非保守FGM杆和板的非线性动力学特性研究
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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ZrO_2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ZrO_2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。
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 ZrO_2 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 ZrO_2 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.
(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ZrO_2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ZrO_2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。
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 ZrO_2 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 ZrO_2 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.
引文
[1] 新野正之等．斜倾机能材料．日本复合材料学会志．1987，13(6)：257-260
[2] T. Mori, K. Tanaka. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta Metallurgica, 1973,21: 571-574.
[3] R. A. Schapery. Thermal expansion coefficients of composite materials based on energy principles[J]: Journal of Composite Materials,1968, 2:380-404
[4] H. Hatta, M.Taya. Effective thermal conductivity of a mis-oriented short fiber composite[J]. Journal of Applied Physics, 1985, 58:2478-2486
[5] T. Reiter, C. Dvorak, V. Tvergaard, Micromechanics model for graded composite materials[J]. Journal of Mechanics and Physics of Solids,1997, 45(8):1281-1302.
[6] F. Erdogen. The crack problem for bonded inhomogeneous materials under anti-plane shear loading[J]. Journal of Applied Mechanics. 1985,52:823-828.
[7] Y. Tanigawa, T. Akai, R. Kawamura, N. Oka. Transient heat conduction and thermal stress problems of a non-homogeneousplate with temperature-dependent material properties[J]. Journal of Thermal Stresses, 1999,19:77-102.
[8] J. N. ReAdy. Analysis of functionally graded plates[J]. International Journal for Numerical Methods in Engineering, 2000, 47:663-684
[9] L. J. Gibson, M.F. Ashby, G N. Karam, et al. Mechanical properties of natural materials Ⅱ. Microstructures for mechanical efficiency[J]. Proceedings of the Royal Society of London Series A. 1995, 450(1938):141-162
[10] Y. Tanigawa. Some basic thermo-elastic problems for non-homogeneous structural materials[J]. Applied Mechanics Review, 1995, 48 (4): 287-300
[11] 王保林，杜善义，韩杰才．功能梯度复合材料的热／机械耦合分析进展[J]，力学进展，1999，29(4)：528-548．
[12] 沈惠申．功能梯度复合材料板壳结构的弯曲、屈曲和振动[J]，力学进展，2004，34(1)：53-60
[13] Noda Naotake, Tsuji Tomoaki. Steady thermal stresses in a plate of functionally gradient material[J], 日本机械学会论文集A编, 1991, 57(535):625-631
[14] R. C. Wetherhold, S. Seehnan and J. Z. Wang. The use of functionally graded materials to eliminate or control thermal deformation[J]. Composites Science and Technology, 1996,56:1099-1104.
[15] Z Q Cheng, R C Batra. Three-dimensional thermoelastic deformations of a functionally graded elliptic plate[J]. Composites Part B: Engineering, 2000, 31:97-106
[16] J.N.Reddy, C.D.Chin. Thermomachanical analysis of functionally graded cylinders and plates[J]. Journal of Thermal stresses, 1998,21:593-626
[17] Y Obata, N Noda. Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded materials[J]. Journal of thermal Stresses, 1994, 17:471-178
[18] Y Obata, N Noda. Unsteady thermal stresses in a functionally gradient material plate (analysis of one-dimensional unsteady heat transfer problem)[J]. Transactions of the Japan Society of Mechanical Engineers, Seres A, 1993, 59(560): 1090-1096
[19] Y. Obata, N. Noda. Transient thermal stresses in a plate of functionally gradient material[J]. Ceramic Transaction 34, Functionally Gradient Materials.American Ceramic Society, 1993:403-410
[20] Y Obata, N Noda. Unsteady thermal stresses in a functionally gradient material, plate[J]. Transactions of the Japan Society of Mechanical Engineers, Series A, 1993, 59(560): 1097-1103
[21] S. S. Vel, R. C. Batra. Three-dimensional analysis of transient thermal stresses in functionally graded plates[J]. International Journal of Solids and Structures. 2003, 40:7181-7196
[22] J. N. Reddy, C. M. Wang, S. Kitipomchal. Axisymmetric bending offunctionally graded circular and annular plates[J]: European Journal of Mechanics A/Solids, 1999,18:185-199
[23] Z. Q. Cheng, R. C. Batra. Three-dimensional thermo-elastic deformations of a functionally graded elliptic plate[J]. Composites Part B: Engineering,2000, 31:97-106
[24] Z. Q. Cheng, R. C. Batra. Deflection relationships between the homogeneous kirchhoff plate theory and different functionally graded plate theories[J], Archives of Mechanics, 2000, 52(1): 143～158
[25] J. N. Reddy, Z. Q. Cheng. Three-Dimensional Thermo-Mechanical Deformation of Functionally Graded Rectangular Plate[J]. European Journal of Mechanics A/Solids, 2001,20:841-855
[26] S. S. Vel, R. C. Batra. Exact solution for thermoelastie deformations of functionally graded thick rectangular plates[J]. AIAA Journal, 2002, 40(7):1421-1433
[27] Y. Y. Yang. Stress analysis in a joint with a functionally graded material under a thermal loading by using the Meilin transform method[J].Int. J. Solids Structure,1998,35(12):1261-1287.
[28] A. Chakraborty, S. Gopalakrishnan. A spectrally formulated finite element for wave propagation analysis in functionally graded rods[J]. International Journal of Solids and Structures, 2003,40:2421-2448.
[29] J. N. Reddy. A simple higher-order theory for laminated composite plates[J]. Journal of Applied Mechanics ASME, 1984,51:745-752
[30] H. S. Shen. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments[J]. International Journal of Mechanical sciences, 2002, 44(3): 561-584
[31] L. S. Ma, T. J. Wang. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings[J]. International Journal of Solids and Structures, 2003, 40, 3311-3330.
[32] 杨杰，沈惠申．热／机械载荷下功能梯度材料矩形厚板的弯曲行为[J]．固体力学学报，2003，24(1)：119-124。
[33 J. Woo, S. A. Meguid. Nonlinear analysis of functionally graded plates and shallow shells[J], International Journal of Solids and structures, 2001, 38:7409-7421
[34] R. Javaberi, M. R. Eslami. Buckling of functionally graded plates under in-plane compressive loading[J]. ZAMM, 2002, 82(4): 277-283
[35] R. Javahefi, M. R. Eslami. Thermal buckling of functionally, graded plates[J]. AIAA Journal, 2002, 40(1): 162-169
[36] R. Javaheri, M. R. Eslami. Thermal buckling of functionally graded plates based on higher order theory[J]. Journal of Thermal Stresses, 2002, 25(7): 603-625
[37] M. M. Najafizadeh, M. R. Eslami. Bucklinganalysis of circular plates of functionally graded materials under uniform radial compression[J]. International Journal of Mechanical Sciences, 2002, 44:2479-2493
[38] M.M. Najafizadeh, M. R. Eslami. First-order-theory-based-thermoelastic stability of functionally graded material circular plates[J]. AIAA Journal, 2002, 40(7): 1444-1450
[39 J. Yang, H. S. Shen. Non-linear analysis of functionally graded plates under transverse and in plane loads[J]. International Journal of Non-linear Mechanics,2003,38(4):467-482
[40] L. S. Ma, T. J. Wang. Axisymmetric post-buckling of a functionally graded circular plate subjected to uniformly distributed radial compression[J]. Materials Science Forum, 2003,423-425:719-724
[41] L H. Wu. Thermal Buckling of Simply Supported Moderately Thick Rectangular FGM Plate[J]. Composite Structures, 2004, 64:211-218
[42] H. S. Shen. Thermal postbuckling behavior of functionally graded cylindrical shells with temperature-dependent properties[j]. International Journal of Solids and Structures, 2004, 41:1961-1974
[43] C. T. Loy, K. Y. Lain, J. N. Reddy. Vibration of functionally graded cylindrical shells[J]. International Journal of Mechanical Sciences, 1999,41(3): 309-324
[44] S. C. Pradhan, C. T. Loy, K. Y. Lain, J. N. Reddy. Vibration characteristics of functionally graded cylindrical shells under various boundary conditions[J]. Applied Acoustics, 2000, 61(1): 119-129
[45] S. W. Gong, K. Y.Lam, J. N. Reddy. The elastic response of functionally graded cylindrical shells to low-velocitY impact[J]. International Journal of Impact Engineering, 1999, 22(4):397-417
[46] J. Yang, H. S. Shen. Dynamic response of initially stressed functionally graded rectangular thin plates[J]. Composite Structures, 2001, 54(4): 497-508
[47] 陈伟球，叶贵如，蔡金标，丁皓江．横观各向同性功能梯度材料矩形板的自由振动[J]．振动工程学报，2001，14(3)：263-267
[48] 陈伟球，叶贵如，蔡金标，丁皓江．球面各向同性功能梯度材料球壳的自由振动．力学学报，2001，33(4)：768-775
[49] 边祖光，王亚明，陈伟球。Winkler地基上功能梯度材料厚板的自由振动[J]。工程力学，2001，(增刊)：181-185
[50] W Q Chen, X Wang, H J Ding. Free vibration fluid-filled hollow sphere of functionally graded material with spherical isotropy[J]. Journal of Acoustical Society of America, 1999, 106(5):2588～2594
[51] X. Han, D. Xu, G. R. Liu. Transient responses in a functionally graded cylindrical shells to a point load[J]. Journal of Sound and Vibration, 2002, 251(5):783-805
[52] J. Yang, H. S. Shen. Vibration characteristics and transient responses of shear deformable functionally graded plates in thermal environment[J]. Journal of Sound and Vibration, 2002, 255(3): 579-602
[53] G. N. Praveen, J. N. Reddy. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates[J]. International Journal of Solids and Structures, 1998, 35(33):4457-4476
[54] Z. Q, Cheng, J. N. Reddy. Frequency of functionally graded plates with three-dimensional asymptotic approach[J]. Journal of Engineering Mechanics ASCE, 2003, 129(8):896-900
[55] T. Y. Ng, K. Y. Lain, K. M. Liew. Effects of FGM materials on the parametric resonance of plate structures[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 190 (8-10): 953-962
[56] T. Y. Ng, K. Y. Lain, K. M. Liew, J. N. Reddy. Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading[J]. International Journal of Solids and Structures, 2001, 38:1295-1309
[57] Liviu Librescu, Sang-Yong Oh, Ohseop Song. Thin-walled rods made of functionally graded materials and operating in a high temperature environment: vibration and stability[J]. Journal of Thermal Stresses, 2005, 28(6):649-712
[58] J Woo, S A Meguid, L SOng. Nonlinear free Vibration behavior of functionally graded plates[J]. Journal of Sound and Vibration, 20061289(3):595-611
[59] S C Chun. Nonlinear vibration of a shear deformable functionally graded plates[j]. Composite Structures, 2005, 68:295-302
[60] 曹志远．功能梯度板的非线性动力分析[J]．固体力学学报，2006，27(1)：21-24
[61] V. V. Bolotin. Non-conservative Problems of the Theory of Elastic Stability[M], Pergamon Press, Oxford, 1963
[62] G. Herrmann, R. W. Bungay. On the stability of elastic systems subjected to nonconservative forces[J]. Journal of Applied Mechanics, 1964, 31:435-440
[63] G. Herrmann. Stability of equilibrium of elastic systems subjected to nonconservative forces[J]. Applied Mechanics Review, 1967, 20(1):103-108.
[64] H. Leipholz. Stability of elastic systems[M]. Alphen aanden Rijthoff & Noordhoff,1980
[65] 刘殿魁，张其浩．弹性理论中非保守问题的一般变分原理[J]．力学学报．1981，(6)：562-570
[66] 熊跃熙．非保守系统的有限变形弹性变分原理[J]．力学学报．1983，15(1)：86-90
[67] 郑泉水．非线性弹性理论的泛变分原理[J]．应用数学和力学．1984，5(2)：205-216
[68] 梁立孚，刘殿魁，宋海燕．非保守系统的两类变量的广义拟变分原理[J]．中国科学．2005，35(2)：201-212
[69] M. P. Paidoussis. Fluid-Structure Interactions: Slender Structures and Axial Flow[M]. London: Academic Press. 1998
[70] M. A. Langthjem, Y. Sugiyama. Dynamic stability of columns subjected to follower loads: a survey[J]. Journal of Sound and Vibration 2000, 238(5): 809-851.
[71] H. Leipholz. On the sufficiency of the energy criterion for the stability of certain nonconservative systems of follower-load type[J]. ASME Journal of Applied Mechanics 1972, 39:717-722.
[72] H. Leipholz. On principles of stationary for non-self-adjoint rod problems[J]. Computer Methods in Applied Mechanics and Engineering 1986, 59: 215-226.
[73] M. P. Paidoussis. Dynamics of cylindrical structures subjected to axial flow[J]. Journal of Sound and Vibration. 1973, 29: 365-385.
[74] A. M. Gasparrini, A. V. Saetta, R. V. Vitaliani. On the stability and nistability regions of non-conservative continuous system under partially follower forces[J]. Computer Methods in Applied Mechanics and Engineering, 1995, 124(1):63-78
[75] L.R. Kumar, P. K.Datta, D. L. Prabhakara. Dynamic instability characteristics of laminated composite plates subjected to partial follower edge load with damping[J]. International Journal of mechanical Sciences, 2003, 45(9): 1429-1448
[76] K. Higuchi, E. H. Dowell. Elect of strueturai damping on flutter of plates with a follower force[J]. AIAA Journal, 1992, 30:820-825
[77] K. Higuchi, E. H. Dowell. Elects of the Poisson ratio and negative thrust on the dynamic stability of a free plate subjected to a follower force[J]. Journal of Sound and Vibration, 1989, 129: 255-269.
[78] K. Higuehi, E. H. Dowell. Dynamic stability of a rectangular plate with four free edges subjected to a follower force[J]. AIAA Journal, 1990, 28: 1300-1305.
[79] J. H. Kim, J. H. Park. On the dynamic stability of rectangular-plates subjected to intermediate follower forces[J]. Journal of Sound and Vibration 1998, 209:882-888
[80] M. N. Bismarck. Dynamic stability of shallow shells subjectedto follower forces[J]. AIAA Journal 1995, 33:355-360
[81] J. H. Kim, H. S. Kim. A study on the dynamic stability of plates under a follower force[J]. Computers and Structures, 2000, 74: 351-363
[82] M. Nishiwaki. Generalized theory of brake noise[J]. Proceedings of the InStitution of Mechanical Engineers, 1993, 207:195-202
[83] J. E. Mottershead, S, N. Chan. Flutter instability of circular discs with frictional follower loads[J]. ASME Journal of Vibration and ACoustics, 1995, 117:161-163
[84] D. Lee, A. M. Waas. Stability analysis of a rotating multi-layer annular plate with a stationary frictional follower load[J]. International Journal of Mechanical Sciences, 1997, 39:1117-1138
[85] 宋志远，万虹，梅占馨．非保守力作用下矩形薄板的稳定问题[J]．固体力学学报．1991，12(4)：359-363
[86] 王忠民，计伊周．矩形薄板在随从力作用下的动力稳定性分析[J]。振动工程学报．1992，5(1)：78-83
[87] 王忠民，赵凤群．弹性非保守简支矩形薄板的后屈曲性态[J]．西安理工大学学报．1997，13(3)：238-242
[88] 莫宵依，计伊周，王忠民．矩形薄板在非保守力作用下的动力稳定性[J]．西安理工大学学报．2000，16(4)：370-375
[89] 王忠民，高教伯，李会侠．非保守圆薄板的轴对称振动和稳定性[J]．固体力学学报，2003，24(2)：155-162
[90] B. Kang, C.A. Tan. Parametric instability of a Leipholz rod due to distributed frictional axial load[J]. International Journal of Mechanical Sciences. 46 (2004) 807-825
[91] T. A. Weisshaar, R. H. Plaut. Alphen aan den Rijn: Sijthoff and Noordhoff. Structural. optimization under nonconservative loading[M]. Optimization of Distributed Parameter Structures, 1981,843-864.
[92] 武际可，苏先樾．弹性系统的稳定性[M]．北京：科学出版社，1994．
[93] R V Vitaliani, A M Gaspadni and A V seatta. Finite element solution of the stability problem for nonlinear undamped and damped system under nonconsevative loading [J]. International Journal of Solids and Structures, 1997, 34:2497-2516
[94] F M Detinko. On the elactic stability of uniform rods and circular arches under non-consevative loading[J]. International Journal of Solids and Structures, 2000,37:5505-5515.
[95] R Sampaio, M P Almeida. Buckling of extensible rods[J]. Revista Brasileira de Ciencias Mecanicas, 1985,7 373-385 (in Portuguese)
[96] C P Filipich, M B Rosales. A further study on the post-buckling of extensible elastic rods[J]. Internet J Non-Linear Mech., 2000, 35(6):997～1022
[97] 李世荣，吴莹，周又和．轴向随动分布载荷作用下简支杆的后屈曲[J]．工程力学，2003，20(3)：94～97
[98] P. L. Vemlere de Irassar, G. M. Fiecadenti and P. A. A. Laura. Dynamic analysis of a rod with an intermediate elastic support[J]. Journal of Sound and Vibration, 1984, 98(3): 381～389
[99] M. S. Ingber, A. L. Pate and J. M. Salazer. Vibration of a clamped plate with concentrated mass and spring attachments[J]. Journal of Sound and Vibration, 1992, 153:142-166
[100] W.Glabisz. Vibration and stability of a rod with elastic supports and concentrated masses under conservative and non-conservative forces[J]. Computers and Structures, 1999,70:305-313
[101] W. Nicholson, L. A. Bergman. Free vibration of combined dynamical systems, Journal of Engineering Mechanics, 1986, 112:1-13
[102] D. N. Manikanahally, M. J. Crocker. Vibration absorbers for hysteretically damped mass-loaded rods[J], ASME Journal of Vibration and Acoustics, 1991, 113:116-122
[103] L. Ercoli, P. A. A. Laura. Analytical and experiment investigation on continuous rods carrying elastically mounted masses[J]. Journal of Sound and Vibration, 1987114:519-533
[104] C. S. Kim, S.M. Dickinson. On the analysis of laterally vibrating slender rods subject to various complicating effects[J]. Journal of Sound and Vibration, 1988,122(3): 441-455
[105] S. Azimi. Free Vibration:of Circular Plates with Elastic or Rigid Interior Support[J]. Journal of Sound and Vibration.1988,120(1):37-52
[106] S. Azimi. A xisymmetric Vibration of Point-Supported Circular Plates[J]. Journal of Sound and Vibration. 1989, 135(2): 177-195
[107] 刘肖莹，金梦石．Green函数法解点支承任意板自由振动频率[J]．固体力学学报，1993，(1)：54-58
[108] 严宗达．结构力学中的富里叶级数解法[M]．天津：天津大学出版社，1989
[109] 冯文杰，邹振祝．点支圆板自由振动[J]．振动与冲击，1998，17(2)：67-70
[110] M.Xu, D.Cheng. A new approach to solving a type of vibration problem[J]. Journal of Sound and Vibration. 1994,177(4): 565-571.
[111] 许明田，程德林．用积分方程法解板的振动问题[J]．应用数学和力学，1996，17(7)：655-660
[112] 王卫东，程泉，许明田．一种求解含振子及弹性支承圆板振动的新方法[J]．应用力学学报，2001，18(2)：86-90
[113] C.J.Wu, R.G.White. Vibrational power transmission in a finite multi-supported rod[J]. Journal of Sound and Vibration, 1995,181(1): 99-114.
[114] C. H. Riedel; C. A. Tan. Dynamic characteristics and mode localization of elastically constrained axially moving string and rod[J]. Journal of Sound and Vibration, 1998,215(3): 455-473.
[115] Y. K. Cheung, D. Zhou. Vibrations of rectangular plates with elastic intermediate line-supports and cons traits[J]. Thin-Walled Structures, 2000, 37: 305～331.
[116] J. K. Sinha, P. M. Mujimdar and R. I. K. Moorthy. Detection of spring support locations in elastic structures using a gradient-based fmite element model updating technique[J]. Journal of Sound and Vibration, 2001,240(3): 499-518.
[117] C. Tosone, A. Maceri Continuous rods with unilateral elastic supports[J]. Journal of Mathematical Analysis and Applications, 2001,259:302～318.
[118] M. H. Huangs, D. P. Thambiratnam. Free vibration analysis of rectangular plates on elastic intermediate supports[J]. Journal of Sound and Vibration, 2001, 240(3): 567～580.
[119] D. Wang. Optimization of support positions to minimize the maximal deflection of structures[J]. International Journal of Solids and structures. 2004, 41: 7445-7458.
[120] 赵凤群，王忠民．点弹性支承的非保守矩形薄板的稳定性[J]．西安理工大学学报1998，14(4)：398-403
[121] 赵凤群，王忠民，常安定．点弹性支承的非保守杆稳定性的积分方程法[J]．西安地质学院学报，1997，19：119-123
[122] H C Lin, S S Chen. Vibration and stability of fluid-conveying pipe[J]. Shock and Vibration, 1976,46(2),267～283
[123] S T Noah, G R Hopking. Dynamic stability of elastically supported pipes conveying fluid[J]. Jounal of. Sound and Vibration. 1980, 71(1):103～116
[124] 金基铎，林影，邹光胜．悬臂输流管道的颤振与混沌运动分析[J]．振动工程学报，1997，10(3)：314～320
[125] J D Jin, Stability and chaotic motion of a restrained pipe conveying fluid[J]. Jounal of. Sound and Vibration. 1997, 28(3): 427～439
[126] 金基铎，邹光胜，张宇飞．悬臂输流管道的运动分岔现象和混沌运动[J]．力学学报，2002，34 (6)：863～873
[127] 倪樵，黄玉盈，陈贻平．微分求积法分析具有弹性支承输液管的临界流速[J]。计算力学学报， 2000，18(2)：146～149
[128] 杨挺青．粘弹性力学[M]．武汉：华中理工大学出版社，1990
[129] 倪振华．振动力学[M]．西安：西安交通大学出版社，1994
[130] 方同，薛璞．振动理论与应用[M]．西安：西北工业大学出版社，1998．
[131] 李世荣，程昌钧．加热弹性杆的热过屈曲分析[J]．应用数学和力学，2000，21(2)：119-125
[2] T. Mori, K. Tanaka. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta Metallurgica, 1973,21: 571-574.
[3] R. A. Schapery. Thermal expansion coefficients of composite materials based on energy principles[J]: Journal of Composite Materials,1968, 2:380-404
[4] H. Hatta, M.Taya. Effective thermal conductivity of a mis-oriented short fiber composite[J]. Journal of Applied Physics, 1985, 58:2478-2486
[5] T. Reiter, C. Dvorak, V. Tvergaard, Micromechanics model for graded composite materials[J]. Journal of Mechanics and Physics of Solids,1997, 45(8):1281-1302.
[6] F. Erdogen. The crack problem for bonded inhomogeneous materials under anti-plane shear loading[J]. Journal of Applied Mechanics. 1985,52:823-828.
[7] Y. Tanigawa, T. Akai, R. Kawamura, N. Oka. Transient heat conduction and thermal stress problems of a non-homogeneousplate with temperature-dependent material properties[J]. Journal of Thermal Stresses, 1999,19:77-102.
[8] J. N. ReAdy. Analysis of functionally graded plates[J]. International Journal for Numerical Methods in Engineering, 2000, 47:663-684
[9] L. J. Gibson, M.F. Ashby, G N. Karam, et al. Mechanical properties of natural materials Ⅱ. Microstructures for mechanical efficiency[J]. Proceedings of the Royal Society of London Series A. 1995, 450(1938):141-162
[10] Y. Tanigawa. Some basic thermo-elastic problems for non-homogeneous structural materials[J]. Applied Mechanics Review, 1995, 48 (4): 287-300
[11] 王保林，杜善义，韩杰才．功能梯度复合材料的热／机械耦合分析进展[J]，力学进展，1999，29(4)：528-548．
[12] 沈惠申．功能梯度复合材料板壳结构的弯曲、屈曲和振动[J]，力学进展，2004，34(1)：53-60
[13] Noda Naotake, Tsuji Tomoaki. Steady thermal stresses in a plate of functionally gradient material[J], 日本机械学会论文集A编, 1991, 57(535):625-631
[14] R. C. Wetherhold, S. Seehnan and J. Z. Wang. The use of functionally graded materials to eliminate or control thermal deformation[J]. Composites Science and Technology, 1996,56:1099-1104.
[15] Z Q Cheng, R C Batra. Three-dimensional thermoelastic deformations of a functionally graded elliptic plate[J]. Composites Part B: Engineering, 2000, 31:97-106
[16] J.N.Reddy, C.D.Chin. Thermomachanical analysis of functionally graded cylinders and plates[J]. Journal of Thermal stresses, 1998,21:593-626
[17] Y Obata, N Noda. Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded materials[J]. Journal of thermal Stresses, 1994, 17:471-178
[18] Y Obata, N Noda. Unsteady thermal stresses in a functionally gradient material plate (analysis of one-dimensional unsteady heat transfer problem)[J]. Transactions of the Japan Society of Mechanical Engineers, Seres A, 1993, 59(560): 1090-1096
[19] Y. Obata, N. Noda. Transient thermal stresses in a plate of functionally gradient material[J]. Ceramic Transaction 34, Functionally Gradient Materials.American Ceramic Society, 1993:403-410
[20] Y Obata, N Noda. Unsteady thermal stresses in a functionally gradient material, plate[J]. Transactions of the Japan Society of Mechanical Engineers, Series A, 1993, 59(560): 1097-1103
[21] S. S. Vel, R. C. Batra. Three-dimensional analysis of transient thermal stresses in functionally graded plates[J]. International Journal of Solids and Structures. 2003, 40:7181-7196
[22] J. N. Reddy, C. M. Wang, S. Kitipomchal. Axisymmetric bending offunctionally graded circular and annular plates[J]: European Journal of Mechanics A/Solids, 1999,18:185-199
[23] Z. Q. Cheng, R. C. Batra. Three-dimensional thermo-elastic deformations of a functionally graded elliptic plate[J]. Composites Part B: Engineering,2000, 31:97-106
[24] Z. Q. Cheng, R. C. Batra. Deflection relationships between the homogeneous kirchhoff plate theory and different functionally graded plate theories[J], Archives of Mechanics, 2000, 52(1): 143～158
[25] J. N. Reddy, Z. Q. Cheng. Three-Dimensional Thermo-Mechanical Deformation of Functionally Graded Rectangular Plate[J]. European Journal of Mechanics A/Solids, 2001,20:841-855
[26] S. S. Vel, R. C. Batra. Exact solution for thermoelastie deformations of functionally graded thick rectangular plates[J]. AIAA Journal, 2002, 40(7):1421-1433
[27] Y. Y. Yang. Stress analysis in a joint with a functionally graded material under a thermal loading by using the Meilin transform method[J].Int. J. Solids Structure,1998,35(12):1261-1287.
[28] A. Chakraborty, S. Gopalakrishnan. A spectrally formulated finite element for wave propagation analysis in functionally graded rods[J]. International Journal of Solids and Structures, 2003,40:2421-2448.
[29] J. N. Reddy. A simple higher-order theory for laminated composite plates[J]. Journal of Applied Mechanics ASME, 1984,51:745-752
[30] H. S. Shen. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments[J]. International Journal of Mechanical sciences, 2002, 44(3): 561-584
[31] L. S. Ma, T. J. Wang. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings[J]. International Journal of Solids and Structures, 2003, 40, 3311-3330.
[32] 杨杰，沈惠申．热／机械载荷下功能梯度材料矩形厚板的弯曲行为[J]．固体力学学报，2003，24(1)：119-124。
[33 J. Woo, S. A. Meguid. Nonlinear analysis of functionally graded plates and shallow shells[J], International Journal of Solids and structures, 2001, 38:7409-7421
[34] R. Javaberi, M. R. Eslami. Buckling of functionally graded plates under in-plane compressive loading[J]. ZAMM, 2002, 82(4): 277-283
[35] R. Javahefi, M. R. Eslami. Thermal buckling of functionally, graded plates[J]. AIAA Journal, 2002, 40(1): 162-169
[36] R. Javaheri, M. R. Eslami. Thermal buckling of functionally graded plates based on higher order theory[J]. Journal of Thermal Stresses, 2002, 25(7): 603-625
[37] M. M. Najafizadeh, M. R. Eslami. Bucklinganalysis of circular plates of functionally graded materials under uniform radial compression[J]. International Journal of Mechanical Sciences, 2002, 44:2479-2493
[38] M.M. Najafizadeh, M. R. Eslami. First-order-theory-based-thermoelastic stability of functionally graded material circular plates[J]. AIAA Journal, 2002, 40(7): 1444-1450
[39 J. Yang, H. S. Shen. Non-linear analysis of functionally graded plates under transverse and in plane loads[J]. International Journal of Non-linear Mechanics,2003,38(4):467-482
[40] L. S. Ma, T. J. Wang. Axisymmetric post-buckling of a functionally graded circular plate subjected to uniformly distributed radial compression[J]. Materials Science Forum, 2003,423-425:719-724
[41] L H. Wu. Thermal Buckling of Simply Supported Moderately Thick Rectangular FGM Plate[J]. Composite Structures, 2004, 64:211-218
[42] H. S. Shen. Thermal postbuckling behavior of functionally graded cylindrical shells with temperature-dependent properties[j]. International Journal of Solids and Structures, 2004, 41:1961-1974
[43] C. T. Loy, K. Y. Lain, J. N. Reddy. Vibration of functionally graded cylindrical shells[J]. International Journal of Mechanical Sciences, 1999,41(3): 309-324
[44] S. C. Pradhan, C. T. Loy, K. Y. Lain, J. N. Reddy. Vibration characteristics of functionally graded cylindrical shells under various boundary conditions[J]. Applied Acoustics, 2000, 61(1): 119-129
[45] S. W. Gong, K. Y.Lam, J. N. Reddy. The elastic response of functionally graded cylindrical shells to low-velocitY impact[J]. International Journal of Impact Engineering, 1999, 22(4):397-417
[46] J. Yang, H. S. Shen. Dynamic response of initially stressed functionally graded rectangular thin plates[J]. Composite Structures, 2001, 54(4): 497-508
[47] 陈伟球，叶贵如，蔡金标，丁皓江．横观各向同性功能梯度材料矩形板的自由振动[J]．振动工程学报，2001，14(3)：263-267
[48] 陈伟球，叶贵如，蔡金标，丁皓江．球面各向同性功能梯度材料球壳的自由振动．力学学报，2001，33(4)：768-775
[49] 边祖光，王亚明，陈伟球。Winkler地基上功能梯度材料厚板的自由振动[J]。工程力学，2001，(增刊)：181-185
[50] W Q Chen, X Wang, H J Ding. Free vibration fluid-filled hollow sphere of functionally graded material with spherical isotropy[J]. Journal of Acoustical Society of America, 1999, 106(5):2588～2594
[51] X. Han, D. Xu, G. R. Liu. Transient responses in a functionally graded cylindrical shells to a point load[J]. Journal of Sound and Vibration, 2002, 251(5):783-805
[52] J. Yang, H. S. Shen. Vibration characteristics and transient responses of shear deformable functionally graded plates in thermal environment[J]. Journal of Sound and Vibration, 2002, 255(3): 579-602
[53] G. N. Praveen, J. N. Reddy. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates[J]. International Journal of Solids and Structures, 1998, 35(33):4457-4476
[54] Z. Q, Cheng, J. N. Reddy. Frequency of functionally graded plates with three-dimensional asymptotic approach[J]. Journal of Engineering Mechanics ASCE, 2003, 129(8):896-900
[55] T. Y. Ng, K. Y. Lain, K. M. Liew. Effects of FGM materials on the parametric resonance of plate structures[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 190 (8-10): 953-962
[56] T. Y. Ng, K. Y. Lain, K. M. Liew, J. N. Reddy. Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading[J]. International Journal of Solids and Structures, 2001, 38:1295-1309
[57] Liviu Librescu, Sang-Yong Oh, Ohseop Song. Thin-walled rods made of functionally graded materials and operating in a high temperature environment: vibration and stability[J]. Journal of Thermal Stresses, 2005, 28(6):649-712
[58] J Woo, S A Meguid, L SOng. Nonlinear free Vibration behavior of functionally graded plates[J]. Journal of Sound and Vibration, 20061289(3):595-611
[59] S C Chun. Nonlinear vibration of a shear deformable functionally graded plates[j]. Composite Structures, 2005, 68:295-302
[60] 曹志远．功能梯度板的非线性动力分析[J]．固体力学学报，2006，27(1)：21-24
[61] V. V. Bolotin. Non-conservative Problems of the Theory of Elastic Stability[M], Pergamon Press, Oxford, 1963
[62] G. Herrmann, R. W. Bungay. On the stability of elastic systems subjected to nonconservative forces[J]. Journal of Applied Mechanics, 1964, 31:435-440
[63] G. Herrmann. Stability of equilibrium of elastic systems subjected to nonconservative forces[J]. Applied Mechanics Review, 1967, 20(1):103-108.
[64] H. Leipholz. Stability of elastic systems[M]. Alphen aanden Rijthoff & Noordhoff,1980
[65] 刘殿魁，张其浩．弹性理论中非保守问题的一般变分原理[J]．力学学报．1981，(6)：562-570
[66] 熊跃熙．非保守系统的有限变形弹性变分原理[J]．力学学报．1983，15(1)：86-90
[67] 郑泉水．非线性弹性理论的泛变分原理[J]．应用数学和力学．1984，5(2)：205-216
[68] 梁立孚，刘殿魁，宋海燕．非保守系统的两类变量的广义拟变分原理[J]．中国科学．2005，35(2)：201-212
[69] M. P. Paidoussis. Fluid-Structure Interactions: Slender Structures and Axial Flow[M]. London: Academic Press. 1998
[70] M. A. Langthjem, Y. Sugiyama. Dynamic stability of columns subjected to follower loads: a survey[J]. Journal of Sound and Vibration 2000, 238(5): 809-851.
[71] H. Leipholz. On the sufficiency of the energy criterion for the stability of certain nonconservative systems of follower-load type[J]. ASME Journal of Applied Mechanics 1972, 39:717-722.
[72] H. Leipholz. On principles of stationary for non-self-adjoint rod problems[J]. Computer Methods in Applied Mechanics and Engineering 1986, 59: 215-226.
[73] M. P. Paidoussis. Dynamics of cylindrical structures subjected to axial flow[J]. Journal of Sound and Vibration. 1973, 29: 365-385.
[74] A. M. Gasparrini, A. V. Saetta, R. V. Vitaliani. On the stability and nistability regions of non-conservative continuous system under partially follower forces[J]. Computer Methods in Applied Mechanics and Engineering, 1995, 124(1):63-78
[75] L.R. Kumar, P. K.Datta, D. L. Prabhakara. Dynamic instability characteristics of laminated composite plates subjected to partial follower edge load with damping[J]. International Journal of mechanical Sciences, 2003, 45(9): 1429-1448
[76] K. Higuchi, E. H. Dowell. Elect of strueturai damping on flutter of plates with a follower force[J]. AIAA Journal, 1992, 30:820-825
[77] K. Higuchi, E. H. Dowell. Elects of the Poisson ratio and negative thrust on the dynamic stability of a free plate subjected to a follower force[J]. Journal of Sound and Vibration, 1989, 129: 255-269.
[78] K. Higuehi, E. H. Dowell. Dynamic stability of a rectangular plate with four free edges subjected to a follower force[J]. AIAA Journal, 1990, 28: 1300-1305.
[79] J. H. Kim, J. H. Park. On the dynamic stability of rectangular-plates subjected to intermediate follower forces[J]. Journal of Sound and Vibration 1998, 209:882-888
[80] M. N. Bismarck. Dynamic stability of shallow shells subjectedto follower forces[J]. AIAA Journal 1995, 33:355-360
[81] J. H. Kim, H. S. Kim. A study on the dynamic stability of plates under a follower force[J]. Computers and Structures, 2000, 74: 351-363
[82] M. Nishiwaki. Generalized theory of brake noise[J]. Proceedings of the InStitution of Mechanical Engineers, 1993, 207:195-202
[83] J. E. Mottershead, S, N. Chan. Flutter instability of circular discs with frictional follower loads[J]. ASME Journal of Vibration and ACoustics, 1995, 117:161-163
[84] D. Lee, A. M. Waas. Stability analysis of a rotating multi-layer annular plate with a stationary frictional follower load[J]. International Journal of Mechanical Sciences, 1997, 39:1117-1138
[85] 宋志远，万虹，梅占馨．非保守力作用下矩形薄板的稳定问题[J]．固体力学学报．1991，12(4)：359-363
[86] 王忠民，计伊周．矩形薄板在随从力作用下的动力稳定性分析[J]。振动工程学报．1992，5(1)：78-83
[87] 王忠民，赵凤群．弹性非保守简支矩形薄板的后屈曲性态[J]．西安理工大学学报．1997，13(3)：238-242
[88] 莫宵依，计伊周，王忠民．矩形薄板在非保守力作用下的动力稳定性[J]．西安理工大学学报．2000，16(4)：370-375
[89] 王忠民，高教伯，李会侠．非保守圆薄板的轴对称振动和稳定性[J]．固体力学学报，2003，24(2)：155-162
[90] B. Kang, C.A. Tan. Parametric instability of a Leipholz rod due to distributed frictional axial load[J]. International Journal of Mechanical Sciences. 46 (2004) 807-825
[91] T. A. Weisshaar, R. H. Plaut. Alphen aan den Rijn: Sijthoff and Noordhoff. Structural. optimization under nonconservative loading[M]. Optimization of Distributed Parameter Structures, 1981,843-864.
[92] 武际可，苏先樾．弹性系统的稳定性[M]．北京：科学出版社，1994．
[93] R V Vitaliani, A M Gaspadni and A V seatta. Finite element solution of the stability problem for nonlinear undamped and damped system under nonconsevative loading [J]. International Journal of Solids and Structures, 1997, 34:2497-2516
[94] F M Detinko. On the elactic stability of uniform rods and circular arches under non-consevative loading[J]. International Journal of Solids and Structures, 2000,37:5505-5515.
[95] R Sampaio, M P Almeida. Buckling of extensible rods[J]. Revista Brasileira de Ciencias Mecanicas, 1985,7 373-385 (in Portuguese)
[96] C P Filipich, M B Rosales. A further study on the post-buckling of extensible elastic rods[J]. Internet J Non-Linear Mech., 2000, 35(6):997～1022
[97] 李世荣，吴莹，周又和．轴向随动分布载荷作用下简支杆的后屈曲[J]．工程力学，2003，20(3)：94～97
[98] P. L. Vemlere de Irassar, G. M. Fiecadenti and P. A. A. Laura. Dynamic analysis of a rod with an intermediate elastic support[J]. Journal of Sound and Vibration, 1984, 98(3): 381～389
[99] M. S. Ingber, A. L. Pate and J. M. Salazer. Vibration of a clamped plate with concentrated mass and spring attachments[J]. Journal of Sound and Vibration, 1992, 153:142-166
[100] W.Glabisz. Vibration and stability of a rod with elastic supports and concentrated masses under conservative and non-conservative forces[J]. Computers and Structures, 1999,70:305-313
[101] W. Nicholson, L. A. Bergman. Free vibration of combined dynamical systems, Journal of Engineering Mechanics, 1986, 112:1-13
[102] D. N. Manikanahally, M. J. Crocker. Vibration absorbers for hysteretically damped mass-loaded rods[J], ASME Journal of Vibration and Acoustics, 1991, 113:116-122
[103] L. Ercoli, P. A. A. Laura. Analytical and experiment investigation on continuous rods carrying elastically mounted masses[J]. Journal of Sound and Vibration, 1987114:519-533
[104] C. S. Kim, S.M. Dickinson. On the analysis of laterally vibrating slender rods subject to various complicating effects[J]. Journal of Sound and Vibration, 1988,122(3): 441-455
[105] S. Azimi. Free Vibration:of Circular Plates with Elastic or Rigid Interior Support[J]. Journal of Sound and Vibration.1988,120(1):37-52
[106] S. Azimi. A xisymmetric Vibration of Point-Supported Circular Plates[J]. Journal of Sound and Vibration. 1989, 135(2): 177-195
[107] 刘肖莹，金梦石．Green函数法解点支承任意板自由振动频率[J]．固体力学学报，1993，(1)：54-58
[108] 严宗达．结构力学中的富里叶级数解法[M]．天津：天津大学出版社，1989
[109] 冯文杰，邹振祝．点支圆板自由振动[J]．振动与冲击，1998，17(2)：67-70
[110] M.Xu, D.Cheng. A new approach to solving a type of vibration problem[J]. Journal of Sound and Vibration. 1994,177(4): 565-571.
[111] 许明田，程德林．用积分方程法解板的振动问题[J]．应用数学和力学，1996，17(7)：655-660
[112] 王卫东，程泉，许明田．一种求解含振子及弹性支承圆板振动的新方法[J]．应用力学学报，2001，18(2)：86-90
[113] C.J.Wu, R.G.White. Vibrational power transmission in a finite multi-supported rod[J]. Journal of Sound and Vibration, 1995,181(1): 99-114.
[114] C. H. Riedel; C. A. Tan. Dynamic characteristics and mode localization of elastically constrained axially moving string and rod[J]. Journal of Sound and Vibration, 1998,215(3): 455-473.
[115] Y. K. Cheung, D. Zhou. Vibrations of rectangular plates with elastic intermediate line-supports and cons traits[J]. Thin-Walled Structures, 2000, 37: 305～331.
[116] J. K. Sinha, P. M. Mujimdar and R. I. K. Moorthy. Detection of spring support locations in elastic structures using a gradient-based fmite element model updating technique[J]. Journal of Sound and Vibration, 2001,240(3): 499-518.
[117] C. Tosone, A. Maceri Continuous rods with unilateral elastic supports[J]. Journal of Mathematical Analysis and Applications, 2001,259:302～318.
[118] M. H. Huangs, D. P. Thambiratnam. Free vibration analysis of rectangular plates on elastic intermediate supports[J]. Journal of Sound and Vibration, 2001, 240(3): 567～580.
[119] D. Wang. Optimization of support positions to minimize the maximal deflection of structures[J]. International Journal of Solids and structures. 2004, 41: 7445-7458.
[120] 赵凤群，王忠民．点弹性支承的非保守矩形薄板的稳定性[J]．西安理工大学学报1998，14(4)：398-403
[121] 赵凤群，王忠民，常安定．点弹性支承的非保守杆稳定性的积分方程法[J]．西安地质学院学报，1997，19：119-123
[122] H C Lin, S S Chen. Vibration and stability of fluid-conveying pipe[J]. Shock and Vibration, 1976,46(2),267～283
[123] S T Noah, G R Hopking. Dynamic stability of elastically supported pipes conveying fluid[J]. Jounal of. Sound and Vibration. 1980, 71(1):103～116
[124] 金基铎，林影，邹光胜．悬臂输流管道的颤振与混沌运动分析[J]．振动工程学报，1997，10(3)：314～320
[125] J D Jin, Stability and chaotic motion of a restrained pipe conveying fluid[J]. Jounal of. Sound and Vibration. 1997, 28(3): 427～439
[126] 金基铎，邹光胜，张宇飞．悬臂输流管道的运动分岔现象和混沌运动[J]．力学学报，2002，34 (6)：863～873
[127] 倪樵，黄玉盈，陈贻平．微分求积法分析具有弹性支承输液管的临界流速[J]。计算力学学报， 2000，18(2)：146～149
[128] 杨挺青．粘弹性力学[M]．武汉：华中理工大学出版社，1990
[129] 倪振华．振动力学[M]．西安：西安交通大学出版社，1994
[130] 方同，薛璞．振动理论与应用[M]．西安：西北工业大学出版社，1998．
[131] 李世荣，程昌钧．加热弹性杆的热过屈曲分析[J]．应用数学和力学，2000，21(2)：119-125
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