得结果可为轮胎的设计与制造提供一定的理论基础.
The vertical dynamic responses of rubber tires with fractional damping under the Kanai-Tajimi noise excitation were investigated with the stochastic averaging method. Firstly,the earthquake wave was approximated with the Kanai-Tajimi noise,and the differential equation for tire vibration was established through combination of the point contact model with the fractional derivative model. Then,the stochastic averaging method was used to solve the stationary probability density analytically. In turn,validity of the proposed method was verified against the Monte-Carlo numerical simulation results. The probability density was applied to determine the mean values and variances of vibration displacements of the 2 kinds of tires made of polybutadiene and butyl B252 rubbers,respectively. The results show that the mean value and variance of vibration displacement increase with the rubber's storage modulus and decrease with its dissipation modulus. That means,the lower the rubber's storage modulus is or the higher its dissipation modulus is,the better the vibration damping effect of the resulting tire will be. The work provides a theoretical reference for the design and manufacture of rubber tires.
The vertical dynamic responses of rubber tires with fractional damping under the Kanai-Tajimi noise excitation were investigated with the stochastic averaging method. Firstly,the earthquake wave was approximated with the Kanai-Tajimi noise,and the differential equation for tire vibration was established through combination of the point contact model with the fractional derivative model. Then,the stochastic averaging method was used to solve the stationary probability density analytically. In turn,validity of the proposed method was verified against the Monte-Carlo numerical simulation results. The probability density was applied to determine the mean values and variances of vibration displacements of the 2 kinds of tires made of polybutadiene and butyl B252 rubbers,respectively. The results show that the mean value and variance of vibration displacement increase with the rubber's storage modulus and decrease with its dissipation modulus. That means,the lower the rubber's storage modulus is or the higher its dissipation modulus is,the better the vibration damping effect of the resulting tire will be. The work provides a theoretical reference for the design and manufacture of rubber tires.
引文
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[15]Agrawal O P.Stochastic analysis of dynamic systems containing fractional derivatives[J].Journal of Sound and Vibration,2001,247(5):927-938.
[16]Ye K,Li L,Tang J.Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative[J].Earthquake Engineering and Engineering Vibration,2003,2(1):133-139.
[17]Huang Z L,Jin X L.Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative[J].Journal of Sound and Vibration,2009,319(3/5):1121-1135.
[18]Liu D,Xu W,Xu Y.Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation[J].Journal of Sound and Vibration,2012,331(17):4045-4056.
[19]Chen L C,Wang W H,Li Z S,Zhu W Q.Stationary response of Duffing oscillator with hardening stiffness and fracional derivative[J].International Journal of Non-Linear Mechanics,2013,48:44-50.
[20]陈林聪,朱位秋.谐和与宽带噪声联合激励下含分数导数型阻尼的Duffing振子的平稳响应[J].应用力学学报,2010,3(27):517-521.(CHEN Lin-cong,ZHU Wei-qiu.Stationary response of Duffing oscillator with fractional derivative damping under combined harmonic and wide band noise excitations[J].Chinese Journal of Applied Mechanics,2010,3(27):517-521.(in Chinese))
[21]张菊辉.有关Kanai-Tajimi模型的统计特征分析[J].世界地震工程,2007,23(1):156-160.(ZHANG Ju-hui.Statistical properties analysis of Kainai-Tajimi model[J].Word Earthquake Engineering,2007,23(1):156-160.(in Chinese))
[22]宦荣华,潘国峰,金伟良,朱位秋.计及列车车体随机振动影响时受电弓的随机动力响应[J].铁道学报,2010,32(3):39-42.(HUAN Rong-hua,PAN Guo-feng,JIN Wei-liang,ZHU Wei-qiu.Stochastic response of pantograph under action of car body random vibration[J].Journal of the China Railway Society,2010,32(3):39-42.(in Chinese))
[23]朱位秋.非线性随机动力学与控制:Hamilton理论体系框架[M].北京:科学出版社,2003.(ZHU Wei-qiu.Nonlinear Stochastic Dynamics and Control-Hamilton Theoretical System Framework[M].Beijing:Science Press,2003.(in Chinese))
[24]王吉忠,庄继德.轮胎胎面单元表观刚度计算[J].农业工程学报,2000,16(2):28-31.(WANG Ji-zhong,ZHUANG Ji-de.Calculation of the apparent stiffness of tyre tread[J].Transaction of the Chinese Society of Agricultural Engineering,2000,16(2):28-31.(in Chinese))
[25]梁超锋,欧进萍.结构阻尼与材料阻尼的关系[J].地震工程与工程振动,2006,26(1):49-55.(LIANG Chao-feng,OU Jin-pin.Relationship between structural damping and material damping[J].Earthquake Engineering and Engineering Dynamics,2006,26(1):49-55.(in Chinese))
[2]Roghaei M,Zabihollah A.An efficient and reliable structural health monitoring system for buildings after earthquake[J].APCBEE Procedia,2014,9:309-316.
[3]张丙强.竖向环境震动对人车路系统耦合震动的影响[J].动力学与控制学报,2012,10(2):186-191.(ZHANG Bing-qiang.Influence of vertical ambient vibration on the body-vehicle-road coupled vibrating system[J].Journal of Dynamics and Control,2012,10(2):186-191.(in Chinese))
[4]张丙强.地震作用对车辆-道路系统耦合震动的影响[J].河南理工大学学报,2012,31(6):728-733.(ZHANG Bing-qiang.Influence of sesmic on the coupled vibration of vehicle-road system[J].Journal of Henan Polytechnic University,2012,31(6):728-733.(in Chinese))
[5]周光泉,刘孝敏.粘弹性理论[M].合肥:中国科学技术大学出版社,1996.(ZHOU Guang-quan,LIU Xiao-min.Viscoelastic Theory[M].Hefei:University of Science and Technology of China Press,1996.(in Chinese))
[6]Bagley R L,Torvik P J.A theoretical basis for the application of fractional calculus to viscoelasticity[J].Journal of Rheology,1983,27(3):201-210.
[7]Bagley R L,Torvik P J.Fractional calculus a different approach to the analysis of viscoelastically damped structures[J].AIAA Journal,1983,21(5):741-748.
[8]Song D Y,Jiang T Q.Study on the constitutive equation with fractional derivative for the viscoelastic fluids-modified Jeffreys model and its application[J].Rheologica Acta,1998,37(5):512-517.
[9]李根国,朱正佑,程昌钧.具有分数导数型本构关系的粘弹性柱的动力稳定性[J].应用数学和力学,2001,22(3):250-258.(LI Gen-guo,ZHU Zheng-you,CHENG Chang-jun.Dynamical stability of viscoelastic column with fractional derivative constitutive relation[J].Applied Mathematics and Mechanics,2001,22(3):250-258.(in Chinese))
[10]吴杰,上官文斌.采用粘弹性分数导数模型的橡胶隔振器动态特性的建模及应用[J].工程力学,2008,25(1):161-166.(WU Jie,SHANGGUAN Wen-bin.Modeling and applications of dynamic characteristics for rubber isolators using viscoelastic fractional[J].Engineering Mechanics,2008,25(1):161-166.(in Chinese))
[11]周超,吴庆呜,张强.粘弹性阻尼隔振体的非线性振动分析[J].工程设计学报,2009,16(3):205-208.(ZHOU Chao,WU Qing-wu,ZHANG Qiang.Nonlinear vibration analysis of viscoelastic isolator[J].Chinese Journal of Engineering Design,2009,16(3):205-208.(in Chinese))
[12]Adhikari S.Damping models for structural vibration[D].Ph D Thesis.Cambridge:University of Cambridge,2000:75-78.
[13]Agrawal O P.Analytical solution for stochastic response of a fractionally damped beam[J].Journal of Vibration and Acoustics,2004,126(4):561-566.
[14]Agrawal O P.Stochastic analysis of a 1-D system with fractional damping of order 1/2[J].Journal of Vibration and Acoustics,2002,124(3):454-460.
[15]Agrawal O P.Stochastic analysis of dynamic systems containing fractional derivatives[J].Journal of Sound and Vibration,2001,247(5):927-938.
[16]Ye K,Li L,Tang J.Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative[J].Earthquake Engineering and Engineering Vibration,2003,2(1):133-139.
[17]Huang Z L,Jin X L.Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative[J].Journal of Sound and Vibration,2009,319(3/5):1121-1135.
[18]Liu D,Xu W,Xu Y.Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation[J].Journal of Sound and Vibration,2012,331(17):4045-4056.
[19]Chen L C,Wang W H,Li Z S,Zhu W Q.Stationary response of Duffing oscillator with hardening stiffness and fracional derivative[J].International Journal of Non-Linear Mechanics,2013,48:44-50.
[20]陈林聪,朱位秋.谐和与宽带噪声联合激励下含分数导数型阻尼的Duffing振子的平稳响应[J].应用力学学报,2010,3(27):517-521.(CHEN Lin-cong,ZHU Wei-qiu.Stationary response of Duffing oscillator with fractional derivative damping under combined harmonic and wide band noise excitations[J].Chinese Journal of Applied Mechanics,2010,3(27):517-521.(in Chinese))
[21]张菊辉.有关Kanai-Tajimi模型的统计特征分析[J].世界地震工程,2007,23(1):156-160.(ZHANG Ju-hui.Statistical properties analysis of Kainai-Tajimi model[J].Word Earthquake Engineering,2007,23(1):156-160.(in Chinese))
[22]宦荣华,潘国峰,金伟良,朱位秋.计及列车车体随机振动影响时受电弓的随机动力响应[J].铁道学报,2010,32(3):39-42.(HUAN Rong-hua,PAN Guo-feng,JIN Wei-liang,ZHU Wei-qiu.Stochastic response of pantograph under action of car body random vibration[J].Journal of the China Railway Society,2010,32(3):39-42.(in Chinese))
[23]朱位秋.非线性随机动力学与控制:Hamilton理论体系框架[M].北京:科学出版社,2003.(ZHU Wei-qiu.Nonlinear Stochastic Dynamics and Control-Hamilton Theoretical System Framework[M].Beijing:Science Press,2003.(in Chinese))
[24]王吉忠,庄继德.轮胎胎面单元表观刚度计算[J].农业工程学报,2000,16(2):28-31.(WANG Ji-zhong,ZHUANG Ji-de.Calculation of the apparent stiffness of tyre tread[J].Transaction of the Chinese Society of Agricultural Engineering,2000,16(2):28-31.(in Chinese))
[25]梁超锋,欧进萍.结构阻尼与材料阻尼的关系[J].地震工程与工程振动,2006,26(1):49-55.(LIANG Chao-feng,OU Jin-pin.Relationship between structural damping and material damping[J].Earthquake Engineering and Engineering Dynamics,2006,26(1):49-55.(in Chinese))
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