Incremental harmonic balance method for multi-harmonic solution of high-dimensional delay differential equations: Application to crossflow-induced nonlinear vibration of steam generator tubes

Diverse science and engineering problems are governed by delay differential equations (DDE). Seeking periodic solutions of DDEs is crucial for many nonlinear dynamic systems. The incremental harmonic balance (IHB) method is an efficient semi-analytical ap- proach for periodic solutions of DDE. Among various DDE systems, flow-induced vibration (FIV) of tube bundles with loose support is unique, in the sense that the added damping cancels out structural damping in the vicinity of critical velocity, making it extremely slow to retain a convergent solution if numerical integration (NI) is utilized. Despite the effort s of developing IHB for various mechanical vibration problems with time-delay, no attempt has been made to employ IHB to nonlinear FIV problems. Here we fill this blank by sepa- rately combining two spatial discretization strategies, i.e., discretization via linear vibration modes or via finite element method (FEM), with IHB to capture limit cycle solutions after the onset of instability. Within the range of gap velocity of interest, the IHB demonstrates excellent convergence by increasing harmonic terms, and good agreement is obtained between IHB results and the results by NI method. An 18 degree-of-freedom (DOF) nonlin- ear DDE system was readily dealt by integrating FEM and IHB, each DOF being approximated by three harmonics. This is the first attempt to employ IHB for multi-harmonic solution of high-dimensional FIV problems, which opens a door for solutions of other DDEs. 

We publicize all the data and codes via the link https://github.com/XJTU-Zhou-group/IHB.