科研之友
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摘要
Mathieu equation is the classic differential equation with periodic coefficients, which governs the response of many physical systems having one degree of freedom. However, the riser is a multi-degree of freedom system. The equation of transverse motion of a riser under parametric excitations is a fourth-order partial differential equation containing a time-dependent coefficient. Firstly, this equation is reduced to Mathieu equation by using the Galerkin method. According to Mathieu instability chart, every mode is analysed in the same manner as the system with one-degree of freedom, and the parametric instability chart of the riser is obtained by mode superposition. In addition, the results are compared with those obtained by numerical simulation based on the Floquet theory. The results show that natural frequencies of the riser are very dense due to the high slenderness ratio, which leads to overlapping of different instability zones. The first instability zones are more important than higher instability zones which are very narrow in the instability charts. Special attention on the system parameters is needed for designers to avoid vibration instability.
