Nonlinear Dynamics of a Duffing-Like Negative Stiffness Oscillator: Modeling and Experimental Characterization

<div>RFS estimation of the restoring force <svg height="11.5564pt" id="M123" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 28.7938 11.5564" width="28.7938pt" 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="M771 669C722 624 660 598 616 598C593 598 565 610 530 629C460 664 429 678 388 678C284 678 211 602 211 527C211 453 270 401 350 401C418 401 468 443 501 508L483 517C450 464 405 431 352 431C288 431 252 473 252 525S306 626 370 626C398 626 435 612 472 592C525 563 556 548 596 548C607 548 623 550 632 553L633 550C510 448 474 360 391 210C314 77 264 16 181 16C121 16 79 48 75 79H77C84 68 97 66 106 66C134 66 150 90 150 115C150 144 130 165 103 165C63 165 40 130 40 91C40 38 96 -15 183 -15C301 -15 418 42 546 321C553 310 560 306 580 306C619 306 620 284 620 275C620 211 535 149 535 70C535 21 581 -12 628 -12C689 -12 755 40 830 132L811 147C736 61 680 18 642 18C621 18 610 29 610 52C610 93 686 163 686 232C686 263 673 286 638 321V323C721 350 782 402 849 522C876 570 898 630 916 646H917C918 613 943 601 962 601C990 601 1004 621 1004 646C1004 674 981 687 953 687C900 687 834 608 741 471C684 385 650 354 609 346C596 358 582 360 567 360C627 465 677 573 784 653L771 669Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.117,0)"><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,-0.013,17.615,0)"><path d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.09,0)"><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> (a) and the corresponding potential <svg height="11.5564pt" id="M124" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 26.6147 11.5564" width="26.6147pt" 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="M805 658H719L589 415C537 319 349 15 237 15C214 15 208 28 208 43C208 59 219 86 247 130L433 417C464 464 471 506 471 541C471 638 384 687 292 687C164 687 36 597 36 474C36 406 83 360 151 360C238 360 296 442 309 509H289C265 440 225 390 150 390C103 390 77 426 77 474C77 582 175 652 283 652C340 652 404 619 404 568C404 541 397 512 368 467L223 240C170 155 153 111 153 73C153 18 186 -15 234 -15C290 -15 367 41 439 138L442 137C427 106 425 83 425 66C425 18 463 -12 511 -12C575 -12 638 42 714 134L693 147C621 63 565 18 525 18C506 18 492 28 492 53C492 60 493 77 509 108L805 658Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.946,0)"><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,-0.013,15.444,0)"><path d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.919,0)"><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> (b). Red dots: stable and unstable equilibrium positions; gray dots: restoring surface for <svg height="12.5612pt" id="M125" style="vertical-align:-3.27287pt" version="1.1" viewbox="-0.0498162 -9.28833 40.2341 12.5612" width="40.2341pt" 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="M162 -163V703H101V-163H162Z"></path></g><g transform="matrix(.013,0,0,-0.013,9.5,-.405)"><path d="M-161 571C-161 600 -183 623 -212 623C-238 623 -262 600 -262 571C-262 544 -238 520 -212 520C-183 520 -161 544 -161 571Z"></path></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"></path></g><g transform="matrix(.013,0,0,-0.013,9.917,0)"><path d="M162 -163V703H101V-163H162Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.968,0)"><path d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.231,0)"><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(.0091,0,0,-0.0091,33.236,3.132)"><path d="M319 325C317 349 306 409 297 431C277 440 250 449 209 449C117 449 57 389 57 319C57 243 122 209 182 182C232 159 261 135 261 91C261 48 227 21 190 21C130 21 85 79 68 145L41 140C41 104 51 36 58 22C75 7 121 -12 172 -12C252 -12 337 35 337 126C337 195 286 231 210 262C166 281 126 304 126 348C126 388 152 417 191 417C240 417 274 378 294 318L319 325Z"></path></g><g transform="matrix(.013,0,0,-0.013,37.086,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g></svg></div>

Shock and Vibration

fig17

Figure 17

Figure 17: Nonlinear Dynamics of a Duffing-Like Negative Stiffness Oscillator: Modeling and Experimental Characterization 