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

Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

This paper presents space-time numerical simulation and validation of analytical predictions for the finite-amplitude forced dynamics of suspended cables. The main goal is to complement analytical and numerical solutions, accomplishing overall quantitative/qualitative comparisons of nonlinear response characteristics. By relying on an approximate, kinematically non-condensed, planar modeling, a simply supported horizontal cable subject to a primary external resonance and a 1:1, or 1:1 vs. 2:1, internal resonance is analyzed. To obtain analytical solution, a second-order multiple scales approach is applied to a complete eigenfunction-based series of nonlinear ordinary-differential equations of cable damped forced motion. Accounting for both quadratic/cubic geometric nonlinearities and multiple modal contributions, local scenarios of cable uncoupled/coupled responses and associated stability are predicted, based on chosen reduced-order models. As a cross-checking tool, numerical simulation of the associated nonlinear partial-differential equations describing the dynamics of the actual infinite-dimensional system is carried out using a finite difference technique employing a hybrid explicit-implicit integration scheme. Based on system control parameters and initial conditions, cable amplitude, displacement and tension responses are numerically assessed, thoroughly validating the analytically predicted solutions as regards the actual existence, the meaningful role and the predominating internal resonance of coexisting/competing dynamics. Some methodological aspects are noticed, along with a discussion on the kinematically approximate versus exact, as well as planar versus non-planar, cable modeling

Wake-induced transverse vibration of two interfering cylinders in tandem arrangement : modelling and analysis

This paper discusses wake-induced transverse vibration of two interfering cylinders in tandem arrangement

Performance analysis and optimization of a box-hull wave energy converter concept

In this paper we consider a wave energy converter concept which is created by linking a box barge to the mechanical reference by linear dampers. The response to incident wave action in terms of power take-off is expressed explicitly as the solution of a linear frequency-domain model. The simplicity of the model combined with the possibility of the application of theory allows for a nested, and so manageable, procedure of optimization. We find that for any geometry, i.e., a combination of e.g. the breadth-to-length and breadth-to-draught aspect ratios of the box, the optimum is characterized by resonance at least in one of the two degrees of freedom, heave or pitch. Furthermore, optimal geometries turn out to be extremal: either long attenuator-type or wide terminator-type devices perform the best. We find also that optimal wavelengths, which are comparable to the device length in case of attenuators, emerge either due to the progressively increasing buoyancy restoring force characteristic, or due to the finite bandwidth of irregular waves. In particular, diffraction forces are more significant under optimal conditions for performance in irregular seas in comparison with conditions necessary for the most intensive displacement response of the free-floating box barge exposed to regular waves

Numerical and experimental comparisons of vortex-induced vibrations of marine risers in uniform/sheared currents

This paper presents a general theoretical reduced-order model capable of evaluating the multi-mode nonlinear dynamics of marine risers subject to uniform and sheared currents. The main objectives are to predict the vortex-induced vibration responses and parametrically compare between numerical and experimental results. The emphasis is placed on the analysis of cross-flow vibrations due to unsteady lift forces. The nonlinear equations governing riser axial/transversal motions are derived based on a top-tensioned beam model with typical pinned-pinned boundary conditions. The riser geometric nonlinearities owing to possible large dynamic displacements and multi-mode interactions are accounted for. To approximate the space-time varying lift force, the empirical hydrodynamic model, based on a nonlinear van der Pol wake oscillator with a distributed diffusive term, is used. A low-dimensional dynamic model and computationally-robust time-domain tool are then developed to evaluate the multi-mode fluid-riser interactions. These are very useful in dealing with large parametric studies involving varying system parameters

Experimental investigation of the flow-induced vibration of a curved cylinder in convex and concave configurations

Experiments have been conducted to investigate the two-degree-of-freedom vortex-induced vibration (VIV) response of a rigid section of a curved circular cylinder with low mass-damping ratio. Two curved configurations, a concave and a convex, were tested regarding the direction of the flow, in addition to a straight cylinder that served as reference. Amplitude and frequency responses are presented versus reduced velocity for a Reynolds number range between 750 and 15 000. Results for the curved cylinders with concave and convex configurations revealed significantly lower vibration amplitudes when compared to the typical VIV response of a straight cylinder. However, the concave cylinder showed relatively higher amplitudes than the convex cylinder which were sustained beyond the typical synchronisation region. We believe this distinct behaviour between the convex and the concave configurations is related to the wake interference taking place in the lower half of the curvature due to perturbations generated in the horizontal section when it is positioned upstream. Particle-image velocimetry (PIV) measurements of the separated flow along the cylinder highlight the effect of curvature on vortex formation and excitation revealing a complex fluid–structure interaction mechanism

Experimental investigation of the flow-induced vibration of a curved circular cylinder

Experiments have been conducted to investigate the vortex-induced vibration (VIV) response of a rigid section of a curved circular cylinder. Two curved configurations, a concave and a convex, were tested regarding the direction of the flow, in addition to a straight cylinder that served as reference. Amplitude and frequency response are presented versus reduced velocity for a Reynolds number range between 750 and 15,000. Results showed that the curved cylinders presented significant less vibration for both concave and convex configurations when compared to the typical VIV response of a straight cylinder. The concave configuration presented relatively high amplitudes of vibration that are sustained beyond the typical synchronisation region. We believe this distinct behaviour between the convex and the concave configurations is related to the wake interference happening in the lower half of the curvature due to perturbations generated in the horizontal section when it is positioned upstream. Particle-image velocimetry (PIV) measurements of the separated flow along the cylinder highlight the effect of curvature on vortex formation and excitation revealing a complex fluid-structure interaction mechanism

Experimental investigation of the vortex-induced vibration of a curved cylinder

This presentation looks at the experimental investigation of the vortex-induced vibration of a curved cylinder

Two-degree-of-freedom VIV of circular cylinder with variable natural frequency ratio : experimental and numerical investigations

Slender offshore structures possess multiple natural frequencies in different directions which can lead to different resonance conditions when undergoing vortex-induced vibration (VIV). This paper presents an experimental and numerical investigation of a two-degree-of-freedom VIV of a flexibly mounted circular cylinder with variable in-line-to-cross-flow natural frequency ratio. A mechanical spring-cylinder system, achieving a low equivalent mass ratio in both in-line and cross-flow directions, is tested in a water towing tank and subject to a uniform steady flow in a sub-critical Reynolds number range of about 2×103–5×104. A generalized numerical prediction model is based on the calibrated Duffing-van der Pol (structure-wake) oscillators which can capture the structural geometrical coupling and fluid-structure interaction effects through system cubic and quadratic nonlinearities. Experimental results for six measurement datasets are compared with numerical results in terms of response amplitudes, lock-in ranges, oscillation frequencies, time-varying trajectories and phase differences of cross-flow/in-line motions. Some good qualitative agreements are found which encourage the use of the implemented numerical model subject to calibration and the sensitivity analysis of empirical coefficients. Moreover, comparisons of the newly-obtained and published experimental results are carried out and discussed, highlighting a good correspondence in both amplitude and frequency responses. Various patterns of figure-of-eight orbital motions associated with dual two-to-one resonances are observed experimentally as well as numerically: the forms of trajectories are noticed to depend on the system mass ratio, damping ratio, reduced velocity parameter and natural frequency ratio of the two-dimensional oscillating cylinder

The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables

This study aims at comparing non-linear modal interactions in shallow horizontal cables with kinematically non-condensed vs. condensed modeling, under simultaneous primary external and internal resonances. Planar 1:1 or 2:1 internal resonance is considered. The governing partial-differential equations of motion of non-condensed model account for spatio-temporal modification of dynamic tension, and explicitly capture non-linear coupling of longitudinal/ vertical displacements. On the contrary, in the condensed model, a single integro-differential equation is obtained by eliminating the longitudinal inertia according to a quasi-static cable stretching assumption, which entails spatially uniform dynamic tension. This model is largely considered in the literature. Based on a multi-modal discretization and a second-order multiple scales solution accounting for higher-order quadratic effects of a infinite number of modes, coupled/uncoupled dynamic responses and the associated stability are evaluated by means of frequency- and force-response diagrams. Direct numerical integrations confirm the occurrence of amplitude-steady or -modulated responses. Non-linear dynamic configurations and tensions are also examined. Depending on internal resonance condition, system elasto-geometric and control parameters, the condensed model may lead to significant quantitative and/or qualitative discrepancies, against the non-condensed model, in the evaluation of resonant dynamic responses, bifurcations and maximal/minimal stresses. Results of even shallow cables reveal meaningful drawbacks of the kinematic condensation and allow us to detect cases where the more accurate non-condensed model has to be used