轴向变速黏弹性梁渐近摄动分析及其数值验证
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摘要
许多工程系统装置,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、单索架空索道等均可模型化为轴向运动梁。因此,轴向运动梁的横向振动问题的研究具有重要的工程应用价值。同时,作为典型的陀螺连续体,轴向运动梁的分析方法可以推广到更复杂的陀螺连续体,如轴向运动或旋转的缆索、板和壳等,因此,对它的研究也具有重要的理论意义。
本文研究了由轴向运动速度振荡引起的黏弹性梁的横向参数振动,发展了轴向运动黏弹性梁的渐近分析的近似解析方法和微分求积的数值方法,分析了轴向运动黏弹性梁的线性横向振动稳定性边界和非线性横向振动稳态幅频响应,并将数值分析结果与近似解析分析结果进行了比较。具体内容包括:
一方面,研究了基于Kelvin模型本构关系描述的轴向运动黏弹性梁横向线性振动的稳定性区域和横向非线性振动的稳态幅频响应。Kelvin模型本构关系引入了物质时间导数。应用渐近摄动法分析了轴向运动黏弹性梁横向参数振动各阶模态主共振和多模态组合共振的线性稳定性区域,推导出了渐近摄动法的可解性条件,并得到了主共振和组合共振的稳定性失稳边界;分析了非线性轴向运动黏弹性梁组合参数共振的稳态幅频响应,推导出渐近摄动法的可解性条件,得到稳态响应非零解的振幅和存在条件,应用Routh-Hurwitz判据得到组合共振稳态响应的零解和非零解稳定性条件。利用微分求积法数值研究了轴向运动黏弹性梁横向振动的稳定性区域以及稳态幅频响应,实现了数值结果与解析结果的相互验证。
另一方面,将三参数黏弹性本构关系引入了轴向运动梁横向振动的线性和非线性控制方程,发展渐近摄动近似解析方法研究了三参数描述的轴向运动黏弹性梁在轴向运动速度振荡引起的的横向振动的线性稳定性区域以及非线性稳态幅频响应,并发展微分求积数值方法对近似解析结果加以验证,数值结果表明,渐近摄动法在轴向运动梁横向振动分析中有着较高精度。
本文主要创新工作包括:
1.首次将渐近摄动法的思路应用于分析轴向运动黏弹性线性梁的稳定性和非线性梁稳态响应;
2.使用微分求积法数值求解轴向运动黏弹性横向振动,并验证上述问题的近似解析分析结果;
3.建立了轴向运动三参数模型黏弹性线性和非线性梁的控制方程,并使用渐近摄动法和微分求积法进行研究;
4.使用解析方法和数值方法重新研究了Kelvin模型的线性稳定性和非线性稳态响应,得到了与多尺度方法研究的相同结果。
Axially moving beams can represent many engineering devices, such as band saws, power transmission belts, aerial cable tramways, crane hoist cables, flexible robotic manipulators, and spacecraft deploying appendages. However, vibrations associated with the devices have limited their applications. Therefore, understanding transverse vibrations of axially moving beams is important for the design of the devices. The investigations on vibrations of axially moving beams have theoretical as well, because an axially moving beam is a typical representative of distributed gyroscopic systems. The method of analyzing vibrations of an axially moving beam can be applied to other more complicated distributed gyroscopic systems. In this paper, an asymptotic perturbation method is proposed to and the differential quadrature scheme is developed to investigate stability and steady-state response of the parameter vibrations of an axially moving viscoelastic beam. Meanwhile, the numerical calculations validate the analytical results.
Stability and steady-state response of an axially moving viscoelastic beam constituted by Kelvin model are investigated. The material time derivative is used in the viscoelastic constitutive relation. Asymptotic perturbation method is applied to analyze stability region of transverse parameter vibration of axially moving beam for the principle and summation resonance. The solvability condition and instability boundary of stability are obtained for the principle and summation resonance via asymptotic analysis. Nonlinear steady-state response is investigated in summation parameter resonance of axially moving viscoelastic beam. Nontrivial amplitude and existence condition of nontrivial solution for steady-state response of parameter vibrations are obtained. Base on the Routh-Hurvitz criterion, stability condition of trivial and nontrivial solutions in summation parameter resonance are obtained when the steady-state response occurs. The differential quadrature scheme is developed to study numerically stability region and steady-state response of axially moving viscoelastic beam. Finally, the numerical and the analytical results are compared.
The standard linear solid model is used to describe viscoelastic material of axially moving beam. Linear and nonlinear governing equations of traverse parameter vibration of beam are created. Asymptotic perturbation method is developed to investigate analytically linear stability region and nonlinear steady-state response of transverse vibration of axially moving viscoelastic beam. Numerical results are validated by the analytical results via differential quadrature scheme. Numerical examples show asymptotic perturbation method is applied to analyze the transverse vibration of axially moving beam with high accuracy.
The main innovations of this dissertation are as follows:
1. For first time, the idea that stability and steady-state response of axially moving viscoelastic beam is investigated via asymptotic perturbation method is proposed.
2. The differential quadrature method is used to solve numerically governing equations of transverse vibration of axially moving viscoelastic beam, and validate the analytical results of above-mentioned problems.
3. Creating the linear and nonlinear equations of an axially moving viscoelastic beam with viscoelastic constitutive relation constituted by the standard linear solid model, then asymptotic perturbation method and differential quadrature method are applied to investigate stability and steady-state response of axially moving viscoelastic beam.
4. Analytical and numerical methods of present dissertation are used to revisit linear stability and steady-state response of an axially moving viscoelastic beam with Kelvin model, and then the resualts is the same as those via method of multiple scales.
本文研究了由轴向运动速度振荡引起的黏弹性梁的横向参数振动,发展了轴向运动黏弹性梁的渐近分析的近似解析方法和微分求积的数值方法,分析了轴向运动黏弹性梁的线性横向振动稳定性边界和非线性横向振动稳态幅频响应,并将数值分析结果与近似解析分析结果进行了比较。具体内容包括:
一方面,研究了基于Kelvin模型本构关系描述的轴向运动黏弹性梁横向线性振动的稳定性区域和横向非线性振动的稳态幅频响应。Kelvin模型本构关系引入了物质时间导数。应用渐近摄动法分析了轴向运动黏弹性梁横向参数振动各阶模态主共振和多模态组合共振的线性稳定性区域,推导出了渐近摄动法的可解性条件,并得到了主共振和组合共振的稳定性失稳边界;分析了非线性轴向运动黏弹性梁组合参数共振的稳态幅频响应,推导出渐近摄动法的可解性条件,得到稳态响应非零解的振幅和存在条件,应用Routh-Hurwitz判据得到组合共振稳态响应的零解和非零解稳定性条件。利用微分求积法数值研究了轴向运动黏弹性梁横向振动的稳定性区域以及稳态幅频响应,实现了数值结果与解析结果的相互验证。
另一方面,将三参数黏弹性本构关系引入了轴向运动梁横向振动的线性和非线性控制方程,发展渐近摄动近似解析方法研究了三参数描述的轴向运动黏弹性梁在轴向运动速度振荡引起的的横向振动的线性稳定性区域以及非线性稳态幅频响应,并发展微分求积数值方法对近似解析结果加以验证,数值结果表明,渐近摄动法在轴向运动梁横向振动分析中有着较高精度。
本文主要创新工作包括:
1.首次将渐近摄动法的思路应用于分析轴向运动黏弹性线性梁的稳定性和非线性梁稳态响应;
2.使用微分求积法数值求解轴向运动黏弹性横向振动,并验证上述问题的近似解析分析结果;
3.建立了轴向运动三参数模型黏弹性线性和非线性梁的控制方程,并使用渐近摄动法和微分求积法进行研究;
4.使用解析方法和数值方法重新研究了Kelvin模型的线性稳定性和非线性稳态响应,得到了与多尺度方法研究的相同结果。
Axially moving beams can represent many engineering devices, such as band saws, power transmission belts, aerial cable tramways, crane hoist cables, flexible robotic manipulators, and spacecraft deploying appendages. However, vibrations associated with the devices have limited their applications. Therefore, understanding transverse vibrations of axially moving beams is important for the design of the devices. The investigations on vibrations of axially moving beams have theoretical as well, because an axially moving beam is a typical representative of distributed gyroscopic systems. The method of analyzing vibrations of an axially moving beam can be applied to other more complicated distributed gyroscopic systems. In this paper, an asymptotic perturbation method is proposed to and the differential quadrature scheme is developed to investigate stability and steady-state response of the parameter vibrations of an axially moving viscoelastic beam. Meanwhile, the numerical calculations validate the analytical results.
Stability and steady-state response of an axially moving viscoelastic beam constituted by Kelvin model are investigated. The material time derivative is used in the viscoelastic constitutive relation. Asymptotic perturbation method is applied to analyze stability region of transverse parameter vibration of axially moving beam for the principle and summation resonance. The solvability condition and instability boundary of stability are obtained for the principle and summation resonance via asymptotic analysis. Nonlinear steady-state response is investigated in summation parameter resonance of axially moving viscoelastic beam. Nontrivial amplitude and existence condition of nontrivial solution for steady-state response of parameter vibrations are obtained. Base on the Routh-Hurvitz criterion, stability condition of trivial and nontrivial solutions in summation parameter resonance are obtained when the steady-state response occurs. The differential quadrature scheme is developed to study numerically stability region and steady-state response of axially moving viscoelastic beam. Finally, the numerical and the analytical results are compared.
The standard linear solid model is used to describe viscoelastic material of axially moving beam. Linear and nonlinear governing equations of traverse parameter vibration of beam are created. Asymptotic perturbation method is developed to investigate analytically linear stability region and nonlinear steady-state response of transverse vibration of axially moving viscoelastic beam. Numerical results are validated by the analytical results via differential quadrature scheme. Numerical examples show asymptotic perturbation method is applied to analyze the transverse vibration of axially moving beam with high accuracy.
The main innovations of this dissertation are as follows:
1. For first time, the idea that stability and steady-state response of axially moving viscoelastic beam is investigated via asymptotic perturbation method is proposed.
2. The differential quadrature method is used to solve numerically governing equations of transverse vibration of axially moving viscoelastic beam, and validate the analytical results of above-mentioned problems.
3. Creating the linear and nonlinear equations of an axially moving viscoelastic beam with viscoelastic constitutive relation constituted by the standard linear solid model, then asymptotic perturbation method and differential quadrature method are applied to investigate stability and steady-state response of axially moving viscoelastic beam.
4. Analytical and numerical methods of present dissertation are used to revisit linear stability and steady-state response of an axially moving viscoelastic beam with Kelvin model, and then the resualts is the same as those via method of multiple scales.
引文
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