含裂纹的非保守粘弹性及压电层合板动力学研究
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摘要
非保守系统的动力学分析在工程中有着广泛的应用,如有阻尼介质中运动的物体、输流管道、桥梁及刹车鼓轮系统等。因而研究非保守结构的动力稳定性具有重要的理论意义和广泛的应用价值。本文以粘弹性及压电层合板为研究对象,系统地研究了含有裂纹的在非保守力作用下的粘弹性及压电层合板的动力特性及其稳定性。具体研究工作如下：
(1)研究了含裂纹粘弹性矩形薄板及线性变厚度矩形薄板的横向振动特性。由于裂纹的存在,导致了板在裂纹两边转角的不连续性,运用断裂力学的理论得到裂纹导致的附加转角。从粘弹性微分型本构关系和薄板理论出发,利用裂纹处的连续性条件推导了含表面贯通直裂纹粘弹性矩形薄板及线性变厚度矩形薄板的运动微分方程,采用微分求积法建立了含裂纹粘弹性矩形板的复特征方程。通过求解广义特征值问题,给出了复频率的变化曲线,研究含有一条裂纹和多条裂纹的情况下薄板的动力特性。
(2)研究了在切向均布随从力作用下含裂纹粘弹性矩形薄板及变厚度矩形薄板的动力特性和稳定性。建立了含裂纹非保守粘弹性矩形薄板、含裂纹非保守线性变厚度矩形薄板和抛物线型变厚度粘弹性矩形薄板的运动微分方程,采用微分求积法,将裂纹处连续性条件用节点替代法处理,导出了体变为弹性、畸变服从Kelvin-Voigt模型的粘弹性非保守矩形板的复特征方程。对含裂纹非保守粘弹性板的广义特征值问题进行了求解,分析板的几何参数、材料的无量纲延滞时间、裂纹参数对非保守粘弹性板的横向振动特性、失稳形式及相应的临界荷载的影响。
(3)研究了内部具有有限多个点弹性支承以及边界上具有弹性约束的弹性矩形薄板的自由振动。将伽辽金无网格法应用于求解具有点弹性支承和弹性约束矩形薄板的横向振动问题。基于弹性动力学Hamilton原理的推广,采用伽辽金无网格法建立了具有有限多个点弹性支承和弹性约束的弹性薄板横向振动的无量纲运动方程和特征方程。通过数值计算给出了薄板的固有频率随点弹性支承的刚性系数、点弹性支承位置和弹性约束刚度的变化曲线,分析了点弹性支承和弹性约束对矩形薄板的横向振动特性的影响。
(4)将伽辽金无网格法(EFGM)应用于求解上下表面对称贴有压电片的含有裂纹矩形薄板的横向振动问题。首先,建立了系统的能量泛函,根据弹性动力学Hamilton原理的推广和引入无量纲量,推导了无量纲化的含裂纹压电层合矩形薄板的变分式；然后用伽辽金无网格法建立了含多裂纹的板横向振动方程和特征方程；给出了振型函数的表达式。通过数值计算,得出了薄板的无量纲固有频率随裂纹和压电片几何参数的变化曲线,分析在改变压电片长度、宽度及厚度的情况下,压电片对矩形薄板的横向振动特性的影响和薄板产生裂纹后固有特性的变化。
(5)研究了含有裂纹的压电层合粘弹性板在切向随从力作用下的动力稳定性问题。粘贴在板上下表面的压电材料对板结构的作用力就是一种典型的非保守力,以粘贴在板上下表面的压电材料和粘弹性材料矩形薄板构成的层合板为研究对象,建立了在随从力作用下压电层合粘弹性薄板的运动微分方程；建立微分方程和边界条件的积分弱形式,再对等效积分弱形式的方程进行离散和求解,用伽辽金无网格方法建立在切向随从力作用下含有裂纹的压电层合粘弹性矩形薄板的特征方程,分析压电片对非保守粘弹性薄板稳定性的影响。
The dynamic analysis of the non-conservative systems are widely applied in the practical engineering, such as moving systems in the damping medium, pipe conveying fluid, bridge, aerofoil and brake drum. Therefore, analyses of the dynamic stability of the non-conservative structure have important theoretical significance and widespread application value. The research objects of this paper are the viscoelastic and piezoelectric composite plate, the dynamic characteristics and dynamic stability of the viscoelastic and piezoelectric composite plate containing cracks subjected to non-conservative forces are analyzed. The main research work is as follows.
(1) The transverse vibration characteristics of a viscoelastic rectangular thin plate and a viscoelastic plate with linearly varying thickness and cracks are investigated. The presence of cracks can result in discontinuity of the slope compatibility condition at the two sides of the crack location. Based on fracture mechanics theory, the additional rotation induced by the crack is given.Based on the thin plate theory and the two-dimensional viscoelastic differential type constitutive relation, the differential equation of motion of the viscoelastic plate and a plate with linearly varying thickness containing the all-over part-through crack are established by continuity conditions at the crack location. The generalized complex eigenvalue of the cracked viscoelastic plate are calculated, and the curves of complex frequencies versus are obtained.The transverse vibration characteristics of the viscoelastic plate containing single and multiple all-over part-through cracks are analyzed.
(2) The vibration characteristics and dynamic stability of the viscoelastic rectangular plate and varying thickness viscoelastic plate with crack and subjected to uniformly distributed tangential follower forces are analyzed. The differential equations of the cracked viscoelastic rectangular, linearly varying thickness viscoelastic plate and parabolicly varying thickness viscoelastic plate subjected to follower force are established. The complex eigenvalue equations of the cracked viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt laws for distortion subjected to follower force are obtained by the differential quadrature method, and the 5 method is adopted at the crack continuity conditions.The general eigenvalue equations of cracked viscoelastic plate subjected to follower force are calculated. The effects of the geometric parameter, the dimensionless delay time and the crack parameters on the transverse vibration characteristics, type of instability and the corresponding critical loads of the non-conservative viscoelastic plates are analyzed.
(3) The free vibration problems of a rectangular thin plate with finite elastic point supports and the edge attached to distributed elastic restraint are studied. The element-free Galerkin method is proposed to solve the transverse vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges. Based on the extended Hamilton's principle for the elastic dynamics system, the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established by the element-free Galerkin method. The eigenvalue equations are presented. Via numerical calculation, the curves of the natural frequency of thin plates versus the spring constant, locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained. As a conclusion, the effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.
(4) The element free Galerkin method is proposed to solve the transverse free vibration of the composite cracked plate with surface bonded piezoelectric sheet. First, the energy function of the system is established, utilizing the extended Hamilton's principle for the elastic dynamics system and introduced the dimensionless variables and parameters, the dimensionless variational expression of the piezoelectric composite plate with cracks is deduced. Then, the dimensionless equations of motion and general eigenvalue equations of transverse vibration of the piezoelectric composite plate with multiple cracks are obtained. The vibration model functions are given. By number calculating the curves of the dimensionless natural frequency of thin plate versus the geometry parameters of cracks and piezoelectric materials are plotted. The effects of piezoelectric sheet on the transverse vibration characteristics of the thin plate and the changes of the natural characteristics of a rectangular plate due to the presence of cracks are analyzed.
(5) The dynamic stability of the piezoelectric composite viscoelastic plate with crack and subjected to uniformly distributed tangential follower forces are analyzed. The active force of surface bonded piezoelectric materials to the plate is the typical non-conservative force.The research object is the composite plate of the viscoelastic plate with surface bonded piezoelectric sheet, the differential equation of motion of the piezoelectric composite viscoelastic plate subjected to follower force are established. The weak integral form of the differential equations and the boundary conditions are obtained, and the equations of equivalent weak integral form are discreted and calculated. The complex eigenvalue equations of the cracked piezoelectric composite viscoelastic plate subjected to follower force are obtained by the element-free Galerkin method. The effects of the piezoelectric sheet with voltage on the dynamic stability of the non-conservative viscoelastic plates are analyzed.
(1)研究了含裂纹粘弹性矩形薄板及线性变厚度矩形薄板的横向振动特性。由于裂纹的存在,导致了板在裂纹两边转角的不连续性,运用断裂力学的理论得到裂纹导致的附加转角。从粘弹性微分型本构关系和薄板理论出发,利用裂纹处的连续性条件推导了含表面贯通直裂纹粘弹性矩形薄板及线性变厚度矩形薄板的运动微分方程,采用微分求积法建立了含裂纹粘弹性矩形板的复特征方程。通过求解广义特征值问题,给出了复频率的变化曲线,研究含有一条裂纹和多条裂纹的情况下薄板的动力特性。
(2)研究了在切向均布随从力作用下含裂纹粘弹性矩形薄板及变厚度矩形薄板的动力特性和稳定性。建立了含裂纹非保守粘弹性矩形薄板、含裂纹非保守线性变厚度矩形薄板和抛物线型变厚度粘弹性矩形薄板的运动微分方程,采用微分求积法,将裂纹处连续性条件用节点替代法处理,导出了体变为弹性、畸变服从Kelvin-Voigt模型的粘弹性非保守矩形板的复特征方程。对含裂纹非保守粘弹性板的广义特征值问题进行了求解,分析板的几何参数、材料的无量纲延滞时间、裂纹参数对非保守粘弹性板的横向振动特性、失稳形式及相应的临界荷载的影响。
(3)研究了内部具有有限多个点弹性支承以及边界上具有弹性约束的弹性矩形薄板的自由振动。将伽辽金无网格法应用于求解具有点弹性支承和弹性约束矩形薄板的横向振动问题。基于弹性动力学Hamilton原理的推广,采用伽辽金无网格法建立了具有有限多个点弹性支承和弹性约束的弹性薄板横向振动的无量纲运动方程和特征方程。通过数值计算给出了薄板的固有频率随点弹性支承的刚性系数、点弹性支承位置和弹性约束刚度的变化曲线,分析了点弹性支承和弹性约束对矩形薄板的横向振动特性的影响。
(4)将伽辽金无网格法(EFGM)应用于求解上下表面对称贴有压电片的含有裂纹矩形薄板的横向振动问题。首先,建立了系统的能量泛函,根据弹性动力学Hamilton原理的推广和引入无量纲量,推导了无量纲化的含裂纹压电层合矩形薄板的变分式；然后用伽辽金无网格法建立了含多裂纹的板横向振动方程和特征方程；给出了振型函数的表达式。通过数值计算,得出了薄板的无量纲固有频率随裂纹和压电片几何参数的变化曲线,分析在改变压电片长度、宽度及厚度的情况下,压电片对矩形薄板的横向振动特性的影响和薄板产生裂纹后固有特性的变化。
(5)研究了含有裂纹的压电层合粘弹性板在切向随从力作用下的动力稳定性问题。粘贴在板上下表面的压电材料对板结构的作用力就是一种典型的非保守力,以粘贴在板上下表面的压电材料和粘弹性材料矩形薄板构成的层合板为研究对象,建立了在随从力作用下压电层合粘弹性薄板的运动微分方程；建立微分方程和边界条件的积分弱形式,再对等效积分弱形式的方程进行离散和求解,用伽辽金无网格方法建立在切向随从力作用下含有裂纹的压电层合粘弹性矩形薄板的特征方程,分析压电片对非保守粘弹性薄板稳定性的影响。
The dynamic analysis of the non-conservative systems are widely applied in the practical engineering, such as moving systems in the damping medium, pipe conveying fluid, bridge, aerofoil and brake drum. Therefore, analyses of the dynamic stability of the non-conservative structure have important theoretical significance and widespread application value. The research objects of this paper are the viscoelastic and piezoelectric composite plate, the dynamic characteristics and dynamic stability of the viscoelastic and piezoelectric composite plate containing cracks subjected to non-conservative forces are analyzed. The main research work is as follows.
(1) The transverse vibration characteristics of a viscoelastic rectangular thin plate and a viscoelastic plate with linearly varying thickness and cracks are investigated. The presence of cracks can result in discontinuity of the slope compatibility condition at the two sides of the crack location. Based on fracture mechanics theory, the additional rotation induced by the crack is given.Based on the thin plate theory and the two-dimensional viscoelastic differential type constitutive relation, the differential equation of motion of the viscoelastic plate and a plate with linearly varying thickness containing the all-over part-through crack are established by continuity conditions at the crack location. The generalized complex eigenvalue of the cracked viscoelastic plate are calculated, and the curves of complex frequencies versus are obtained.The transverse vibration characteristics of the viscoelastic plate containing single and multiple all-over part-through cracks are analyzed.
(2) The vibration characteristics and dynamic stability of the viscoelastic rectangular plate and varying thickness viscoelastic plate with crack and subjected to uniformly distributed tangential follower forces are analyzed. The differential equations of the cracked viscoelastic rectangular, linearly varying thickness viscoelastic plate and parabolicly varying thickness viscoelastic plate subjected to follower force are established. The complex eigenvalue equations of the cracked viscoelastic plate constituted by elastic behavior in dilatation and the Kelvin-Voigt laws for distortion subjected to follower force are obtained by the differential quadrature method, and the 5 method is adopted at the crack continuity conditions.The general eigenvalue equations of cracked viscoelastic plate subjected to follower force are calculated. The effects of the geometric parameter, the dimensionless delay time and the crack parameters on the transverse vibration characteristics, type of instability and the corresponding critical loads of the non-conservative viscoelastic plates are analyzed.
(3) The free vibration problems of a rectangular thin plate with finite elastic point supports and the edge attached to distributed elastic restraint are studied. The element-free Galerkin method is proposed to solve the transverse vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges. Based on the extended Hamilton's principle for the elastic dynamics system, the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established by the element-free Galerkin method. The eigenvalue equations are presented. Via numerical calculation, the curves of the natural frequency of thin plates versus the spring constant, locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained. As a conclusion, the effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.
(4) The element free Galerkin method is proposed to solve the transverse free vibration of the composite cracked plate with surface bonded piezoelectric sheet. First, the energy function of the system is established, utilizing the extended Hamilton's principle for the elastic dynamics system and introduced the dimensionless variables and parameters, the dimensionless variational expression of the piezoelectric composite plate with cracks is deduced. Then, the dimensionless equations of motion and general eigenvalue equations of transverse vibration of the piezoelectric composite plate with multiple cracks are obtained. The vibration model functions are given. By number calculating the curves of the dimensionless natural frequency of thin plate versus the geometry parameters of cracks and piezoelectric materials are plotted. The effects of piezoelectric sheet on the transverse vibration characteristics of the thin plate and the changes of the natural characteristics of a rectangular plate due to the presence of cracks are analyzed.
(5) The dynamic stability of the piezoelectric composite viscoelastic plate with crack and subjected to uniformly distributed tangential follower forces are analyzed. The active force of surface bonded piezoelectric materials to the plate is the typical non-conservative force.The research object is the composite plate of the viscoelastic plate with surface bonded piezoelectric sheet, the differential equation of motion of the piezoelectric composite viscoelastic plate subjected to follower force are established. The weak integral form of the differential equations and the boundary conditions are obtained, and the equations of equivalent weak integral form are discreted and calculated. The complex eigenvalue equations of the cracked piezoelectric composite viscoelastic plate subjected to follower force are obtained by the element-free Galerkin method. The effects of the piezoelectric sheet with voltage on the dynamic stability of the non-conservative viscoelastic plates are analyzed.
引文
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