Page 11 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

... CONTENTS
VI11
5.5 Pipes with supported ends ............................... 302
5.5.1 The effect of amplitude on frequency .................. 302
5.5.2 The post-divergence dynamics ....................... 303
5.5.3 Pipes with an axially sliding downstream end ............. 314
5.5.4 Impulsively excited 3-D motions ..................... 315
5.6 Articulated cantilevered pipes ............................ 316
5.6.1 Cantilever with constrained end ...................... 316
5.6.2 Unconstrained cantilevers .......................... 317
5.6.3 Concluding comment ............................. 327
5.7 Cantilevered pipes .................................... 328
5.7.1 2-D limit-cycle motions ........................... 328
5.7.2 3-D limit-cycle motions ........................... 333
5.7.3 Dynamics under double degeneracy conditions ............ 336
5.7.4 Concluding comment ............................. 348
5.8 Chaotic dynamics .................................... 348
5.8.1 Loosely constrained pipes .......................... 348
5.8.2 Magnetically buckled pipes ......................... 366
5.8.3 Pipe with added mass at the free end .................. 368
5.8.4 Chaos near double degeneracies ...................... 387
5.8.5 Chaos in the articulated system ...................... 392
5.9 Nonlinear parametric resonances .......................... 394
5.9.1 Pipes with supported ends .......................... 394
5.9.2 Cantilevered pipes ............................... 402
5.10 Oscillation-induced flow ................................ 412
5.11 Concluding remarks ................................... 413

6 Curved Pipes Conveying Fluid 415
6.1 Introduction ........................................ 415
6.2 Formulation of the problem .............................. 417
6.2.1 Kinematics of the system .......................... 417
6.2.2 The equations of motion ........................... 421
6.2.3 The boundary conditions ........................... 426
6.2.4 Nondimensional equations .......................... 426
6.2.5 Equations of motion of an inextensible pipe .............. 428
6.2.6 Equations of motion of an extensible pipe ............... 429
6.3 Finite element analysis ................................. 430
6.3.1 Analysis for inextensible pipes ....................... 431
6.3.2 Analysis for extensible pipes ........................ 436
6.4 Curved pipes with supported ends ......................... 437
6.4.1 Conventional inextensible theory ..................... 438
6.4.2 Extensible theory ............................... 440
6.4.3 Modified inextensible theory ........................ 446
6.4.4 More intricate pipe shapes and other work ............... 448
6.4.5 Concluding remarks .............................. 451
6.5 Curved cantilevered pipes ............................... 452
6.5.1 Modified inextensible and extensible theories ............. 452
6.5.2 Nonlinear and chaotic dynamics ...................... 457