【作者】 赵凤群；

【导师】 刘宏昭； 王忠民；

【作者基本信息】 西安理工大学 ， 机械工程， 2007， 博士

【摘要】 功能梯度材料（FGM）是近二十年来材料科学领域出现的一种新型复合材料，在航空航天、生物医学、核工业等领域有着广阔的应用前景。因而研究非保守FGM结构的力学性能具有重要的意义。本文在已有研究的基础上，分别对非保守杆在各种支承条件下的振动和稳定性问题、非保守FGM杆的过屈曲特性、FGM杆在热载荷作用下的过屈曲特性，以及非保守FGM矩形板的线性振动和非线性振动特性进行了研究。具体研究工作如下。（1）用积分方程法研究了具有多个点弹性支承的Kelvin型粘弹性简支杆在切向均布随从力作用下的动力特性和稳定性问题。对于该微分方程的复特征值问题，先用叠加原理求出核函数，将微分方程化为积分方程；再利用退化核特性，从积分方程导出复特征方程。最后给出算例，分析了点弹性支承的弹性系数、支承位置和材料的无量纲延滞时间对杆的自振频率和稳定性的影响。算例表明，该法能有效地处理广义δ函数及变系数的微分方程的复特征值问题。（2）研究了具有多个点弹性支承的弹性简支杆在切向均布随从力作用下的动力特性和稳定性问题。对于所出现的支承情形，提出以分段表示的运动微分方程及连续性条件和边界条件去描述，采用有限差分法导出了递推格式，分析了弹性支承的弹性系数、支承位置和转动惯量对非保守杆的振动频率和稳定性的影响。（3）研究了具有可移动弹性支承的输流管道的稳定性问题。建立了分段表示的运动微分方程以及弹性支承处的连续性条件和边界条件，采用有限差分法导出了复特征方程，分析了弹性支承的弹性常数、支承位置对输流管道振动频率和稳定性的影响，给出了稳定性区域图，指出了发散和颤振区域。（4）对受切向均布随从力作用的FGM杆，基于轴线可伸长杆的大变形理论建立了非线性控制微分方程组，对由金属和陶瓷所构成的右端可移简支和右端不可移简支FGM杆的后屈曲特性，分别用Runge-Kutta法结合打靶法进行了数值分析，并与保守FGM杆的后屈曲特性进行了比较。给出了不同梯度指标下FGM杆的后屈曲特征曲线，并与金属和陶瓷两种单相材料杆的相应特性进行了比较，讨论了FGM杆长高比对其后屈曲特性的影响。（5）对陶瓷—金属FGM杆建立了在热载荷作用下的非线性控制微分方程，采用打靶法分析了由ZrO₂和Ti-6Al-4V两种材料组成的两端不可移简支和夹支FGM杆的热后屈曲行为。首先给出了在均匀温度场中不同梯度指标的FGM杆的热后屈曲平衡路径，并与ZrO₂和Ti-6Al-4V两种均质材料杆的相应特性进行了比较，同时讨论了不同端部转角下梯度指标对FGM杆稳定性的影响；然后分别研究了在温差一定、下表面温度变化时和在下表面温度一定、温差变化时FGM杆的热后屈曲特性，也与两种均质材料杆的后屈曲特性进行了比较。（6）对受均布随从力作用的FGM矩形板，引入应力函数，得到了以应力函数和挠度函数表示的耦合运动微分方程组。用Fourier级数法研究了四边简支FGM非保守矩形板的稳定性，给出了不同边长比和不同梯度指标下频率和发散载荷的变化曲线，以及梯度指标变化对频率和发散载荷的影响。（7）研究了切向均布随从力作用下简支FGM矩形板的非线性振动问题。按照材料组份体积分数的简单幂率分布规律，考虑了FGM板的材料常数仅沿厚度连续变化。由大挠度的von Karman理论建立了以应力函数和挠度函数表示的运动偏微分方程组，再由Galerkin法转化成非线性常微分方程。对随从力作用下的四边简支陶瓷／金属矩形板，讨论了随从力、梯度指标和边长比对板的动力特性的影响，得到了各种条件下板中心振幅与非线性基频的关系。更多还原

【Abstract】 Functionally graded material （FGM） is a new type of composite materials arisen in material science field in recent twenty years, and has wider application foreground in many fields such as spaceflight, biomedicine, and nuclear industry. Therefore analyses of the dynamic behavior of non-conservative FGM structure have important significance. On the basis of the existing research, the vibration and stability of non-conservative rod with various elastic supports, the post-buckling behavior of non-conservative FGM rod, the post-buckling behavior of FGM rod subjected to thermal loads, and the linear and nonlinear vibration behavior of non-conservative rectangular plate made of FGM are analyzed respectively in this paper. The main research work is as follows.（1） The dynamic behaviors and stability of Kelvin’s viscoelastic rods with interior point elastic supports subjected to uniformly distributed tangential follower forces are investigated by integral equation theory. In solving the complex characteristic problem of the differential equation, firstly, nucleus function is obtained by using superposition principle, and the differential equation of mode shape is reduced to an integral equation. Then, based on the integral equation, complex characteristic equation is derived in accidence with the properties of degenerative nucleus. At last, the effect of the elastic constant of the elastic point supports, location of point support and dimensionless delay time of Kelvin’s materials on complex frequencies and stability of non-conservative rods are analyzed by the examples. It is shown that the proposed method is effective and practical in solving the complex eigenvalue problem of differential equation with generalizedδ-function and complex variable coefficients.（2） The dynamic behaviors and stability of simply supported elastic rods with point elastic supports subjected to uniformly distributed tangential follower force are investigated. For the case of point elastic supports, it is described by the differential equations of mode shape in piecewise and continuous condition as well as boundary conditions. The recurrence formula is obtained by using finite difference method, and the effect of elastic constant of the linear elastic supports, location of elastic support and moment of inertia on vibration frequencies and stability of non-conservative rods are analyzed.（3） The stability of pipes conveying fluid with removable elastic support is studied. A differential equation of the pipe in piecewise, continuous conditions at movable elastic support and boundary conditions are established. The recurrence formulas are obtained by using the finite difference method. The effect of elastic constant of the elastic support, location of elastic support on vibration frequencies and stability of elastic pipe conveying fluid are analyzed, and the stability regional diagram （divergence region and flutter region） are plotted.（4） Based on the large deformation theory of extensible elastic rod, nonlinear governing differential equations of FGM rod subjected to a uniform distributed tangential load along the central axis are established. By using Runge-Kutta method and shooting method, the post-buckling behaviors of movable hinge and fixed hinge FGM rod made of metal and ceramic are analyzed,and compared with the properties of conservative FGM rod. The post-buckling characteristics curves of FGM rod under the different gradient index are plotted, and compared with the properties of metal and ceramic material rods. The influence of ratio of the length of the rod to the height of rectangular cross-section on the post-buckling behaviors of FGM rod is discussed.（5） The non-linear governing differential equations of fixed hinge /clamped FGM rod subjected to thermal loads were derived. The thermal post-buckling behaviors of FGM rod made of ZrO₂ and Ti-6Al-4Vwere analyzed by shooting method. Firstly, the thermal post-buckling equilibrium paths of the FGM rod with different gradient index in the uniform temperature field were plotted, and compared with the behaviors of the homogeneous rods made of ZrO₂ and Ti-6A1-4V materials respectively. For given value of end rotation angles, the influence of gradient index on the thermal post-buckling behaviors of FGM rod was discussed. Secondly, the thermal post-buckling characteristics of the FGM rod were analyzed when the temperature difference parameter is changing while the bottom temperature parameter remains constant, and when the bottom temperature parameter is changing while the temperature difference parameter remains constant, and compared with the characteristics of the two homogeneous material rods.（6） The coupled differential equations of motion of FGM rectangular plates under the action of uniformly distributed tangential follower forces are derived, which are expressed by introducing stress function and deflection function. The stability of simply supported FGM rectangular plates subjected to non-conservative forces is analyzed by using Fourier series method. The variation curves of vibration frequency and divergence load for different aspect ratio and gradient index are plotted, and the effects of gradient index on vibration frequency and divergence load are investigated.（7） Nonlinear vibration of simply supported FGM rectangular plates subjected to uniformly distributed tangential follower forces is presented. The material properties of a FGM plate were graded continuously in the direction of thickness, according to a simply power-law distribution of the volume fraction of the constituents. The governing nonlinear partial differential equations which expressed by stress function and deflection function are obtained using the von Karman theory. Then the nonlinear partial differential equations are transformed into nonlinear ordinary differential equation using Galerkin method. For simply supported ceramic/metal FGM plates under the action of uniformly distributed tangential follower forces, the effect of follower force, gradient index and aspect ratio on the dynamic behavior of the plates is discussed. The relation between central amplitude and nonlinear fundamental frequency for different parameter are derived.更多还原