本文研究线弹性均质材料杆的固有振动对偶问题,即两种杆在怎样的截面变化和齐次边界下具有相同固有频率. 首先,通过纵向位移和内力的对偶描述,给出两种杆异截面对偶的截面变化条件和边界条件,并将其分类为固定固定杆与自由自由杆对偶,固定自由杆与自由固定杆对偶等. 上述对偶杆具有相同固有频率,而两者的位移振型互为位置坐标的导数. 其次,限定两种对偶杆的截面变化相同,给出杆的截面积函数表达式. 此时,固定固定杆与自由自由杆构成同截面对偶,而固定自由杆和自由固定杆的同截面对偶彼此为镜像；等截面杆也具有上述对偶性质. 最后,将上述研究推广到材料性质沿轴向变化的变截面杆固有振动对偶问题. 文中所有结论均适用于圆轴在齐次边界条件下的扭转固有振动对偶问题.
The paper deals with the dual problem of a pair of rods made of a linear elastic and homogeneous material in natural vibrations. That is to answer what kind of crosssectional variation and homogeneous boundaries the two rods should have so that they have the same natural frequencies. Based on the dual of displacement description and internal force description, the paper presents the crosssectional variation and homogeneous boundaries for a dual of rods with different crosssections and the classification of all rods, including the dual of a fixedfixed rod and a freefree rod and the dual of a fixedfree rod and a freefixed rod. The two rods in a dual have the same natural frequencies while their displacement mode shapes are the position derivatives of each other. Then, the paper gives the formula of crosssectional area for a dual of rods with identical crosssections. In such a case, a fixedfixed rod and a freefree rod are a dual while a fixedfree rod and a freefixed rod are a pair of mirrors. The rods with uniform crosssections have the above dual properties by nature. Finally, the paper extends the above studies to the dual problem of a pair of arbitrary rods with both crosssection and material properties varying along their axes. The conclusions in the paper hold also true for the duality relations of circular shafts with homogeneous boundaries in natural vibrations.