An Analytical-Based Lightning-Induced Damage Model for an Aircraft Wing Exposed to Pressure Loading of Lightning

<table class="table-group" id="tab9"><tr><td><table class="table"><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr class="thead"><td class="align_left"></td><td class="align_center">Vibration freq. (rad/s) <svg height="12.5794pt" id="M219" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 22.303 12.5794" width="22.303pt" 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="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.467,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 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635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g></svg>)</span></td><td class="align_center"><span style="width: 32.497ptpx;"><svg height="13.2528pt" id="M223" style="vertical-align:-3.29896pt" version="1.1" viewbox="-0.0498162 -9.95384 32.497 13.2528" width="32.497pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,2.989,-.666)"><path d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.495,-.666)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,14.993,-.666)"><path d="M594 629L561 652L159 -9L190 -32L594 629ZM194 635C92 635 49 534 49 453C49 374 92 272 193 272C295 272 338 374 338 454S295 635 194 635ZM193 605C257 605 265 504 265 453C265 405 257 302 194 302C133 302 122 403 122 454S133 605 193 605ZM567 351C465 351 421 250 421 169C421 90 465 -12 566 -12S710 90 710 170S667 351 567 351ZM566 322C628 322 637 221 637 169C637 121 628 18 567 18S494 119 494 170C494 223 506 322 566 322Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.821,-.666)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td></tr><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr><td class="align_left" rowspan="6">Goland wing</td><td class="align_center"><i>ω</i><sub>1</sub></td><td class="align_center">49.6</td><td class="align_center">49.6</td><td class="align_center">0.00</td></tr><tr><td class="align_center"><i>ω</i><sub>2</sub></td><td class="align_center">97.0</td><td class="align_center">97.0</td><td class="align_center">0.00</td></tr><tr><td class="align_center"><i>ω</i><sub>3</sub></td><td class="align_center">248.9</td><td class="align_center">248.9</td><td class="align_center">0.00</td></tr><tr><td class="align_center"><i>ω</i><sub>4</sub></td><td class="align_center">355.6</td><td class="align_center">355.6</td><td class="align_center">0.00</td></tr><tr><td class="align_center"><i>ω</i><sub>5</sub></td><td class="align_center">451.5</td><td class="align_center">451.5</td><td class="align_center">0.00</td></tr><tr><td class="align_center"><i>ω</i><sub>6</sub></td><td class="align_center">610.1</td><td class="align_center">610.3</td><td class="align_center">0.03</td></tr><tr class="table-tr"><td colspan="5"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>The first six vibration frequencies of the aircraft wing.</div>

Mathematical Problems in Engineering

tab9

Table 9

Table 9: An Analytical-Based Lightning-Induced Damage Model for an Aircraft Wing Exposed to Pressure Loading of Lightning 