橡胶隔振器动力学建模及动态特性研究
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摘要
隔振器是飞机APU(辅助动力装置)不可缺少的部件之一，为保证APU工作安全，APU系统都采用安装橡胶隔振器的方式以隔离APU与飞机振动的相互传递。为满足先进航空器研制的需要，进行橡胶隔振器动力学特性的研究十分必要。橡胶材料具有较强的非线性动力学特性，本文对APU橡胶隔振器动力学特性开展研究，以其为橡胶隔振器的理论研究和工程应用提供依据和参考。主要研究内容和创新成果如下：
(1)提出了一种频率相关性力和振幅相关性力并联的橡胶隔振器非线性动力学五参数分数导数模型：频率相关性力用五参数分数导数粘弹性力描述；振幅相关性力用摩擦力描述。利用数值求解的方法给出定量结果。仿真计算和试验验证表明：该模型能很好地拟合试验结果，可以在宽广频率范围内更好地描述APU橡胶隔振器的动态特性。研究了动力学模型参数对动刚度影响，为橡胶隔振器的设计与应用提供了理论基础。本文提出的五参数分数导数模型在理论上具有学术创新价值。
(2)提出采用五参数分数导数建立隔振器橡胶材料非线性动力学本构模型。通过试验结果拟合获得非线性动力学本构模型参数。利用本构模型建立了橡胶隔振器非线性动力学有限元方程。依据分数导数因子特性，通过合理的简化，迭代求解了非线性动力学有限元方程。仿真分析了本构模型参数对隔振器动态响应特性的影响。通过试验验证表明：该有限元方程能很好地预测隔振器的动力学特性。通过本构模型参数对隔振器动态响应特性影响分析，能更好地把握粘弹性材料的非线性特性，为分析橡胶隔振器的多向隔振应用和隔振器设计提供了有效的理论分析基础。本文提出非线性动力学本构模型及非线性动力学有限元方程在理论上有学术创新价值。
(3)建立了橡胶隔振器-质量系统的非线性动力学有限元方程。分析了本构模型中各个参数及隔振器结构参数变化对隔振器轴向传递率特性和径向传递率特性的影响，提出了隔振器结构设计方法。试验验证结果表明：该非线性动力学有限元方程和数值计算方法能更好地预测含橡胶隔振器振动系统的动力学特性。通过模型参数对隔振器-质量系统的传递率特性影响的分析，能更好地把握隔振器非线性特性，预测含橡胶隔振器的复杂结构系统动力学响应特性。为APU系统的隔振器结构设计和安装布局提供了理论基础，具有重要工程价值。
(4)分析了线性分数导数模型在分析冲击振动响应特性上的局限性，提出了橡胶隔振器在冲击激励下的非线性分数导数动力学模型。在冲击振动试验的基础上，研究了二种不同的非线性分数导数因子项模型与试验数据的拟合效果：第一种模型假设非线性出现在分数导数因子项的系数上；第二种模型假设非线性出现在分数导数因子项被求导函数上。研究了冲击激励下橡胶隔振器-质量系统加速度响应特性，利用数值仿真分析了各参数对冲击激励下橡胶隔振器响应衰减速率特性和响应频率的影响。研究结果表明：本文所提非线性分数导数模型能更好地拟合试验数据，与传统的动力学模型相比，该模型能用较少的参数准确描述APU隔振器在冲击激励下的动态特性。本文所提非线性分数导数模型在理论上具有学术创新价值。
综上所述，本文较系统的研究了橡胶隔振器非线性动力学模型、非线性本构模型、非线性有限元方程、橡胶隔振器-质量系统动力特性、非线性分数导数动力学模型等问题。研究结果对指导橡胶隔振系统设计与应用具有重要的理论和工程价值。
Isolators were one indispensable part of APU (auxiliary power unit) for an aircraft. Rubberisolators were installed in APU systems to isolate the vibration transmission between APU and theaircraft for APU operating safety. It is necessary to study the rubber isolator’s dynamic characteristicsfor meeting to need advanced aircraft development. In this paper, dynamic characteristics of the APUrubber isolator are studied for rubber material with strong nonlinear dynamic characteristics, whichprovides basis and reference for rubber isolator’s theoretical research and engineering applications.The main contents and innovations were researched as following:
(1) A nonlinear dynamic five-parameter fractional derivative model of rubber isolator waspresented about the frequency and amplitude parallel connection: the former were described withfive-parameter fractional derivative and the latter were described with friction force respectively.Quantitative results were given by numerical simulation. Simulation and experimental results showedthat the model fit well with test results and could describ much better dynamic characteristics of theAPU rubber isolator in a wide frequency range. The influence of model parameters for dynamicstiffness was also studied in the paper so as to provid theoretical basis for design and applicayion ofrubber isolator. The proposed five-parameter fractional derivative model possessed academicinnovation value.
(2) A nonlinear dynamics constitutive model of isolator’s rubber material was established makinguse of five-parameter fractional derivative. Constitutive model parameters were obtained by fitting theexperimental results. Nonlinear dynamics finite element equations about the rubber isolator wereestablished with this model. These equations were solved by iteration according to reasonablesimplification and features of the fractional derivative factors. The influence of model parametersabout dynamic response characteristics of the isolator was analyzed with simulating. The results byexperimental verification showed that the finite element equations could predict much better thedynamic characteristics of the isolators. Characteristics of the nonlinear viscoelastic material weregrasped making use of influence analysis of model parameters for dynamic response of the isolator,and theory analytical basis was provided effectively for rubber design and application of isolator. Theproposed nonlinear dynamics constitutive model and finite element equations possessed academicinnovation value.
(3) Nonlinear dynamics finite element equations of the rubber isolator-mass system were established. Influence of the transmiting characteristics of axial and radial directions were analyzedabout constitutive model parameters and structural parameters, and the design method of the isolatorwere given. Experimental results showed that dynamic characteristics of the rubber isolator could bepredicted well making use of finite element equations and the numerical method. Nonlinearcharacteristics can be grasped, and dynamic response of complex structure system containing therubber isolator can be predicted making use of influence analysis of model parameters abouttransmiting characteristics of the isolator-mass system. The theoretical basis was provided for designmethod and installation layout of the APU system isolator, which possessed an important engineeringvalue.
(4) The limitations were analyzed about linear fractional derivative model on analyzing impactvibration. The nonlinear fractional derivative dynamics model was presented about the rubber isolatorunder impact exciting. Two different nonlinear fractional derivative models were studied and were fitwith results of experimental data basing on the impact exciting: the first model was assumed asnonlinear in the coefficient about fractional derivatives factor; and the second model in the derivativefactor. Acceleration response characteristics under impact exciting were studied about the rubberisolator-mass system, and response characteristics are analyzed about various parameters effecting ondecay rate and response frequency of the rubber isolator with numerical simulations. The resultsshowed that the nonlinear fractional derivative model proposed in the paper could fit the experimentaldata better, and the dynamic characteristics under impact exciting could be described accurately with afewer parameters comparing with traditional dynamic model. The nonlinear fractional derivativemodel possessed academic innovation value in theory.
In conclusion, nonlinear dynamics model, nonlinear constitutive model and nonlinear finiteelement equations of the rubber isolator, dynamic characteristics of the isolator-mass system andnonlinear fractional derivative model were studied systematically in the paper. The results possessedimportant significance for to conduct the rubbe isolator design and application in the theoretical andengineering.
(1)提出了一种频率相关性力和振幅相关性力并联的橡胶隔振器非线性动力学五参数分数导数模型：频率相关性力用五参数分数导数粘弹性力描述；振幅相关性力用摩擦力描述。利用数值求解的方法给出定量结果。仿真计算和试验验证表明：该模型能很好地拟合试验结果，可以在宽广频率范围内更好地描述APU橡胶隔振器的动态特性。研究了动力学模型参数对动刚度影响，为橡胶隔振器的设计与应用提供了理论基础。本文提出的五参数分数导数模型在理论上具有学术创新价值。
(2)提出采用五参数分数导数建立隔振器橡胶材料非线性动力学本构模型。通过试验结果拟合获得非线性动力学本构模型参数。利用本构模型建立了橡胶隔振器非线性动力学有限元方程。依据分数导数因子特性，通过合理的简化，迭代求解了非线性动力学有限元方程。仿真分析了本构模型参数对隔振器动态响应特性的影响。通过试验验证表明：该有限元方程能很好地预测隔振器的动力学特性。通过本构模型参数对隔振器动态响应特性影响分析，能更好地把握粘弹性材料的非线性特性，为分析橡胶隔振器的多向隔振应用和隔振器设计提供了有效的理论分析基础。本文提出非线性动力学本构模型及非线性动力学有限元方程在理论上有学术创新价值。
(3)建立了橡胶隔振器-质量系统的非线性动力学有限元方程。分析了本构模型中各个参数及隔振器结构参数变化对隔振器轴向传递率特性和径向传递率特性的影响，提出了隔振器结构设计方法。试验验证结果表明：该非线性动力学有限元方程和数值计算方法能更好地预测含橡胶隔振器振动系统的动力学特性。通过模型参数对隔振器-质量系统的传递率特性影响的分析，能更好地把握隔振器非线性特性，预测含橡胶隔振器的复杂结构系统动力学响应特性。为APU系统的隔振器结构设计和安装布局提供了理论基础，具有重要工程价值。
(4)分析了线性分数导数模型在分析冲击振动响应特性上的局限性，提出了橡胶隔振器在冲击激励下的非线性分数导数动力学模型。在冲击振动试验的基础上，研究了二种不同的非线性分数导数因子项模型与试验数据的拟合效果：第一种模型假设非线性出现在分数导数因子项的系数上；第二种模型假设非线性出现在分数导数因子项被求导函数上。研究了冲击激励下橡胶隔振器-质量系统加速度响应特性，利用数值仿真分析了各参数对冲击激励下橡胶隔振器响应衰减速率特性和响应频率的影响。研究结果表明：本文所提非线性分数导数模型能更好地拟合试验数据，与传统的动力学模型相比，该模型能用较少的参数准确描述APU隔振器在冲击激励下的动态特性。本文所提非线性分数导数模型在理论上具有学术创新价值。
综上所述，本文较系统的研究了橡胶隔振器非线性动力学模型、非线性本构模型、非线性有限元方程、橡胶隔振器-质量系统动力特性、非线性分数导数动力学模型等问题。研究结果对指导橡胶隔振系统设计与应用具有重要的理论和工程价值。
Isolators were one indispensable part of APU (auxiliary power unit) for an aircraft. Rubberisolators were installed in APU systems to isolate the vibration transmission between APU and theaircraft for APU operating safety. It is necessary to study the rubber isolator’s dynamic characteristicsfor meeting to need advanced aircraft development. In this paper, dynamic characteristics of the APUrubber isolator are studied for rubber material with strong nonlinear dynamic characteristics, whichprovides basis and reference for rubber isolator’s theoretical research and engineering applications.The main contents and innovations were researched as following:
(1) A nonlinear dynamic five-parameter fractional derivative model of rubber isolator waspresented about the frequency and amplitude parallel connection: the former were described withfive-parameter fractional derivative and the latter were described with friction force respectively.Quantitative results were given by numerical simulation. Simulation and experimental results showedthat the model fit well with test results and could describ much better dynamic characteristics of theAPU rubber isolator in a wide frequency range. The influence of model parameters for dynamicstiffness was also studied in the paper so as to provid theoretical basis for design and applicayion ofrubber isolator. The proposed five-parameter fractional derivative model possessed academicinnovation value.
(2) A nonlinear dynamics constitutive model of isolator’s rubber material was established makinguse of five-parameter fractional derivative. Constitutive model parameters were obtained by fitting theexperimental results. Nonlinear dynamics finite element equations about the rubber isolator wereestablished with this model. These equations were solved by iteration according to reasonablesimplification and features of the fractional derivative factors. The influence of model parametersabout dynamic response characteristics of the isolator was analyzed with simulating. The results byexperimental verification showed that the finite element equations could predict much better thedynamic characteristics of the isolators. Characteristics of the nonlinear viscoelastic material weregrasped making use of influence analysis of model parameters for dynamic response of the isolator,and theory analytical basis was provided effectively for rubber design and application of isolator. Theproposed nonlinear dynamics constitutive model and finite element equations possessed academicinnovation value.
(3) Nonlinear dynamics finite element equations of the rubber isolator-mass system were established. Influence of the transmiting characteristics of axial and radial directions were analyzedabout constitutive model parameters and structural parameters, and the design method of the isolatorwere given. Experimental results showed that dynamic characteristics of the rubber isolator could bepredicted well making use of finite element equations and the numerical method. Nonlinearcharacteristics can be grasped, and dynamic response of complex structure system containing therubber isolator can be predicted making use of influence analysis of model parameters abouttransmiting characteristics of the isolator-mass system. The theoretical basis was provided for designmethod and installation layout of the APU system isolator, which possessed an important engineeringvalue.
(4) The limitations were analyzed about linear fractional derivative model on analyzing impactvibration. The nonlinear fractional derivative dynamics model was presented about the rubber isolatorunder impact exciting. Two different nonlinear fractional derivative models were studied and were fitwith results of experimental data basing on the impact exciting: the first model was assumed asnonlinear in the coefficient about fractional derivatives factor; and the second model in the derivativefactor. Acceleration response characteristics under impact exciting were studied about the rubberisolator-mass system, and response characteristics are analyzed about various parameters effecting ondecay rate and response frequency of the rubber isolator with numerical simulations. The resultsshowed that the nonlinear fractional derivative model proposed in the paper could fit the experimentaldata better, and the dynamic characteristics under impact exciting could be described accurately with afewer parameters comparing with traditional dynamic model. The nonlinear fractional derivativemodel possessed academic innovation value in theory.
In conclusion, nonlinear dynamics model, nonlinear constitutive model and nonlinear finiteelement equations of the rubber isolator, dynamic characteristics of the isolator-mass system andnonlinear fractional derivative model were studied systematically in the paper. The results possessedimportant significance for to conduct the rubbe isolator design and application in the theoretical andengineering.
引文
[1]. Singh MP, Moreschi LM. Optimal placement of dampers for passive response control [J].Earthquake Eng Struct Dyn2002,(3):955～976.
[2]. Shukla AK, Datta TK. Optimal use of viscoelastic dampers in building frames for seismicforce [J]. Struct Eng1999,(1):401～409.
[3]. Singh MP, Chang TS. Seismic analysis of structures with viscoelastic dampers [J]. EngMech2009,(5):571～580.
[4]. Miehe C, Keck J. Superimposed finite elastic-viscoelatic-plastoelastic stress response withdamage in filled rubbery polymers, Experiments, modeling and algorithmic implementation[J]. Journal of the Mechanics and Physics of Solids,2000,(8):323~365
[5]. Chang TS, Singh MP. Mechanical model parameters for viscoelastic dampers [J]. Eng Mech2009,(1):581～584.
[6].杨国辉. APU橡胶隔振器特性研究.[D].南京:南京航空航天大学硕士学位论文，2011.
[7].方建敏.橡胶材料的分数导数型本构模型研究及动力学应用.[D].南京:南京航空航天大学硕士学位论文，2012.
[8]. M. Sjo¨berg, Rubber isolators—measurements and modelling using fractional derivativeand friction [D]. SAE,2000.
[9]. N. Gil-Negrete, On the Modeling and Dynamic Stiffness Prediction of Rubber Isolators,PhD Thesis [D]. University of Navarra, Spain,2004.
[10].J. Lubliner, A model of rubber viscoelasticity [J]. Mechanics Research Communications,1985,(12)93～99.
[11].R.Bagley, P.Torvik, Fractional calculus—a different approach to the analysis ofviscoelastically damped structures.[J]. AIAA Journal,1983,(2)741～748.
[12].王锐,李世其.隔振橡胶本构建模研究.[J].振动与冲击,2007,(1):77～79.
[13].王锐.橡胶隔振器动力学性能及设计方法研究.[D].武汉:华中科技大学博士学位论文，2007.
[14].孙德伟，张广玉.橡胶隔振器迟滞阻尼特性识别的新方法.[J].振动与冲击,2010,(4):164～168.
[15].赵广，刘健，刘占生.橡胶隔振器非线性动力学模型理论与实验研究.[J].振动与冲击，2010,(1):173～177.
[16].韩德宝，宋希庚.橡胶隔振器刚度和阻尼本构关系的试验研究.[J].振动与冲击，2009,(1):156～160.
[17].吴瑞安，冯加权，王玉军，仲政.橡胶隔振系统非线性振动参数识别与特性研究.[J].力学季刊，2007,(2):281～285.
[18].赵荣国，徐友拒，陈忠富，胡绍全，黄西成，刘青林.一个新的非线性迟滞隔振系统动力学模型.[J].机械工程学报，2004,(2):185～188.
[19].周相荣，王强，孙洪军.一种估算橡胶隔振器机械阻抗特性的简化方法.[J].舰船科学技术，2006,(2):93～97.
[20].陈安华，刘德顺，朱萍玉.被动隔振体的非线性振动分析.[J].机械工程学报，2001,(6):99～105.
[21].T. Yajima. H. Nagahama. Differential geometry of viscoelastic models with fractional-orderderivatives.[J]. Phys. A: Math. Theor.2010,(4):385207.
[22].Ross, B., A brief history and exposition of the fundamental theory of fractional calculus.[J].Fractional Calculus and Its Applications, Lecture Notes in Mathematics, Springer-Verlag,Berlin,1975,(1)1～36.
[23].Oldham K B, Spanier J. The Fractional Calculus.[D]. New York-London: Academic Press,1974.
[24].Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives.[D]. Theory andApplications. Switzerland:Gordon and Breach Science Publishers,1993.
[25].Miller K S, Ross B. An introduction to the fractional calculus and fractional differentialequations.[D]. New York:John Wiley&Sons Inc,1993.
[26].M.CAPUTO, F.MAINARDI. Linear models of dissipation in anelastic solids.[J]. RivistaDel Nuovo Cimento.1971,(1):161-198.
[27].R.L.BAGLEY. Application of generalized derivates to viscoelasticity.[J]. Air Force Instituteof Technology,1979, TR-79-4103.
[28].Schiessel H, Friedrich C, Blumen A. Applications to problems in polymer physics andrheology.[J]. Applications of Fractional Calculus in Physics. Singapore: World ScientificPublishing Co Pte Ltd,2000,(1):331～376.
[29].Gl ckle W G, Nonenmacher T F. Fractional integral operators and fox function in the theoryof viscoelasticity.[J]. Macromolecules,1991,(24):6426～6434.
[30].Nonnenmacher T F, Metzler R. On the Riemann-Liouville fractional calculus and somerecent applications.[J]. Fractals,1995,(3):557～566.
[31].Friedrich C. Relaxation and retardation functions of the Maxwell model with fractionalderivatives.[J]. Rheol Acta,1991,(3):151～158.
[32].Schiessel H, Metzler R, Nonnenmacher T F. Generalized viscoelastic models: Theirfractional equations with solutions.[J]. Phys A: Math Gen,1995,(28):6567～6584.
[33].Gl ckle W G, Nonnenmacher T F. Fox function representation of Non-Debye relaxationprocesses.[J]. Stat Phys,1993,(1):741～757.
[34].Xu M Y, Tan W C. Representation of the constitutive equation of viscoelastic materials bythe generalized fractional element networks and its generalized solutions.[J]. Science inChina, Series G,2003,(2):145～157.
[35].Lam L. Introduction to Nonlinear Physics.[M]. New York: Springer-Verlag Inc.,1997.
[36].Papoulia KD, Panoskaltsis VP, Korovajchuk I, Kurup NV. Rheological representation offractional derivative models in linear viscoelasticity.[J]. Rheol Acta; in press，381～400.
[37].Makris N, Constantinou MC. Fractional-derivative Maxwell model for viscous dampers.[J].Struct Eng1991,(17):2708～2724.
[38].Pritz T. Analysis of four-parameter fractional derivative model of real solid materials.[J].Sound Vib1996,(1):103～115.
[39].Atanackovic TM. A modified Zener model of a viscoelastic body.[J]. Continuum MechThermodyn2002,(14):137～148.
[40].Song DY, Jiang TQ. Study on the constitutive equation with fractional derivative for theviscoelastic fluids modified Jeffreys model and its application.[J]. Rheol Acta1998,(5):512～517.
[41].Park SW. Analytical modeling of viscoelastic dampers for structural and vibration control.[J]. Int J Solids Struct2001,(1):8065～92.
[42].Schmidt A, Gaul L. Finite element formulation of viscoelastic constitutive equations usingfractional time derivatives.[J]. Nonlinear Dyn2002,(1):37～55.
[43].Pritz T. Five-parameter fractional derivative model for polymeric damping materials.[J].Sound Vib2003,(1):935～52.
[44].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,(1):161～166.
[45].潘孝勇，柴国钟.含有橡胶隔振器振动系统时域响应的测试与计算分析.[J].振动与冲击,2007,(12):32～35.
[46].潘孝勇，上官文斌.基于超弹性、分数导数和摩擦模型的碳黑填充橡胶隔振器动态建模.[J].振动与冲击,2007,(10):6～15.
[47].潘孝勇，上官文斌，柴国钟，黄志.橡胶隔振器动态特性计算方法的研究.[J].振动工程学报，2009,(4):345～350.
[48].潘孝勇.橡胶隔振器动态特性计算与建模方法的研究.[D].杭州:浙江工业大学博士学位论文，2009.
[49].刘林超.分数导数型粘弹性材料的力学行为及在结构减振中的应用研究.[D].暨南大学硕士学位论文，2005年5月.
[50].孙海忠.高分子材料的分数导数型本构关系及应用.[D].暨南大学，2006年5月.
[51].夏亚东.高分子材料的线性与非线性分数算子本构关系研究.[D].暨南大学，2010年11月.
[52].Yu. A, Rossikhin, M.V. Shitikova. Analysis of damped vibrations of linear viscoelasticplates with damping modeled with fractional derivatives.[J]. Signal Processing,2006,(1):2703～2711.
[53].Djamel Ouis. Combination of a standard viscoelastic model and fractional derivate calculusto the characterization of polymers.[J]. Mat Res Innovat,2003,(7):42～46.
[54].Nicole Heymans. Constitutive equations for polymer viscoelasticity derived fromhierarchical models in cases of failure of time-temperature superposition.[J]. SignalProcessing83,2003,(1):2345～2357.
[55].Mattias Sjoberg. Rubber Isolators-Measurements and Modelling using FractionalDerivatives and Friction.[J]. Society of Automotive Engineers,2000,(1):01～3518.
[56].Mattias Sjoberg, Leif Kari. Non-Linear Behavior of a Rubber Isolator System UsingFractional Derivatives.[J]. Vehicle System Dynamics.2002,(1):217～236.
[57].Mattias Sjoberg. Nonlinear Isolator Dynamics at Finite Deformations: An EffectiveHyperelastic, Fractional Derivative, Generalized Friction Model.[J]. Nonlinear Dynamics.2003,(1):323～336.
[58].Ya Pu, Hartono Sumali. Modeling of Nonlinear Elastoneric Mounts. Part1: DynamicTesting and Parameter Identification.[J]. Society of Automotive Engineers.2001,01-0042.
[59].S. W. Park. Analytical modeling of viscoelastic dampers for structural and vibration control.[J]. International Journal of Solids and Structures2001,(1):8065～8092.
[60].Nicole Heymans. Fractional Calculus Description of Non-Linear Viscoelastic Behaviour ofPolymers.[J]. Nonlinear Dynamics.2004,(1):221～231.
[61].SIEGMAR KEMPFLE, INGO SCH FER. Fractional Calculus via Functional Calculus:Theory and Applications.[J]. Nonlinear Dynamics2002,(1):99～127.
[62].M. Kohandel, S. Sivaloganathan. Frequency dependence of complex moduli of brain tissueusing a fractional Zener model.[J]. Phys. Med. Biol.2005,(1):2799～2805.
[63].P. Haupt, A. Lion. On the dynamic behaviour of polymers under finite strains: constitutivemodelling and identification of parameters.[J]. International Journal of Solids andStructures.2000,(1):3633～3646.
[64].M. Fukunaga. N. Shimizu. A nonlinear fractional derivative model of impulse motion forviscoelastic materials.[J]. Phys. Scr.2009, T136,014010.
[65].M Fukunaga, N. Shimizu. Nonlinear Fractional Derivative Models of Viscoelastic ImpactDynamics Based on Entropy Elasticity and Generalized Maxwell Law.[J]. Journal ofComputational and Nonlinear Dynamics.2011, Vol.6/021005-1.
[66].唐振寰，罗贵火.橡胶隔振器粘弹性5参数分数导数并联动力学模型.[J].航空动力学报.2013,(1):99～105.
[67].Desalvo G J, Swanson J A..ANSYS engineering analysis system-theoretical manual(Version5.3).[M]. Houston: Swanson Analysis System Inc,1994,4～9.
[68].贝塞J K.非线性有限元分析——ADINA理论文本，张圣坤译.[M].上海：上海交通大学出版社，1986，5～10.
[69].陈莲，周海亭.计算橡胶隔振器静态特性的数值分析方法.[J].振动与冲击,2005,(3):120～123.
[70].吴斌，张驰，骞永博.某弹载结构橡胶隔振器动态特性研究.[J].机械科学与技术，2006,(9):1114～1116.
[71].王晓侠，刘德立，周海亭.橡胶隔振器参数计算与分析.[J].噪声与振动控制2008,(4):9～12.
[72].吴恒亮，代会军.橡胶隔振器设计开发研究.[J].噪声与振动控制，2009,(1):114～121.
[73].王锐，李世其，宋少云.橡胶隔振器系列化设计方法研究.[J].噪声与振动控制，2006,(4):11～13.
[74].王利荣，吕振华.橡胶隔振器有限元建模技术及静态弹性特性分析.[J].汽车工程，2002,(6):480～485.
[75].姚利锋，周海亭，陶杰.有限元在大载荷橡胶隔振器设计中的应用研究.[J].噪声与振动控制，2006,(6):16～18.
[76].N.Gil-Negrete, J.vinolas, L.Kari. A Nonlinear Rubber Material Model Combining FractionalOrder Viscoelasticity and Amplitude Dependent Effects.[J]. Journal of Applied Mechanics,2009, Vol.76/011009-1.
[77].Andre Schmidt, Lothar Gaul. Finite Element Formulation of Viscoelastic ConstitutiveEquations Using Fractional Time Derivatives.[J]. Nonlinear Dynamics,2002,(1):37～55.
[78].Sun-Yong Kim, Doo-Ho Lee. Identification of fractional-derivative-model parameters ofviscoelastic materials from measured FRFs.[J]. Journal of Sound and Vibration.2009,324:570～586.
[79].F.J. Jurado, A. Mateo, N. Gil-Negrete, J, L. Kari. Testing and FE modelling of the dynamicproperties of carbon black filled rubber.[J]. Proceedings of the EAEC, Conference I,Barcelona, June–July1999,(1):119～126.
[80].G.D.Dean, J.C. Duncan, A.F. Johnson, Determination of nonlinear dynamic properties ofcarbon-black filled rubbers.[J]. Polymer Testing4,1984,(1):225～249.
[81].M.J. Wang, Effect of polymer-filler and filler-filler interactions on dynamic properties offilled vulcanizates.[J]. Rubber Chemistry and Technology71,1998,(1):520～589.
[82].何春明,郑慕侨.测定橡胶Mooney_Rivlin模型常数的一种新方法.[J].北京理工大学学报,1997,(2):142～145.
[83].冯筱蓉.隔振橡胶的研究和应用.[J].特种橡胶制品,1983,(1):50～56.
[84].柯维，郑建华.基于橡胶硬度的隔振器性能判定.[J].船海工程，2008,(4):142～144.
[85].石菲，童宗鹏，龚丽琴，柳贡民，朱卫华.橡胶隔振器老化寿命的预测.[J].船舶工程，2009,(4):38～40.
[86].王强，周强.橡胶隔振器蠕变特性研究.[J].噪声与振动控制，2001,(4):19～22.
[87].N.Gil-Negrete, J.vinolas, L.Kari. A simplified methodology to predict the dynamic stiffnessof carbon-black filled rubber isolators using a finite element code [J]. Journal of Sound andVibration,2006,(1):757～776.
[88].C. M. Richards, R. Singh. Characterization of Rubber Isolator Nonlinearities in the Contextof Single and Multi-degree of Freedom Experimental Systems.[J]. Journal of Sound andVibration,2001,(5):807～834.
[89].Leif Kari. On the dynamic stiffness of preloaded vibration isolators in the audible frequencyrange: Modeling and experiments.[J]. Acoust. Soc. Am.2003,113(4), Pt.1.
[90].L. Kari. Pitchfork Phase Bifurcation of Isolator Stiffness.[J]. Journal of Sound andvibration,2002,(2):373～376.
[91].Beda T, Chevalier Y. New method for identifying rheological parameter for fractionalderivative modeling of viscoelastic behavior.[J]. Mech Time-Depend Mater,2004,(8):105～18.
[92].Soula M, Vinh T, Chevalier Y. Transient responses of polymers and elastomers deducedfrom harmonic responses.[J]. Sound Vibr1997,(3):185～203.
[93].Gaul L, Schmidt A. Parameter identification and FE implementation of a viscoelasticconstitutive equation using fractional derivatives.[J]. PAMM (Proc Appl Math Mech)2002,(1):153～154.
[94].Eldred LB, Baker WP, Palazotto AN. Kelvin–Voigt vs fractional derivative model asconstitutive relations for viscoelastic materials.[J]. AIAA J1995,(3):547～550.
[95].Makris N. Complex-parameter Kelvin model for elastic foundations.[J]. Earthquake EngStruct Dyn1994,(3):251～264.
[96].Vasques CMA, Moreira RAS. Experimental identification of GHM and ADF parameters forviscoelastic damping modeling.[J]. In: Mota Soares CA et al., editors. Proceedings of theIII European conference on computational mechanics. Solids, structures and coupledproblems in engineering, Lisbon, Portugal; June2008,(3):1～25.
[97].Gerlach S, Matzenmiller A. Comparison of numerical methods for identification ofviscoelastic line spectra from static test data.[J]. Int J Numer Meth Eng2005,(4):428～54.
[98].Syed Mustapha SMFD, Philips TN. A dynamic nonlinear regression method for thedetermination of the discrete relaxation spectrum.[J]. Phys D2000,(12):19～29.
[99].Gerlach S, Matzenmiller A. On parameter identification for material and microstructuralproperties.[J]. GAMM-Mitt2007,(3):481～505.
[100]. Satwinder Jit Singh, Anindya Chatterjee. Beyond fractional derivatives: localapproximation of other convolution integrals.[J]. Proc. R. Soc. A,2010,(7):563～581.
[101]. J.A. Tenreiro Machado. Fractional derivatives: Probability interpretation and frequencyresponse of rational approximations.[J]. Commun Nonlinear Sci Numer Simulat.2009,(14):3492～3497.
[102]. Attila Palfalvi. Efficient solution of a vibration equation involving fractionalderivatives.[J]. International Journal of Non-Linear Mechanics.2010,(5):169～175.
[103]. Q. Yang, F. Liu, I. Turner. Numerical methods for fractional partial differentialequations with Riesz space fractional derivatives.[J]. International Journal of Non-LinearMechanics44(2009)727-734Applied Mathematical Modelling.2010,(7):200～218.
[104]. A. Le Méhauté. A. El. Kaabouchi. Riemann's conjecture and a fractional derivative.[J].Computers and Mathematics with Applications.2010,(3):1610～1613.
[105]. M. D. Ortigueira. The fractional quantum derivative and its integral representations.[J].Commun Nonlinear Sci Numer Simulat.2010,(15):956～962.
[106]. S Sorrentino, A Fasana Finite element analysis of vibrating linear systems withfractional derivative viscoelastic models.[J]. Journal of Sound and Vibration,2007,(2):839～853.
[2]. Shukla AK, Datta TK. Optimal use of viscoelastic dampers in building frames for seismicforce [J]. Struct Eng1999,(1):401～409.
[3]. Singh MP, Chang TS. Seismic analysis of structures with viscoelastic dampers [J]. EngMech2009,(5):571～580.
[4]. Miehe C, Keck J. Superimposed finite elastic-viscoelatic-plastoelastic stress response withdamage in filled rubbery polymers, Experiments, modeling and algorithmic implementation[J]. Journal of the Mechanics and Physics of Solids,2000,(8):323~365
[5]. Chang TS, Singh MP. Mechanical model parameters for viscoelastic dampers [J]. Eng Mech2009,(1):581～584.
[6].杨国辉. APU橡胶隔振器特性研究.[D].南京:南京航空航天大学硕士学位论文，2011.
[7].方建敏.橡胶材料的分数导数型本构模型研究及动力学应用.[D].南京:南京航空航天大学硕士学位论文，2012.
[8]. M. Sjo¨berg, Rubber isolators—measurements and modelling using fractional derivativeand friction [D]. SAE,2000.
[9]. N. Gil-Negrete, On the Modeling and Dynamic Stiffness Prediction of Rubber Isolators,PhD Thesis [D]. University of Navarra, Spain,2004.
[10].J. Lubliner, A model of rubber viscoelasticity [J]. Mechanics Research Communications,1985,(12)93～99.
[11].R.Bagley, P.Torvik, Fractional calculus—a different approach to the analysis ofviscoelastically damped structures.[J]. AIAA Journal,1983,(2)741～748.
[12].王锐,李世其.隔振橡胶本构建模研究.[J].振动与冲击,2007,(1):77～79.
[13].王锐.橡胶隔振器动力学性能及设计方法研究.[D].武汉:华中科技大学博士学位论文，2007.
[14].孙德伟，张广玉.橡胶隔振器迟滞阻尼特性识别的新方法.[J].振动与冲击,2010,(4):164～168.
[15].赵广，刘健，刘占生.橡胶隔振器非线性动力学模型理论与实验研究.[J].振动与冲击，2010,(1):173～177.
[16].韩德宝，宋希庚.橡胶隔振器刚度和阻尼本构关系的试验研究.[J].振动与冲击，2009,(1):156～160.
[17].吴瑞安，冯加权，王玉军，仲政.橡胶隔振系统非线性振动参数识别与特性研究.[J].力学季刊，2007,(2):281～285.
[18].赵荣国，徐友拒，陈忠富，胡绍全，黄西成，刘青林.一个新的非线性迟滞隔振系统动力学模型.[J].机械工程学报，2004,(2):185～188.
[19].周相荣，王强，孙洪军.一种估算橡胶隔振器机械阻抗特性的简化方法.[J].舰船科学技术，2006,(2):93～97.
[20].陈安华，刘德顺，朱萍玉.被动隔振体的非线性振动分析.[J].机械工程学报，2001,(6):99～105.
[21].T. Yajima. H. Nagahama. Differential geometry of viscoelastic models with fractional-orderderivatives.[J]. Phys. A: Math. Theor.2010,(4):385207.
[22].Ross, B., A brief history and exposition of the fundamental theory of fractional calculus.[J].Fractional Calculus and Its Applications, Lecture Notes in Mathematics, Springer-Verlag,Berlin,1975,(1)1～36.
[23].Oldham K B, Spanier J. The Fractional Calculus.[D]. New York-London: Academic Press,1974.
[24].Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives.[D]. Theory andApplications. Switzerland:Gordon and Breach Science Publishers,1993.
[25].Miller K S, Ross B. An introduction to the fractional calculus and fractional differentialequations.[D]. New York:John Wiley&Sons Inc,1993.
[26].M.CAPUTO, F.MAINARDI. Linear models of dissipation in anelastic solids.[J]. RivistaDel Nuovo Cimento.1971,(1):161-198.
[27].R.L.BAGLEY. Application of generalized derivates to viscoelasticity.[J]. Air Force Instituteof Technology,1979, TR-79-4103.
[28].Schiessel H, Friedrich C, Blumen A. Applications to problems in polymer physics andrheology.[J]. Applications of Fractional Calculus in Physics. Singapore: World ScientificPublishing Co Pte Ltd,2000,(1):331～376.
[29].Gl ckle W G, Nonenmacher T F. Fractional integral operators and fox function in the theoryof viscoelasticity.[J]. Macromolecules,1991,(24):6426～6434.
[30].Nonnenmacher T F, Metzler R. On the Riemann-Liouville fractional calculus and somerecent applications.[J]. Fractals,1995,(3):557～566.
[31].Friedrich C. Relaxation and retardation functions of the Maxwell model with fractionalderivatives.[J]. Rheol Acta,1991,(3):151～158.
[32].Schiessel H, Metzler R, Nonnenmacher T F. Generalized viscoelastic models: Theirfractional equations with solutions.[J]. Phys A: Math Gen,1995,(28):6567～6584.
[33].Gl ckle W G, Nonnenmacher T F. Fox function representation of Non-Debye relaxationprocesses.[J]. Stat Phys,1993,(1):741～757.
[34].Xu M Y, Tan W C. Representation of the constitutive equation of viscoelastic materials bythe generalized fractional element networks and its generalized solutions.[J]. Science inChina, Series G,2003,(2):145～157.
[35].Lam L. Introduction to Nonlinear Physics.[M]. New York: Springer-Verlag Inc.,1997.
[36].Papoulia KD, Panoskaltsis VP, Korovajchuk I, Kurup NV. Rheological representation offractional derivative models in linear viscoelasticity.[J]. Rheol Acta; in press，381～400.
[37].Makris N, Constantinou MC. Fractional-derivative Maxwell model for viscous dampers.[J].Struct Eng1991,(17):2708～2724.
[38].Pritz T. Analysis of four-parameter fractional derivative model of real solid materials.[J].Sound Vib1996,(1):103～115.
[39].Atanackovic TM. A modified Zener model of a viscoelastic body.[J]. Continuum MechThermodyn2002,(14):137～148.
[40].Song DY, Jiang TQ. Study on the constitutive equation with fractional derivative for theviscoelastic fluids modified Jeffreys model and its application.[J]. Rheol Acta1998,(5):512～517.
[41].Park SW. Analytical modeling of viscoelastic dampers for structural and vibration control.[J]. Int J Solids Struct2001,(1):8065～92.
[42].Schmidt A, Gaul L. Finite element formulation of viscoelastic constitutive equations usingfractional time derivatives.[J]. Nonlinear Dyn2002,(1):37～55.
[43].Pritz T. Five-parameter fractional derivative model for polymeric damping materials.[J].Sound Vib2003,(1):935～52.
[44].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,(1):161～166.
[45].潘孝勇，柴国钟.含有橡胶隔振器振动系统时域响应的测试与计算分析.[J].振动与冲击,2007,(12):32～35.
[46].潘孝勇，上官文斌.基于超弹性、分数导数和摩擦模型的碳黑填充橡胶隔振器动态建模.[J].振动与冲击,2007,(10):6～15.
[47].潘孝勇，上官文斌，柴国钟，黄志.橡胶隔振器动态特性计算方法的研究.[J].振动工程学报，2009,(4):345～350.
[48].潘孝勇.橡胶隔振器动态特性计算与建模方法的研究.[D].杭州:浙江工业大学博士学位论文，2009.
[49].刘林超.分数导数型粘弹性材料的力学行为及在结构减振中的应用研究.[D].暨南大学硕士学位论文，2005年5月.
[50].孙海忠.高分子材料的分数导数型本构关系及应用.[D].暨南大学，2006年5月.
[51].夏亚东.高分子材料的线性与非线性分数算子本构关系研究.[D].暨南大学，2010年11月.
[52].Yu. A, Rossikhin, M.V. Shitikova. Analysis of damped vibrations of linear viscoelasticplates with damping modeled with fractional derivatives.[J]. Signal Processing,2006,(1):2703～2711.
[53].Djamel Ouis. Combination of a standard viscoelastic model and fractional derivate calculusto the characterization of polymers.[J]. Mat Res Innovat,2003,(7):42～46.
[54].Nicole Heymans. Constitutive equations for polymer viscoelasticity derived fromhierarchical models in cases of failure of time-temperature superposition.[J]. SignalProcessing83,2003,(1):2345～2357.
[55].Mattias Sjoberg. Rubber Isolators-Measurements and Modelling using FractionalDerivatives and Friction.[J]. Society of Automotive Engineers,2000,(1):01～3518.
[56].Mattias Sjoberg, Leif Kari. Non-Linear Behavior of a Rubber Isolator System UsingFractional Derivatives.[J]. Vehicle System Dynamics.2002,(1):217～236.
[57].Mattias Sjoberg. Nonlinear Isolator Dynamics at Finite Deformations: An EffectiveHyperelastic, Fractional Derivative, Generalized Friction Model.[J]. Nonlinear Dynamics.2003,(1):323～336.
[58].Ya Pu, Hartono Sumali. Modeling of Nonlinear Elastoneric Mounts. Part1: DynamicTesting and Parameter Identification.[J]. Society of Automotive Engineers.2001,01-0042.
[59].S. W. Park. Analytical modeling of viscoelastic dampers for structural and vibration control.[J]. International Journal of Solids and Structures2001,(1):8065～8092.
[60].Nicole Heymans. Fractional Calculus Description of Non-Linear Viscoelastic Behaviour ofPolymers.[J]. Nonlinear Dynamics.2004,(1):221～231.
[61].SIEGMAR KEMPFLE, INGO SCH FER. Fractional Calculus via Functional Calculus:Theory and Applications.[J]. Nonlinear Dynamics2002,(1):99～127.
[62].M. Kohandel, S. Sivaloganathan. Frequency dependence of complex moduli of brain tissueusing a fractional Zener model.[J]. Phys. Med. Biol.2005,(1):2799～2805.
[63].P. Haupt, A. Lion. On the dynamic behaviour of polymers under finite strains: constitutivemodelling and identification of parameters.[J]. International Journal of Solids andStructures.2000,(1):3633～3646.
[64].M. Fukunaga. N. Shimizu. A nonlinear fractional derivative model of impulse motion forviscoelastic materials.[J]. Phys. Scr.2009, T136,014010.
[65].M Fukunaga, N. Shimizu. Nonlinear Fractional Derivative Models of Viscoelastic ImpactDynamics Based on Entropy Elasticity and Generalized Maxwell Law.[J]. Journal ofComputational and Nonlinear Dynamics.2011, Vol.6/021005-1.
[66].唐振寰，罗贵火.橡胶隔振器粘弹性5参数分数导数并联动力学模型.[J].航空动力学报.2013,(1):99～105.
[67].Desalvo G J, Swanson J A..ANSYS engineering analysis system-theoretical manual(Version5.3).[M]. Houston: Swanson Analysis System Inc,1994,4～9.
[68].贝塞J K.非线性有限元分析——ADINA理论文本，张圣坤译.[M].上海：上海交通大学出版社，1986，5～10.
[69].陈莲，周海亭.计算橡胶隔振器静态特性的数值分析方法.[J].振动与冲击,2005,(3):120～123.
[70].吴斌，张驰，骞永博.某弹载结构橡胶隔振器动态特性研究.[J].机械科学与技术，2006,(9):1114～1116.
[71].王晓侠，刘德立，周海亭.橡胶隔振器参数计算与分析.[J].噪声与振动控制2008,(4):9～12.
[72].吴恒亮，代会军.橡胶隔振器设计开发研究.[J].噪声与振动控制，2009,(1):114～121.
[73].王锐，李世其，宋少云.橡胶隔振器系列化设计方法研究.[J].噪声与振动控制，2006,(4):11～13.
[74].王利荣，吕振华.橡胶隔振器有限元建模技术及静态弹性特性分析.[J].汽车工程，2002,(6):480～485.
[75].姚利锋，周海亭，陶杰.有限元在大载荷橡胶隔振器设计中的应用研究.[J].噪声与振动控制，2006,(6):16～18.
[76].N.Gil-Negrete, J.vinolas, L.Kari. A Nonlinear Rubber Material Model Combining FractionalOrder Viscoelasticity and Amplitude Dependent Effects.[J]. Journal of Applied Mechanics,2009, Vol.76/011009-1.
[77].Andre Schmidt, Lothar Gaul. Finite Element Formulation of Viscoelastic ConstitutiveEquations Using Fractional Time Derivatives.[J]. Nonlinear Dynamics,2002,(1):37～55.
[78].Sun-Yong Kim, Doo-Ho Lee. Identification of fractional-derivative-model parameters ofviscoelastic materials from measured FRFs.[J]. Journal of Sound and Vibration.2009,324:570～586.
[79].F.J. Jurado, A. Mateo, N. Gil-Negrete, J, L. Kari. Testing and FE modelling of the dynamicproperties of carbon black filled rubber.[J]. Proceedings of the EAEC, Conference I,Barcelona, June–July1999,(1):119～126.
[80].G.D.Dean, J.C. Duncan, A.F. Johnson, Determination of nonlinear dynamic properties ofcarbon-black filled rubbers.[J]. Polymer Testing4,1984,(1):225～249.
[81].M.J. Wang, Effect of polymer-filler and filler-filler interactions on dynamic properties offilled vulcanizates.[J]. Rubber Chemistry and Technology71,1998,(1):520～589.
[82].何春明,郑慕侨.测定橡胶Mooney_Rivlin模型常数的一种新方法.[J].北京理工大学学报,1997,(2):142～145.
[83].冯筱蓉.隔振橡胶的研究和应用.[J].特种橡胶制品,1983,(1):50～56.
[84].柯维，郑建华.基于橡胶硬度的隔振器性能判定.[J].船海工程，2008,(4):142～144.
[85].石菲，童宗鹏，龚丽琴，柳贡民，朱卫华.橡胶隔振器老化寿命的预测.[J].船舶工程，2009,(4):38～40.
[86].王强，周强.橡胶隔振器蠕变特性研究.[J].噪声与振动控制，2001,(4):19～22.
[87].N.Gil-Negrete, J.vinolas, L.Kari. A simplified methodology to predict the dynamic stiffnessof carbon-black filled rubber isolators using a finite element code [J]. Journal of Sound andVibration,2006,(1):757～776.
[88].C. M. Richards, R. Singh. Characterization of Rubber Isolator Nonlinearities in the Contextof Single and Multi-degree of Freedom Experimental Systems.[J]. Journal of Sound andVibration,2001,(5):807～834.
[89].Leif Kari. On the dynamic stiffness of preloaded vibration isolators in the audible frequencyrange: Modeling and experiments.[J]. Acoust. Soc. Am.2003,113(4), Pt.1.
[90].L. Kari. Pitchfork Phase Bifurcation of Isolator Stiffness.[J]. Journal of Sound andvibration,2002,(2):373～376.
[91].Beda T, Chevalier Y. New method for identifying rheological parameter for fractionalderivative modeling of viscoelastic behavior.[J]. Mech Time-Depend Mater,2004,(8):105～18.
[92].Soula M, Vinh T, Chevalier Y. Transient responses of polymers and elastomers deducedfrom harmonic responses.[J]. Sound Vibr1997,(3):185～203.
[93].Gaul L, Schmidt A. Parameter identification and FE implementation of a viscoelasticconstitutive equation using fractional derivatives.[J]. PAMM (Proc Appl Math Mech)2002,(1):153～154.
[94].Eldred LB, Baker WP, Palazotto AN. Kelvin–Voigt vs fractional derivative model asconstitutive relations for viscoelastic materials.[J]. AIAA J1995,(3):547～550.
[95].Makris N. Complex-parameter Kelvin model for elastic foundations.[J]. Earthquake EngStruct Dyn1994,(3):251～264.
[96].Vasques CMA, Moreira RAS. Experimental identification of GHM and ADF parameters forviscoelastic damping modeling.[J]. In: Mota Soares CA et al., editors. Proceedings of theIII European conference on computational mechanics. Solids, structures and coupledproblems in engineering, Lisbon, Portugal; June2008,(3):1～25.
[97].Gerlach S, Matzenmiller A. Comparison of numerical methods for identification ofviscoelastic line spectra from static test data.[J]. Int J Numer Meth Eng2005,(4):428～54.
[98].Syed Mustapha SMFD, Philips TN. A dynamic nonlinear regression method for thedetermination of the discrete relaxation spectrum.[J]. Phys D2000,(12):19～29.
[99].Gerlach S, Matzenmiller A. On parameter identification for material and microstructuralproperties.[J]. GAMM-Mitt2007,(3):481～505.
[100]. Satwinder Jit Singh, Anindya Chatterjee. Beyond fractional derivatives: localapproximation of other convolution integrals.[J]. Proc. R. Soc. A,2010,(7):563～581.
[101]. J.A. Tenreiro Machado. Fractional derivatives: Probability interpretation and frequencyresponse of rational approximations.[J]. Commun Nonlinear Sci Numer Simulat.2009,(14):3492～3497.
[102]. Attila Palfalvi. Efficient solution of a vibration equation involving fractionalderivatives.[J]. International Journal of Non-Linear Mechanics.2010,(5):169～175.
[103]. Q. Yang, F. Liu, I. Turner. Numerical methods for fractional partial differentialequations with Riesz space fractional derivatives.[J]. International Journal of Non-LinearMechanics44(2009)727-734Applied Mathematical Modelling.2010,(7):200～218.
[104]. A. Le Méhauté. A. El. Kaabouchi. Riemann's conjecture and a fractional derivative.[J].Computers and Mathematics with Applications.2010,(3):1610～1613.
[105]. M. D. Ortigueira. The fractional quantum derivative and its integral representations.[J].Commun Nonlinear Sci Numer Simulat.2010,(15):956～962.
[106]. S Sorrentino, A Fasana Finite element analysis of vibrating linear systems withfractional derivative viscoelastic models.[J]. Journal of Sound and Vibration,2007,(2):839～853.