Analytical solution to a coupled system including tuned liquid damper and single degree of freedom under free vibration with modal decomposition method

Mahdiyar Khanpour, A. Mohammadian, H. Shirkhani, Reza Kianoush
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Abstract

This research focuses on employing a linear analytical approach to transform free surface waves and velocities into mode coordinates, with the aim of investigating the free vibration behavior of a coupled system consisting of a Single Degree of Freedom and a sloshing tank. Through a series of manipulations and simplifications of the coupled equations, a fourth-order ordinary differential equation is derived to showcase the overall response of the system, highlighting the contribution of each odd mode. Key concepts explored include system stability, mode-specific natural periods, establishment of initial boundary conditions, and formulation of the complete system response. The analytical method applied to study Tuned Liquid Dampers, a type of elevated sloshing tank, reveals that in higher modes, the lower frequency aligns with the structural natural frequency, while the higher frequency is approximately n times the structural natural frequency (where n is the odd mode number). This approach also elucidates why the system's response does not exhibit a higher-frequency component in higher modes. The study further investigates concepts such as employing an initial perturbation to excite higher frequencies and the potential for approximating the system through the first mode. Additionally, a numerical model was developed using variable separation and modal decomposition methods to complement and validate the analytical approach. Finally, further verification of the model was performed using the Preismann scheme applied to the relevant equations and the central upwind applied to nonlinear equations.
用模态分解法分析自由振动下包括调谐液体阻尼器和单自由度的耦合系统
本研究的重点是采用线性分析方法将自由表面波和速度转换为模态坐标,目的是研究由单自由度和滑动槽组成的耦合系统的自由振动行为。通过对耦合方程的一系列处理和简化,得出了一个四阶常微分方程,以展示系统的整体响应,突出每个奇数模式的贡献。探讨的关键概念包括系统稳定性、特定模式的自然周期、初始边界条件的建立以及完整系统响应的表述。在研究调谐液体阻尼器(一种高架荡槽)时采用的分析方法显示,在较高的模式中,较低的频率与结构固有频率一致,而较高的频率大约是结构固有频率的 n 倍(其中 n 为奇数模式数)。这种方法还解释了为什么在较高模态下系统响应没有表现出较高频率成分。研究还进一步探究了一些概念,如采用初始扰动来激发更高频率,以及通过第一模态近似系统的可能性。此外,还利用变量分离和模态分解方法开发了一个数值模型,以补充和验证分析方法。最后,利用应用于相关方程的 Preismann 方案和应用于非线性方程的中央上风法对该模型进行了进一步验证。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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