L-Stable Method for Differential-Algebraic Equations of Multibody System Dynamics

<table class="fixed-width table-group" id="tab5"><tr><td><table class="table"><colgroup><col style="width:8.37em"/><col style="width:5.46em"/><col style="width:5.59em"/><col style="width:11.23em"/><col style="width:10.39em"/><col style="width:12.91em"/></colgroup><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr class="thead"><td class="align_left"> </td><td class="align_center">Runtime/s</td><td class="align_center"><span style="width: 47.049ptpx;"><svg height="11.439pt" id="M125" style="vertical-align:-2.39208pt" version="1.1" viewbox="-0.0498162 -9.04692 47.049 11.439" width="47.049pt" 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="M797 0V26C739 32 732 36 732 103V296C732 394 682 449 605 449C576 449 550 437 529 423C504 407 475 389 446 366C425 418 382 449 334 449C303 449 279 437 253 421C222 403 201 385 180 371V452C135 432 85 419 41 411V388C99 379 102 374 102 310V103C102 38 93 32 27 26V0H238V26C189 32 180 38 180 103V338C210 363 250 390 289 390C351 390 377 348 377 275V103C377 37 368 32 306 26V0H520V26C465 32 456 38 456 101V296C456 314 455 326 453 338C491 369 529 390 565 390C628 390 653 345 653 274V107C653 36 642 32 583 26V0H797Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.608,0)"><path d="M433 39L423 65C413 59 399 54 387 54C370 54 352 69 352 114V299C352 352 342 392 307 422C285 440 255 449 225 449C168 437 102 399 75 379C56 365 44 353 44 339C44 315 69 296 87 296C101 296 111 303 116 319C124 349 133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.315,0)"><path d="M474 0V26C414 34 401 43 364 100L267 248C300 297 324 332 345 358C381 400 394 405 455 411V437H272V411C316 406 323 401 305 370C287 337 267 306 247 276L188 369C169 397 173 405 215 411V437H16V411C71 404 83 396 114 348L201 212C171 167 144 127 116 92C77 42 66 34 4 26V0H190V26C139 34 136 43 156 77C175 113 198 150 220 183L294 66C311 39 302 31 260 26V0H474Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.74,.241)"><path d="M162 -163V703H101V-163H162Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.172,0)"><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(.013,0,0,-0.013,33.503,0)"><path d="M729 650H468V622C546 615 554 610 554 527V365H213V527C213 610 221 616 297 622V650H38V622C120 616 128 611 128 527V123C128 39 120 34 40 28V0H303V28C221 34 213 40 213 123V322H554V123C554 39 546 34 460 28V0H728V28C647 34 639 39 639 123V527C639 610 647 615 729 622V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,43.461,.241)"><path d="M162 -163V703H101V-163H162Z"></path></g></svg></span></td><td class="align_center">displacement constraints</td><td class="align_center">speed-level constraints</td><td class="align_center">acceleration-level constraints</td></tr><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td class="align_left">equidistant nodes</td><td class="align_center">5.6094</td><td class="align_center">0.0011</td><td class="align_center">2.7756e-15</td><td class="align_center">5.9405e-04</td><td class="align_center">0.9854</td></tr><tr><td class="align_left">Chebyshev nodes</td><td class="align_center">5.2188</td><td class="align_center">4.3245e-05</td><td class="align_center">2.9976e-15</td><td class="align_center">1.8094e-04</td><td class="align_center">0.6880</td></tr><tr><td class="align_left">Legendre nodes</td><td class="align_center">5.1094</td><td class="align_center">2.5327e-04</td><td class="align_center">2.8866e-15</td><td class="align_center">2.9047e-04</td><td class="align_center">0.7560</td></tr><tr class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Comparison of the results of the L-stable method taking uniform and nonuniform nodes.</div>

Mathematical Problems in Engineering

tab5

Table 5

Table 5: L-Stable Method for Differential-Algebraic Equations of Multibody System Dynamics 