Duality relations of rods in natural vibrations

The paper deals with the dual problem of a pair of rods made of a linear elastic and homogeneous ma? terial in natural vibrations. That is to answer what kind of cross,sectional variation and homogeneous boundaries the two rods should have so that they have the same natural frequencies. Based on the dual of displacement de, scription and internal force description, the paper presents the cross,sectional variation and homogeneous bounda, ries for a dual of rods with different cross,sections and the classification of all rods, including the dual of a fixed, fixed rod and a free,free rod and the dual of a fixed,free rod and a free,fixed 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 cross,sectional area for a dual of rods with identical cross,sections. In such a case, a fixed,fixed rod and a free,free rod are a dual while a fixed,free rod and a free,fixed rod are a pair of mirrors. The rods with uniform cross,sections 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 cross,section and material properties varying along their axes. The conclusions in the paper hold also true for the duality relations of circular shafts with homo, geneous boundaries in natural vibrations.