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 (a) and vertical (b) displacement bifurcation behaviors in response to the spinning speed detuning <svg height="6.34998pt" id="M196" style="vertical-align:-0.2063899pt" version="1.1" viewbox="-0.0498162 -6.14359 7.47218 6.34998" width="7.47218pt" 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="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"></path></g></svg> at <span class="nowrap"><svg height="12.7178pt" id="M197" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 39.4241 12.7178" width="39.4241pt" 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="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.984,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.247,0)"><path d="M409 0V34C329 39 322 56 322 125V635C234 611 160 593 94 583V554L145 551C186 547 190 541 190 488V125C190 54 181 39 102 34V0H409Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.747,0)"><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,-0.013,32.711,0)"><path d="M252 635C85 635 28 460 28 311C28 167 82 -12 250 -12C419 -12 472 166 472 312C472 457 419 635 252 635ZM248 594C314 594 335 455 335 314C335 174 318 29 252 29S165 180 165 315S181 594 248 594Z"></path></g></svg>.</span></div>

Shock and Vibration

fig11

Figure 11

Figure 11: 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

Figure 11 | Multiple Time-Scales Analysis to Predict the Quasiperiodic Oscillatory Response of a Thin-Walled Beam Subjected to  Simultaneous Resonance 