Mechanical Behaviors of Electrostatic Microbeams with Nonideal Supports

<table class="table-group" id="tab2"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left"> </td><td class="align_center">Torsional spring stiffness value <svg height="9.18808pt" id="M104" style="vertical-align:-3.22717pt" version="1.1" viewbox="-0.0498162 -5.96091 12.7239 9.18808" width="12.7239pt" 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="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M639 21C616 28 593 39 566 68C542 92 523 117 486 170C457 215 415 282 397 314C472 346 524 398 524 485C524 545 497 590 456 617C417 641 369 650 296 650H43V617C122 612 130 608 130 523V129C130 45 122 38 38 33V0H306V33C225 38 216 45 216 129V282H249C302 282 319 270 336 243C365 195 401 132 436 84C483 18 525 -5 597 -8C610 -9 622 -9 635 -9L639 21ZM216 319V559C216 586 219 596 227 603S250 613 281 613C356 613 429 576 429 469C429 416 413 379 374 348C349 327 316 319 265 319H216Z"></path></g></svg></td><td class="align_center">Translational spring stiffness value <svg height="9.14241pt" id="M105" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -5.96091 12.6235 9.14241" width="12.6235pt" 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="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M601 496C596 555 591 627 591 675H567C551 656 544 650 516 650H120C91 650 79 651 64 675H41C38 619 34 555 29 492H63C75 537 83 566 96 582C110 604 128 612 207 612H272V126C272 46 264 38 170 33V0H464V33C368 38 359 48 359 126V612H430C502 612 520 605 535 582C547 566 556 541 567 492L601 496Z"></path></g></svg></td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left">Case 1</td><td class="align_center"><span style="width: 52.0202ptpx;"><svg height="13.8595pt" id="M106" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -11.5914 52.0202 13.8595" width="52.0202pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><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(.0091,0,0,-0.0091,12.527,-5.741)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,16.959,-5.741)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.922,0)"><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,-0.013,26.42,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.05,0)"><path d="M983 225C983 348 891 448 766 448C646 448 579 377 521 276C458 383 382 448 256 448C165 448 37 373 37 208C37 98 119 -12 256 -12C375 -12 440 60 498 161C561 53 639 -12 766 -12C865 -12 983 64 983 225ZM478 196C428 106 365 21 266 21C183 21 129 114 129 240C129 353 183 415 244 415C352 415 420 301 478 196ZM889 204C889 66 838 21 776 21C666 21 598 135 540 241C590 331 655 415 754 415C842 415 889 320 889 204Z"></path></g><g transform="matrix(.013,0,0,-0.013,47.311,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td><td class="align_center"><span style="width: 52.0202ptpx;"><svg height="13.8595pt" id="M107" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -11.5914 52.0202 13.8595" width="52.0202pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><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 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transform="matrix(.013,0,0,-0.013,34.05,0)"><path d="M983 225C983 348 891 448 766 448C646 448 579 377 521 276C458 383 382 448 256 448C165 448 37 373 37 208C37 98 119 -12 256 -12C375 -12 440 60 498 161C561 53 639 -12 766 -12C865 -12 983 64 983 225ZM478 196C428 106 365 21 266 21C183 21 129 114 129 240C129 353 183 415 244 415C352 415 420 301 478 196ZM889 204C889 66 838 21 776 21C666 21 598 135 540 241C590 331 655 415 754 415C842 415 889 320 889 204Z"></path></g><g transform="matrix(.013,0,0,-0.013,47.311,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td></tr><tr><td class="align_left">Case 2</td><td class="align_center"><span style="width: 17.5728ptpx;"><svg height="11.7978pt" id="M108" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.5914 17.5728 11.7978" width="17.5728pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><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(.0091,0,0,-0.0091,12.527,-5.741)"><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"></path></g></svg></span></td><td class="align_center"><span style="width: 17.5728ptpx;"><svg height="11.7978pt" id="M109" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.5914 17.5728 11.7978" width="17.5728pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><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(.0091,0,0,-0.0091,12.527,-5.741)"><path d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z"></path></g></svg></span></td></tr><tr><td class="align_left">Case 3</td><td class="align_center"><span style="width: 52.0202ptpx;"><svg height="13.8595pt" id="M110" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -11.5914 52.0202 13.8595" width="52.0202pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><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(.0091,0,0,-0.0091,12.527,-5.741)"><path d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"></path></g><g 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-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(.0091,0,0,-0.0091,35.143,-5.741)"><path d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z"></path></g></svg></span></td></tr><tr><td class="align_left">Case 4</td><td class="align_center"><span style="width: 17.7806ptpx;"><svg height="9.92533pt" id="M112" style="vertical-align:-1.57651pt" version="1.1" viewbox="-0.0498162 -8.34882 17.7806 9.92533" width="17.7806pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M95 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class="tbody-hr"/></td></tr></table></td></tr></table>

Shock and Vibration

tab2

Table 2

Table 2: Mechanical Behaviors of Electrostatic Microbeams with Nonideal Supports 