Free Vibration Analysis of Arbitrary-Shaped Plates Based on the Improved Rayleigh–Ritz Method

<table class="table-group" id="tab6"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td class="align_left" rowspan="2">Mode</td><td class="align_center" colspan="3"><span style="width: 83.9474ptpx;"><svg height="15.1937pt" id="M36" style="vertical-align:-3.6023pt" version="1.1" viewbox="-0.0498162 -11.5914 83.9474 15.1937" width="83.9474pt" 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="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.473,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.736,0)"><path d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.031,0)"><path d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,39.611,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.558,-.336)"><path d="M778 808H735L485 -141H483L284 324L133 247L149 215L244 263L484 -298H486L778 808Z"></path></g><rect height="0.65243" width="29.1378" x="54.5921" y="-10.8794"></rect><g transform="matrix(.013,0,0,-0.013,54.592,0)"><path d="M516 277C516 378 460 448 376 448C335 448 288 434 240 401C158 345 127 277 107 159C53 -161 38 -221 23 -247L28 -261C43 -260 95 -250 114 -226C122 -216 129 -199 146 -75C152 -29 157 0 164 31C182 9 206 -12 237 -12C267 -12 325 8 376 46C451 102 516 179 516 277ZM432 259C432 134 366 40 259 40C236 40 204 48 171 76L195 203C219 327 253 400 326 400C398 400 432 336 432 259Z"></path></g><g transform="matrix(.013,0,0,-0.013,61.602,0)"><path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"></path></g><g transform="matrix(.013,0,0,-0.013,68.31,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,73.672,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z"></path></g></svg></span></td></tr><tr class="thead"><td class="align_center">Present</td><td class="align_center">Ref. [<a href="/journals/ace/2019/7041592/#B26" target="_blank">26</a>]</td><td class="align_center">Error (%)</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td class="align_left">1</td><td class="align_center">3.471</td><td class="align_center">3.470</td><td class="align_center">0.039</td></tr><tr><td class="align_left">2</td><td class="align_center">8.508</td><td class="align_center">8.504</td><td class="align_center">0.045</td></tr><tr><td class="align_left">3</td><td class="align_center">21.296</td><td class="align_center">21.279</td><td class="align_center">0.078</td></tr><tr><td class="align_left">4</td><td class="align_center">27.217</td><td class="align_center">27.201</td><td class="align_center">0.059</td></tr><tr><td class="align_left">5</td><td class="align_center">30.965</td><td class="align_center">30.948</td><td class="align_center">0.054</td></tr><tr><td class="align_left">6</td><td class="align_center">54.219</td><td class="align_center">54.185</td><td class="align_center">0.063</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Natural frequency of free vibration on the rectangular plate under the condition of one side clamped and three sides free (Hz).</div>

Advances in Civil Engineering

tab6

Table 6

Table 6: Free Vibration Analysis of Arbitrary-Shaped Plates Based on the Improved Rayleigh–Ritz Method 