A Series Solution for the Vibration of Mindlin Rectangular Plates with Elastic Point Supports around the Edges

<div>The first six mode shapes of square rectangular plates with completely elastic boundary conditions, <span class="nowrap"><svg height="16.9324pt" id="M199" style="vertical-align:-5.341pt" version="1.1" viewbox="-0.0498162 -11.5914 75.3489 16.9324" width="75.3489pt" 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="M425 682C425 693 422 699 407 699C369 699 308 677 260 646C213 651 191 674 191 701C191 715 195 726 200 734L183 740C153 730 118 709 118 681C118 644 163 619 218 615C168 582 115 534 115 463C115 418 147 382 180 369C102 329 23 261 23 163C23 74 74 38 158 14C183 7 201 2 221 -2C279 -15 290 -33 290 -58C290 -94 251 -145 230 -169L252 -189C280 -171 367 -103 367 -25C367 20 333 51 266 68C245 73 218 80 191 89C139 106 107 135 107 192C107 261 161 329 221 355C248 350 281 347 314 347C330 347 370 388 370 406C370 418 366 422 352 422C321 422 280 413 255 407C218 413 192 448 192 493C192 544 229 588 266 610C309 610 354 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xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M425 682C425 693 422 699 407 699C369 699 308 677 260 646C213 651 191 674 191 701C191 715 195 726 200 734L183 740C153 730 118 709 118 681C118 644 163 619 218 615C168 582 115 534 115 463C115 418 147 382 180 369C102 329 23 261 23 163C23 74 74 38 158 14C183 7 201 2 221 -2C279 -15 290 -33 290 -58C290 -94 251 -145 230 -169L252 -189C280 -171 367 -103 367 -25C367 20 333 51 266 68C245 73 218 80 191 89C139 106 107 135 107 192C107 261 161 329 221 355C248 350 281 347 314 347C330 347 370 388 370 406C370 418 366 422 352 422C321 422 280 413 255 407C218 413 192 448 192 493C192 544 229 588 266 610C309 610 354 615 376 623C397 631 425 651 425 682Z" id="g113-236"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.499,3.132)"><path d="M771 650H519L511 619L537 615C578 610 579 600 539 564C444 476 379 428 336 396C312 378 281 357 252 347L283 514C300 605 304 610 388 619L395 650H134L127 619C212 610 216 605 199 514L130 140C112 41 103 38 28 31L18 0H281L288 31C201 40 198 41 216 140L246 311C274 328 289 324 315 284C393 167 458 86 528 0H686L695 31C635 37 612 45 570 97C521 156 420 285 361 367L596 549C663 601 684 609 764 620L771 650Z" id="g50-76"></path></g><g transform="matrix(.0065,0,0,-0.0065,12.024,5.233)"><path d="M256 606C256 635 239 657 207 657C171 657 147 619 147 593C147 563 173 543 194 543C234 543 256 573 256 606ZM233 98L216 125C191 101 156 75 139 75C133 75 132 82 139 107L214 377C226 420 224 454 198 454C169 454 98 415 31 351L46 323C81 353 110 372 116 372C129 372 126 356 121 338L57 101C31 5 47 -12 72 -12C115 -12 197 52 233 98Z" id="g176-106"></path></g><g transform="matrix(.0065,0,0,-0.0065,13.472,5.233)"><path d="M405 606C405 634 387 657 355 657C316 657 294 621 294 593C294 563 319 543 346 543C377 543 405 573 405 606ZM371 354C383 412 387 454 360 454C324 454 253 412 182 345L199 312C221 333 269 364 278 364C288 364 287 354 279 310L195 -97C185 -139 166 -179 137 -179C119 -179 94 -169 79 -164C68 -160 59 -160 52 -168C42 -177 25 -197 25 -216C25 -238 50 -253 74 -253C91 -253 124 -242 189 -187C255 -131 292 -40 316 81L371 354Z" id="g176-107"></path></g><g transform="matrix(.013,0,0,-0.013,20.976,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,32.239,0)"><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>,</span> calculated by the present method.</div>

Shock and Vibration

fig5

Figure 5

Figure 5: A Series Solution for the Vibration of Mindlin Rectangular Plates with Elastic Point Supports around the Edges 