Modelling of coupled cross-flow/in-line vortex-induced vibrations using double Duffing and van der Pol oscillators

Many studies have typically applied a linear structural spring–mass–damper oscillator and a van der Pol wake oscillator to model a one-dimensional cross-flow vortex-induced vibration (VIV). In this study, an advanced model for predicting a two-dimensional coupled cross-flow/in-line VIV of a flexibly mounted circular cylinder in a uniform flow is proposed and validated. The ensuing dynamical system is based on double Duffing–van der Pol (structural-wake) oscillators with the two structural equations containing both cubic and quadratic nonlinear terms. The cubic nonlinearities capture the geometrical coupling of cross-flow/in-line displacements excited by hydrodynamic lift/drag forces whereas the quadratic nonlinearities allow the wake–cylinder interactions. Some empirical coefficients are calibrated against published experimental results to establish a new generic analytical function accounting for the dependence of VIV on a physical mass and/or damping parameter. By varying flow velocities in the numerical simulations, the derived low-order model captures several important VIV characteristics including a two-dimensional lock-in, hysteresis phenomenon and figure-of-eight trajectory tracing the periodically coupled in-line/cross-flow oscillations with their tuned two-to-one resonant frequencies. By making use of a newly derived empirical formula, the predicted maximum cross-flow/in-line VIV amplitudes and associated lock-in ranges compare well with several experimental results for cylinders with low/high mass or damping ratios. Moreover, the parametric studies highlight the important effect of geometrical nonlinearities through new displacement coupling terms and the ratio of in-line to cross-flow natural frequencies of the freely vibrating cylinder.