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First, based on the modified Hamiltonian principle, the nonlinear dynamic control equation of the deep-sea coarse-grained ore vertical lifting pipeline system was established. Then, the equation was discretized using the Galerkin method, converting the partial differential equation into a system of ordinary differential equations. Finally, numerical solutions were obtained for the stability characteristics of the pipeline using the eigenvalue method and Newton-Raphson iteration method. The system analyzed the influence of four key dimensionless parameters, namely the particle feed concentration, terminal lumped mass, gravity coefficient of the pipeline, and fluid velocity, on the critical flow velocity of the structure. Results show that when the feed concentration is Cv is 0.1, Hopf bifurcations occur only in the first three modes. When the feed concentration increases to Cv = 0.3, a Hopf bifurcation also occurs in the fourth mode, and the minimum critical flow velocity ucr decreases significantly. When the internal flow velocity is below the critical value, the imaginary parts of the complex eigenfrequencies remain positive, indicating decaying structural vibrations. Once the internal flow velocity exceeds the critical threshold, the imaginary part of the first complex eigenfrequency becomes negative, signifying flutter instability. When the gravity coefficient α is held constantly, the critical flow velocity ucr increases progressively with the dimensionless terminal mass κ5. Conversely, with fixed κ5, an increase in the gravity coefficient leads to a significant rise in critical flow velocity, exhibiting an approximately parabolic growth trend.
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| 科 Family | 属数 Number of genus | 种数 Number of species | 占总种数比例 Percentage of total species (%) | 属 Genus | 种数 Number of species | 占总种数比例 Percentage of total species (%) |
|---|---|---|---|---|---|---|
| 鹅膏菌科Amanitaceae | 2 | 11 | 5.26 | 鹅膏菌属 Amanita | 10 | 4.78 |
| 小菇科 Mycenaceae | 2 | 12 | 5.74 | 丝盖伞属 Inocybe | 5 | 2.39 |
| 多孔菌科 Polyporaceae | 8 | 14 | 6.70 | 蜡蘑属 Laccaria | 5 | 2.39 |
| 红菇科 Russulaceae | 3 | 23 | 11.00 | 小皮伞属 Marasmius | 6 | 2.87 |
| 小菇属 Mycena | 11 | 5.26 | ||||
| 光柄菇属 Pluteus | 5 | 2.39 | ||||
| 红菇属 Russula | 17 | 8.13 | ||||
| 栓菌属 Trametes | 5 | 2.39 |
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