弹性薄板与流体耦合作用的力学分析
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摘要
弹性薄板是工程结构中常见的元件,它与流体的耦合作用在不同的工程领域中均有应用。薄板在流体载荷作用下产生变形或运动,而薄板的变形或运动又反过来影响到流场,从而改变流体载荷的分布和大小。国内外对这一领域的研究主要集中在耦合系统动静态问题的数值分析方面,并且已经有了很多研究成果。相比之下,流固耦合理论分析的发展却较为缓慢,现有的理论研究成果也比较少,仍有很大的发展空间和发展需要。
采用相容拉格朗日-欧拉法,根据流体弹性力学分类准则,导出了弹性薄板在理想不可压缩的势流不间断横向绕流条件下,发生小弯曲变形的接触面运动学方程和动力学方程,给出了薄板的弯曲平衡方程。应用偶极子理论,给出相应于刚体的流体势函数,采用均流与偶极子叠加的方法,最终求解出小弯曲变形的挠度函数以及应力表达式。以简支弹性平板和固支弹性圆平板为例,对其进行了弯曲变形的挠度、应力以及流体速度的分析,讨论了挠度、应力以及流体速度随系统参数的变化规律。
采用相容拉格朗日-欧拉法,推导了弹性薄板在理想不可压缩的势流不间断横向绕流条件下,发生大弯曲变形的接触面运动学方程和动力学方程。给出了弹性薄板发生大弯曲变形时的弯曲平衡方程。为了求解该复杂的非线性方程,将薄板的面内位移和曲率改变量均用挠度表示,求解出薄板大弯曲变形的挠度,进而求出面内位移及应力表达式,并给出了算例。
对于粘性流体,在低速流动小雷诺数下,忽略惯性力项,将流体运动的纳维-斯托克斯方程近似为斯托克斯方程。采用相容拉格朗日-欧拉法,研究了在低速流动中的弹性平板以行波形式振动对流动状态的影响。此外,分析了两个固体壁之间不可压缩粘性流体的晃动问题,给出了液体表面位移对于内部流体流速的影响规律。
应用单一拉格朗日法,讨论了位于不同密度的理想流体层间弹性隔层板的静力学问题和动力学问题。在给出用单一拉格朗日法表示的流体和隔层板的动力方程和边界条件的基础上,求解了弹性隔层板发生对称变形的挠度函数及隔层板在流体自由表面处给一扰动时的振动解。通过具体算例,求解了固定薄板弯曲的挠度和应力函数以及不同扰动时薄板的振动状态。讨论有关参数的变化对薄板变形、应力以及固有频率的影响,绘出了相应的关系曲线。
采用有限元软件ANSYS分别对相应问题进行数值模拟。最终将理论解与数值模拟的结果进行比较,验证解的可信度,并分析了产生误差的原因。
证明了采用相容拉格朗日-欧拉法和单一拉格朗日法,可以有效地求解弹性薄板在流体作用下的变形与应力,为新的数值分析方法打下了理论基础。
Thin elastic plate is one kind of common project components, and the plate and fluid coupling theory is usually used in different engineering field. The thin plate’s deformation or motion under the pressure of fluid affect the flow field and alter the fluid load distribution and size. Researches on FSI at home and abroad focus on numerical analysis on static and dynamic problems of coupling systems, and have issued many results. On the contrary, theoretical analysis on FSI has comparatively slow development and has rare available theoretical results. So they have vast potential and great need of development.
The thin elastic plate is in a continuous incompressible ideal cross-flow. The kinematic equation and dynamic equation of fluid-solid contact surfaces suitable for small deflection are put forward by united Lagrangian-Eulerian method and the criteria classification of nonlinear FSI problems. Then the partial differential equations are derived for the bending problem of the plate. The velocity potential function corresponding to the rigid plate is presented using doublet theory. And by applying superposition principle, the deflection and stress expressions are derived for small deformation elastic plate. Taken simply supported rectangular plate and clamped circular plate as examples, the deflection and stress of plate, also the velocity of fluid are analyzed. The influences of system parameters on them are discussed.
As the plate in a continuous incompressible ideal cross-flow, the kinematic equation and dynamic equation of fluid-solid contact surfaces suitable for large deflection are put forward by united Lagrangian-Eulerian method. Then the partial differential equations are derived for the large deflection problem of the plate. For the solutions of nonlinear equations, the in-plane displacement and the curvature change of thin plate are described as flection. The deflection, in-plane displacement and stress of plate are expressed.
For simplicity about viscous fluid, the Reynolds number is assumed small. Because the Reynolds number is the ratio of inertia force and viscous drag force, the inertia force can be neglected at low Reynolds number. And Navier-Stokes equation of fluid movement is almost equal to the Stokes equation. By united Lagrangian-Eulerian method, the influences of plate’s vibration in a travelling wave manner on the fluid flow are analyzed in a state of slow viscous flow. Besides, the sloshing problem of incompressible viscous fluid between two rigid solid wall is considered. The influences of surface displacement on internal fluid velocity are given.
Using single Lagrangian method, static and dynamic forces analysis of elastic inner plate between different density ideal fluid are discussed. After giving the kinematic equation of fluid and plate as well as boundary condition described by single Lagrangian method, the symmetrical deflection function of inner plate is solved. The vibrations of clamped plate by dynamic disturbance are analyzed. The deflection and stress functions of plate and the vibration of plate by different disturbance are presented by examples. And the influences of parameters on the deformation, stress and natural frequency are given.
Applying ANSYS simulate the corresponding problem. The theoretical results are compared with numerical solutions, and error analysis is given. It is shown that the theoretical results effectively.
United Lagrangian-Eulerian method and single Lagrangian method for deformation and stress of plate acted by fluid are effective, and lay a good theoretic foundation for new numerical simulate.
采用相容拉格朗日-欧拉法,根据流体弹性力学分类准则,导出了弹性薄板在理想不可压缩的势流不间断横向绕流条件下,发生小弯曲变形的接触面运动学方程和动力学方程,给出了薄板的弯曲平衡方程。应用偶极子理论,给出相应于刚体的流体势函数,采用均流与偶极子叠加的方法,最终求解出小弯曲变形的挠度函数以及应力表达式。以简支弹性平板和固支弹性圆平板为例,对其进行了弯曲变形的挠度、应力以及流体速度的分析,讨论了挠度、应力以及流体速度随系统参数的变化规律。
采用相容拉格朗日-欧拉法,推导了弹性薄板在理想不可压缩的势流不间断横向绕流条件下,发生大弯曲变形的接触面运动学方程和动力学方程。给出了弹性薄板发生大弯曲变形时的弯曲平衡方程。为了求解该复杂的非线性方程,将薄板的面内位移和曲率改变量均用挠度表示,求解出薄板大弯曲变形的挠度,进而求出面内位移及应力表达式,并给出了算例。
对于粘性流体,在低速流动小雷诺数下,忽略惯性力项,将流体运动的纳维-斯托克斯方程近似为斯托克斯方程。采用相容拉格朗日-欧拉法,研究了在低速流动中的弹性平板以行波形式振动对流动状态的影响。此外,分析了两个固体壁之间不可压缩粘性流体的晃动问题,给出了液体表面位移对于内部流体流速的影响规律。
应用单一拉格朗日法,讨论了位于不同密度的理想流体层间弹性隔层板的静力学问题和动力学问题。在给出用单一拉格朗日法表示的流体和隔层板的动力方程和边界条件的基础上,求解了弹性隔层板发生对称变形的挠度函数及隔层板在流体自由表面处给一扰动时的振动解。通过具体算例,求解了固定薄板弯曲的挠度和应力函数以及不同扰动时薄板的振动状态。讨论有关参数的变化对薄板变形、应力以及固有频率的影响,绘出了相应的关系曲线。
采用有限元软件ANSYS分别对相应问题进行数值模拟。最终将理论解与数值模拟的结果进行比较,验证解的可信度,并分析了产生误差的原因。
证明了采用相容拉格朗日-欧拉法和单一拉格朗日法,可以有效地求解弹性薄板在流体作用下的变形与应力,为新的数值分析方法打下了理论基础。
Thin elastic plate is one kind of common project components, and the plate and fluid coupling theory is usually used in different engineering field. The thin plate’s deformation or motion under the pressure of fluid affect the flow field and alter the fluid load distribution and size. Researches on FSI at home and abroad focus on numerical analysis on static and dynamic problems of coupling systems, and have issued many results. On the contrary, theoretical analysis on FSI has comparatively slow development and has rare available theoretical results. So they have vast potential and great need of development.
The thin elastic plate is in a continuous incompressible ideal cross-flow. The kinematic equation and dynamic equation of fluid-solid contact surfaces suitable for small deflection are put forward by united Lagrangian-Eulerian method and the criteria classification of nonlinear FSI problems. Then the partial differential equations are derived for the bending problem of the plate. The velocity potential function corresponding to the rigid plate is presented using doublet theory. And by applying superposition principle, the deflection and stress expressions are derived for small deformation elastic plate. Taken simply supported rectangular plate and clamped circular plate as examples, the deflection and stress of plate, also the velocity of fluid are analyzed. The influences of system parameters on them are discussed.
As the plate in a continuous incompressible ideal cross-flow, the kinematic equation and dynamic equation of fluid-solid contact surfaces suitable for large deflection are put forward by united Lagrangian-Eulerian method. Then the partial differential equations are derived for the large deflection problem of the plate. For the solutions of nonlinear equations, the in-plane displacement and the curvature change of thin plate are described as flection. The deflection, in-plane displacement and stress of plate are expressed.
For simplicity about viscous fluid, the Reynolds number is assumed small. Because the Reynolds number is the ratio of inertia force and viscous drag force, the inertia force can be neglected at low Reynolds number. And Navier-Stokes equation of fluid movement is almost equal to the Stokes equation. By united Lagrangian-Eulerian method, the influences of plate’s vibration in a travelling wave manner on the fluid flow are analyzed in a state of slow viscous flow. Besides, the sloshing problem of incompressible viscous fluid between two rigid solid wall is considered. The influences of surface displacement on internal fluid velocity are given.
Using single Lagrangian method, static and dynamic forces analysis of elastic inner plate between different density ideal fluid are discussed. After giving the kinematic equation of fluid and plate as well as boundary condition described by single Lagrangian method, the symmetrical deflection function of inner plate is solved. The vibrations of clamped plate by dynamic disturbance are analyzed. The deflection and stress functions of plate and the vibration of plate by different disturbance are presented by examples. And the influences of parameters on the deformation, stress and natural frequency are given.
Applying ANSYS simulate the corresponding problem. The theoretical results are compared with numerical solutions, and error analysis is given. It is shown that the theoretical results effectively.
United Lagrangian-Eulerian method and single Lagrangian method for deformation and stress of plate acted by fluid are effective, and lay a good theoretic foundation for new numerical simulate.
引文
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