绳摆一般振动

 研究了由重绳和质点构成的绳摆一般振动，给出了任意初始条件下解析解。一般振动包含基波与高次谐波，后者是绳摆区别于复摆的主要特征。证明了本征函数族的广义正交性，计算了本征函数模，由初始条件计算出一般解中各系数。在质点与重绳质量比值远大于1和远小于1两种情况下，求出了各本征频率解析近似结果。在前一情况下，随着质量比值的增大，高次谐波频率按其平方根增大，而基波频率则趋于单摆频率；在后一情况下，各谐波频率均接近重绳本征频率。给出了一般参数下前五次谐波本征频率数值结果。指出了相关文献中一个错误。

General Oscillation of a Pendulum with a Massive Cord

The general oscillation of a pendulum consists of a particle and a massive cord is studied. Analytic solutions for the displacement function under arbitrary initial conditions are given. The general oscillation involves higher modes as well as the ground mode. Existence of higher modes is the main feature of the cord pendulum in comparison with the compound pendulum. The generalized orthogonality of the eigenfunctions is proved and their modules are calculated. Then all coefficients in the general solution are worked out in terms of the initial conditions. Analytic eigenfrequencies are obtained approximately when the mass ratio of the particle to the cord is much larger or much smaller than 1. In the former case,the eigenfrequencies of higher modes increase in roughly the same way as the square root of the ratio, while the eigenfrequency of the ground mode tends to that of the simple pendulum. In the latter case,the eigenfrequencies are close to those of a massive cord. Numerical results for the first five eigenfrequencies are presented for general mass ratio. An error in the literature is pointed out.