Fault Diagnosis of Beam-Like Structure Using Modified Fuzzy Technique

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/aav/volume-2014/491510/figures/491510.fig.0012a.svgz" name="491510.fig.0012a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>rcd versus rnf<sub>1</sub> and rnf<sub>2</sub></td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/aav/volume-2014/491510/figures/491510.fig.0012b.svgz" name="491510.fig.0012b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>rcd versus rnf<sub>1</sub> and rnf<sub>3</sub></td></tr><tr class="fig-image" id="c"><td><object data="https://static.hindawi.com/articles/aav/volume-2014/491510/figures/491510.fig.0012c.svgz" name="491510.fig.0012c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>rcd versus rnf<sub>1</sub> and rm<sub>1</sub></td></tr><tr class="fig-image" id="d"><td><object data="https://static.hindawi.com/articles/aav/volume-2014/491510/figures/491510.fig.0012d.svgz" name="491510.fig.0012d" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(d) </b>rcd versus rnf<sub>1</sub> and rm<sub>2</sub></td></tr><tr class="fig-image" id="e"><td><object data="https://static.hindawi.com/articles/aav/volume-2014/491510/figures/491510.fig.0012e.svgz" name="491510.fig.0012e" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(e) </b>rcd versus rnf<sub>1</sub> and rm<sub>3</sub></td></tr></table>

<div>Surface plots for relative crack depth versus relative natural frequency <svg height="14.2303pt" id="M96" style="vertical-align:-4.532pt" version="1.1" viewbox="-0.0498162 -9.6983 53.5004 14.2303" width="53.5004pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><path d="M181 342V451C133 431 89 419 40 411V388C98 381 102 377 102 311V104C102 38 95 32 33 26V0H263V26C186 32 181 38 181 104V287C203 343 235 372 261 372C277 372 289 366 304 352C310 346 318 345 330 350C349 359 362 379 362 399C362 422 338 449 304 449C256 449 213 393 183 342H181Z" id="g190-115"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="371" vert-adv-y="371"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,9.796,0)"><path d="M524 0V26C466 32 460 36 460 104V297C460 393 411 449 331 449C302 449 276 437 248 419C223 402 201 387 181 372V451C137 432 90 420 42 411V388C96 378 102 374 102 310V104C102 38 97 33 29 26V0H246V26C187 32 181 36 181 104V339C211 365 250 390 290 390C357 390 381 345 381 276V109C381 40 374 32 315 26V0H524Z" id="g190-111"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="547" vert-adv-y="547"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,17.206,0)"><path d="M54 437L27 408L31 397H101V103C101 37 94 32 30 26V0H266V25C187 33 180 36 180 110V397H288C299 404 304 428 298 437H180V477C179 562 190 610 203 630C214 647 230 659 256 659C289 659 318 641 337 622C346 612 355 612 364 619C374 627 380 635 383 643C388 655 387 667 378 678C362 697 333 710 299 712C260 707 225 689 189 659C135 613 119 563 112 541S101 490 101 458V437H54Z" id="g190-103"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="296" vert-adv-y="296"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,21.278,3.264)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z" id="g50-106"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="280" vert-adv-y="280"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,23.934,3.264)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,29.729,3.264)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,34.347,3.264)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z" id="g50-45"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,36.595,3.264)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,41.213,3.264)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z" id="g50-45"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,43.461,3.264)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z" id="g50-52"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,48.678,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg> and relative mode shape differences <svg height="14.2099pt" id="M97" style="vertical-align:-4.532pt" version="1.1" viewbox="-0.0498162 -9.6779 53.174 14.2099" width="53.174pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><path d="M181 342V451C133 431 89 419 40 411V388C98 381 102 377 102 311V104C102 38 95 32 33 26V0H263V26C186 32 181 38 181 104V287C203 343 235 372 261 372C277 372 289 366 304 352C310 346 318 345 330 350C349 359 362 379 362 399C362 422 338 449 304 449C256 449 213 393 183 342H181Z" id="g190-115"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="371" vert-adv-y="371"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,9.796,0)"><path d="M797 0V26C739 32 732 36 732 103V296C732 394 682 449 605 449C576 449 550 437 529 423C504 407 475 389 446 366C425 418 382 449 334 449C303 449 279 437 253 421C222 403 201 385 180 371V452C135 432 85 419 41 411V388C99 379 102 374 102 310V103C102 38 93 32 27 26V0H238V26C189 32 180 38 180 103V338C210 363 250 390 289 390C351 390 377 348 377 275V103C377 37 368 32 306 26V0H520V26C465 32 456 38 456 101V296C456 314 455 326 453 338C491 369 529 390 565 390C628 390 653 345 653 274V107C653 36 642 32 583 26V0H797Z" id="g190-110"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="819" vert-adv-y="819"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,20.952,3.264)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z" id="g50-106"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="280" vert-adv-y="280"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,23.608,3.264)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,29.402,3.264)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,34.021,3.264)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z" id="g50-45"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,36.268,3.264)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,40.887,3.264)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z" id="g50-45"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,43.135,3.264)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z" id="g50-52"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,48.352,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg>.</div>

Advances in Acoustics and Vibration

fig12

Figure 12

Figure 12: Fault Diagnosis of Beam-Like Structure Using Modified Fuzzy Technique 