Cylindrical Shell with Junctions: Uncertainty Quantification of Free Vibration and Frequency Response Analysis

<div>Frequency response: comparison of sparse grids: <svg height="12.1292pt" id="M339" style="vertical-align:-0.2454996pt" version="1.1" viewbox="-0.0498162 -11.8837 23.6692 12.1292" width="23.6692pt" 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="M571 489C571 598 494 687 374 687C259 687 169 605 169 498C169 421 218 372 294 372C365 372 419 422 442 503L423 508C402 449 368 402 297 402C237 402 210 444 210 497C210 590 282 654 371 654C456 654 495 603 495 506C495 437 458 340 407 249C336 125 246 15 167 15C119 15 90 30 73 69C78 66 91 63 98 63C126 63 145 89 145 116C145 141 121 163 94 163C59 163 36 130 36 91C36 40 74 -15 150 -15C253 -15 324 44 392 141H394C387 126 385 98 385 86C385 19 422 -14 486 -14C568 -14 622 51 691 139L671 154C596 67 552 30 515 30S465 56 465 95C465 162 505 261 532 323C620 520 713 654 786 654C774 642 769 626 769 614C769 582 794 568 819 568C846 568 873 588 873 624C873 669 839 687 799 687C730 687 653 632 573 489H571Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.583,-5.741)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,18.09,-5.741)"><path d="M495 163C480 117 462 85 444 65C421 39 387 34 332 34C290 34 256 36 236 47C218 57 213 77 213 131V526C213 612 222 616 301 622V650H40V622C122 616 128 611 128 526V126C128 41 120 34 36 28V0H489C498 31 519 126 525 157L495 163Z"></path></g></svg> vs <span class="nowrap"><svg height="12.1292pt" id="M340" style="vertical-align:-0.2454996pt" version="1.1" viewbox="-0.0498162 -11.8837 24.4365 12.1292" width="24.4365pt" 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="M571 489C571 598 494 687 374 687C259 687 169 605 169 498C169 421 218 372 294 372C365 372 419 422 442 503L423 508C402 449 368 402 297 402C237 402 210 444 210 497C210 590 282 654 371 654C456 654 495 603 495 506C495 437 458 340 407 249C336 125 246 15 167 15C119 15 90 30 73 69C78 66 91 63 98 63C126 63 145 89 145 116C145 141 121 163 94 163C59 163 36 130 36 91C36 40 74 -15 150 -15C253 -15 324 44 392 141H394C387 126 385 98 385 86C385 19 422 -14 486 -14C568 -14 622 51 691 139L671 154C596 67 552 30 515 30S465 56 465 95C465 162 505 261 532 323C620 520 713 654 786 654C774 642 769 626 769 614C769 582 794 568 819 568C846 568 873 588 873 624C873 669 839 687 799 687C730 687 653 632 573 489H571Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.583,-5.741)"><path d="M614 175C564 76 510 21 408 21C256 21 146 149 146 336C146 488 235 629 402 629C510 629 570 586 597 480L626 488C620 541 614 582 606 638C578 643 510 665 429 665C206 665 44 527 44 316C44 157 153 -15 402 -15C474 -15 558 5 586 11C604 45 629 119 643 165L614 175Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,17.698,-5.741)"><path d="M614 175C564 76 510 21 408 21C256 21 146 149 146 336C146 488 235 629 402 629C510 629 570 586 597 480L626 488C620 541 614 582 606 638C578 643 510 665 429 665C206 665 44 527 44 316C44 157 153 -15 402 -15C474 -15 558 5 586 11C604 45 629 119 643 165L614 175Z"></path></g></svg>;</span> maximal transverse deflection <svg height="9.25658pt" id="M341" style="vertical-align:-3.295671pt" version="1.1" viewbox="-0.0498162 -5.96091 25.6565 9.25658" width="25.6565pt" 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="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.931,3.132)"><path d="M810 0V30C751 35 744 40 744 106V296C744 396 694 451 614 451C583 451 558 438 537 425C512 409 485 391 454 368C434 420 390 451 342 451C309 451 285 439 260 424C228 405 206 387 185 372V453C141 434 90 421 43 413V386C102 378 105 373 105 309V106C105 41 96 35 29 30V0H243V30C194 35 185 41 185 106V335C216 360 256 389 297 389C357 389 384 348 384 275V106C384 41 374 35 312 30V0H530V30C475 35 465 41 465 104V294C465 313 464 324 463 336C502 367 539 389 576 389C637 389 663 346 663 274V109C663 41 653 35 593 30V0H810Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,16.493,3.132)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,20.579,3.132)"><path d="M482 0V30C420 38 407 48 370 103L273 248C308 299 331 332 352 357C387 397 401 404 463 409V439H277V409C318 405 324 400 307 369C290 337 270 307 251 279L194 367C175 396 180 402 220 409V439H18V409C74 401 87 395 117 349L204 213C173 168 147 130 119 95C78 46 67 38 5 30V0H195V30C144 38 143 47 162 79C181 112 202 149 225 181L297 70C314 43 307 35 264 30V0H482Z"></path></g></svg> vs frequency; dashed line represents the added standard deviation. (a) Gauss <svg height="12.1292pt" id="M342" style="vertical-align:-0.2454996pt" version="1.1" viewbox="-0.0498162 -11.8837 23.6692 12.1292" width="23.6692pt" 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="M571 489C571 598 494 687 374 687C259 687 169 605 169 498C169 421 218 372 294 372C365 372 419 422 442 503L423 508C402 449 368 402 297 402C237 402 210 444 210 497C210 590 282 654 371 654C456 654 495 603 495 506C495 437 458 340 407 249C336 125 246 15 167 15C119 15 90 30 73 69C78 66 91 63 98 63C126 63 145 89 145 116C145 141 121 163 94 163C59 163 36 130 36 91C36 40 74 -15 150 -15C253 -15 324 44 392 141H394C387 126 385 98 385 86C385 19 422 -14 486 -14C568 -14 622 51 691 139L671 154C596 67 552 30 515 30S465 56 465 95C465 162 505 261 532 323C620 520 713 654 786 654C774 642 769 626 769 614C769 582 794 568 819 568C846 568 873 588 873 624C873 669 839 687 799 687C730 687 653 632 573 489H571Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.583,-5.741)"><path d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,18.09,-5.741)"><path d="M495 163C480 117 462 85 444 65C421 39 387 34 332 34C290 34 256 36 236 47C218 57 213 77 213 131V526C213 612 222 616 301 622V650H40V622C122 616 128 611 128 526V126C128 41 120 34 36 28V0H489C498 31 519 126 525 157L495 163Z"></path></g></svg><i> p</i> = 2, 41 points. (b) Clenshaw–Curtis <svg height="12.1292pt" id="M343" style="vertical-align:-0.2454996pt" version="1.1" viewbox="-0.0498162 -11.8837 24.4365 12.1292" width="24.4365pt" 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="M571 489C571 598 494 687 374 687C259 687 169 605 169 498C169 421 218 372 294 372C365 372 419 422 442 503L423 508C402 449 368 402 297 402C237 402 210 444 210 497C210 590 282 654 371 654C456 654 495 603 495 506C495 437 458 340 407 249C336 125 246 15 167 15C119 15 90 30 73 69C78 66 91 63 98 63C126 63 145 89 145 116C145 141 121 163 94 163C59 163 36 130 36 91C36 40 74 -15 150 -15C253 -15 324 44 392 141H394C387 126 385 98 385 86C385 19 422 -14 486 -14C568 -14 622 51 691 139L671 154C596 67 552 30 515 30S465 56 465 95C465 162 505 261 532 323C620 520 713 654 786 654C774 642 769 626 769 614C769 582 794 568 819 568C846 568 873 588 873 624C873 669 839 687 799 687C730 687 653 632 573 489H571Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.583,-5.741)"><path d="M614 175C564 76 510 21 408 21C256 21 146 149 146 336C146 488 235 629 402 629C510 629 570 586 597 480L626 488C620 541 614 582 606 638C578 643 510 665 429 665C206 665 44 527 44 316C44 157 153 -15 402 -15C474 -15 558 5 586 11C604 45 629 119 643 165L614 175Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,17.698,-5.741)"><path d="M614 175C564 76 510 21 408 21C256 21 146 149 146 336C146 488 235 629 402 629C510 629 570 586 597 480L626 488C620 541 614 582 606 638C578 643 510 665 429 665C206 665 44 527 44 316C44 157 153 -15 402 -15C474 -15 558 5 586 11C604 45 629 119 643 165L614 175Z"></path></g></svg><i> p</i> = 2, 41 points. (c) Combined.</div>

Shock and Vibration

fig9

Figure 9

Figure 9: Cylindrical Shell with Junctions: Uncertainty Quantification of Free Vibration and Frequency Response Analysis 