Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions

<div>Finite strip element representation of a rectangular finite plate element subjected to arbitrary boundary conditions at <svg height="14.8173pt" id="M70" style="vertical-align:-5.52897pt" version="1.1" viewbox="-0.0498162 -9.28833 56.0622 14.8173" width="56.0622pt" 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="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z" id="g113-122"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="579" vert-adv-y="579"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.161,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.424,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,30.084,0)"><path d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z" id="g113-77"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="553" vert-adv-y="553"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,38.378,3.132)"><path d="M563 395C563 428 543 451 519 451C500 451 480 438 472 423C468 415 465 403 470 394C477 381 482 368 482 347C482 272 386 136 342 74H340C334 163 323 257 312 337C302 410 290 451 257 451C214 451 160 385 127 338L145 310C169 341 203 373 209 373C219 373 225 369 232 325C249 215 269 62 271 -15C220 -80 135 -165 18 -231L26 -257L142 -230C254 -105 279 -74 342 13C394 85 484 211 526 286C553 334 563 365 563 395Z" id="g50-122"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,44.237,0)"><path d="M368 703H309L44 -163H104L368 703Z" id="g113-48"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="412" vert-adv-y="412"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,49.598,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> and <svg height="14.8173pt" id="M71" style="vertical-align:-5.52897pt" version="1.1" viewbox="-0.0498162 -9.28833 25.924 14.8173" width="25.924pt" 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="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z" id="g113-77"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="553" vert-adv-y="553"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,8.294,3.132)"><path d="M563 395C563 428 543 451 519 451C500 451 480 438 472 423C468 415 465 403 470 394C477 381 482 368 482 347C482 272 386 136 342 74H340C334 163 323 257 312 337C302 410 290 451 257 451C214 451 160 385 127 338L145 310C169 341 203 373 209 373C219 373 225 369 232 325C249 215 269 62 271 -15C220 -80 135 -165 18 -231L26 -257L142 -230C254 -105 279 -74 342 13C394 85 484 211 526 286C553 334 563 365 563 395Z" id="g50-122"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,14.154,0)"><path d="M368 703H309L44 -163H104L368 703Z" id="g113-48"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="412" vert-adv-y="412"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,19.515,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> (●: nodes; grey circles: nodal values).</div>

Mathematical Problems in Engineering

fig3

Figure 3

Figure 3: Frequency Domain Spectral Element Model for the Vibration Analysis of a Thin Plate with Arbitrary Boundary Conditions 