中文摘要：

推导两端支承梁弯曲振动的频率方程和振型函数的解析表达式。利用频率方程讨论两端扭转弹簧刚度变化对梁的前两阶弯曲振动特征值的影响。以两端支承梁的振型函数为假设振型导出两端支承输流管道在定常流作用下临界速度的解析表达式，为今后分析这类系统的动态响应提供理论依据。利用临界流速公式系统地分析和讨论扭转刚度、重力系数和轴向预紧力对管道临界流速的影响特性。研究结果表明，量纲-扭转弹簧刚度在0到50区间内变化时对临界流速的影响较大，但大于50时影响明显减弱。当重力系数和轴向预紧力增大时，临界流速也随着增大。一般而言，两端扭转弹簧刚度越大也会增大相应的临界流速值。

英文摘要：

The characteristic equation and the eigenfunctions of the supported beam are derived. Using the characteristic equation derived, the effect of the spring constants on the eigenvalues of the beam is discussed. The analytical expressions of critical flow velocity of the supported pipes, which may be used in the analysis of the dynamic behavior of the system, are obtained using the eigenfunctions derived for beam with the same end conditions. The effect of some parameters of the system on the values of critical flow velocity is discussed according to the analytical expressions derived. The results obtained show that the effect of the spring constants on the value of critical flow velocity is very great if their dimensionless values are taken in region 0-50, whereas the value of critical flow velocity is very little affected by the values of spring constants when the dimensionless values are greater than 50. Generally, as the gravity, tension and spring constants increase the value of critical flow velocity increase also.