Vortex-Induced Vibration Response Bifurcation Analysis of Top-Tensioned Riser Based on the Model of Variable Lift Coefficient

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/mpe/volume-2018/6491517/figures/6491517.fig.006a.svgz" name="6491517.fig.006a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Amplitude-frequency response of isolated solution (<i>C</i><sub><i>L</i>0</sub>=0.01,<i> C</i><sub>1</sub>=8, and <svg height="6.37608pt" id="M97" style="vertical-align:-0.2324901pt" version="1.1" viewbox="-0.0498162 -6.14359 6.02377 6.37608" width="6.02377pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g></svg>=825)</td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/mpe/volume-2018/6491517/figures/6491517.fig.006b.svgz" name="6491517.fig.006b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Amplitude-frequency response of single solution (C<sub>L0</sub>=0.06, C<sub>1</sub>=2, and <svg height="6.37608pt" id="M98" style="vertical-align:-0.2324901pt" version="1.1" viewbox="-0.0498162 -6.14359 6.02377 6.37608" width="6.02377pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g></svg>=825)</td></tr><tr class="fig-image" id="c"><td><object data="https://static.hindawi.com/articles/mpe/volume-2018/6491517/figures/6491517.fig.006c.svgz" name="6491517.fig.006c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>Vector field diagram of isolated solution (<i>C</i><sub><i>L</i>0</sub>=0.01,<i> C</i><sub>1</sub>=8, and <svg height="6.37608pt" id="M99" style="vertical-align:-0.2324901pt" version="1.1" viewbox="-0.0498162 -6.14359 6.02377 6.37608" width="6.02377pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g></svg>=825)</td></tr><tr class="fig-image" id="d"><td><object data="https://static.hindawi.com/articles/mpe/volume-2018/6491517/figures/6491517.fig.006d.svgz" name="6491517.fig.006d" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(d) </b>Vector field diagram of general single solution (C<svg height="6.256pt" id="M100" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.0498162 -2.96489 10.0515 6.256" width="10.0515pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0091,0,0,-0.0091,0,3.132)"><path d="M499 166C484 119 466 88 447 68C424 43 390 38 338 38C297 38 262 40 242 51C223 62 218 80 218 133V522C218 607 226 612 309 617V650H41V617C124 612 131 605 131 522V129C131 45 123 38 38 33V0H497C505 30 529 128 534 160L499 166Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,4.987,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"></path></g></svg>=0.06, C<sub>1</sub>=2, and <svg height="6.37608pt" id="M101" style="vertical-align:-0.2324901pt" version="1.1" viewbox="-0.0498162 -6.14359 6.02377 6.37608" width="6.02377pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g></svg>=825)</td></tr></table>

<div>Comparisons of amplitude-frequency response and vector field diagram between the isolated solution and single solution at dimensionless frequency <svg height="6.37608pt" id="M96" style="vertical-align:-0.2324901pt" version="1.1" viewbox="-0.0498162 -6.14359 6.02377 6.37608" width="6.02377pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g></svg>=825.</div>

Mathematical Problems in Engineering

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Figure 6

Figure 6: Vortex-Induced Vibration Response Bifurcation Analysis of Top-Tensioned Riser Based on the Model of Variable Lift Coefficient 