Nonconservative Systems Research Articles

CALCULATION OF REINFORCED CONCRETE STRUCTURES: CONSIDERATIONS ON DEVELOPING NEW THEORY

Introduction. Current regulatory documents in the field of concrete and reinforced concrete are based mainly on research results obtained in the past century.Aim. This article addresses the approaches to improve the theory of calculation of concrete and reinforced concrete structures, in the light of the accumulated knowledge.Materials and methods. Since concrete structures belong to non-conservative systems, it is inaccurate to determine the forces arising in them using standard methods of structural mechanics. To achieve this goal, the deformation and relaxation problems for an axially compressed concrete fiber were solved following the A.A. Gvozdev and K.Z. Galustov’s two-component theory of creep, where deformations are categorized on the basis of reversibility.Results. Historical background was provided. Proposed by the authors of this article is a solution to the deformation and relaxation problems for the elementary part of a concrete structure for the case of axial compression, since such an event is impossible. In this regard, it was suggested that estimating the loss of stability by Euler is also incorrect. It was observed that the achieved maximum load-bearing capacity of the construction is equivalent to zero resistibility, with a progressive collapse occurring. For further research, it was proposed to use the hypothesis of a straight normal, assuming that this law is valid in the absence of tangential stresses in the studied sections of the structure.Conclusions. To improve the theory of calculation of concrete and reinforced concrete structures, it is necessary to revise drastically the existing approach to scientific research.

Singular cotangent models in fluids with dissipation

In this article we analyze several mathematical models with singularities where the classical cotangent model is replaced by a b-cotangent model. We provide physical interpretations of the singular symplectic geometry underlying in b-cotangent bundles featuring two models: the canonical (or non-twisted) model and the twisted one. The canonical one models systems on manifolds with boundary and the twisted one represents Hamiltonian systems with a singularity on the fiber. The twisted cotangent model includes (for linear potentials) the case of fluids with dissipation. We prove (non)-existence of cotangent lift dynamics and show the existence of an infinite number of escape orbits in this model. We also discuss more general physical interpretations of the twisted and non-twisted b-symplectic models. Twisted b-symplectic models yield in a natural way escape orbits that go to the critical set. Under compactness assumptions those escape orbits are continued as singular periodic orbits in the sense of Miranda and Oms (2021) and Miranda (2020). These models offer a Hamiltonian formulation for systems which are dissipative, extending the horizons of Hamiltonian dynamics and opening a new approach to study non-conservative systems.

Two-dimensional bright soliton in exciton-polariton condensate

In non-conservative nonlinear systems, the basic physical mechanics of soliton generation is that the kinetic energy and nonlinear terms of the system, as well as the gain and dissipation terms reach a double dynamic balance. How to generate stable free high-dimensional solitons in such a system is currently a challenging topic in soliton theory. In this article, we propose a theoretical scheme for realizing two-dimensional free bright solitons in exciton-polariton Bose-Einstein condensates, which proposes a physical mechanism for generating stable two-dimensional free space bright solitons through time periodic modulation interactions and a dual balance between gain and dissipation. In this end, firstly, we obtain the dynamic equations of two-dimensional bright soliton parameters through the Lagrange variational method, and obtain its dynamically stable parameter space. Secondly, the evolution of the generalized dissipative Gross-Pitaveskii equation is numerically simulated to verify the stability of two-dimensional bright solitons. Finally, we add Gaussian noise to simulate a real experimental environment and find that two-dimensional bright solitons are also stable within the observable time range of the experiment. Our experimental scheme opens the door to the study of bright solitons in high-dimensional free space in non-conservative systems.

From Generalized Hamilton Principle to Generalized Schrodinger Equation

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.

Asymptotic stability of planar rarefaction wave to a multi-dimensional two-phase flow

<p style='text-indent:20px;'>We are concerned with the time-asymptotic stability of planar rarefaction wave to a non-conservative two-phase flow system described by two-dimentional compressible Euler and Navier-Stokes equations through drag force. In this paper, we show the planar rarefaction wave to a non-conservative compressible two-phase model is asymptotically stable under small initial perturbation in <inline-formula><tex-math id="M1">\begin{document}$ H^3 $\end{document}</tex-math></inline-formula>. The main difficulties overcome in this paper come from the non-viscosity of one fluid and the interaction between two fluids caused by drag force. The stability result is proved by the energy method.</p>

MODIFIED HOMOTOPY PERTURBATION METHOD BASED SOLUTION OF LINEARLY DAMPED DUFFING OSCILLATOR AND COMPARISON WITH SIMULATED SOLUTION*

The Dung oscillator provides a basis for studying nonlinear dynamics as its phase space trajectory is fairly complex and depends on the parameter of the system viz., initial amplitude, phase, frequency, linear damping coefficient and non-linearity parameter. In order to understand the complexity of the system, three variable effective expansions have been introduced in the usual homotopy perturbation framework to obtain the solution of damped Dung system which nds application in several areas in engineering sciences such as vibration of bars, plates and electronic circuits, etc. The necessity of the extended homotopy frame work has been further discussed for non-conservative system. Simulation results for different parameters of the systems, such as, linear damping coefficient (μ), amplitude (α) and nonlinearity parameter (ε) are compared with the corresponding results based on perturbative homotopy analysis up to third order by changing (i) the magnitude of linear damping coefficient (μ), (ii) the magnitude of the nonlinearity of the system (ε). Even though the simulated result matches satisfactorily with the perturbative solution over the entire evolutionary time scale, noticeable divergence and phase shift are observed only lately for increased value μ and ε, respectively.

A pendulum-based absorber-harvester with an embedded hybrid vibro-impact electromagnetic-dielectric generator

In this paper, a novel hybrid vibro-impact electromagnetic-dielectric generator (VI EDG) is proposed, which is further embedded into a pendulum structure to form a pendulum-based absorber-harvester (PAH) system. The PAH system can convert the vibration energy into electrical one with the help of the VI EDG, and has the potential to reduce the swing amplitude of the pendulum. The physical model of the PAH system is first introduced, and its governing equations involving the dynamical and electrical parts are derived based on the Euler-Lagrange’s equation of a non-conservative system. Next, the governing equations of the PAH system are validated experimentally by measuring the swing motion of the pendulum under a given harmonic excitation. On this basis, the system energy harvesting performance under different excitation parameters (amplitude and frequency) and dimensional parameters (the rod’s length and the distance between two dielectric elastomer membranes) is fully investigated and explained through numerical simulations. Finally, the vibration absorption performance of the VI EDG is discussed by comparing the peak-to-peak values of the swing motions between the proposed PAH system and the conventional pendulum without embedded VI EDG. Research results show that the proposed VI EDG can not only effectively harvest ambient vibration energy from the pendulum system, but also has the potential to absorb the vibration of the system.

Prestressed elasticity of amorphous solids

Prestress in amorphous solids bears the memory of their formation and plays a profound role in their mechanical properties. Here we develop a set of mathematical tools to investigate mechanical response of prestressed systems, using stress rather than strain as the fundamental variable. This theory allows microscopic prestress to vary for the same bond or contact configuration and is particularly convenient for nonconservative systems, such as granular packings and jammed suspensions, where there is no well-defined reference state, invalidating conventional elasticity. Using prestressed nonconservative triangular lattices and a computational model of amorphous solids, we show that drastically different mechanical responses can show up in amorphous materials at the same density, due to nonconservative interactions which evolve over time, or different preparation protocols. In both cases, the information is encoded in the prestress of the network and not visible at all from the configurations of the network in the case of nonconservative interactions.

Boundary control of vibration and dynamic tension fluctuation in a lifting system during the loading process

This article focuses on the boundary control for a lifting system during the loading process. A lifting system during the loading process is essentially a non-conservative system. The system dynamics are modelled based on the extended Hamilton’s principle. We consider the unmeasurable impact force caused by the uncertain materials falling velocity as a boundary disturbance and design a disturbance observer to eliminate its effect. The control system needs to guarantee the output constraints that the boundary vibration and dynamic tension fluctuation are both suppressed in a constrained scope of the desired value. Two appropriate boundary backstepping control schemes are proposed by actuators acting on different boundaries to realize the output constraints and to ensure the stability of the closed-loop system in theory. Numerical simulations are provided to illustrate the effectiveness of the proposed control laws in suppressing the cage vibration and rope dynamic tension fluctuation amplitude.

Unified theory of thermodynamics and stochastic thermodynamics for nonlinear Langevin systems driven by non-conservative forces

We construct a unified theory of thermodynamics and stochastic thermodynamics for classical nonequilibrium systems driven by non-conservative forces, using the recently developed covariant Ito-Langevin theory. The thermodynamic forces are split into a conservative part and a non-conservative part. Thermodynamic functions are defined using the reference conservative system. Work and heat are partitioned into excess parts and house-keeping parts, which are due to, respectively, conservative forces and non-conservative forces. Excess entropy production (EP) and house-keeping EP are analogously defined. The splitting of thermodynamic forces is subjected to an arbitrariness resembling a gauge symmetry, with each gauge defining a reference conservative Langevin system. In the special Gibbs gauge, the nonequilibrium steady state (NESS) is characterized by Gibbs canonical distribution, the excess heat agrees with that defined by Hatano and Sasa, and the excess EP agrees with that of Glansdorff and Prigogine, i.e., it is the time rate of the second-order variation of system entropy near the NESS. Adopting the Gibbs gauge, and focusing on the excess parts of thermodynamic quantities, a complete analogy between thermodynamics of non-conservative systems and that of conservative systems is established. One important consequence of this analogy is that both the free energy and excess EP are minimized at NESS. Our theory therefore constitutes a statistical foundation both for the steady-state thermodynamics theory due to Sasa and Tasaki and for the stability theory of NESS due to Glansdorff and Prigogine. These results are valid even if the system is far from equilibrium. By studying detailed fluctuation theorem, we find striking differences between systems with symmetric kinetic matrices and those with asymmetric kinetic matrices. For systems with asymmetric kinetic matrices, the total EP is the sum of house-keeping EP, excess EP, and pumped entropy. Entropy pumping is an exchange of entropy between the system and environment without necessarily involving dissipation. In the presence of entropy pumping, the system may behave as either a demon or an antidemon. Fluctuation theorems and work relations are derived both for total work and for excess work. For systems with symmetric kinetic matrices, there is no entropy pumping, yet in the Gibbs gauge, the excess work and house-keeping work each satisfies a separate fluctuation theorem. We illustrate our theory using many concrete examples.

Well-balanced path-conservative central-upwind schemes based on flux globalization

In this paper, we introduce a new approach for constructing robust well-balanced (WB) finite-volume methods for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. The WB property, namely, the ability of the scheme to exactly preserve physically relevant steady-state solutions is enforced using a flux globalization approach according to which a studied system is rewritten in an equivalent quasi-conservative form with global fluxes. To this end, one needs to incorporate nonconservative product terms into the global fluxes. The resulting system can then be solved using a Riemann-problem-solver-free central-upwind (CU) scheme. However, a straightforward integration of the nonconservative terms would result in a scheme capable of exactly preserving very simple smooth steady states only and failing to preserve discontinuous steady states naturally arising in the nonconservative models.In order to ameliorate the flux globalization based CU scheme, we evaluate the integrals of the nonconservative product terms using the technique introduced in [5], where a path-conservative central-upwind scheme (PCCU) was introduced. This results in a new flux globalization based WB PCCU scheme, which is much more accurate and robust than both the original PCCU scheme and the straightforward flux globalization based CU scheme. This is illustrated using two nonconservative systems: a system describing fluid flows in nozzles with variable cross-sections and the two-layer shallow water equations. We demonstrate superiority of the proposed flux globalization based WB PCCU scheme on a number of challenging examples.

New solitary wave and computational solitons for Kundu–Eckhaus equation

The goal of this research is to find novel optical solutions to the Kundu–Eckhaus equation, which possess crucial roles in the field of nonlinear optics. A collective variable (CV) strategy is adopted to solve governing equation including the Raman effect and quintic nonlinearity. This method is a suitable to deal with both conservative and non-conservative systems by exposing a set of equations of motion regardless of nonlinearities or dissipative components. The parameters employed in this approach are chirp, temporal position, phase, amplitude, frequency and width, namely, collective variables. The fourth order Runge–Kutta technique is a well-known numerical scheme that aims towards the solution of the resulting system of ordinary differential equations representing the variables involved in the pulse ansatz. This technique presents the evolution of pulse parameters with regard to propagation variables. The graphical profiles at suitable values of pulse parameters are also provided. The unified technique is also applied to find soliton solutions. The obtained solution is a periodic solitary wave, showed graphically. The results developed in this article are found to be new in the literature and the approach utilized, can be applied to solve a variety of nonlinear problems in the mathematical sciences.

Closed-Form Solutions to a Forced Damped Rotational Pendulum Oscillator

In this investigation, some analytical solutions to both conserved and non-conserved rotational pendulum systems are reported. The exact solution to the conserved oscillator (unforced, undamped rotational pendulum oscillator), is derived in the form of a Jacobi elliptical function. Moreover, an approximate solution for the conserved case is obtained in the form of a trigonometric function. A comparison between both exact and approximate solutions to the conserved oscillator is examined. Moreover, the analytical approximations to the non-conserved oscillators including the unforced, damped rotational pendulum oscillator and forced, damped rotational pendulum oscillator are obtained. Furthermore, all mentioned oscillators (conserved and non-conserved oscillators) are linearized, and their exact solutions are derived. In addition, all obtained approximations are compared with the four-order Runge–Kutta (RK4) numerical approximations and with the exact solutions to the linearized oscillators. The obtained results can help several authors for discussing and interpreting their results.

High order numerical methods for flows with hysteretic fluxes

A second order method and a limited variation WENO numerical scheme are proposed for the Scanning Hysteresis Model (SHM) which describes continuum flows with hysteretic looping fluxes with local memory. The SHM is a nonconservative hyperbolic system with discontinuous coefficients. The limited variation WENO method is fifth order accurate in spatial coordinate and third order accurate in time steps in smooth and monotone regions. The proposed high resolution methods produce the desired solutions more efficiently than the first order upwinding method. These methods are total variation diminishing for the flux, and monotonicity-preserving for the flux and the hysteresis state variable. The flux and the hysteresis state variable produced by these two methods satisfy maximum principle. These two methods also have invariant regions.

Enhancing performance and stability of gain-scheduling control system using evolutionary algorithms: A case study on transport aircraft

An enhanced gain-scheduling control strategy was presented to establish a smooth adaptation mechanism for the longitudinal motion of B747 between different flight envelopes. The focal objective is to accomplish a smooth and non-conservative gain-scheduling control system that has high performance and stability characteristics for different flight conditions and transitions between them. The proposed approach ensured a successful gain scheduling control system by allocating a specific error performance index that should be minimized for all flight conditions. Then all required performance indices were cohered according to the flight envelope in the form of a multi-objective optimization problem that could be solved using evolutionary algorithms. The performance and stability of the transition points between different flight conditions were preserved through the sets of feasible solutions obtained by an evolutionary algorithm which are known as a set of Pareto optimal solutions. The proposed gain-scheduling system was built based on 2DoF-PID (Two-degree-of-freedom Proportional Integral Derivative) controller. An auto-decision-maker was developed to adjust the parameters of the 2DoF-PID gain-scheduling control system with the current flight condition. The effectiveness of the proposed gain-scheduling control structure was studied for two different multiobjective evolutionary-based optimizers known as fast sorting and elite multi-objective genetic algorithm.II (NSGA.II), and sub-population genetic algorithm.II (SPGA.II). The proposed methodology was evaluated by simulation results according to the quality of the pareto-front and transient response characteristics of the proposed gain-scheduling controller for the chosen flight conditions in normal flight conditions and 50% loss of actuator effectiveness.

Analytical Scheme of Stability Analysis for 4-DoF Mechanical System Subjected to Friction-Induced Vibrations

Purpose:The stability problem for non-conservative multi-parameter dynamical system is usually associated with labor-intensive calculations, and numerical methods do not always allow one to obtain the desired information. The presence of circulatory forces often leads to the so-called ”destabilization effect” of the system under the influence of small dissipative forces. In this regard, it seems important to develop analytical approaches that make it possible to use a simplified scheme for checking the stability conditions.Methods:When obtaining and analyzing stability conditions, the algebra of polynomials and elements of mathematical analysis are applied. To obtain a simplified scheme for checking the stability conditions, an asymptotic method is used.Results and Conclusion:A mechanical system with four degrees of freedom which is under the action of dissipative, potential and non-conservative potential (circulatory) forces is considered. The stability problem of friction-induced vibrations is studying. In the case of weak damping an analytical approach is proposed that makes it possible to simplify the analysis of stability conditions, which, due to the presence of many uncertain parameters, are very cumbersome. With the help of numerical testing, the adequacy of the results obtained for the reduced conditions and full stability conditions was established. The results of the analysis make it possible to single out the ”advantageous” regions in the space of dimensionless parameters, which makes it possible to improve the design of the system to increase its reliability.

Modal Analysis of Non-Conservative Combined Dynamic Systems

Abstract The emergence of the use of mechanical metamaterials for vibration suppression and the creation of frequency gaps in structures require an understanding of the fundament underlying dynamics partial differential equations coupled to ordinary differential equations. Essentially periodic structures consist of a distributed parameter structure connected (embedded) to a series of spring-mass-dampers. Such systems in the past have been studied as combined dynamical systems. This work deals with the modal analysis of non-conservative combined dynamic systems formed by assembling distributed parameter structures and linear, viscously damped, lumped parameter oscillators. The mathematical model of the forced response of such dynamic systems is presented via differential operators. The related non-linear eigenproblem is formulated next and a proper solution is provided. Furthermore, the orthogonality of the eigenfunctions is studied and the completeness of the generated solution space is verified. Coupled modal coordinate differential equations result through modal analysis, thus revealing the non-proportional damping configuration, while the proportional damping conditions are also derived and discussed. The theory is applied to non-conservative Euler–Bernoulli beams subject to different types of boundary conditions and coupled to linear, viscously damped oscillators. A numerical example yields interesting conclusions about the non-proportionality and the applicability of the associated methods to solving the coupled differential equations.

A bedload transport in shallow water model applied for sediment transport in open channel flows

This paper presents the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modelled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a non-conservative hyperbolic system. To discretize it, first we consider a Non-Homogeneous Riemann Solver scheme as well as a variant based on the use of flux limiters. In order to develop second-order scheme, we use a MUSCL method incorporating slope limiters in the spatial approximation and a two-step Runge-Kutta method for time integration. The comparison between results based on the proposed scheme and analytical results shows good agreement..

Fifth-order A-WENO schemes based on the path-conservative central-upwind method

We develop fifth-order A-WENO finite-difference schemes based on the path-conservative central-upwind method for nonconservative one- and two-dimensional hyperbolic systems of nonlinear PDEs. The main challenges in development of accurate and robust numerical methods for the studied systems come from the presence of nonconservative products. Semi-discrete second-order finite-volume path-conservative central-upwind (PCCU) schemes recently proposed in Castro Díaz et al. (2019) [8] provide one with a reliable Riemann-problem-solver-free numerical method for nonconservative hyperbolic system. In this paper, we extend the PCCU schemes to the fifth-order of accuracy in the framework of A-WENO finite-difference schemes.We apply the developed schemes to the two-layer shallow water equations. We ensure that the developed schemes are well-balanced in the sense that they are capable of exactly preserving “lake-at-rest” steady states. We illustrate the performance of the new fifth-order schemes on a number of one- and two-dimensional examples, where one can clearly see that the proposed fifth-order schemes clearly outperform their second-order counterparts.

Steady states and coarsening in one-dimensional driven Allen-Cahn system.

We study the steady states and the coarsening dynamics in a one-dimensional driven nonconserved system modeled by the so-called driven Allen-Cahn equation, which is the standard Allen-Cahn equationwith an additional driving force. In particular, we derive equationsof motion for the phase boundaries in a phase ordering system obeying this equationusing a nearest-neighbor interaction approach. Using the equationsof motion we explore kink binary and ternary interactions and analyze how the average domain size scale with respect to time. Further, we employ numerical techniques to perform a bifurcation analysis of the one-period stationary solutions of the equation. We then investigate the linear stability of the two-period solutions and thereby identify and study various coarsening modes.