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中文摘要：
准确答案在三个不同结束条件下面为有限地长的、机能上地分级的空柱体的短暂扭力的回答被获得，即免费免费，免费修理并且修理修理。有它修理的外部表面的柱体在内部表面受到一个动态砍压力。材料性质被假定尽最大努力在光线的方向变化法律形式，当时在轴的方向保持不变。与在轴的方向的扩大以三角法的系列，为关于光线的并列 r 和时间 t 的未知功能的管理方程被推出。由连续地使用可变替换技术，重叠方法和变量的分离，方程的系列形式解决方案被获得。自然频率和短暂扭力的回答最后为机能上地分级的有限空柱体被讨论。
英文摘要：
Exact solutions are obtained for transient torsio- nal responses of a finitely long, functionally graded hollow cylinder under three different end conditions, i.e. free-free, free-fixed and fixed-fixed. The cylinder with its external surface fixed is subjected to a dynamic shearing stress at the internal surface. The material properties are assumed to vary in the radial direction in a power law form, while keep invariant in the axial direction. With expansion in the axial direction in terms of trigonometric series, the governing equations for the unknown functions about the radial coordinate r and time t are deduced. By applying the variable substitution technique, the superposition method and the separation of variables consecutively, series-form solutions of the equations are obtained. Natural frequencies and the transient torsional responses are finally discussed for a functionally graded finite hollow cylinder.
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