运动粘弹性板的横向振动及稳定性研究
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摘要
轴向运动系统在机械、纺织、电子、航空和航天等领域有着非常广泛的应用。因此,轴向运动系统的横向振动和稳定性的研究有着重要的实际应用价值。然而,已有的研究大都仅限于研究对象为轴向运动的粘弹性弦线和粘弹性梁的一维系统,鲜有对轴向运动粘弹性板的研究。本文分别对轴向运动等厚度粘弹性板和轴向运动变厚度粘弹性板的横向振动及稳定性问题、轴向运动非保守粘弹性板的稳定性问题、轴向加速运动粘弹性板的动力稳定性问题进行了研究。具体研究工作如下。
(1)研究了轴向运动等厚度粘弹性矩形薄板的横向振动和稳定性问题。基于薄板理论和二维粘弹性微分型本构关系,推导了轴向运动粘弹性矩形薄板在Laplace域内的微分方程,该方程适用于任一微分型本构关系的粘弹性模型。继而得到体变为弹性、畸变服从Kelvin-Voigt模型的轴向运动粘弹性板在时域内的运动微分方程。采用微分求积法,建立系统的复特征方程。求解复特征方程,得到系统的前三阶复频率的实部和虚部与速度的变化关系曲线以及临界速度和失稳类型。
(2)对轴向运动线性及抛物线型变厚度粘弹性矩形薄板的横向振动和稳定性问题进行了分析。建立了轴向运动变厚度粘弹性矩形薄板在时域内的运动微分方程,通过引入无量纲量,得到轴向运动变厚度粘弹性板振型微分方程。采用微分求积法对振型方程和边界条件进行离散,得到该问题的复特征方程。运用Matlab语言编程,求解系统的前三阶模态,并分析了薄板的长宽比、无量纲运动速度、材料的无量纲延滞时间、板的厚度比的变化对其横向振动及稳定性的影响。
(3)对于均布切向随从力作用下的粘弹性矩形薄板,研究了非保守粘弹性板的失稳类型及临界载荷。给出三种不同边界条件下,粘弹性板的复频率与随从力的关系曲线,讨论了各个因素对非保守粘弹性板的失稳类型及临界载荷的影响。
(4)对于轴向运动非保守粘弹性板,其运动微分方程为四阶变系数偏微分方程,方程中的变系数是由切向随从力产生的。采用Levy法结合幂级数法,得到微分方程的复特征值问题。求解复特征方程,得到轴向运动非保守粘弹性板的复频率及失稳形式,从而分析无量纲延滞时间、非保守力及无量纲轴向运动速度对轴向运动非保守粘弹性的稳定性的影响。
(5)研究了轴向加速运动粘弹性板的参数振动特性。设粘弹性板的轴向运动速度为常平均速度与简谐涨落的叠加,建立轴向加速运动粘弹性矩形薄板的运动微分方程。采用微分求积法对方程中的空间变量进行离散,得到仅含有时间变量的三阶周期系数微分方程组。引入状态变量,得到一阶状态方程,采用隐式Runge-Kutta法进行求解。根据Floquet理论确定了粘弹性板的动力不稳定区域,并讨论平均轴向运动速度,轴向运动速度涨落幅值对轴向加速运动粘弹性板动力稳定性的影响。
Axially moving systems have extensive application in many fields such as machinery, textile, electron and spaceflight. Therefore, the study of the tranverse vibration and stability of the axially moving systems is of great significance. However, the existent studies have been mostly confined to unidimension problem, that is, the research objects are viscoelastic column and viscoelastic beam, few papers have been presented on the axially moving viscoelastic plate. The tranverse vibration and stability of the axially moving uniform plate and plate with varying thickness, the stability of axially moving non-conservative plate and the parametric vibration of the axially accelerating viscoelastic plate are analyzed in this paper, respectively. The main research work is as follows.
(1) The transverse vibration and stability of the axially moving uniform viscoelastic plate are investigated. On the basis of the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of the axially moving viscoelastic rectangular plate in the Laplace domain is deduced, the equation is suitable for various viscoelastic differential models. Then, the differential equation of motion of the viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt model for distortion in time domain is derived. The complex eigen-equation of axially moving viscoelastic plate is established by the differential quadrature method. The curves of real parts and imaginary parts of the first three order dimensionless complex frequencies versus dimensionless axially moving speed, the critical speed and the instability type are obtained by solving the complex eigen-equation.
(2) The transverse vibration and stability of the axially moving viscoelastic rectangular plate with linearly and parabolically varying thickness in the direction orthogonal to the direction of motion are analyzed. The differential equation of motion of the axially moving viscoelastic plate with varying thickness in time domain is established, by introducing dimensionless variables, the mode of vibration equation of axially moving viscoelastic plate with varying thickness is obtained. The complex eigen-equation is derived by the discretization of the mode of vibration equation and boundary conditions using the differential quadrature method. The first three modes of the system is solved by Matlab program and effects of the aspect ratio, thickness ratio, the dimensionless moving speed and the dimensionless delay time on the transverse vibration and stability of the axially moving viscoelastic plate are analyzed.
(3) The instability type and the critical load of viscoelastic rectangular plate subjected to uniformly distributed tangential follower force are studied. The curves of the dimensionless complex frequencies versus dimensionless follower force under three different boundary conditions are given, the factors influencing the instability type and critical load of the non-conservative visoelastic rectangular plate are discussed.
(4) The differential equation of motion of axially moving non-conservative viscoelastic plate is a four order partial differential equation with variable coefficients, and the variable coefficients issue from tangential follower force. By the Levy method and power series method, the complex eigenvalue problem of the differential equation is derived. The complex frequency and the instability type of the axially moving non-conservative is obtained, thereby effects of the dimensionless delay time, non-conservative force and dimensionless axially moving speed on the stability of axially moving non-conservative viscoelastic plate is analyzed.
(5) The parametric vibration of the axially accelerating viscoelastic plate is studied. The axially moving speed is a simple harmonic fluctuation about a constant mean speed, the differential equation of motion of axially accelerating viscoelastic plate is established. By the discretization of spatial variable in the equation, a third-order differential system of equations containing periodic time-varying coefficient is derived. Introducing the state vector, the first-order state equation is obtained, and it is solved by implicit expression of Runge-Kutta method. Based on the Floquet theory, the dynamic stability regions of the axially moving viscoelastic plate is determined, effects of the constant mean speed and the amplited of the simple harmonic fluctuation on the dynamic stability of the axially accelerating viscoelastic plate are discussed.
(1)研究了轴向运动等厚度粘弹性矩形薄板的横向振动和稳定性问题。基于薄板理论和二维粘弹性微分型本构关系,推导了轴向运动粘弹性矩形薄板在Laplace域内的微分方程,该方程适用于任一微分型本构关系的粘弹性模型。继而得到体变为弹性、畸变服从Kelvin-Voigt模型的轴向运动粘弹性板在时域内的运动微分方程。采用微分求积法,建立系统的复特征方程。求解复特征方程,得到系统的前三阶复频率的实部和虚部与速度的变化关系曲线以及临界速度和失稳类型。
(2)对轴向运动线性及抛物线型变厚度粘弹性矩形薄板的横向振动和稳定性问题进行了分析。建立了轴向运动变厚度粘弹性矩形薄板在时域内的运动微分方程,通过引入无量纲量,得到轴向运动变厚度粘弹性板振型微分方程。采用微分求积法对振型方程和边界条件进行离散,得到该问题的复特征方程。运用Matlab语言编程,求解系统的前三阶模态,并分析了薄板的长宽比、无量纲运动速度、材料的无量纲延滞时间、板的厚度比的变化对其横向振动及稳定性的影响。
(3)对于均布切向随从力作用下的粘弹性矩形薄板,研究了非保守粘弹性板的失稳类型及临界载荷。给出三种不同边界条件下,粘弹性板的复频率与随从力的关系曲线,讨论了各个因素对非保守粘弹性板的失稳类型及临界载荷的影响。
(4)对于轴向运动非保守粘弹性板,其运动微分方程为四阶变系数偏微分方程,方程中的变系数是由切向随从力产生的。采用Levy法结合幂级数法,得到微分方程的复特征值问题。求解复特征方程,得到轴向运动非保守粘弹性板的复频率及失稳形式,从而分析无量纲延滞时间、非保守力及无量纲轴向运动速度对轴向运动非保守粘弹性的稳定性的影响。
(5)研究了轴向加速运动粘弹性板的参数振动特性。设粘弹性板的轴向运动速度为常平均速度与简谐涨落的叠加,建立轴向加速运动粘弹性矩形薄板的运动微分方程。采用微分求积法对方程中的空间变量进行离散,得到仅含有时间变量的三阶周期系数微分方程组。引入状态变量,得到一阶状态方程,采用隐式Runge-Kutta法进行求解。根据Floquet理论确定了粘弹性板的动力不稳定区域,并讨论平均轴向运动速度,轴向运动速度涨落幅值对轴向加速运动粘弹性板动力稳定性的影响。
Axially moving systems have extensive application in many fields such as machinery, textile, electron and spaceflight. Therefore, the study of the tranverse vibration and stability of the axially moving systems is of great significance. However, the existent studies have been mostly confined to unidimension problem, that is, the research objects are viscoelastic column and viscoelastic beam, few papers have been presented on the axially moving viscoelastic plate. The tranverse vibration and stability of the axially moving uniform plate and plate with varying thickness, the stability of axially moving non-conservative plate and the parametric vibration of the axially accelerating viscoelastic plate are analyzed in this paper, respectively. The main research work is as follows.
(1) The transverse vibration and stability of the axially moving uniform viscoelastic plate are investigated. On the basis of the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of the axially moving viscoelastic rectangular plate in the Laplace domain is deduced, the equation is suitable for various viscoelastic differential models. Then, the differential equation of motion of the viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt model for distortion in time domain is derived. The complex eigen-equation of axially moving viscoelastic plate is established by the differential quadrature method. The curves of real parts and imaginary parts of the first three order dimensionless complex frequencies versus dimensionless axially moving speed, the critical speed and the instability type are obtained by solving the complex eigen-equation.
(2) The transverse vibration and stability of the axially moving viscoelastic rectangular plate with linearly and parabolically varying thickness in the direction orthogonal to the direction of motion are analyzed. The differential equation of motion of the axially moving viscoelastic plate with varying thickness in time domain is established, by introducing dimensionless variables, the mode of vibration equation of axially moving viscoelastic plate with varying thickness is obtained. The complex eigen-equation is derived by the discretization of the mode of vibration equation and boundary conditions using the differential quadrature method. The first three modes of the system is solved by Matlab program and effects of the aspect ratio, thickness ratio, the dimensionless moving speed and the dimensionless delay time on the transverse vibration and stability of the axially moving viscoelastic plate are analyzed.
(3) The instability type and the critical load of viscoelastic rectangular plate subjected to uniformly distributed tangential follower force are studied. The curves of the dimensionless complex frequencies versus dimensionless follower force under three different boundary conditions are given, the factors influencing the instability type and critical load of the non-conservative visoelastic rectangular plate are discussed.
(4) The differential equation of motion of axially moving non-conservative viscoelastic plate is a four order partial differential equation with variable coefficients, and the variable coefficients issue from tangential follower force. By the Levy method and power series method, the complex eigenvalue problem of the differential equation is derived. The complex frequency and the instability type of the axially moving non-conservative is obtained, thereby effects of the dimensionless delay time, non-conservative force and dimensionless axially moving speed on the stability of axially moving non-conservative viscoelastic plate is analyzed.
(5) The parametric vibration of the axially accelerating viscoelastic plate is studied. The axially moving speed is a simple harmonic fluctuation about a constant mean speed, the differential equation of motion of axially accelerating viscoelastic plate is established. By the discretization of spatial variable in the equation, a third-order differential system of equations containing periodic time-varying coefficient is derived. Introducing the state vector, the first-order state equation is obtained, and it is solved by implicit expression of Runge-Kutta method. Based on the Floquet theory, the dynamic stability regions of the axially moving viscoelastic plate is determined, effects of the constant mean speed and the amplited of the simple harmonic fluctuation on the dynamic stability of the axially accelerating viscoelastic plate are discussed.
引文
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