振动流化床的动态特性及稳定性研究
详细信息    本馆镜像全文|  推荐本文 |  |   获取CNKI官网全文
摘要
随着综采机械化的广泛应用,原煤中0~6mm细粒级煤炭含量逐渐上升。大多数缺水干旱地区对细粒煤不加处理直接进入市场造成严重的资源浪费,因而亟需研究简单、可靠、高效的细粒煤分选技术及配套设备。振动流化床干法分选技术将振动能量引入传统的流化床中,形成均匀稳定的流化床层,为实现细粒煤炭的干法分选提供了有效的方法。然而,良好的分选特性需要稳定的振动环境作为保障。至今为止仍缺少对振动流化床振动特性以及复杂振动条件下的流化特性的相关研究。因此,本文针对振动机械的基本运动特征,并结合颗粒物料流态化基本过程,对直线摇摆复合振动下的流化特性和振动流化床简化力学模型的动态特性以及稳定性展开了研究,为振动流化床的工业化和大型化提供理论参考。
     根据细粒煤振动流化床干法分选机的分选原理和振动装置的特点,提出了新型结构振动流化床主机,确定了振动流化床主机的基本参数,初步选择了振动流化床的工艺参数,完成了新型结构振动流化床的系统设计。
     对振动流化床直线摇摆复合振动下的流化特性进行了试验研究。设计了具有直线摇摆复合运动的振动流化床试验平台,并利用高速动态拍摄系统分析了颗粒物料按密度分层的基本过程以及扰动条件下的物料反混过程。以振动频率、振动强度和摇摆扰动条件为试验因素进行单因素试验。发现中低频振动条件下具有较好的分选效果而高频振动无法实现分层;在摇摆扰动条件下床层内出现大气泡的产生和大范围环流,导致已分层的物料发生反混;将竖直和摇摆运动的相位差控制在0°附近可以有效避免大范围环流的产生。
     采用拉格朗日方程建立了振动流化床简化模型的运动方程,并利用多尺度法得到该方程的一次近似解和一次派生系统的平均方程。分析了非共振条件和1:1:2:2内共振条件下的自由振动特性。在内共振条件下,一次派生系统系数的响应幅值和频率都随初始条件的变化而变化,竖直和摇摆运动之间存在能量交换现象。利用李雅普诺夫第一方法分析了系统的运动稳定性,发现内共振条件下存在较多的不稳定因素。在振动流化床的设计过程中应该避免内共振发生。
     利用高速动态拍摄系统验证了平面内带有双转子的振动系统存在0°和180°相位差自同步运动。在自同步条件下,揭示了该类振动系统的受迫振动特性。求解了非共振和主共振条件下的系统受迫响应,并利用李雅普诺夫第一方法分析了系统的运动稳定性。考虑摇摆运动的高次项,利用多尺度法得到了系统在主共振条件下的频响方程,发现大幅度摇摆会导致频响曲线峰值发生偏移,从而使稳定区域缩小甚至消失。
     对振动流化床中试装置进行了模态分析和动力响应试验研究。利用试验模态分析获得了振动流化床的前6阶固有频率。在模态分析的基础上,测试获得了负载和空载下的振动流化床动力响应,并获得了启动、停机和稳态运行的时域和频域信号。测试结果表明启动和停机共振区附近会出现倍频振动和混沌运动,稳态运行时沿床宽方向存在一定的摆动。消除摆动和避免共振是保证良好流化环境的基本条件。
The0~6mm level fine-grained coal content of raw coal gradually increase with theextensive application of fully mechanized mining, and directly accessing to marketwithout any processing of fine coal cause serious waste of resources in most dry area,thus, the research on a simple, reliable, and efficient separation technology andequipment for fine coal is significance and urgent. The vibrating energy is introducedinto the traditional fluidized bed to develop a vibrating fluidized bed (VFB) dryseparation technology, which provides a uniform and stable fluidized bed layer and aneffective way for fine coal dry separation. However, stable vibration environment is theguarantee of good separation performance. So far, relevant research about vibrationcharacteristics of VFB and fluidization characteristics under the condition of complexvibration nearly remains blank. According to the basic movement characteristics of thevibrating machine and the basic process of granular material’s fluidization, this thesispresents a study on the fluidization characteristics under the condition of liner andswing composite vibration and dynamic characteristics of simplified VFB mechanicsmodel, providing theoretical reference for the industrialization and large-scale sizeVFB.
     According to the separation principle of fine coal VFB dry separation machine andcharacteristics of vibration device, a novel structure of VFB is proposed, the basicparameters of VFB are determined, the preliminary process parameters of VFB areselected and system design of the VFB is completed.
     Experimental study is carried out to study the fluidization characteristics of VFBunder the condition of linear and swing complex vibration. The VFB testing platformwith linear and swing complex motion is designed. The basic process of densitystratification and back mixing under the disturbance condition are analyzed by using thehigh-speed dynamic system. The single-factor experiment is conducted with testingfactor of vibration frequency, vibration intensity and swing disturbance conditions,respectively, from which the conclusions are found out that better separation efficiencybrought out under conditions of low and medium frequency vibration while nostratification brought out under the condition of high-frequency vibration, big bubblesand a wide range of circulation are formed inside the bed under the swing disturbancecondition, leading the stratified materials back mixed and keeping the phase differencebetween vertical motion and swing motion around zero degree can effectively avoid theoccurrence of a wide range of circulation.
     The motion equations of VFB simplified model are established using Lagrangeequation. An approximate solution of the motion equations and the average equation offirst order derived system are obtained through the multiple scales method. Freevibration characteristics of non-resonance conditions and1:1:2:2internal resonanceconditions are analyzed. Under the resonance condition, both response amplitude andfrequency of coefficient of first order derived system vary with the change of initialconditions, and there exists a phenomenon of energy exchange between the vertical andswing motion. Stability of the system is analyzed with first method of Lyapunov,finding out that many factors of instability exist under the resonance condition. Theresonance condition should be avoided when the VFB is designed.
     The high speed dynamic system is used to verify that the in-plane vibration systemwith double rotors can achieve synchronized movement with0°and180°phasedifference. Under the condition of self-synchronization, the forced vibrationcharacteristics of this system are revealed. The forced response of system under bothnon-resonant and main resonance conditions are solved, and stability of the system isanalyzed by the first method of Lyapunov. Considering higher order terms of swingmotion, the frequency response equations under the condition of main resonance areobtained by using multiple scales method, and significant swing causing shifted peak ofthe frequency response curve is found out, so that the stable region shrink or evendisappear.
     Modal analysis and dynamic response experiment are conducted on the VFB pilotplant testing device. Experimental modal analysis is used to obtain the first six ordernatural frequency of VFB. Based on modal experimental analysis, the dynamicresponses of loaded and unloaded VFB are acquired, the time domain and frequencydomain signals in the starting, stopping and steady state operating period are obtained.Testing results indicate that there may be frequency-doubled oscillation and chaoticmotion nearby the resonance region of starting and stopping period, and a certain degreeof swing motion in the steady state operation period along the bed width direction.Eliminating swing motion and avoiding resonance are the basic conditions to guaranteethe good fluidization environment.
引文
[1]胡珺.2013年全国煤炭产量37亿吨[N].中国能源报,2014-1-20(3).
    [2]国家能源局.国家能源科技―十二五‖规划[R].北京:国家能源局,2011.
    [3]陈清如.发展清洁煤技术推动节能减排[J].中国高校科技与产业化,2008(3):65-67.
    [4]赵跃民.煤炭资源综合利用手册[M],北京:科学出版社,2004.
    [5]谢广元.选矿学[M].徐州:中国矿业大学出版社,2001.
    [6] Luo Z, Fan M, Zhao Y, et al. Density-dependent separation of dry fine coal in a vibratedfluidized bed [J]. Powder Technology,2008,187(2):119-23.
    [7] Yang X, Zhao Y, Luo Z, et al. Fine coal dry beneficiation using autogenous medium in avibrated fluidized bed [J]. International Journal of Mineral Processing,2013,125(0):86-91.
    [8] Yang X, Zhao Y, Luo Z, et al. Fine coal dry cleaning using a vibrated gas-fluidized bed [J]. FuelProcessing Technology,2013,106(0):338-43.
    [9] Chen W H, Chen C J, Hung C I. Taguchi approach for co-gasification optimization of torrefiedbiomass and coal [J]. Bioresource Technology,2013,144(0):615-22.
    [10] Emami-Taba L, Irfan M F, Wan Daud W M A, et al. Fuel blending effects on the co-gasificationof coal and biomass–A review [J]. Biomass and Bioenergy,2013,57(0):249-63.
    [11] Wang L.-Q., Chen Z.-S. Gas generation by co-gasification of biomass and coal in anautothermal fluidized bed gasifier [J]. Applied Thermal Engineering,2013,59(1–2):278-82.
    [12] Pei X, He B, Yan L, et al. Process simulation of oxy-fuel combustion for a300 MWpulverized coal-fired power plant using Aspen Plus [J]. Energy Conversion and Management,2013,76(0):581-7.
    [13] Skorek-Osikowska A, Bartela L, Kotowicz J, et al. Thermodynamic and economic analysis ofthe different variants of a coal-fired,460MW power plant using oxy-combustion technology [J].Energy Conversion and Management,2013,76(0):109-20.
    [14] W E Council, Water for Energy[R], London,2010.
    [15] CHEN Q, YANG Y. Development of dry beneficiation of coal in China[J]. Coal Preparation:139A Multinational Journal,2003,23(1-2):3-12.
    [16]陈清如,陈尉.空气重介质流化床干法选煤技术发展现状与进展[J].煤炭科技,1999(1):4-6.
    [17] LUO Z, ZHAO Y, TAO X, et al. Progress in dry coal cleaning using air-dense medium fluidizedbeds[J]. Coal Preparation: A Multinational Journal,2003,23(1-2):13-20.
    [18] A K SAHU, S K BISWAL, PARIDA A. Development of air dense medium fluidized bedtechnology for dry beneficiation of coal_A review[J]. International Journal of Coal Preparationand Utilization,2009,29(4):216-241.
    [19] Chen Q, Wei L. Coal dry beneficiation technology in china: the state-of-the-art [J]. ChinaParticuology,2003,1(2):52-6.
    [20] Chen Q, Wei L. Development of coal dry beneficiation with air-dense medium fluidized bed inchina [J]. China Particuology,2005,3(1–2):42.
    [21] Luo Z F, Zhu J F, Fan M M, et al. Low Density Dry Coal Beneficiation Using an Air DenseMedium Fluidized Bed [J]. Journal of China University of Mining and Technology,2007,17(3):306-9.
    [22] Zhenfu L, Qingru C. Dry beneficiation technology of coal with an air dense-medium fluidizedbed [J]. International Journal of Mineral Processing,2001,63(3):167-75.
    [23]李功民,杨云松.复合式干法选煤技术在中国的应用[J].煤炭加工与综合利用,2006,05:33-36.
    [24] Sampaio C H, Aliaga W, Pacheco E T, et al. Coal beneficiation of Candiota mine by dry jigging[J]. Fuel Processing Technology,2008,89(2):198-202.
    [25]骆振福,Maoming FAN,陈清如,等.振动参数对流化床分选性能的影响[J].中国矿业大学学报,2006,02:209-213.
    [26] Daud W R W. Fluidized Bed Dryers—Recent Advances [J]. Advanced Powder Technology,2008,19(5):403-18.
    [27] Xiang L, Shuyan W, Huilin L, et al. Numerical simulation of particle motion in vibratedfluidized beds [J]. Powder Technology,2010,197(1–2):25-35.
    [28] Cicek T. Dry cleaning of Turkish coal [J]. Energy Sources Part a-Recovery Utilization andEnvironmental Effects,2008,30(7):593-605.
    [29] Dwari R K, Rao K H. Dry beneficiation of coal-A review [J]. Mineral Processing andExtractive Metallurgy Review,2007,28(3):177-234.
    [30] Sahu A K, Biswal S K, Parida A. Development of air dense medium fluidized bed technologyfor bendficiation of coal–a review [J]. International Journal of Coal Preparation and Utilization,2009,29(4):216-41.
    [31] Haibin L, Zhenfu L, Yuemin Z, et al. Cleaning of South African coal using a compound drycleaning apparatus [J]. Mining Science and Technology (China),2011,21(1):117-21.
    [32]陈清如,骆振福.干法选煤评述[J].选煤技术,2003,06:34-40+18.
    [33] WEITKAMPER L, WOTRUBA H. Development and Performance of a New Separator for theDry Gravity Sepatation of Fines: proceedings of the XXIV Internatioal Mineral ProcessingCongress, Beijing,2008[C]. Beijing: Science Press,2008.
    [34]沈丽娟,陈建中.复合式干法分选机中床层物料的运动分析[J].中国矿业大学学报,2005,04:447-451.
    [35]剧殿臣,康华,林井祥.复合式干法选煤的现状及经济分析[J].煤炭技术,2010,10:109-111.
    [36]赵跃民,李功民,骆振福,等.模块式气固流化床干法选煤技术的研究[J].选煤技术,2009,06:1-5+1.
    [37] KELLY E G, SPOTTISWOOD D J. The theory of electrostatic separations: A review part III.The separation of particles[J]. Minerals Engineering,1989,2(3):337-349.
    [38] KELLY E G, SPOTTISWOOD D J. The theory of electrostatic separations: A review Part I.Fundamentals[J]. Minerals Engineering,1989,2(1):33-46.
    [39] KELLY E G, SPOTTIWOOD D J. The theory of electrostatic separations: A review part II.Particle charging[J]. Minerals Engineering,1989,2(2):193-205.
    [40] DWARI R K, HANUMANTHA RAO K. Fine coal preparation using novel tribo-electrostaticseparator[J]. Minerals Engineering,2009,22(2):119-127.
    [41]章新喜,段超红,于凤芹,等.微粉煤的电性质及摩擦带电研究[J].中国矿业大学学报(自然科学版),2005,34(6):694-697.
    [42]马瑞欣,石常省,章新喜.煤与单一矿物质在摩擦电选过程中的分离[J].中国矿业大学学报(自然科学版),2010,39(2):270-274.
    [43] FAN M, CHEN Q, ZHAO Y, et al. Magnetically stabilized fluidized beds for fine coalseparation[J]. Powder Technology,2002,123(2-3):208-211.
    [44] FAN M, CHEN Q, ZHAO Y, et al. Fine coal (6-1mm) separation in magnetically stabilizedfluidized beds[J]. International Journal of Mineral Processing,2001,63(4):225-232.
    [45] FAN M, CHEN Q, ZHAO Y, et al. Fundamentals of a Magnetically Stabilized Fluidized Bedfor Coal Separation[J]. International Journal of Coal Preparation and Utilization,2003,23(1):47-55.
    [46] LUO Z, FAN M, ZHAO Y, et al. Density-dependent separation of dry fine coal in a vibratedfluidized bed[J]. Powder Technology,2008,187(2):119-123.
    [47]骆振福, FAN M,赵跃民,等.物料在振动力场流化床中的分离[J].中国矿业大学学报(自然科学版),2007(01):27-32.
    [48]骆振福,陈清如.振动流化床的分选特性[J].中国矿业大学学报(自然科学版),2000,29(6):566-570.
    [49]骆振福,陈清如,陶秀祥.振动流化床的形成机理[J].中国矿业大学学报,2000,03:8-12.
    [50] WEITKAEMPER L, WOTRUBA H, SAMPAIO C H. Effective Dry Density Beneficiation ofFine Coal Using a New Developed Fluidized Bed Separator: proceedings of the XVIInternatIonal Coal Preparation Congress, Lexington,2010[C]. Littleton, Colorado: SME,2010.
    [51]骆振福,樊茂明,陈清如.磁场流化床的稳定性研究[J].中国矿业大学学报,2001,04:30-33.
    [52]樊茂明,陈清如,赵跃民,等.磁稳定流化床干法选煤试验研究[J].选煤技术,2002(03):6-8+1.
    [53]宋树磊.空气重介磁稳定流化床分选细粒煤的基础研究[D];中国矿业大学,2009.
    [54]张敏郎,方全利.振动流动干燥机的研究和设计[J].浙江化工,1988,04:14-19.
    [55]杨旭亮.细粒煤振动流态化的能量作用及分离机制的研究[D].中国矿业大学,2013.
    [56] Yue Min Z, Chu Sheng L, Xiao Mei H, et al. Dynamic design theory and application of largevibrating screen [J]. Procedia Earth and Planetary Science,2009,1(1):776-84.
    [57]贺孝梅,刘初升,张成勇.基于多频约束和解析灵敏度法的大型振动筛优化设计[J].中南大学学报(自然科学版),2011,03:664-670.
    [58]闫俊霞.高速大型振动筛结构动态特性及可靠性基础研究[D].中国矿业大学,2011.
    [59]杨俊哲.基于Workbench多倾角型振动筛的模态分析[J].煤炭学报,2012,S1:240-244.
    [60]张德臣,孙艳平.大型振动筛动态仿真和模态分析实验综述[J].鞍山科技大学学报,2003,01:1-3.
    [61]曹金红.基于Lanczos模态分析法的振动筛固有频率分析[J].辽东学院学报(自然科学版),2013,02:125-128.
    [62] Liu C S, Zhang S M, Zhou H P, et al. Dynamic analysis and simulation of four-axis forcedsynchronizing banana vibrating screen of variable linear trajectory [J]. Journal of Central SouthUniversity of Technology (English Edition),2012,19(6):1530-6.
    [63]张阳,刘初升,魏群,等.大型圆振动筛固有振动特性及动力响应分析[J].矿山机械,2010,21:90-93.
    [64]彭晨宇.振动筛结构随机动力学研究[D].辽宁工程技术大学,2012.
    [65]傅志方,华宏星.模态分析理论与应用[M].上海:上海交通大学出版社,2000.
    [66]曾攀.有限元分析及应用[M].北京:清华大学出版社,2004.
    [67]李仁宪.有限元法基础[M].北京:国防工业出版社,2002.
    [68]何正嘉,陈雪峰.小波有限元理论研究与工程应用的进展[J].机械工程学报,2005,03:1-11.
    [69]陈雪峰,杨胜军,马军星,何正嘉.小波有限元的研究及其工程应用[J].西安交通大学学报,2003,01:1-4.
    [70]尤琼,史治宇,罗绍湘.基于小波有限元的移动荷载识别[J].振动工程学报,2010,02:188-193.
    [71]沈鹏程,何沛祥.计算力学中的样条有限元法的进展[J].力学进展,2000,02:191-199.
    [72]秦剑,黄克服,张清东.几何非线性样条有限元法[J].工程力学,2011,S1:1-4.
    [73]沈鹏程,河沛祥.多变量样条有限元法[J].固体力学学报,1994,03:234-243.
    [74] Ewins D J. Modal Testing: Theory and Practice[M]. RSPLTD,1986.
    [75] Allen M S, Ginsberg J H. A global, single-input–multi-output (SIMO) implementation of thealgorithm of mode isolation and application to analytical and experimental data [J]. MechanicalSystems and Signal Processing,2006,20(5):1090-111.
    [76] Hanson D, Randall R B, Antoni J, et al. Cyclostationarity and the cepstrum for operationalmodal analysis of mimo systems—Part I: Modal parameter identification [J]. MechanicalSystems and Signal Processing,2007,21(6):2441-58.
    [77] Moore S M, Lai J C S, Shankar K. ARMAX modal parameter identification in the presence ofunmeasured excitation—II: Numerical and experimental verification [J]. Mechanical Systemsand Signal Processing,2007,21(4):1616-41.
    [78] U Füllekrug, J M Sinapius. Identification of modal parameters, generalized and effectivemasses during base-driven tests[J]. Aerospace Science and Technology,1998,2(7):469-480.
    [79] M Hagara, F Trebuňa, R Huňady, et al. Experimental Identification of Modal Parameters ofThin Metal Sheets by using of DIC[J]. Procedia Engineering,2012,48(0):180-188.
    [80] S L James Hu, X Bao, H Li. Model order determination and noise removal for modal parameterestimation[J]. Mechanical Systems and Signal Processing,2010,24(6):1605-1620.
    [81] J Lardies, S Gouttebroze. Identification of modal parameters using the wavelet transform[J].International Journal of Mechanical Sciences,2002,44(11):2263-2283.
    [82] H Li, P Li, S L J Hu. Modal parameter estimation for jacket-type platforms using noisyfree-vibration data: Sea test study[J]. Applied Ocean Research,2012,37(0):45-53.
    [83] E Reynders, R Pintelon, G De Roeck. Uncertainty bounds on modal parameters obtained fromstochastic subspace identification[J]. Mechanical Systems and Signal Processing,2008,22(4):948-969.
    [84] G Zhang, B Tang, G Tang. An improved stochastic subspace identification for operationalmodal analysis[J]. Measurement,2012,45(5):1246-1256.
    [85]杨前进,张培强,李川奇.多输入——多输出实验模态参数识别(3)——频域识别法及其内在联系[J].实验力学,1992,01:60-69.
    [86] Eritenel T, Parker R G. Modal properties of three-dimensional helical planetary gears [J].Journal of Sound and Vibration,2009,325(1–2):397-420.
    [87]王峰,刘恋华.筛分机的动态特性分析[J].矿山机械,1989,10:41-43.
    [88]周广林,张维屏.大型振动筛的强度分析[J].矿山机械,1990(9):29-31.
    [89]王正浩,张兆汉,李云洋. ZS1756振动筛动态分析[J].沈阳建筑工程学报,1996,12(3):284-287.
    [90]刘杰,谢广平,纪盛青.大型热矿振动筛动态特性的有限元分析[J].东北大学学报,1997,18(3):316-319.
    [91]李向东. YR_924振动筛筛框结构理论模态分析[J].矿山机械,1997(10):42-44.
    [92]舒仲周,等.运动稳定性[M].北京:中国铁道出版社,2001.
    [93]徐明友.用李雅普诺夫直接法建立弹箭飞行动稳定条件[J].南京理工大学学报,2001,01:10-12.
    [94]朱如曾,谈镐生.连续系统稳定性问题中的李雅普诺夫直接法[J].中国科学(A辑数学物理学天文学技术科学),1984,(4):51-61.
    [95]周雪鸥.关于李雅普诺夫第二方法中稳定性及不稳定性定理的推广[J].四川大学学报(自然科学版),1960,01:51-61.
    [96]王联.广义李雅普诺夫函数的作法[J].中国科学,1980,(9):821-829.
    [97]时培明,李纪召,刘彬,等.一类准周期参激非线性相对转动动力系统的稳定性与时滞反馈控制[J].物理学报,2011,09:441-450.
    [98]符文彬,唐驾时.基于状态反馈参数激励系统的超谐共振分岔控制[J].物理学报,2004,09:2889-2893.
    [99] Baisheng Wu, Weipeng Sun, Zhengguang Li, et al. Circular whirling and stability due tounbalanced magnetic pull and eccentric force [J]. Journal of Sound and Vibration,2011,(21):4949-4954.
    [100]戴余良,王长湖,苗海,等.潜艇水下运动稳定性非线性分析研究[J].船舶力学,2011,(08):844-852.
    [101]陈予恕.非线性动力学[M].北京:高等教育出版社,2002.
    [102] Blekhman I I, Fradkov A L, Tomchina O P, et al. Self-synchronization and controlledsynchronization: general definition and example design [J]. Mathematics and Computers inSimulation,2002,58(4–6):367-84.
    [103]闻邦椿,赵春雨.机械系统的振动同步与控制同步[M].北京:科学出版社,2003.
    [104]赵春雨,闻邦椿,赵广耀.同向回转双机传动振动系统相位差的模糊监督控制[J].振动工程学报,2001,14(3):42-46.
    [105]赵春雨,王得刚,张昊,等.同向回转双机驱动振动系统的频率俘获[J].应用力学学报,2009,26(2):283-287.
    [106]赵春雨,刘戡,叶小芬,等.反向回转双机驱动振动系统的自同步理论[J].机械工程学报,2009,45(9):24-30.
    [107] Li J, Liu C S, Peng L P, et al. Free vibration of vibrating device coupling two pendulums usingmultiple time scales method [J]. Journal of Central South University,2013,20(8):2134-41.
    [108]何松林,黄焱.用改进平均法求解自由衰减振动[J].振动与冲击,2011,(01):227-229.
    [109]张宝善.求解一类非线性方程的改进的平均法[J].应用数学和力学,1994,(12):1119-1126.
    [110]韩景龙,朱德懋.非线性振动系统的规范形与平均法[J].振动工程学报,1996,04:51-57.
    [111]陈立群,吴哲民.多自由度非线性振动分析的平均法[J].振动与冲击,2002,03:65-66+96.
    [112]龙钢.复内积平均法及其在转子系统中的应用[D].哈尔滨工业大学,2007.
    [113]杨翊仁,倪樵.结构非线性颤振分析的KBM法及实验对比[J].振动工程学报,1995,(04):351-355.
    [114]唐续俞.拟线性系统振动问题的KBM法[J].华南理工大学学报(自然科学版),1999,(09):70-74.
    [115]侯东晓,刘彬,时培明.一类滞后相对转动动力学方程的分岔特性及其解析近似解[J].物理学报,2009,09:5942-5949.
    [116]吴晓,罗佑新,杨立军.基础位移激励下斜置弹簧减振系统的共振特性[J].北京工业大学学报,2010,12:1637-1641.
    [117]刘志军,陈国平.考虑抗弯刚度影响的斜拉索面内非线性自由振动分析[J].振动工程学报,2007,01:57-60.
    [118]任传波,周继磊.分段线性耦合动力系统的周期解及稳定性分析[J].应用力学学报,2011,05:527-531+558.
    [119]韩景龙,朱德懋.一种单元谐波平衡法[J].力学学报,1999,(06):753-760.
    [120]巫世晶,刘振皓,王晓笋,等.基于谐波平衡法的复合行星齿轮传动系统非线性动态特性[J].机械工程学报,2011,01:55-61.
    [121]蔡铭,刘济科,李军.多自由度强非线性颤振分析的增量谐波平衡法[J].应用数学和力学,2006,07:833-838.
    [122]梁峰,杨晓东,闻邦椿.基于增量谐波平衡法的两端固定输流管参数共振[J].机械工程学报,2009,07:126-130.
    [123]王威,宋玉玲,李瑰贤.基于增量谐波平衡法的汽车转向系非线性动力学特性研究[J].振动工程学报,2010,03:355-360.
    [124]王本利,张相盟,卫洪涛.基于谐波平衡法的含Iwan模型干摩擦振子非线性振动[J].航空动力学报,2013,01:1-9.
    [125] Buryak A V. Optical solitons due to quadratic nonlinearities: from basic physics to futuristicapplications[J]. Physics Reports,2002,370(2),63-235.
    [126] Ghayesh M H, S Kazemirad, M A Darabi. A general solution procedure for vibrations ofsystems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions[J].Journal of Sound and Vibration,2011,330(22),5382-5400.
    [127] Hegazy U H.3:1Internal resonance of a string-beam coupled system with cubicnonlinearities[J]. Communications in Nonlinear Science and NumericalSimulation,2010,15(12),4219-4229.
    [128] O Coulaud. Multiple time scales and perturbation methods for high frequencyelectromagnetic-hydrodynamic coupling in the treatment of liquid metals[J]. NonlinearAnalysis: Theory, Methods&Applications,1997,30(6):3637-3643.
    [129] K D Mease. Multiple time-scales in nonlinear flight mechanics: diagnosis and modeling[J].Applied Mathematics and Computation,2005,164(2):627-648.
    [130] T Sch fer, R O Moore. A path integral method for coarse-graining noise in stochasticdifferential equations with multiple time scales[J]. Physica D: NonlinearPhenomena,2011,240(1):89-97.
    [131] A Abdulle, G A Pavliotis. Numerical methods for stochastic partial differential equations withmultiple scales[J]. Journal of Computational Physics,2012,231(6):2482-2497.
    [132] H Demiray. Multiple time scale formalism and its application to long water waves[J]. AppliedMathematical Modelling,2010,34(5):1187-1193.
    [133] G S Ladde, S G Rajalakshmi. Singular perturbations of linear systems with multiparametersand multiple time scales[J]. Journal of Mathematical Analysis andApplications,1988,129(2):457-481.
    [134] B Leimkuhler, S Reich. A Reversible Averaging Integrator for Multiple Time-ScaleDynamics[J]. Journal of Computational Physics,2001,171(1):95-114.
    [135] A Lucia. Multi-scale methods and complex processes: A survey and look ahead[J]. Computers&Chemical Engineering,2010,34(9):1467-1475.
    [136]梅树立,张森文.基于精细积分技术的非线性动力学方程的同伦摄动法[J].计算力学学报,2005,(06):665-670.
    [137] G M Abd El-Latif. A homotopy technique and a perturbation technique for non-linearproblems[J]. Applied Mathematics and Computation,2005,169(1):576-588.
    [138] J H He. Homotopy perturbation technique[J]. Computer Methods in Applied Mechanics andEngineering,1999,178(3–4):257-262.
    [139] J H He. Homotopy perturbation method: a new nonlinear analytical technique[J]. AppliedMathematics and Computation,2003,135(1):73-79.
    [140] S L Mei,S W Zhang. Coupling technique of variational iteration and homotopy perturbationmethods for nonlinear matrix differential equations[J]. Computers&Mathematics withApplications,2007,54(7–8):1092-1100.
    [141] Z M Odibat. A new modification of the homotopy perturbation method for linear and nonlinearoperators[J]. Applied Mathematics and Computation,2007,189(1):746-753.
    [142]俞翔,朱石坚,刘树勇.多自由度非线性隔振系统建模及其非共振响应[J].振动与冲击,2007,07:69-73+100+181-182.
    [143]李欣业,陈予恕,吴志强,等.多自由度内共振系统非线性模态的分岔特性[J].力学学报,2002,03:401-407.
    [144] A H Nayfeh, D T Mook. Nonlinear Oscillations[M]. New York:John Wiley,1979.
    [145] A H Nayfeh. Introduction to Perturbation Techniques[M]. New York:John Wiley,1981.
    [146]赵啦啦,刘初升,闫俊霞,等.颗粒分层过程三维离散元法模拟研究[J].物理学报,2010,59(3):1870-07.
    [147]杨旭亮,赵跃民,骆振福,等.振动流态化的能量传递机制及对细粒煤的分选研究[J].中国矿业大学学报,2013,42(02):266-270.
    [148] WEN C Y, YU Y H. A generalized method for predicting the minimum fluidization velocity[J].AIChE Journal,1966,12(3):610-612.
    [149]吴德荣.化工工艺设计手册[M].北京:化学工业出版社,2009(6):356~358.
    [150]严峰,刘焕胜.筛分机械[M].北京:煤炭工业出版社,1995.
    [151]李珺,刘初升,赵跃民,等.新型振动流化床的动态特性及试验研究[J].煤炭学报,2013,10(38):1882.
    [152]胡海岩.应用非线性动力学[M].北京:航空工业出版社,2000.
    [153] C Huygens. Horoloqium Oscilatorium[M]. Parisiis, France,1673.
    [154]闫俊霞.高速大型振动筛结构动态特性及可靠性基础研究[D].中国矿业大学,2012:87-88.