Numerical stability study on vibration responses of casing string systems

The classical transfer matrix method (CTMM) is widely employed for analyzing structural vibrations. However, when applying the CTMM to solve the vibration responses of long casing string systems, severe numerical instability issues arise. To mitigate this problem, this paper introduces a hybrid energy transfer matrix method (HETMM) tailored for solving the vibration responses of casing string systems with distributed elastic supports. This approach significantly enhances the numerical stability compared to CTMM. Firstly, the dynamic model of the casing string system is established, and the correctness and accuracy of the model are verified by tests and numerical calculations. Subsequently, a systematic analysis is conducted to evaluate the influence of various parameters on the stability of the numerical solution for the system's lateral vibration. The research findings indicate that the system length is the primary factor contributing to the instability of the numerical solution. Thus, based on the idea of reducing the characteristic length of the casing string system, the HETMM is proposed to address the lateral vibration of the casing string system. The stability and efficiency of this method in calculating high-frequency and long casing string systems are verified by comparing with other calculation methods. Finally, the frequency domain response of a casing string system with a length of 1450 m is calculated and analyzed. It is found that the number of unit divisions should not be too many or too few, otherwise, it will result in unstable numerical solutions. The appropriate number of units should be determined based on the frequency range of interest. The improved HETMM proposed in this paper is sufficient for solving the lateral vibration response of cementing casing sections ranging from several hundred to several thousand meters. This research provides theoretical guidance for obtaining the response of the long casing string system and optimizing the vibration cementing equipment.