Free Vibration Analysis of the Unified Functionally Graded Shallow Shell with General Boundary Conditions

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/sv/volume-2017/7025190/figures/7025190.fig.004a.svgz" name="7025190.fig.004a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Plate</td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/sv/volume-2017/7025190/figures/7025190.fig.004b.svgz" name="7025190.fig.004b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Cylindrical shell</td></tr><tr class="fig-image" id="c"><td><object data="https://static.hindawi.com/articles/sv/volume-2017/7025190/figures/7025190.fig.004c.svgz" name="7025190.fig.004c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>Spherical shell</td></tr><tr class="fig-image" id="d"><td><object data="https://static.hindawi.com/articles/sv/volume-2017/7025190/figures/7025190.fig.004d.svgz" name="7025190.fig.004d" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(d) </b>Hyperbolic paraboloidal shell</td></tr></table>

<div>Variations of the first frequency parameters <svg height="8.68572pt" id="M347" style="vertical-align:-0.0498209pt" version="1.1" viewbox="-0.0498162 -8.6359 9.97754 8.68572" width="9.97754pt" 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" id="g113-153"></path></g></svg> of the Al/Al<sub>2</sub>O<sub>3</sub> shallow shells with E<sup>3</sup>E<sup>3</sup>E<sup>3</sup>E<sup>3</sup> and different material types: Type 1: <svg height="8.55521pt" id="M348" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 27.889 8.55521" width="27.889pt" 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="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" id="g113-98"></path></g><g transform="matrix(.013,0,0,-0.013,10.212,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.475,0)"><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" id="g113-50"></path></g></svg>, <svg height="9.49473pt" id="M349" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 47.6273 9.49473" width="47.6273pt" 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="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z" id="g113-99"></path></g><g transform="matrix(.013,0,0,-0.013,9.733,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,20.996,0)"><path d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z" id="g113-100"></path></g><g transform="matrix(.013,0,0,-0.013,29.91,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,41.173,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path></g></svg>; 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Shock and Vibration

fig4

Figure 4

Figure 4: Free Vibration Analysis of the Unified Functionally Graded Shallow Shell with General Boundary Conditions 