Haider N. Arafat

MULTIMODE INTERACTIONS IN SUSPENDED CABLES

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the phases and amplitudes of the interacting modes. Then, we investigate the behavior of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied and determine the bifurcations they may undergo.

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.

Modeling Steady-state and Transient Forces on a Cylinder

Numerical simulations of the flow past a stationary circular cylinder at different Reynolds numbers (Re) have been performed using a computational fluid dynamics (CFD) solver that is based on the unsteady Reynolds-averaged Navier—Stokes equations (RANS). The results obtained are used to develop reduced-order models for the lift and drag coefficients. The models not only match the numerical simulation results in the time domain, but also in the spectral domain. They capture the steady-state region with excellent accuracy. Further, the models are verified by comparing their results in the transient region with their counterparts from the CFD simulations and very good agreement is found. The work performed here is a step towards building models for vortex-induced vibrations (VIV) encountered in risers, spars, and other offshore structures.

MODELING THE TRANSIENT AND STEADY-STATE FLOW OVER A STATIONARY CYLINDER

A reduced-order model for the two-dimensional flow over a stationary circular cylinder is examined. The lift is modeled with the van der Pol equation with three parameters; it models self-excited self-limiting systems. The drag is modeled as the sum of a mean term and a time-varying term proportional to the product of the lift and its time derivative. The transient and steady-state flows are calculated using a CFD code based on the unsteady Reynolds-averaged Navier-Stokes equations. The steady-state lift and drag CFD results are used to identify the three parameters in the lift model using a combination of higher-order spectral techniques and perturbation methods. The model is validated using steady-state numerical simulations for three cases describing low, moderate, and high Reynolds number flows. Then, the model is shown to reproduce the transient lift and drag calculated with the CFD code.Copyright

Nonlinear response of cantilever beams to combination and subcombination resonances

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinarydifferential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

Investigation of subcombination internal resonances in cantilever beams

Activation of subcombination internal resonances in transversely excited cantilever beams is investigated. The effect of geometric and inertia nonlinearities, which are cubic in the governing equation of motion, is considered. The method of time-averaged Lagrangian and virtual work is used to determine six nonlinear ordinary-differential equations governing the amplitudes and phases of the three interacting modes. Frequency- and force-response curves are generated for the case ω ≈ ω4 ≈ 1/2(ω2

Nonlinear Dynamics of Closed Spherical Shells

We investigate the axisymmetric dynamics of forced closed spherical shells. The nonlinear equations of motion are formulated using a variational approach and surface analysis. First, we revisit the linear eigenvalue problem. Then, using the method of multiple scales, we assess the possibility of the activation of two-to-one internal resonances between the different types of modes. Lastly, we examine the shell’s nonlinear responses to an axisymmetric primary-resonance excitation and analyze their bifurcations.Copyright

Modal Interactions in a Thermally Loaded Annular Plate

We investigate the nonlinear forced vibrations of a thermally loaded annular plate with clamped-clamped immovable boundary conditions in the presence of a three-to-one internal resonance between the first and second axisymmetric modes. We consider the in-plane thermal load to be axisymmetric and excite the plate externally by a harmonic force near primary resonance of the second mode. We then use the nonlinear von Karman plate equations to model the behavior of the system and apply the method of multiple scales to investigate its responses. We found that the response can be periodic oscillations consisting of both modes, with a large component from the first mode. Moreover, the periodic solutions may undergo Hopf bifurcations which lead to aperiodic oscillations of the plate.Copyright

Modeling Vortex Shedding Over a Stationary Circular Cylinder

Numerical simulations of flow past a stationary circular cylinder at different Reynolds numbers have been performed using a computational fluid dynamics (CFD) solver that is based on the Reynolds-averaged Navier-Stokes equations (RANS). The results obtained are used to develop reduced-order models for the lift and drag coefficients. The models do not only match the numerical simulation results in the time domain, but also in the spectral domain. They capture the steady-state region with excellent accuracy. Further, the models are verified by comparing their results in the transient region with their counterparts from the CFD simulations and a very good agreement is found. The work performed here is a step towards building models for vortex-induced vibrations (VIV) encountered in offshore structures, such as risers and spars.Copyright

Nonlinear Interactions in the Responses of Heated Annular Plates

Nonlinear modal interactions in the forced vibrations of a thermally loaded annular plate with clampedclamped immovable boundary conditions are investigated. The mechanism responsible for the interaction is a subcombination internal resonance involving the natural frequencies of the three lowest axisymmetric modes. The in-plane thermal load acting on the plate is assumed to be axisymmetric and the plate is externally excited