单位分数跨集中阻尼弦本征解的结构及其性质

Structure and Properties of Eigensolutions of a Taut String with a Lumped Damping at a Unit-Fraction-Span

针对斜拉索-阻尼(器)系统,推导出阻尼(器)位于任意单位分数跨时(拉索1/n跨位置)系统超越频率方程的代数形式,根据代数基本定理讨论了系统本征解的结构,并结合4个算例分析解的性质.结果表明:1)本征解可归为n-1个解支.2)对于同一解支,各阶本征值实部(其相反数即单位时间对数衰减率)均相同,各阶本征值虚部(即频率)构成等差数列.3)根据频率随阻尼系数变化的特点,解支可分为三类:第一类解支的频率均依赖于阻尼;第二类解支的频率均不受阻尼影响;第三类解支的频率随阻尼系数的不同,具有第一类解支或第二类解支的特点,即随阻尼系数的增大,频率先随阻尼系数变化,到达某一临界值后为常数.

For the cable-damper system,the algebraic form of the transcendental frequency equation of the sys⁃tem is derived when the damper is located at a unit-fraction-span(1/n span of the cable).According to the funda⁃mental theorem of algebra,the structure of the eigensolutions of the system is discussed,and the properties of the so⁃lution are analyzed with four examples.The results show that:1)The eigensolutions can be divided into n-1 branches.2)Within one solution branch,all eigenvalues take an identical value in their real part(as an additive in⁃verse of the logarithmic decrement ratio per unit time),while their imaginary parts(meaning in physics,the fre⁃quency)form an arithmetic sequence.3)According to the way that the frequencies vary with the damping,the solu⁃tion branches can be classified as three types:The frequency of type 1 solutions is related on damping;The fre⁃quency of type 2 solutions is not affected by damping.The frequency of type 3 solutions may or may not vary with the damping,which means that a type 3 solution may behave like a type 1 or a type 2 solution,depending on the damp⁃ing is under or over some certain critical value.