非均匀变截面梁动力响应的一般解
The General Solution for Dynamic Response of Nonhomogeneous Beam with Variable Cross Section
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摘要: 本文利用精确解析法[1]给出非均匀变截面梁在任意谐振荷载和边界条件下的动力响应的一般解.问题最后归结为求解一个非正定微分方程.对于这一问题用一般变分法求解失败,利用本文的方法,这个问题的一般解可以写成解析的形式.因此对优化特别方便.本文给出收敛性证明.文末给出的算例表明本文的方法可获得满意的结果.Abstract: In this paper by means of the exact analytic method [1],the general solution for dynamic response of nonhomogeneous beam with variable cross section is obtained under arbitrary resonant load and boundary conditions.The problem is reduced to solvea non-positive differential equation.Generally,it is not solved by variational method.By the present method,the general solution for this problem may be written as an ana-lytic form.Hence,it is convenient for structure optimizing problem.In this paper,its convergence is proved.Numerical examples are given at the end of the paper.which in-dicates satisfactory results can be obtained.
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[1] 纪振义、叶开沅,任意变系数微分方程的精确解析法.应用数学和力学,10(10)(1989),841-852. [2] 叶开沅,非均匀变厚度弹性力学若干问题的一般解,非均匀变厚度梁的弯曲,稳定和自由振动,兰州大学学报(自然科学版),(力学专号),(1979),133-157. [3] 叶开沅、丁延岭,非均匀变厚度连续梁的稳定性和自由振动,兰州大学学报(自然科学版),18(2)(1982),42-47. [4] 叶开沅、郑书英,非均匀弹性地基上的非均匀变截面梁的稳定性,兰州大学学报(自然科学版),(力学专号),19(1983),37-57. [5] 叶开沅、纪振义,非均匀变截面梁的稳定和自由振动的阶梯折算法,兰州大学学报(自然科学版),26(2)(1990),40-46, [6] 王光远,《建筑结构的振动》.科学出版社.北京（1978),150-153. -
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