轴向运动梁横向非线性振动建模、分析和仿真
|
摘要
多种工程系统如动力传送带、磁带、纸带、纺织纤维、带锯、空中缆车索道、高楼升降机缆绳、单索架空索道等都可以模型化为轴向运动弦线和梁。因此轴向运动弦线和梁横向振动的研究中有着重要的工程应用价值。同时,轴向运动连续体作为典型陀螺连续系统,其振动分析也有着重要的理论意义。
本文研究轴向运动黏弹性弦线和梁的建模、分析和仿真。本文建立了轴向运动弦线和梁的耦合振动模型,并考察了不同的横向运动简化模型。基于横向运动模型,应用直接多尺度方法分析了轴向运动黏弹性梁的稳定性和稳态响应。发展了轴向运动黏弹性梁数值仿真的差分法和微分求积法,并将数值仿真结果与解析结果进行了比较。基于数值仿真结果用非线性动力学时间序列分析方法研究了分岔和混沌。具体内容包括下列几方面。
建立轴向运动弦线和梁的横向运动控制方程并通过数值仿真结果进行研究。若忽略耦合效应而仅考虑横向运动,平面耦合的非线性控制方程退化为横向非线性偏微分模型。若将扰动张力用在弦线或者梁上的平均值代替,就从非线性偏微分模型推导出非线性偏微分—积分模型。运用有限差分法分别从时间历程、稳态的幅频响应以及暂态的时间历程的角度比较弦线和梁自由振动、轴向运动黏弹性梁的非线性受迫振动以及速度带轴向变速运动黏弹性梁的非线性参数振动的非线性耦合方程,非线性偏微分模型,以及非线性偏微分—积分模型。
建立描述轴向运动黏弹性梁的横向振动模型。在控制方程的推导中,对黏弹性本构关系采用物质导数,而不是通常采用的只对时间取偏导数的本构关系。将多尺度方法应用于描述梁横向运动的偏微分方程和偏微分—积分方程。确定轴向运动黏弹性梁的横向振动共振特性,并通过数值方法加以验证。建立轴向变速运动黏弹性梁的横向线性参数振动的稳定性条件、导出轴向运动黏弹性梁横向非线性受迫振动和轴向变速黏弹性梁横向非线性参数振动的稳态响应并确定其稳定性,从理论上证明不参与共振的模态对主谐波共振或者亚谐波共振没有影响,并将结果与只对时间取偏导数的传统的Kelvin黏弹性本构关系相比较。发展微分求积法和有限差分法计算轴向变速运动黏弹性梁控制方程的数值解仿真线性亚谐波共振的失稳区域、非线性主共振的稳态幅频曲线以及非线性亚谐波共振的稳态幅频曲线。数值结果与多尺度方法导出的结果一致。
基于描述梁横向运动的非线性偏微分方程和非线性偏微分—积分方程的微分求积数值仿真结果,进行了非线性动力学行为分析。计算了加速轴向运动梁横向振动随速度扰动幅值的分岔图。运用非线性动力学时间序列分析方法计算了周期和混沌运动的Lyapunov指数。还应用相平面图,Poincaré映射和时间历程的初值敏感性考察振动的周期特性和混沌特性。
本文主要创新性工作包括下列几个方面:
1.发展数值方法首次从不同角度比较了轴向运动的弦线和梁的非线性耦合模型、非线性偏微分模型以及非线性偏微分—积分模型;
2.首次从理论上证明了在轴向运动黏弹性梁的受迫振动以及参数振动中,不参与共振的模态不影响主谐波共振和亚谐波共振的稳定性或稳态响应;
3.首次在轴向运动系统的横向振动的研究中,分别发展有限差分法和微分求积法对多尺度方法的近似解析结果加以验证;
4.首次在轴向运动系统的横向振动的研究中,基于数值结果采用非线性动力学时间序列分析方法识别动力学行为。
A variety of engineering systems with axially moving strings and beams involve power transmission chains, band saw blades, aerial cableways and paper sheets during processing. The study of the transverse vibration of the axially moving strings and beams is of great significance. The research of the transverse vibration in that case may pay contribution to the context of continuous gyroscopic systems.
The modeling, analysis and simulation of axially moving viscoelastic strings and beams are investigated in this dissertation. The governing equations of coupled planar of axially moving strings and beams are modeled, and nonlinear models of transverse motion of axially moving strings and beams are computationally investigated. The method of multiple scales is applied to mathematical models to calculate the stability and the steady-state response. The finite difference schemes and the differential quadrature schemes are respectively developed to numerically solve the equations of axially accelerating viscoelastic beams. And the numerical results confirm the results derived from the method of multiple scales. By analyzing the time series based on the numerical solutions of the differential equations, the nonlinear dynamical behaviors like bifurcation and chaos are identified. The main points of the concrete content are as follows:
Nonlinear models of transverse motion of axially moving strings and beams are computationally investigated. The governing equations of coupled planar is reduced to the partial differential equation of transverse motion by neglecting longitudinal terms partial differential equations. The integro-partial-differential equation of transverse motion is derived from the partial differential equation by averaging of the string disturbed tension. The finite difference schemes are developed to numerically solve the coupled equations, the the partial differential equation, and the integro-partial-differential equation. Free vibration of the strings and beams, forced vibration of the axially moving viscoelastic beams, and transverse parametric vibration of axially accelerating viscoelastic beams from the partial differential equations and the integro-partial-differential equations are respectively compared with the transverse component calculated from the coupled equation from the angles of transverse responses, the stability of steady-state responses and the transient transverse responses.
Modeling transverse vibration of nonlinear beams are investigated via numerical solutions of partial differential equations and an integro-partial-differential equation. The governing equation is derived from the viscoelastic constitution relation by using material derivative, not simply by the partial time derivative. The method of multiple scales is applied directly to the governing equations without discretization to calculate the instability regions and the steady-state response for axially accelerating viscoelastic linear beam, nonlinear forced vibration and nonlinear parametric resonance respectively. Neighboring mode has not affected primary resonance, and subharmonic resonance is theoretically proved. Results are compared to previous work in which the partial time derivative was used in the viscoelastic constitutive relation. The finite difference schemes and the differential quadrature schemes are respectively developed to numerically solve the equations of axially accelerating viscoelastic beams for numerical simulations the instability regions of linear subharmonic resonance, the steady-state response of nonlinear primary resonance and nonlinear subharmonic resonance. And the numerical results confirm the results derived from the method of multiple scales.
By analysis of the time series based on the numerical solutions to the partial differential equations and the integro-partial-differential equations calculated by the differential quadrature method, the nonlinear dynamical behaviors like bifurcation and chaos are identified. The bifurcation diagrams are presented in the case that the amplitude of speed fluctuation is varied while other parameters are fixed. The phase plane, the Poincare map, the Lyapunov exponent, and initial value sensitivity are used to identify the periodic motions or chaotic motions occurring in the transverse vibrations of the axially accelerating viscoelastic beam.
The main innovation of this dissertation are as follows:
1. For the first time it compares the coupled equations, the the partial differential equation, and the integro-partial-differential equation from many aspects based on the numerical solutions of the differential equations calculated by the finite difference method;
2. For the first time it theoretically proves that neighboring mode has not affected primary resonance in the transverse forced vibration and subharmonic resonance in the transverse parametric vibration of the axially accelerating viscoelastic beam;
3. For the first time it respectively develops the finite difference schemes and the differential quadrature schemes to numerically solve the equations of axially accelerating viscoelastic beams for confirming the results derived from the method of multiple scales.
本文研究轴向运动黏弹性弦线和梁的建模、分析和仿真。本文建立了轴向运动弦线和梁的耦合振动模型,并考察了不同的横向运动简化模型。基于横向运动模型,应用直接多尺度方法分析了轴向运动黏弹性梁的稳定性和稳态响应。发展了轴向运动黏弹性梁数值仿真的差分法和微分求积法,并将数值仿真结果与解析结果进行了比较。基于数值仿真结果用非线性动力学时间序列分析方法研究了分岔和混沌。具体内容包括下列几方面。
建立轴向运动弦线和梁的横向运动控制方程并通过数值仿真结果进行研究。若忽略耦合效应而仅考虑横向运动,平面耦合的非线性控制方程退化为横向非线性偏微分模型。若将扰动张力用在弦线或者梁上的平均值代替,就从非线性偏微分模型推导出非线性偏微分—积分模型。运用有限差分法分别从时间历程、稳态的幅频响应以及暂态的时间历程的角度比较弦线和梁自由振动、轴向运动黏弹性梁的非线性受迫振动以及速度带轴向变速运动黏弹性梁的非线性参数振动的非线性耦合方程,非线性偏微分模型,以及非线性偏微分—积分模型。
建立描述轴向运动黏弹性梁的横向振动模型。在控制方程的推导中,对黏弹性本构关系采用物质导数,而不是通常采用的只对时间取偏导数的本构关系。将多尺度方法应用于描述梁横向运动的偏微分方程和偏微分—积分方程。确定轴向运动黏弹性梁的横向振动共振特性,并通过数值方法加以验证。建立轴向变速运动黏弹性梁的横向线性参数振动的稳定性条件、导出轴向运动黏弹性梁横向非线性受迫振动和轴向变速黏弹性梁横向非线性参数振动的稳态响应并确定其稳定性,从理论上证明不参与共振的模态对主谐波共振或者亚谐波共振没有影响,并将结果与只对时间取偏导数的传统的Kelvin黏弹性本构关系相比较。发展微分求积法和有限差分法计算轴向变速运动黏弹性梁控制方程的数值解仿真线性亚谐波共振的失稳区域、非线性主共振的稳态幅频曲线以及非线性亚谐波共振的稳态幅频曲线。数值结果与多尺度方法导出的结果一致。
基于描述梁横向运动的非线性偏微分方程和非线性偏微分—积分方程的微分求积数值仿真结果,进行了非线性动力学行为分析。计算了加速轴向运动梁横向振动随速度扰动幅值的分岔图。运用非线性动力学时间序列分析方法计算了周期和混沌运动的Lyapunov指数。还应用相平面图,Poincaré映射和时间历程的初值敏感性考察振动的周期特性和混沌特性。
本文主要创新性工作包括下列几个方面:
1.发展数值方法首次从不同角度比较了轴向运动的弦线和梁的非线性耦合模型、非线性偏微分模型以及非线性偏微分—积分模型;
2.首次从理论上证明了在轴向运动黏弹性梁的受迫振动以及参数振动中,不参与共振的模态不影响主谐波共振和亚谐波共振的稳定性或稳态响应;
3.首次在轴向运动系统的横向振动的研究中,分别发展有限差分法和微分求积法对多尺度方法的近似解析结果加以验证;
4.首次在轴向运动系统的横向振动的研究中,基于数值结果采用非线性动力学时间序列分析方法识别动力学行为。
A variety of engineering systems with axially moving strings and beams involve power transmission chains, band saw blades, aerial cableways and paper sheets during processing. The study of the transverse vibration of the axially moving strings and beams is of great significance. The research of the transverse vibration in that case may pay contribution to the context of continuous gyroscopic systems.
The modeling, analysis and simulation of axially moving viscoelastic strings and beams are investigated in this dissertation. The governing equations of coupled planar of axially moving strings and beams are modeled, and nonlinear models of transverse motion of axially moving strings and beams are computationally investigated. The method of multiple scales is applied to mathematical models to calculate the stability and the steady-state response. The finite difference schemes and the differential quadrature schemes are respectively developed to numerically solve the equations of axially accelerating viscoelastic beams. And the numerical results confirm the results derived from the method of multiple scales. By analyzing the time series based on the numerical solutions of the differential equations, the nonlinear dynamical behaviors like bifurcation and chaos are identified. The main points of the concrete content are as follows:
Nonlinear models of transverse motion of axially moving strings and beams are computationally investigated. The governing equations of coupled planar is reduced to the partial differential equation of transverse motion by neglecting longitudinal terms partial differential equations. The integro-partial-differential equation of transverse motion is derived from the partial differential equation by averaging of the string disturbed tension. The finite difference schemes are developed to numerically solve the coupled equations, the the partial differential equation, and the integro-partial-differential equation. Free vibration of the strings and beams, forced vibration of the axially moving viscoelastic beams, and transverse parametric vibration of axially accelerating viscoelastic beams from the partial differential equations and the integro-partial-differential equations are respectively compared with the transverse component calculated from the coupled equation from the angles of transverse responses, the stability of steady-state responses and the transient transverse responses.
Modeling transverse vibration of nonlinear beams are investigated via numerical solutions of partial differential equations and an integro-partial-differential equation. The governing equation is derived from the viscoelastic constitution relation by using material derivative, not simply by the partial time derivative. The method of multiple scales is applied directly to the governing equations without discretization to calculate the instability regions and the steady-state response for axially accelerating viscoelastic linear beam, nonlinear forced vibration and nonlinear parametric resonance respectively. Neighboring mode has not affected primary resonance, and subharmonic resonance is theoretically proved. Results are compared to previous work in which the partial time derivative was used in the viscoelastic constitutive relation. The finite difference schemes and the differential quadrature schemes are respectively developed to numerically solve the equations of axially accelerating viscoelastic beams for numerical simulations the instability regions of linear subharmonic resonance, the steady-state response of nonlinear primary resonance and nonlinear subharmonic resonance. And the numerical results confirm the results derived from the method of multiple scales.
By analysis of the time series based on the numerical solutions to the partial differential equations and the integro-partial-differential equations calculated by the differential quadrature method, the nonlinear dynamical behaviors like bifurcation and chaos are identified. The bifurcation diagrams are presented in the case that the amplitude of speed fluctuation is varied while other parameters are fixed. The phase plane, the Poincare map, the Lyapunov exponent, and initial value sensitivity are used to identify the periodic motions or chaotic motions occurring in the transverse vibrations of the axially accelerating viscoelastic beam.
The main innovation of this dissertation are as follows:
1. For the first time it compares the coupled equations, the the partial differential equation, and the integro-partial-differential equation from many aspects based on the numerical solutions of the differential equations calculated by the finite difference method;
2. For the first time it theoretically proves that neighboring mode has not affected primary resonance in the transverse forced vibration and subharmonic resonance in the transverse parametric vibration of the axially accelerating viscoelastic beam;
3. For the first time it respectively develops the finite difference schemes and the differential quadrature schemes to numerically solve the equations of axially accelerating viscoelastic beams for confirming the results derived from the method of multiple scales.
引文
[1]Aiken J,An account of some experiments on rigidity produced by centrifugal force,The London,Edinburgh,and Dublin Philosophical Magazine and Journal of Science,1878,5(29)81-105
[2]Mote CDJr,Dynamic stability of axially moving materials.The Shock and Vibration Digest,1972,4(4):2-11
[3]Ulsoy AG,Mote CDJr.Band saw vibration and stability.The Shock and Vibration Digest,1978,10(1):3-15
[4]Ulsoy AG,Mote CDJr,Syzmani R.Proncipal developments in band saw vibration and stability research.Holz als Roh-und Werkstoff,,1978,36:273-280
[5]D'Angelo CⅢ,Slvarado NT,Wang KW,Mote CDJr.Current research on circular saw and band saw vibration and stability.The Shock and Vibration Digest,1985,17(5):11-23
[6]Wickert JA,Mote CDJr.Current research on the vibration and stability of axially-moving materials.The Shock and Vibration Digest,1988.20(5):3-13
[7]Wang KW,Liu SP.On the noise and vibration of chain drive systems.The Shock and Vibration Digest,1991,23(4):8-13
[8]Abrate AS.Vibration of belts and belt drives.Mechanism and Machine Theory,1992,27(6):645-659
[9]Wickert,JA,Mote C.D.Jr.Current research on the vibration and stability of axially moving materials.Shock Vib.Dig,1988,20(5):3-13
[10]陈立群,Zu JW.轴向运动弦线的纵向振动及其控制.力学进展,2001,31(4):535-546
[11]陈立群,Zu JW.平带驱动系统振动分析研究进展.力学与实践,2001,23(4):8-12转18
[12]Chen LQ,Analysis and control of transverse vibrations of axially moving strings.ASME Applied Mechanics Reviews,2005,58:p91-116
[13]Rega G.Nonlinear dynamics of suspended cables-part Ⅰ:modeling and analysis.ASME Applied Mechanics Reviews,2004,57(6):443-478
[14]Rega G.Nonlinear dynamics of suspended cables-part Ⅱ:deterministic phenomena.ASME Applied Mechanics Reviews,2004,57(6):479-513
[15]Kirchhoff G.Vorlesungen fiber Mathematische Physik:Mechanik.Druck und Verlag von B.G.Teubner,Leipzig,1867
[16]Carrier G.F.On the nonlinear vibration problem of the elastic string.Q.J.Appl.Math.1945, 3:157-165
[17]Carrier G.F.A note on the vibrating string.Q.J.Appl.Math.1949,7:97-101
[18]Narasirnha R.Nonlinear vibration of an elastic strings.J.Sound Vib.1968,8:134-155
[19]Oplinger DW:Frequency response of a nonlinear stretched string.J.Acoust.Soc.Am.1960,32:1529-1538
[20]Motleno TC,Tufillaro N.B.An experimental investigation into the dynamics of a string.Am.J.Phys.2004,72:1157-1167
[21]Christie IA Galerkin method for a nonlinear integro-differential wave system.Comput.Methods Appl.Mech.Eng.1984,44:229-237
[22]Bilbao S,Smith,J.O.Energy conserving finite difference schemes for nonlinear strings.Acustica,2005,91:299-311
[23]Liu IS,Rincon M.A.Effect of moving boundaries on the vibrating elastic string.Appl.Numer.Math.2003,47:159-172
[24]Peradze J.A numerical algorithm for the nonlinear Kirchhoff string equation.Numer.Math.2005,102:311-342
[25]Irvine H.M.,Caughey,T.K.The linear theory of free vibrations of a suspended cable.Proc.R.Soc.Lond.A 1974,341:299-315
[26]Shahruz S.M.Boundary control of Kirchhoff's non-linear string.Int.J.Control 1999,72:560-563
[27]Murthy GSS,Ramakrishna BS.Nonlinear character of resonance in stretched strings.J.Acoust.Soc.Am.1965,38:461-471
[28]Mote CD Jr,On the nonlinear oscillation of an axially moving string.J.Appl.Mech.1966,33:463-464
[29]LQ Chen,Modeling nonlinear vibration of an axially accelerating viscoelastic string undergoing 3-dimentional motion,The Second Internatiaonal conference on Dynamics,Vibration and Control,2006,23-26
[30]Chen LQ and W.J.Zhao,The energetics and the stability of axially moving Kirchhoff strings.Journal of the Acoustical Society of America,2005,117:55-58
[31]Chen LQ,Ding Hu,On two nonlinear models of the transversely vibrating string,Archive of Applied Mechanics,DOI:10.1007/s00419-007-0164-7(已录用)
[32]Wicket JA.Non-linear vibration of a traveling tensioned beam.International Journal of Non-Linear Mechanics,1991,27:503-517
[33]Yang XD,Chen LQ.Non-linear forced vibration of axially moving viscoelastic beams.Acta Mechanica Solida Sinica,2006,19(4):365-373
[34]LH Chen,W Zhang and YQ Liu,Modeling of nonlinear osicillations for viscoelastic moving belt using Hamilton's principle,preprints
[35]刘伟,张劲夫,粘弹性传送带的分岔特性和混沌振动分析,动力学与控制学报,2005,3(3):63-67
[36]Zhang L,Zu JW.Nonlinear vibration of parametrically excited moving belts,part Ⅰ:dynamic response,ASME Journal of Applied Mechanics,1999,66(2):396-402
[37]Zhang L,Zu JW.Nonlinear vibration of parametrically excited moving belts,part Ⅱ:stability analysis,ASME Journal of Applied Mechanics,1999,66(2):403-409
[38]Pellicano F,Fregolent A,Bertuzzi A,Vestroni F.Primary and parametric non-linear resonances of a power transmission belt:Experimental and theoretical analysis.Journal of Sound and Vibration,2001,244(4):669-684
[39]Suweken G,Van Horssen WT.On the weakly nonlinear,transversal vibrations of a conveyor belt with a low and time-varying velocity,Nonlinear Dynamics,2003,31(2):197-223
[40]Yao MH,Zhang W.Multi-pulse homoclinic orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt.International Journal of Nonlinear Sciences and Numerical Simulation,2005,6(1):37-45
[41]M Yao and W Zhang,Multi-pulse Shilnikov orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt,International Journal of Nonlinear Sciences and Numerical Simulation,2005,6:37-46
[42]Oz HR,Pakdemirli M,Boyaci H.Non-linear vibrations and stability of an axially moving beam with time-dependent velocity.International Journal of Non-Linear Mechanics,2001,36(1):107-115
[43]Chen LQ,Yang XD.Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation.Chaos,Solitons and Fractals,2006,27(3):748-757
[44]Chen LQ,Yang XD.Steady-state response of axially moving viscoelastic beams with pulsating speed:comparison of two nonlinear models.International Journal of Solids and Structures,2005,42(1):37-50
[45]Marynowski K,Kapitaniak T.Kelvin-Voigt versus Burgers internal damping in modeling of axially moving viscoelastic web.International Journal of Non-Linear Mechanics,2002,37(7):1147-1161
[46]Parker RG,Lin Y.Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations.ASME Journal of Applied Mechanics,2001,68(1):49-57
[47]G.Rega,W Lacarbonara,AH Nayfeh and CM Chin,Multiple resonances in suspended cables:direct versus reduced-order models,International Journal of Non-Linear Mechanics,1999,31:901-924
[48]Chen LQ.Principal parametric resonance of axially accelerating viscoelastic strings constituted by the Boltzmann superposition principle.Proc.R.Soc.Lond.A 2005,461:2701-2720
[49]王建军,邹西凤,李其权.轴向移动系统的参数振动问题研究.应用力学学报,2003,20:33-37
[50]M.Pakdemirli and AG Ulsoy,Stability analysis of an axially accelerating string,Journal of Sound and Vibration,1997,203:815-832
[51]Oz HR,Pakdemidi M.Vibrations of an axially moving beam with time dependent velocity.Journal of Sound and Vibration,1999,227:239-257
[52]Oz HR.On the vibrations of an axially traveling beam on fixed supports with variable velocity.Journal of Sound and Vibration,2001,239:556-564
[53]Chen LQ,Yang XD and Cheng CJ,Dynamic stability of an axially accelerating viscoelastic beam.European Journal of Mechanics A/Solids,2004,23:659-666
[54]LQ Chen and XD Yang,Stability in parametric resonances of an axially accelerating beam constituted by Boltzmann's superposition principle,Journal of Sound and Vibration,2006,289:54-65
[55]LQ Chen,XD Yang.Stability in parametric resonance of an axially moving viscoelastic beam with time-dependent velocity.Journal of Soundand Vibration,2005,284:879-891
[56]Zajaczkowski J,Lipinski J.Instability of the motion of a beam of periodically varying length.Journal of Sound and Vibration.1979,63:9-18
[57]Zajaczkowski J,Yamada G,Further results on the motion of a beam of periodically varying length.Journal of Sound and Vibration.1980,68:173-180
[58]Ariartnam ST,Asokanthan SF.Torsional oscillations in moving bands.ASME Journal of Vibration,Acoustics,Stress,and Reliability in Design,1988,110(3):350-355
[59]Oz HR,Pakdemirli M.Vibrations of an axially moving beam with time dependent velocity.Journal of Sound and Vibration,1999,227:239-257
[60]Ding Hu,Chen LQ,Stability of axially accelerating viscoelastic beams multi-scale analysis with numerical confirmations,European Journal of Mechanics A/Solids,EJMSOL-D-07-00101R1(已录用)
[61]Nayfeh AH,Mook DT.Nonlinear Oscillations,Wiley,New York,1979
[62]Ibrahim RA,Afaneh AA and Lee BH.Structural modal multifurcation with internal resonance Part 1:deterministic approach,ASME Journal of Vibration and Acoustics.1993,115:182-192
[63]Lee CL,Perkins NC.Nonlinear oscillations of suspended cables containing a two-to-one internal resonance.Nonlinear Dynamics,1992,3:465-490
[64]Riedel CH,Tan CA.Coupled,forced response of an axially moving strip with internal resonance,International Journal of Non-Linear Mechanics,2002,37:101-116
[65]张伟,温洪波,姚明辉.黏弹性传动带1:3内共振时的周期和混沌运动.力学学报,2004,36:443-454
[66]Parker RG,Lin Y.Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations.ASME Journal of Applied Mechanics,2001,68(1):49-57
[67]Oz HR,Pakdemirli M,Boyaci.H Non-linear vibrations and stability of an axially moving beam with time-dependent velocity.International Journal of Non-Linear Mechanics,2001,36(1):107-115
[68]冯志华,胡海岩.内共振条件下直线运动梁的动力稳定性.力学学报,2002,34:389-400
[69]Rezaee M,Khadem SE.Non-linear free vibration analysis of a string under bending moment effects using the perturbation method.Journal of Sound and Vibration,2002,254(4):677-691
[70]冯志华,胡海岩.基础直线运动柔性梁的非线性动力学.振动工程学报,2004,17:126-131
[71]陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.,力学学报,2005,37:57-63
[72]Kong L,Parker RG.Vibration of an axially moving beam wrapping on fixed pulleys.Journal of Sound and Vibration,2005,280:1066-1074
[73]Sze KY,Chen SH,Huang JL.The incremental harmonic balance method for nonlinear vibration of axially moving beams.Journal of Sound and Vibration,2005,281:611-626
[74]陈树辉,黄建亮.轴向运动体系横向非线性振动的联合共振.振动工程学报,2005,18:19-23
[75]Chen SH,Huang JL,Sze K.Multi-dimensional Lindstedt-Poincare method for nonlinear vibration of axially moving beams.Journal of Sound and Vibration,2007,in press
[76]Marynowski K.Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension.Chaos,Solitons and Fractals,2004,21:481-490
[77]H.R.Oz,M.Pakdemirli and H.Boyaci,Non-linear vibrations and stability of an axially moving beam with time dependent velocity.International Journal of Non-Linear Mechanics,2001,36:107-115
[78]Yang XD,Chen LQ.Bifurcation and chaos of an axially accelerating viscoelastic beam.Chaos,Solitons and Fractals.2005,23:249-258
[79]Tan CA and Chung CH),Transfer function formulation of constrained distributed parameter systems,part 1:theory,ASME Journal of Applied Mechanics,1993,60(4) 1004-1011
[80]Yang B and Tan CA,Transfer functions of one-dimensional distributed parameter systems,ASME Journal of Applied Mechanics,1992,59(4):1009-1014
[81]Yue MG.Moving contact and coupling in belt strand vibration.ASME Structural Dynamics of Large Scale and Complex Systems,1993,59:139-144
[82]Yue XG.Belt vibration considering moving contact and parametric excitation.ASME Journal of Mechanical Design,1995,117:1024-1030
[83]Takikonda BO,Baruh H.Dynamics and control of a translating flexible beam with a prismatic joint.ASME Journal of Dynamic Systems,Measurement,and Control,1992,114:422-427
[84]Chen LQ,Yang XD.Steady-state response of axially moving viscoelastic beam with pulsating speed:comparison of two nonlinear models.International Journal of Solids and Structures.2005,42(1):37-50
[85]Yang XD,Chen LQ.Bifurcation and chaos of an axially accelerating viscoelastic beam.Chaos,Solitons and Fractals.2005,23:249-258
[86]张红星,张伟,姚明辉.粘弹性传输带非线性振动实验研究.动力学与控制学报,2007,5(4):361-364
[87]Zhang L,Zu JW.Non-linear vibrations of viscoelastic moving belts,part 1:free vibration analysis.Journal of Sound and Vibration,1998,216(1):75-91
[88]Zhang L,Zu JW.Non-linear vibrations of viscoelastic moving belts,part 2:forced vibration analysis.Journal of Sound and Vibration,1998,216(1):93-103
[89]Zhang L,Zu JW.Nonlinear vibration of parametrically excited moving belts.ASME Journal of Applied Mechanics,1998,66:396-409
[90]Chen LQ,Wu J,Zu JW.Asymptotic nonlinear behaviors in transverse vibration of an axially accelerating viscoelastic string.Nonlinear Dynamics,2004,35(4):347-360
[91]Zhang W,Yao MH.Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt.Chaos,Solitons and Fractals,2006,28:42-66
[92]Zhang NH.Dynamic analysis of an axially moving viscoelastic string by the Galerkin method using translating string eigenfunctions.Chaos,Solitons,and Fractals,2007,corrected proof online
[93]Chen LH,Zhang W,Liu YQ.Modeling of nonlinear oscillations for viscoelastic moving belt using Hamilton's principle.ASME Journal of Vibration and Acoustics,2007,129(1):128-132
[94]李映辉,高庆,蹇开林,殷学纲.黏弹性带动力响应分析.应用数学和力学,2003,24:1191-1196
[95]Mockenstrum EM,Guo JP.Nonlinear vibration of parametrically excited,viscoelastic,axially moving strings.ASME Journal of Applied Mechanics,2005,72(3):374-380
[96]Marynowski K.Two-dimensional rheological element in modeling of axially moving viscoelastic web.European Journal of Mechanics A/Solid,2006,25:729-744
[97]丁虎,陈立群.黏弹性变速度轴向运动梁的稳定性分析,机械强度,已录用
[98]JA Wickert,CD Mote Jr,Response and discretization methods for axially moving materials,Appl.Mech.Rev.1991,44:279-284
[99]G.Chakraborty,A.K.Mallik,Wave propagation in and vibration of a traveling beam with and without non-linear effects,part 2:forced vibration,J Sound Vib,2000,236:291-305
[100]Malatkar P,Nayfeh AH.Calculation of the jump frequencies in the response of s.d.o.f.non-linear systems.Journal of Sound and Vibration.2002,254(5):1005-1011
[101]G.Chakraborty,AK Mallik,H.Hatwal,Non-linear vibration of a travelling beam,Int.J.Non-Linear Mech.1999,34:655-670
[102]Zhang L,Zu JW.Nonlinear vibration of viscoelastic moving belts,Part Ⅱ:forced vibration analysis.Journal of Sound and Vibration.1998,216(1):93-105
[103]Pellicano F,Vestroni F.Complex dynamic of high-speed axially moving systems.Journal of Sound and Vibration,2002,258(1):31-44
[104]Chen LQ,Yang XD.Vibration and stability of an axially moving viscoelastic beam with hybrid supports.European Journal of Mechanics A/Solids,2006,25(6),996-1008
[105]Chen LQ,Yang XD.Nonlinear free vibration of an axially moving beam:comparison of two models.Journal of Sound and Vibration,2007,299(1):348-354
[106]陈健,尹晓春.回转梁动力学方程求解的数值方法研究.机械强度,2004,26:484-488
[107]周洪刚,朱凌,郭乙木.轴向加速度运动弦线横向振动的数值计算方法.机械强度,2004.26:16-19
[108]Wicket JA.Non-linear vibration of a traveling tensioned beam.International Journal of Non-Linear Mechanics,1991,27:503-517
[109]Wicket JA,Mote C.D.Jr..Classical vibration analysis of axially moving continua.ASME Journal of Applied Mechanics,1990,57:738-744
[110]Nayfeh AH,Nayfeh JF and Mook DT.On methods for continuous systems with quadratic and cubic nonlinearities.Nonlinear Dynamics.1992,3:145-162
[111]Pakdemirli M,Nayfeh SA and Nayfeh AH.Analysis of non-to-one autoparametric resonances in cables-discretization vs.direct treatment.Nonlinear Dynamics.1995,8:65-83
[112]Nayfeh AH,Nayfeh SA and Jpakdemirli M.in Nonlinear Dynamics and Stochastic Mechanics.N.S.Namachchivaya and W.Kliemann editors:CRC Press,1995,175-200
[113]刘延柱,陈文良,陈立群,振动力学.高等教育出版社,北京,2000
[114]胡海岩,应用非线性动力学,航空工业出版社,北京,2000
[115]陈予恕,非线性振动,高等教育出版社,北京,2002
[116]刘延柱,陈立群,非线性振动,高等教育出版社,北京,2003
[117]陈立群,刘延柱,非线性动力学,高等教育出版社,北京,2007
[118]陈立群,戈新生,徐凯宇,薛纭,理论力学,清华大学出版社,北京,2006.
[119]陈树辉,刘守圭,黄建亮,关于轴向运动梁科氏加速度的注释,力学与实践28(1),p79-80,2006.
[120]Arthur P.Boresi,Ken P.Chong,Sunil Saigal,Approximate Solution Methods in Engineering Mechanics[M],Wiley,New York,2003,42-47
[121]Jones DI,Handbook of viscoelastic vibration damping[M],Wiley,New York,2001
[122]李晓军,陈立群,轴向运动简支-固支梁的横向振动和稳定性,机械强度,2006,28:654-657
[123]Chen LQ,Zu JW.Solvability condition in multi-scale analysis of gyroscopic continua.Journal of Sound and Vibration,accepted
[124]Bert CW,Malik M.The differential quadrature method in computational mechanics:A review[J],Appl Mech Rev.1996.49:1-28
[125]Wang X,Bert CW.A new approach in applying differntial quadrature to static and free vibration analyses of beams and plates.Journal of Sound and Vibration,1993,162:566-572
[126]Wolf A,Swift J,Swinney H,et al.Determining Lyapunov Exponents from a Time Series[J].Physica D,1985,16:285-317
[2]Mote CDJr,Dynamic stability of axially moving materials.The Shock and Vibration Digest,1972,4(4):2-11
[3]Ulsoy AG,Mote CDJr.Band saw vibration and stability.The Shock and Vibration Digest,1978,10(1):3-15
[4]Ulsoy AG,Mote CDJr,Syzmani R.Proncipal developments in band saw vibration and stability research.Holz als Roh-und Werkstoff,,1978,36:273-280
[5]D'Angelo CⅢ,Slvarado NT,Wang KW,Mote CDJr.Current research on circular saw and band saw vibration and stability.The Shock and Vibration Digest,1985,17(5):11-23
[6]Wickert JA,Mote CDJr.Current research on the vibration and stability of axially-moving materials.The Shock and Vibration Digest,1988.20(5):3-13
[7]Wang KW,Liu SP.On the noise and vibration of chain drive systems.The Shock and Vibration Digest,1991,23(4):8-13
[8]Abrate AS.Vibration of belts and belt drives.Mechanism and Machine Theory,1992,27(6):645-659
[9]Wickert,JA,Mote C.D.Jr.Current research on the vibration and stability of axially moving materials.Shock Vib.Dig,1988,20(5):3-13
[10]陈立群,Zu JW.轴向运动弦线的纵向振动及其控制.力学进展,2001,31(4):535-546
[11]陈立群,Zu JW.平带驱动系统振动分析研究进展.力学与实践,2001,23(4):8-12转18
[12]Chen LQ,Analysis and control of transverse vibrations of axially moving strings.ASME Applied Mechanics Reviews,2005,58:p91-116
[13]Rega G.Nonlinear dynamics of suspended cables-part Ⅰ:modeling and analysis.ASME Applied Mechanics Reviews,2004,57(6):443-478
[14]Rega G.Nonlinear dynamics of suspended cables-part Ⅱ:deterministic phenomena.ASME Applied Mechanics Reviews,2004,57(6):479-513
[15]Kirchhoff G.Vorlesungen fiber Mathematische Physik:Mechanik.Druck und Verlag von B.G.Teubner,Leipzig,1867
[16]Carrier G.F.On the nonlinear vibration problem of the elastic string.Q.J.Appl.Math.1945, 3:157-165
[17]Carrier G.F.A note on the vibrating string.Q.J.Appl.Math.1949,7:97-101
[18]Narasirnha R.Nonlinear vibration of an elastic strings.J.Sound Vib.1968,8:134-155
[19]Oplinger DW:Frequency response of a nonlinear stretched string.J.Acoust.Soc.Am.1960,32:1529-1538
[20]Motleno TC,Tufillaro N.B.An experimental investigation into the dynamics of a string.Am.J.Phys.2004,72:1157-1167
[21]Christie IA Galerkin method for a nonlinear integro-differential wave system.Comput.Methods Appl.Mech.Eng.1984,44:229-237
[22]Bilbao S,Smith,J.O.Energy conserving finite difference schemes for nonlinear strings.Acustica,2005,91:299-311
[23]Liu IS,Rincon M.A.Effect of moving boundaries on the vibrating elastic string.Appl.Numer.Math.2003,47:159-172
[24]Peradze J.A numerical algorithm for the nonlinear Kirchhoff string equation.Numer.Math.2005,102:311-342
[25]Irvine H.M.,Caughey,T.K.The linear theory of free vibrations of a suspended cable.Proc.R.Soc.Lond.A 1974,341:299-315
[26]Shahruz S.M.Boundary control of Kirchhoff's non-linear string.Int.J.Control 1999,72:560-563
[27]Murthy GSS,Ramakrishna BS.Nonlinear character of resonance in stretched strings.J.Acoust.Soc.Am.1965,38:461-471
[28]Mote CD Jr,On the nonlinear oscillation of an axially moving string.J.Appl.Mech.1966,33:463-464
[29]LQ Chen,Modeling nonlinear vibration of an axially accelerating viscoelastic string undergoing 3-dimentional motion,The Second Internatiaonal conference on Dynamics,Vibration and Control,2006,23-26
[30]Chen LQ and W.J.Zhao,The energetics and the stability of axially moving Kirchhoff strings.Journal of the Acoustical Society of America,2005,117:55-58
[31]Chen LQ,Ding Hu,On two nonlinear models of the transversely vibrating string,Archive of Applied Mechanics,DOI:10.1007/s00419-007-0164-7(已录用)
[32]Wicket JA.Non-linear vibration of a traveling tensioned beam.International Journal of Non-Linear Mechanics,1991,27:503-517
[33]Yang XD,Chen LQ.Non-linear forced vibration of axially moving viscoelastic beams.Acta Mechanica Solida Sinica,2006,19(4):365-373
[34]LH Chen,W Zhang and YQ Liu,Modeling of nonlinear osicillations for viscoelastic moving belt using Hamilton's principle,preprints
[35]刘伟,张劲夫,粘弹性传送带的分岔特性和混沌振动分析,动力学与控制学报,2005,3(3):63-67
[36]Zhang L,Zu JW.Nonlinear vibration of parametrically excited moving belts,part Ⅰ:dynamic response,ASME Journal of Applied Mechanics,1999,66(2):396-402
[37]Zhang L,Zu JW.Nonlinear vibration of parametrically excited moving belts,part Ⅱ:stability analysis,ASME Journal of Applied Mechanics,1999,66(2):403-409
[38]Pellicano F,Fregolent A,Bertuzzi A,Vestroni F.Primary and parametric non-linear resonances of a power transmission belt:Experimental and theoretical analysis.Journal of Sound and Vibration,2001,244(4):669-684
[39]Suweken G,Van Horssen WT.On the weakly nonlinear,transversal vibrations of a conveyor belt with a low and time-varying velocity,Nonlinear Dynamics,2003,31(2):197-223
[40]Yao MH,Zhang W.Multi-pulse homoclinic orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt.International Journal of Nonlinear Sciences and Numerical Simulation,2005,6(1):37-45
[41]M Yao and W Zhang,Multi-pulse Shilnikov orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt,International Journal of Nonlinear Sciences and Numerical Simulation,2005,6:37-46
[42]Oz HR,Pakdemirli M,Boyaci H.Non-linear vibrations and stability of an axially moving beam with time-dependent velocity.International Journal of Non-Linear Mechanics,2001,36(1):107-115
[43]Chen LQ,Yang XD.Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation.Chaos,Solitons and Fractals,2006,27(3):748-757
[44]Chen LQ,Yang XD.Steady-state response of axially moving viscoelastic beams with pulsating speed:comparison of two nonlinear models.International Journal of Solids and Structures,2005,42(1):37-50
[45]Marynowski K,Kapitaniak T.Kelvin-Voigt versus Burgers internal damping in modeling of axially moving viscoelastic web.International Journal of Non-Linear Mechanics,2002,37(7):1147-1161
[46]Parker RG,Lin Y.Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations.ASME Journal of Applied Mechanics,2001,68(1):49-57
[47]G.Rega,W Lacarbonara,AH Nayfeh and CM Chin,Multiple resonances in suspended cables:direct versus reduced-order models,International Journal of Non-Linear Mechanics,1999,31:901-924
[48]Chen LQ.Principal parametric resonance of axially accelerating viscoelastic strings constituted by the Boltzmann superposition principle.Proc.R.Soc.Lond.A 2005,461:2701-2720
[49]王建军,邹西凤,李其权.轴向移动系统的参数振动问题研究.应用力学学报,2003,20:33-37
[50]M.Pakdemirli and AG Ulsoy,Stability analysis of an axially accelerating string,Journal of Sound and Vibration,1997,203:815-832
[51]Oz HR,Pakdemidi M.Vibrations of an axially moving beam with time dependent velocity.Journal of Sound and Vibration,1999,227:239-257
[52]Oz HR.On the vibrations of an axially traveling beam on fixed supports with variable velocity.Journal of Sound and Vibration,2001,239:556-564
[53]Chen LQ,Yang XD and Cheng CJ,Dynamic stability of an axially accelerating viscoelastic beam.European Journal of Mechanics A/Solids,2004,23:659-666
[54]LQ Chen and XD Yang,Stability in parametric resonances of an axially accelerating beam constituted by Boltzmann's superposition principle,Journal of Sound and Vibration,2006,289:54-65
[55]LQ Chen,XD Yang.Stability in parametric resonance of an axially moving viscoelastic beam with time-dependent velocity.Journal of Soundand Vibration,2005,284:879-891
[56]Zajaczkowski J,Lipinski J.Instability of the motion of a beam of periodically varying length.Journal of Sound and Vibration.1979,63:9-18
[57]Zajaczkowski J,Yamada G,Further results on the motion of a beam of periodically varying length.Journal of Sound and Vibration.1980,68:173-180
[58]Ariartnam ST,Asokanthan SF.Torsional oscillations in moving bands.ASME Journal of Vibration,Acoustics,Stress,and Reliability in Design,1988,110(3):350-355
[59]Oz HR,Pakdemirli M.Vibrations of an axially moving beam with time dependent velocity.Journal of Sound and Vibration,1999,227:239-257
[60]Ding Hu,Chen LQ,Stability of axially accelerating viscoelastic beams multi-scale analysis with numerical confirmations,European Journal of Mechanics A/Solids,EJMSOL-D-07-00101R1(已录用)
[61]Nayfeh AH,Mook DT.Nonlinear Oscillations,Wiley,New York,1979
[62]Ibrahim RA,Afaneh AA and Lee BH.Structural modal multifurcation with internal resonance Part 1:deterministic approach,ASME Journal of Vibration and Acoustics.1993,115:182-192
[63]Lee CL,Perkins NC.Nonlinear oscillations of suspended cables containing a two-to-one internal resonance.Nonlinear Dynamics,1992,3:465-490
[64]Riedel CH,Tan CA.Coupled,forced response of an axially moving strip with internal resonance,International Journal of Non-Linear Mechanics,2002,37:101-116
[65]张伟,温洪波,姚明辉.黏弹性传动带1:3内共振时的周期和混沌运动.力学学报,2004,36:443-454
[66]Parker RG,Lin Y.Parametric instability of axially moving media subjected to multifrequency tension and speed fluctuations.ASME Journal of Applied Mechanics,2001,68(1):49-57
[67]Oz HR,Pakdemirli M,Boyaci.H Non-linear vibrations and stability of an axially moving beam with time-dependent velocity.International Journal of Non-Linear Mechanics,2001,36(1):107-115
[68]冯志华,胡海岩.内共振条件下直线运动梁的动力稳定性.力学学报,2002,34:389-400
[69]Rezaee M,Khadem SE.Non-linear free vibration analysis of a string under bending moment effects using the perturbation method.Journal of Sound and Vibration,2002,254(4):677-691
[70]冯志华,胡海岩.基础直线运动柔性梁的非线性动力学.振动工程学报,2004,17:126-131
[71]陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.,力学学报,2005,37:57-63
[72]Kong L,Parker RG.Vibration of an axially moving beam wrapping on fixed pulleys.Journal of Sound and Vibration,2005,280:1066-1074
[73]Sze KY,Chen SH,Huang JL.The incremental harmonic balance method for nonlinear vibration of axially moving beams.Journal of Sound and Vibration,2005,281:611-626
[74]陈树辉,黄建亮.轴向运动体系横向非线性振动的联合共振.振动工程学报,2005,18:19-23
[75]Chen SH,Huang JL,Sze K.Multi-dimensional Lindstedt-Poincare method for nonlinear vibration of axially moving beams.Journal of Sound and Vibration,2007,in press
[76]Marynowski K.Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension.Chaos,Solitons and Fractals,2004,21:481-490
[77]H.R.Oz,M.Pakdemirli and H.Boyaci,Non-linear vibrations and stability of an axially moving beam with time dependent velocity.International Journal of Non-Linear Mechanics,2001,36:107-115
[78]Yang XD,Chen LQ.Bifurcation and chaos of an axially accelerating viscoelastic beam.Chaos,Solitons and Fractals.2005,23:249-258
[79]Tan CA and Chung CH),Transfer function formulation of constrained distributed parameter systems,part 1:theory,ASME Journal of Applied Mechanics,1993,60(4) 1004-1011
[80]Yang B and Tan CA,Transfer functions of one-dimensional distributed parameter systems,ASME Journal of Applied Mechanics,1992,59(4):1009-1014
[81]Yue MG.Moving contact and coupling in belt strand vibration.ASME Structural Dynamics of Large Scale and Complex Systems,1993,59:139-144
[82]Yue XG.Belt vibration considering moving contact and parametric excitation.ASME Journal of Mechanical Design,1995,117:1024-1030
[83]Takikonda BO,Baruh H.Dynamics and control of a translating flexible beam with a prismatic joint.ASME Journal of Dynamic Systems,Measurement,and Control,1992,114:422-427
[84]Chen LQ,Yang XD.Steady-state response of axially moving viscoelastic beam with pulsating speed:comparison of two nonlinear models.International Journal of Solids and Structures.2005,42(1):37-50
[85]Yang XD,Chen LQ.Bifurcation and chaos of an axially accelerating viscoelastic beam.Chaos,Solitons and Fractals.2005,23:249-258
[86]张红星,张伟,姚明辉.粘弹性传输带非线性振动实验研究.动力学与控制学报,2007,5(4):361-364
[87]Zhang L,Zu JW.Non-linear vibrations of viscoelastic moving belts,part 1:free vibration analysis.Journal of Sound and Vibration,1998,216(1):75-91
[88]Zhang L,Zu JW.Non-linear vibrations of viscoelastic moving belts,part 2:forced vibration analysis.Journal of Sound and Vibration,1998,216(1):93-103
[89]Zhang L,Zu JW.Nonlinear vibration of parametrically excited moving belts.ASME Journal of Applied Mechanics,1998,66:396-409
[90]Chen LQ,Wu J,Zu JW.Asymptotic nonlinear behaviors in transverse vibration of an axially accelerating viscoelastic string.Nonlinear Dynamics,2004,35(4):347-360
[91]Zhang W,Yao MH.Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt.Chaos,Solitons and Fractals,2006,28:42-66
[92]Zhang NH.Dynamic analysis of an axially moving viscoelastic string by the Galerkin method using translating string eigenfunctions.Chaos,Solitons,and Fractals,2007,corrected proof online
[93]Chen LH,Zhang W,Liu YQ.Modeling of nonlinear oscillations for viscoelastic moving belt using Hamilton's principle.ASME Journal of Vibration and Acoustics,2007,129(1):128-132
[94]李映辉,高庆,蹇开林,殷学纲.黏弹性带动力响应分析.应用数学和力学,2003,24:1191-1196
[95]Mockenstrum EM,Guo JP.Nonlinear vibration of parametrically excited,viscoelastic,axially moving strings.ASME Journal of Applied Mechanics,2005,72(3):374-380
[96]Marynowski K.Two-dimensional rheological element in modeling of axially moving viscoelastic web.European Journal of Mechanics A/Solid,2006,25:729-744
[97]丁虎,陈立群.黏弹性变速度轴向运动梁的稳定性分析,机械强度,已录用
[98]JA Wickert,CD Mote Jr,Response and discretization methods for axially moving materials,Appl.Mech.Rev.1991,44:279-284
[99]G.Chakraborty,A.K.Mallik,Wave propagation in and vibration of a traveling beam with and without non-linear effects,part 2:forced vibration,J Sound Vib,2000,236:291-305
[100]Malatkar P,Nayfeh AH.Calculation of the jump frequencies in the response of s.d.o.f.non-linear systems.Journal of Sound and Vibration.2002,254(5):1005-1011
[101]G.Chakraborty,AK Mallik,H.Hatwal,Non-linear vibration of a travelling beam,Int.J.Non-Linear Mech.1999,34:655-670
[102]Zhang L,Zu JW.Nonlinear vibration of viscoelastic moving belts,Part Ⅱ:forced vibration analysis.Journal of Sound and Vibration.1998,216(1):93-105
[103]Pellicano F,Vestroni F.Complex dynamic of high-speed axially moving systems.Journal of Sound and Vibration,2002,258(1):31-44
[104]Chen LQ,Yang XD.Vibration and stability of an axially moving viscoelastic beam with hybrid supports.European Journal of Mechanics A/Solids,2006,25(6),996-1008
[105]Chen LQ,Yang XD.Nonlinear free vibration of an axially moving beam:comparison of two models.Journal of Sound and Vibration,2007,299(1):348-354
[106]陈健,尹晓春.回转梁动力学方程求解的数值方法研究.机械强度,2004,26:484-488
[107]周洪刚,朱凌,郭乙木.轴向加速度运动弦线横向振动的数值计算方法.机械强度,2004.26:16-19
[108]Wicket JA.Non-linear vibration of a traveling tensioned beam.International Journal of Non-Linear Mechanics,1991,27:503-517
[109]Wicket JA,Mote C.D.Jr..Classical vibration analysis of axially moving continua.ASME Journal of Applied Mechanics,1990,57:738-744
[110]Nayfeh AH,Nayfeh JF and Mook DT.On methods for continuous systems with quadratic and cubic nonlinearities.Nonlinear Dynamics.1992,3:145-162
[111]Pakdemirli M,Nayfeh SA and Nayfeh AH.Analysis of non-to-one autoparametric resonances in cables-discretization vs.direct treatment.Nonlinear Dynamics.1995,8:65-83
[112]Nayfeh AH,Nayfeh SA and Jpakdemirli M.in Nonlinear Dynamics and Stochastic Mechanics.N.S.Namachchivaya and W.Kliemann editors:CRC Press,1995,175-200
[113]刘延柱,陈文良,陈立群,振动力学.高等教育出版社,北京,2000
[114]胡海岩,应用非线性动力学,航空工业出版社,北京,2000
[115]陈予恕,非线性振动,高等教育出版社,北京,2002
[116]刘延柱,陈立群,非线性振动,高等教育出版社,北京,2003
[117]陈立群,刘延柱,非线性动力学,高等教育出版社,北京,2007
[118]陈立群,戈新生,徐凯宇,薛纭,理论力学,清华大学出版社,北京,2006.
[119]陈树辉,刘守圭,黄建亮,关于轴向运动梁科氏加速度的注释,力学与实践28(1),p79-80,2006.
[120]Arthur P.Boresi,Ken P.Chong,Sunil Saigal,Approximate Solution Methods in Engineering Mechanics[M],Wiley,New York,2003,42-47
[121]Jones DI,Handbook of viscoelastic vibration damping[M],Wiley,New York,2001
[122]李晓军,陈立群,轴向运动简支-固支梁的横向振动和稳定性,机械强度,2006,28:654-657
[123]Chen LQ,Zu JW.Solvability condition in multi-scale analysis of gyroscopic continua.Journal of Sound and Vibration,accepted
[124]Bert CW,Malik M.The differential quadrature method in computational mechanics:A review[J],Appl Mech Rev.1996.49:1-28
[125]Wang X,Bert CW.A new approach in applying differntial quadrature to static and free vibration analyses of beams and plates.Journal of Sound and Vibration,1993,162:566-572
[126]Wolf A,Swift J,Swinney H,et al.Determining Lyapunov Exponents from a Time Series[J].Physica D,1985,16:285-317