OC6 Phase Ia: CFD Simulations of the Free-Decay Motion of the DeepCwind Semisubmersible

Currently, the design of floating offshore wind systems is primarily based on mid-fidelity models with empirical drag forces. The tuning of the model coefficients requires data from either experiments or high-fidelity simulations. As part of the OC6 (Offshore Code Comparison Collaboration, Continued, with Correlation, and unCertainty (OC6) is a project under the International Energy Agency Wind Task 30 framework) project, the present investigation explores the latter option. A verification and validation study of computational fluid dynamics (CFD) models of the DeepCwind semisubmersible undergoing free-decay motion is performed. Several institutions provided CFD results for validation against the OC6 experimental campaign. The objective is to evaluate whether the CFD setups of the participants can provide valid estimates of the hydrodynamic damping coefficients needed by mid-fidelity models. The linear and quadratic damping coefficients and the equivalent damping ratio are chosen as metrics for validation. Large numerical uncertainties are estimated for the linear and quadratic damping coefficients; however, the equivalent damping ratios are more consistently predicted with lower uncertainty. Some difference is observed between the experimental and CFD surge-decay motion, which is caused by mechanical damping not considered in the simulations that likely originated from the mooring setup, including a Coulomb-friction-type force. Overall, the simulations and the experiment show reasonable agreement, thus demonstrating the feasibility of using CFD simulations to tune mid-fidelity models.

Some possible Lagrangians for the quadratically damped system

In this paper, some possible Lagrangians for the quadratically damped systems are investigated. The corresponding classical Hamiltonians are investigated with Hamilton equations. The quantum Hamiltonians are also constructed so that they may be Hermitian. As an example, the quantum mechanics for the harmonic oscillator with a small quadratic damping is discussed.

Energy analysis of Duffing oscillators with quadratic damping: exact solutions

Oscillators with a Duffing-type restoring force and quadratic damping are dealt with in this paper. Four characteristic cases of this restoring force are analysed: hardening, softening, bistable and a pure cubic one. Their energy-displacement relationships are considered, and the corresponding closed-form exact solutions are obtained in terms of the incomplete Gamma function, which represent new results. Such results provide insight into damped dynamics of the class of system, including finding the phase trajectories as well as the comparison between these cases from the viewpoint of the energy loss per cycle.

Harmonic and non-periodic solutions of velocity-dependent conservative equations

We study in this paper a quadratic damping Helmholtz equation presumed to be velocity-dependent conservative nonlinear oscillator. We show that under the usual conditions of existence of particular and exact harmonic solutions, the equation can also exhibit exact and general non-periodic solutions. We show finally the existence of exact and explicit general harmonic and isochronous solutions without requiring that the system Hamiltonian must be identically zero.

Optimal control of drop-induced shock mitigation using magnetorheological energy absorbers considering quadratic damping

The optimal control of a magnetorheological energy absorber (MREA) shock mitigation system is investigated considering quadratic damping in the MREA. To this end, the equation of motion of a single-degree-of-freedom (SDOF) shock suspension system using an MREA with quadratic damping is analyzed. To achieve a soft landing and to maintain stroking load below a maximum allowable value, it is required that the payload comes to rest after fully utilizing the available stroke. For low sink rates, a generalized Bingham number (quadratic) or GBN-Q control algorithm is developed that achieves a soft landing by selecting an initial magnetorheological (MR) force level or generalized Bingham number (GBN) for the quadratic damping at the initial sink rate. To cope with the cases above a critical sink rate, where the deceleration exceeds a maximum allowable threshold when using the GBN-Q control only, a minimum duration deceleration exposure-quadratic (MDDE-Q) controller is developed. This controller seeks to maintain the stroking load at its maximum allowable threshold until the payload slows such that the GBN-Q controller can be used to achieve the soft landing condition. The switching methodology between the GBN-Q controller and the MDDE-Q controller is discussed. Each control method relies on an optimal GBN that is computed to ensure a soft landing. Results show that the MDDE-Q controller can successfully minimize the exposure of the payload to the maximum allowable stroking load.

An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator

The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.