Free Vibration Analysis of Rings via Wave Approach

<table class="fixed-width table-group" id="tab4"><tr><td><table class="table"><colgroup><col style="width:4.46em"/><col style="width:8.57em"/><col style="width:10.35em"/><col style="width:6.89em"/></colgroup><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td class="align_left">Method</td><td class="align_center">Density <svg height="9.39034pt" id="M112" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 7.13289 9.39034" width="7.13289pt" 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="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" id="g113-239"></path></g></svg> (kg/m<sup>3</sup>)</td><td class="align_center">Young modulus <svg height="8.68572pt" id="M113" style="vertical-align:-0.0498209pt" version="1.1" viewbox="-0.0498162 -8.6359 8.13765 8.68572" width="8.13765pt" 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="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z" id="g113-70"></path></g></svg> (Pa)</td><td class="align_center">Poisson’s ratio</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td class="align_left">PMMA</td><td class="align_center">1062</td><td class="align_center">0.32 × 10<sup>10</sup></td><td class="align_center">0.3333</td></tr><tr><td class="align_left">Al</td><td class="align_center">2799</td><td class="align_center">7.21 × 10<sup>10</sup></td><td class="align_center">0.3451</td></tr><tr><td class="align_left">Pb</td><td class="align_center">11600</td><td class="align_center">4.08 × 10<sup>10</sup></td><td class="align_center">0.3691</td></tr><tr><td class="align_left">Ti</td><td class="align_center">4540</td><td class="align_center">11.7 × 10<sup>10</sup></td><td class="align_center">0.32</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Shock and Vibration

tab4

Table 4

Table 4: Free Vibration Analysis of Rings via Wave Approach 