中文摘要：

针对井下钻柱运动的复杂性,基于动力学理论,建立了井下钻柱纵向和横向耦合振动的数学模型,并进行数值求解及分析.根据井下钻柱的实际工况,以整个井下钻柱为研究对象,提出了钻柱纵向和横向耦合振动的动力方程,并利用解析法和无量纲法分别求解出其动刚度和动阻尼的表达式,以及钻柱前两阶振动的固有频率.分析结果表明：当井下钻柱振动频率增大时,其动刚度呈幅值衰减的周期性变化,而其动阻尼呈幅值增强的周期性变化;井下钻柱长度和横截面面积越大,其动刚度和动阻尼的幅值越小;井下钻柱的Poisson（泊松）比对其振动的动刚度、动阻尼和前两阶固有频率没有影响;同时,井下钻柱的第二阶固有频率始终大于第一阶固有频率.该文的研究方法和模型为井下钻柱钻具分析和结果优化提供了理论参考和实际意义.

英文摘要：

Aimed at the complicated motion of the downhole drill string and based on the existing dynamics theory, the longitudinal and lateral coupled vibration model for the drill strings was established, and the numerical solution was obtained. According to the actual working condition of the downhole drill string, the entire drill string was regarded as the object of study. The concrete expressions of the dynamic stiffness, dynamic damping and the 1st 2 orders of natural frequencies were deduced with the analytical method and the dimensionless method in view of the coupled vibration characteristics of the downhole drill string. The research findings reveal that when the vibration frequency of the downhole drill string increases, the dynamic stiffness will periodically change with amplitude attenuation and the dynamic damping will periodically change with amplitude enhancement. The greater the length and the cross-sectional area of the downhole drill string are, the smaller the vibration amplitudes of the dynamic stiffness and the dynamic damping will be. Moreover, the Poisson＇ s ratio of the downhole drill string has no effect on the dynamic stiffness, the dynamic damping and the 1 st 2 orders of natural frequencies. Meanwhile, the 2nd-order natural frequency of the drill string is always greater than the 1st. The proposed methods and model provide theoretical references and practical significances for further analysis and design optimization of the bottom-hole assembly （BHA）.