【作者】 丁虎；

【导师】 陈立群；

【作者基本信息】 上海大学 ， 一般力学与力学基础， 2008， 博士

【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。更多还原

【Abstract】 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.更多还原