有质量弹簧的振动解

The Solutions of Spring Which Owns Mass

从有质量弹簧的波动方程出发 ,运用牛顿第二定律和胡克定律及微元分析法 ,给出定解问题 ;然后分离变量 ,直接求解波动方程 ,得到分离变数形式的解和频率ω所满足的本征值方程 ;再将tanω/ωm 展开成麦克劳林 (Maclaurin)级数形式 ,并采用迭代法解出弹簧振子的本征频率 ,导出弹簧的有效质量的渐近级数表达式 ;最后由初始条件解出其对应的振幅 。

Applying Newton second law and Hooks law and little yuans analytic method,the problem of definite solution are written out,beginning with the wave equation of spring which owns mass.Then the variables are separated and the wave equation is solved directly.The solution of separating parameter form and intrinsic equation that frequency meets are got.Then  tan ω/ω m is changed into McLaurin Series form and the intrinsic frequency of spring oscillator is solved by adopting iterative law.The asymptotic series expression of the effective mass of spring is concluded.According to the initial conditions,the corresponding amplitude is solved and then the solution of spring whose mass can not be ignored oscillator system is got.