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

<div>Estimation of the restoring force <svg height="11.5564pt" id="M172" 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 damping force <svg height="11.5564pt" id="M173" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 27.1758 11.5564" width="27.1758pt" 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="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.505,0)"><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,-0.013,22.084,-.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,-0.013,16.003,0)"><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,-0.013,22.501,0)"><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). Gray dots: sliced restoring surface; orange line: RFS estimation (only in a); blue line: NSI estimation.</div>

Shock and Vibration

fig22

Figure 22

Figure 22: Nonlinear Dynamics of a Duffing-Like Negative Stiffness Oscillator: Modeling and Experimental Characterization 