Free Vibration Investigations of Rotating FG Beams Resting on Elastic Foundation with Initial Geometrical Imperfection in Thermal Environments

<div>The dependence of the first three nondimensional frequencies of the C-F geometrically imperfect beam (S-S) on the rotational speed and <i>n</i>, <span class="nowrap"><svg height="11.6399pt" id="M110" style="vertical-align:-3.29108pt" version="1.1" viewbox="-0.0498162 -8.34882 52.9827 11.6399" width="52.9827pt" 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="M384 393C384 425 346 448 294 448C244 448 200 431 150 399C75 351 23 271 23 176C23 90 52 45 132 19C150 13 166 9 195 2C245 -10 260 -27 260 -55C260 -92 234 -127 218 -145L237 -159C271 -137 333 -93 333 -18C333 43 283 62 228 74C199 80 186 84 172 89C116 109 101 150 101 210C101 318 164 400 241 400C276 400 301 390 327 356C334 347 338 343 346 345C367 349 384 374 384 393Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,4.966,3.132)"><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 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class="nowrap"><svg height="14.0461pt" id="M112" style="vertical-align:-3.9436pt" version="1.1" viewbox="-0.0498162 -10.1025 37.2172 14.0461" width="37.2172pt" 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="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.817,-5.741)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.23,3.784)"><path d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.528,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,30.791,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g></svg>.</span> (a) The first frequency. (b) The second frequency. (c) The third frequency.</div>

Mathematical Problems in Engineering

fig5

Figure 5

Figure 5: Free Vibration Investigations of Rotating FG Beams Resting on Elastic Foundation with Initial Geometrical Imperfection in Thermal Environments 

Figure 5 | Free Vibration Investigations of Rotating FG Beams Resting on Elastic Foundation with Initial Geometrical Imperfection in Thermal Environments 