Multiple Time-Scales Analysis to Predict the Quasiperiodic Oscillatory Response of a Thin-Walled Beam Subjected to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2725pt" id="M1" height="11.4049pt" version="1.1" viewBox="-0.0657574 -11.1324 46.2811 11.4049" width="46.2811pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g187-213" d="M345 0V33C260 38 258 46 258 122V635C174 612 105 594 35 583V554L87 550C124 546 130 539 130 494V122C130 46 127 38 42 33V0H345Z"/></g><g transform="matrix(.017,0,0,-0.017,11.167,0)"><path id="g113-59" d="M116 412C83 412 59 388 59 355C59 321 83 297 115 297S170 321 170 355C170 388 147 412 116 412ZM114 -12C147 -12 171 11 171 46C171 78 147 103 115 103C84 103 60 78 60 46C60 11 84 -12 114 -12Z"/></g><g transform="matrix(.017,0,0,-0.017,19.874,0)"><use xlink:href="#g187-213"/></g><g transform="matrix(.017,0,0,-0.017,31.041,0)"><use xlink:href="#g113-59"/></g><g transform="matrix(.017,0,0,-0.017,39.748,0)"><use xlink:href="#g187-213"/></g></svg> Simultaneous Resonance

<div>The beam’s horizontal and vertical vibration amplitudes <svg height="11.9479pt" id="M184" style="vertical-align:-3.181499pt" version="1.1" viewbox="-0.0498162 -8.7664 30.3747 11.9479" width="30.3747pt" 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="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.993,3.132)"><path d="M345 0V33C260 38 258 46 258 122V635C174 612 105 594 35 583V554L87 550C124 546 130 539 130 494V122C130 46 127 38 42 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183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,25.185,3.132)"><path d="M442 172C405 109 393 108 333 108H171L289 222C383 315 430 374 430 457C430 569 349 635 249 635C187 635 134 610 97 573L34 478L61 455C93 502 136 542 196 542C265 542 300 491 300 422C300 346 261 273 191 190C142 132 84 76 31 25V0H432C443 42 455 105 471 163L442 172Z"></path></g></svg> (a, c) and phases <svg height="12.3916pt" id="M185" style="vertical-align:-3.42948pt" version="1.1" viewbox="-0.0498162 -8.96212 32.4364 12.3916" width="32.4364pt" 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="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 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Shock and Vibration

fig8

Figure 8

Figure 8: Multiple Time-Scales Analysis to Predict the Quasiperiodic Oscillatory Response of a Thin-Walled Beam Subjected to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2725pt" id="M1" height="11.4049pt" version="1.1" viewBox="-0.0657574 -11.1324 46.2811 11.4049" width="46.2811pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g187-213" d="M345 0V33C260 38 258 46 258 122V635C174 612 105 594 35 583V554L87 550C124 546 130 539 130 494V122C130 46 127 38 42 33V0H345Z"/></g><g transform="matrix(.017,0,0,-0.017,11.167,0)"><path id="g113-59" d="M116 412C83 412 59 388 59 355C59 321 83 297 115 297S170 321 170 355C170 388 147 412 116 412ZM114 -12C147 -12 171 11 171 46C171 78 147 103 115 103C84 103 60 78 60 46C60 11 84 -12 114 -12Z"/></g><g transform="matrix(.017,0,0,-0.017,19.874,0)"><use xlink:href="#g187-213"/></g><g transform="matrix(.017,0,0,-0.017,31.041,0)"><use xlink:href="#g113-59"/></g><g transform="matrix(.017,0,0,-0.017,39.748,0)"><use xlink:href="#g187-213"/></g></svg> Simultaneous Resonance

Shock and Vibration

Multiple Time-Scales Analysis to Predict the Quasiperiodic Oscillatory Response of a Thin-Walled Beam Subjected to Simultaneous Resonance

Figure 8