Page 12 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONTENTS ix

6.6 Curved pipes with an axially sliding end ..................... 459
6.6.1 Transversely sliding downstream end .................. 460
6.6.2 Axially sliding downstream end ...................... 460

Appendices
A First-principles Derivation of the Equation of Motion of a Pipe
Conveying Fluid 463

B Analytical Evaluation of b... ... and d.. 466

C Destabilization by Damping: T . Brooke Benjamin’s Work 468

D Experimental Methods for Elastomer Pipes 471
D.l Materials. equipment and procedures ........................ 471
D.2 Short pipes. shells and cylinders ........................... 473
D.3 Flexural rigidity and damping constants ...................... 474
D.4 Measurement of frequencies and damping .................... 476

E The Timoshenko Equations of Motion and Associated Analysis 478
E.l The equations of motion ................................ 478
E.2 The eigenfunctions of a Timoshenko beam .................... 480
E.3 The integrals Zkn ..................................... 481

F Some of the Basic Methods of Nonlinear Dynamics 483
F.l Lyapunov method .................................... 483
F.l . 1 The concept of Lyapunov stability .................... 483
F . 1.2 Linearization .................................. 484
F . 1.3 Lyapunov direct method ........................... 486
F.2 Centre manifold reduction ............................... 487
F.3 Normal forms ....................................... 489
F.4 The method of averaging ............................... 491
F.5 Bifurcation theory and unfolding parameters ................... 493
F.6 Partial differential equations ............................. 495
F.6.1 The method of averaging revisited .................... 495
F.6.2 The Lyapunov-Schmidt reduction .................... 498
F.6.3 The method of alternate problems ..................... 500

G Newtonian Derivation of the Nonlinear Equations of Motion of a Pipe
Conveying Fluid 502
G.l Cantilevered pipe .................................... 502
G.2 Pipe fixed at both ends ................................. 503

H Nonlinear Dynamics Theory Applied to a Pipe Conveying Fluid 506
H.l Centre manifold ..................................... 506
H.2 Normal form ....................................... 507
H.2.1 Dynamic instability .............................. 507
H.2.2 Static instability ................................ 515