Creating Nodes at Selected Locations in a Harmonically Excited Structure Using Feedback Control and Green’s Function

<div>The steady-state deformed shape of a collocated controlled (solid line) and uncontrolled (dashed line) damped cantilever beam when <i>c</i><sub>1</sub> = 0.1, <i>c</i><sub>2</sub> = 0.001, <i>ω</i> = 22, <i>x</i><sub>f</sub> = 1, <span class="nowrap"><svg height="13.6311pt" id="M289" style="vertical-align:-5.28228pt" version="1.1" viewbox="-0.0498162 -8.34882 46.9695 13.6311" width="46.9695pt" 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="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 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133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,11.212,5.233)"><path d="M452 139L416 148C386 81 373 77 319 77H141L282 220C372 312 414 373 414 464C414 569 332 635 240 635C181 635 132 609 101 574L42 493L70 467C98 506 143 562 209 562C280 562 324 515 324 435C324 347 261 272 197 197C148 139 89 84 34 29V0H417L452 139Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.063,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,30.326,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,36.566,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,39.53,0)"><path d="M244 635C114 635 38 519 38 422C38 317 111 240 217 240C236 240 255 244 277 256L345 292C311 140 203 39 59 15L64 -15C89 -15 150 -5 204 17C339 72 440 202 440 386C440 521 368 635 244 635ZM228 602C326 602 352 479 352 390C352 370 351 347 348 324C327 308 293 296 258 296C174 296 124 369 124 458C124 517 152 602 228 602Z"></path></g></svg>.</span> The inset shows the detailed plot of the controlled shape.</div>

Shock and Vibration

fig12

Figure 12

Figure 12: Creating Nodes at Selected Locations in a Harmonically Excited Structure Using Feedback Control and Green’s Function 