Nonlinear Damping Term Research Articles

Model predictive control of a wave-to-wire wave energy converter system with non-linear dynamics and non-linear constraints using a tailored pseudo-spectral method

A novel pseudo-spectral method is proposed to tackle the non-linear control problem of a wave energy converter (WEC) integrated with an all-electrical power-take-off (PTO). The WEC hydrodynamic model is constructed using linear wave theory together with a non-linear viscous damping term. The PTO model encompasses a permanent magnet synchronous generator (PMSG) whose local field weakening mechanism imposes non-linear operational constraints on the WEC’s motion and PTO torque. The energy-maximizing control problem of this wave-to-wire model is non-linear and has not been investigated before. In comparison to pseudo-spectral methods previously studied, the proposed pseudo-spectral method is exempt from the assumption of ideal incoming wave prediction. This is made possible by the introduction of a novel mapping function to convert the finite control horizon onto the Gaussian quadrature interval. This mapping function increases more slowly with time at the beginning of the control horizon where wave prediction attains greater precision. As a result, the collocation points are utilized to the advantage of the better approximation of the objective function, thus better overall control performance.

Existence, Decay, and Blow-up of Solutions for a Weighted m-Biharmonic Equation with Nonlinear Damping and Source Terms

In this paper, we consider the weighted m-biharmonic equation with nonlinear damping and source terms. We proved the global existence of solutions. Later, the decay of the energy is established by using Nakao’s inequality. Finally, we proved the blow-up of solutions in finite time.

A piezoelectric-electromagnetic hybrid energy harvester with frequency-up conversion mechanism towards low-frequency-low-intensity applications

This paper proposes a piezoelectric-electromagnetic hybrid energy harvester (PEHEH) with frequency-up conversion (FUC) mechanisms toward harvesting low-frequency-low-intensity, and wideband vibrational energy. The PEHEH consists of a stator with embedded coils, an eccentric rotor embedded with permanent magnets, and a fan-folded piezoelectric beam fixed to the stator. The eccentric rotor converts low-frequency straight vibration into rotational motion, and generates currents across the coils. The rotor could collide with the fan-folded beam, resulting in voltage generation due to the deformation of the fan-folded beam. The FUC mechanism broadens the bandwidth and harvests additional energy through the piezoelectric effect. The fan-folded beam can be considered as a nonlinear oscillator of hardening type. The electro-magnetic-mechanical system is modeled as a driven pendulum that impacts a mass-spring-damper system. The electromagnetic coupling effect is theoretically determined and equivalent to a nonlinear damping term. The nonlinear dynamical system is analyzed theoretically, numerically and experimentally. The results show that the bandwidth of the PEHEH is broadened than that of the energy harvester without the FUC mechanism. The maximum power reaches 26.4 mW at 4 Hz. The PEHEH can power an acceleration sensor at 4 Hz in laboratory environments and a temperature-humidity sensor in actual working conditions of an agricultural tractor's spray rod.

A parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations with a nonlinear damping term

AbstractIn this work, we propose a parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations attached with a nonlinear damping term, using a fully overlapping domain decomposition approach. In the proposed algorithm, we calculate a local solution in a defined subdomain on a global composite mesh which is fine around the defined subdomain and coarse in other regions. The algorithm is simple to carry out on the basis of available sequential solvers. By a local a priori estimate of the finite element solution, we deduce error bounds of the approximations from our presented algorithm. We perform also some numerical experiments to verify the effectiveness of the proposed algorithm.

Performance characterization and modeling of an oscillating surge wave energy converter

Testing wave energy converters in the ocean could be expensive and complex, which necessitates the use of numerical modeling. However, accurately modeling the response of wave energy converters with high-fidelity simulations can be computationally intensive in the design stage where different configurations must be considered. Reduced-order models based on simplified equations of motion can be very useful in the design, optimization, or control of wave energy converters. Given the complex dynamics of wave energy converters, accurate representation, and evaluation of relative contributions by different forces are required. This effort is concerned with a performance characterization of the hydrodynamic response of an oscillating surge wave energy converter that is based on a reduced-order model. A state-space model is used to represent the radiation damping term. Morison’s representation of unsteady forces is used to account for the nonlinear damping. Wave tank tests are performed to validate simulations. A free response simulation is used to determine the coefficients of the state-space model. Torque-forced simulations are used to identify the coefficients of the nonlinear damping term for different amplitudes and wave frequencies. The impact of varying these coefficients on the response is investigated. An assessment of the capability of the model in predicting the hydrodynamic response under irregular forcing is performed. The results show that the maximum error is 3% when compared with high-fidelity simulations. It is determined that the nonlinear damping is proportional to the torque amplitude and its effects are more pronounced as the amplitude of the flap oscillations increases.

Stability Results for a System of Nonlinear Viscoelastic Plate Equations with Nonlinear Frictional Damping and Logarithmic Source Terms

Stability Results for a System of Nonlinear Viscoelastic Plate Equations with Nonlinear Frictional Damping and Logarithmic Source Terms

Regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term

<abstract><p>This paper was devoted to establishing some regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term. We focused on considering the role of the damping term in regularity criteria and the global existence brought by these criteria.</p></abstract>

The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term

<abstract><p>This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When $ \beta > 3 $, with initial data $ (u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2} $ and external forces $ (f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0}) $, some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the $ ((V_{1}\times V_{2})\times(\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})), V_{1}\times V_{2}) $-continuity of the family of processes $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is demonstrated. Meanwhile, the $ (V_{1}\times V_{2}, \mathbf{H}^2(\Omega)\times\mathbf{H}^2(\Omega)) $-uniform compactness of $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is proved. Finally, the existence of a $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and a $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are obtained. Furthermore, the $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and the $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are verified to be the same.</p></abstract>

Weak Galerkin method for the Navier-Stokes equation with nonlinear damping term

<abstract><p>The primary focus of this research was to investigate the weak Galerkin (WG) finite element method for the Navier-Stokes equations with damping. We established the weak Galerkin finite element numerical scheme and demonstrated the existence and uniqueness of the weak Galerkin numerical solution. Additionally, optimal errors estimates for the velocity and pressure were obtained. Eventually, numerous numerical examples were reported to validate the theoretical analysis.</p></abstract>

Global well-posedness to the 3D incompressible magneto-micropolar Bénard system with damping and zero thermal conductivity

We are concerned with the three-dimensional (3D) Cauchy problem of the incompressible magneto-micropolar Bénard system with a nonlinear damping term α|u|β−1u(α>0andβ≥1) in the momentum equations. By energy method, we derive a unique global strong solution for such a model when β≥4. Our result extends previous related works.

Existence and stability results of nonlinear swelling equations with logarithmic source terms

<abstract><p>We considered a swelling porous-elastic system characterized by two nonlinear variable exponent damping and logarithmic source terms. Employing the Faedo-Galerkin method, we established the local existence of weak solutions under suitable assumptions on the variable exponents functions. Furthermore, we proved the global existence utilizing the well-depth method. Finally, we established several decay results by employing the multiplier method and the Logarithmic Sobolev inequality. To the best of our knowledge, this represents the first study addressing swelling systems with logarithmic source terms.</p></abstract>

Asymptotic behavior of the wave equation solution with nonlinear boundary damping and source term of variable exponent-type

<p>In this study, a nonlinear damped wave equation within a bounded domain was considered. We began by demonstrating the global existence of solutions through the application of the well-depth method. Following this, a general decay rate for the solutions was established using the multiplier method alongside key properties of convex functions. Notably, these results were derived without the imposition of restrictive growth assumptions on the frictional damping, making this work an improvement and extension of previous findings in the field.</p>

Adaptive Regulation With Global KL Guarantees

In the absence of persistency of excitation (PE), referring to adaptive control systems as “uniformly asymptotically stable” typically indicates insufficient understanding of stability concepts. While the state is indeed regulated to zero and the parameter estimate has some limit, namely, the overall state converges to some equilibrium, the equilibrium reached is not unique (and not even necessarily stable) but is dependent on the initial condition. The equilibrium set in the absence of PE is not uniformly attractive (from an open set containing the equilibrium set); hence, uniform asymptotic stability does not hold and KL-estimates are unavailable for the full state. In this paper we pursue adaptive control design with KL guarantees on the regulated state, something that is possible but previously unachieved with smooth, time-invariant and non-hybrid adaptive controllers. This property is referred to as Uniform Global Asymptotic Output Stability, where the regulated state is thought of as a system output. We provide designs for (i) systems with a matched uncertainty and (ii) systems in the parametric strict feedback form. We assume no a priori known bound for the parameters. To guarantee KL-estimates in the absence of PE, our designs employ time-invariant nonlinear damping terms, which depend both on the state and the parameter estimate. With an example, we illustrate the theory.

A parallel finite element post-processing algorithm for the damped Stokes equations

This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method and post-processing technique. The most valuable highlights of the present algorithm are that (1) a global continuous approximate solution is generated via the partition of unity method; (2) by adding an extra coarse grid correction step, the smoothness of the approximate solution is improved; (3) it has a good parallel performance since there requires little communication in solving a series of residual problems in the subdomain of interest. We theoretically derive the L2-error estimates both for the approximate velocity and pressure and H1-error estimate for the velocity under some necessary conditions. Meanwhile, we numerically perform various test examples to validate the theoretically predicted convergence rate and illustrate the high efficiency of the proposed algorithm.

A general stability result for swelling porous elastic media with nonlinear damping and nonlinear delay term

A general stability result for swelling porous elastic media with nonlinear damping and nonlinear delay term

Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity

In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.

Well-posedness and general energy decay of solutions for a nonlinear damping piezoelectric beams system with thermal and magnetic effects

In this article, we study the piezoelectric beams with thermal and magnetic effects in the presence of a nonlinear damping term acting on the mechanical equation. First, we prove that the system is well-posed in the sense of semigroup theory. And by constructing a suitable Liapunov functional, we show a general decay result of the solution for the system from which the polynomial and exponential decay are only special cases. Furthermore, our result does not depend on any relationship between system parameters.

General decay and blow-up of solution for a viscoelastic wave equation with nonlinear degenerate damping and source terms

We are concerned with the initial-boundary value problem for a nonlinear viscoelastic problem with the degenerate damping and the nonlinear source term in this paper. Under appropriate assumptions on the initial data, memory kernel and parameters, we establish several results regarding the global solution, blow-up properties and general decay in the case of positive initial energy. We overcome the difficulties brought by positive initial energy by using the potential well theory.

Dynamics of nonlinear Reissner–Mindlin–Timoshenko plate systems

AbstractThe problem of Reissner–Mindlin–Timoshenko plate systems with nonlinear damping terms is considered. The main result is the existence of global attractors. By showing the system is gradient and asymptotic smoothness via a stabilizability inequality, we establish the existence of global attractors with finite fractal dimension. The continuity of global attractors regarding the parameter in a residual dense set is also proved. The above results allow the damping terms with polynomial growth.

Dynamics of a class of extensible beams with degenerate and non-degenerate nonlocal damping

This work is concerned with new results on long-time dynamics of a class of hyperbolic evolution equations related to extensible beams with three distinguished nonlocal nonlinear damping terms. In the first possibly degenerate case, the results feature the existence of a family of compact global attractors and a thickness estimate for their Kolmogorov’s $\varepsilon$-entropy. Then, in the non-degenerate context, the structure of the helpful nonlocal damping leads to the existence of finite-dimensional compact global and exponential attractors. Lastly, in a degenerate and critical framework, it is proved the existence of a bounded closed global attractor but not compact. To the proofs, we provide several new technical results by means of refined estimates that open up perspectives for a new branch of nonlinearly damped problems.