杆在固有振动中的对偶关系  

DUALITY RELATIONS OF RODS IN NATURAL VIBRATIONS

作　　者：胡海岩[1] Hu Haiyan(MOE Key Lab of Dynamics and Control of Flying Vehicles,School of Aerospace Engineering,Beijing Institute of Technology,Beijing 100081,China)

机构地区：[1]北京理工大学宇航学院飞行器动力学与控制教育部重点实验,北京100081

出　　处：《动力学与控制学报》2020年第2期1-8,共8页Journal of Dynamics and Control

基　　金：国家自然科学基金重点项目(11832005)。

摘　　要：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 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 description and internal force description,the paper presents the cross-sectional variation and homogeneous boundaries 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 homogeneous boundaries in natural vibrations.

关 键 词：变截面杆 等截面杆 对偶 固有振动 内力振型 

分 类 号：O327[理学—一般力学与力学基础]