集中阻尼弦本征解的性质

利用Dirac δ函数，在全域建立并求解集中阻尼弦的动力学方程，导出其本征方程组、频率方程和本征函数的一般形式，推导了单项阻尼下本征函数的具体形式，并分析了中点阻尼对本征解的影响.同时，讨论了混合动力学系统在频率 阻尼关系、衰减率和完全抑制振动的最优阻尼3个方面既不同于连续系统，又不同于离散系统的特性:1）系统频率与其阻尼无关；2）各阶本征函数在单位时间内的衰减率都相同，衰减率与本征值的阶次无关；3）当阻尼取2时，系统衰减率趋于无穷大，系统不能发生任何有阻尼振动.

Properties of the Eigen Solution of Taut Strings With Concentrated Damping

By means of the Dirac delta function, the free-vibration equation of motion for taut strings with concentrated damping, namely the damping hybrid string system, was established and solved. The analytic solution to the eigen problem was obtained for the system with only one single damping dashpot at the midspan, and the properties of the eigen value and the eigen function were analyzed. The dynamic behaviors of the damping hybrid string system, including the frequency-damping relationship, the decay ratio and the full suppression of the motion at the optimal damping, which distinctly differentiate the hybrid system from a continuous system or a discrete system, were identified: 1) The frequency of the hybrid string system is independent of its damping ratio; 2) The decay ratios keep the same for different orders of eigen functions; 3) The decay ratios approach infinity when the damping ratio equals 2, which indicates any damped vibration of the system will be fully suppressed.