Nonconservative Systems Research Articles

On Integrability of Certain Classes of Variable Dissipation Systems

We study the non-conservative systems for which the methods for studying, for example, Hamiltonian systems is not applicable in general. Therefore, for such systems, it is necessary, in some sense, to “directly” integrate the main equation of dynamics. Herewith, we offer more universal interpretation of both obtained cases and new ones of complete integrability in transcendental functions in dynamics in a non-conservative force field.

Wave propagation for the isentropic compressible Navier–Stokes/Allen–Cahn system

We study the Cauchy problem for the three‐dimensional isentropic compressible Navier–Stokes/Allen–Cahn system, which models the phase transitions in two‐component patterns interacting with a compressible fluid. Under the assumption that the initial perturbation is small and decays spatially, we establish the global existence and the pointwise behavior of strong solutions to this nonconserved system. To deal with the source terms involving the phase variable, we employ the Green's function and space‐time weighted estimates. The analysis shows that the phase variable mainly contains the diffusion wave with exponential decaying amplitude over time, and consequently the density and momentum of the compressible fluid adhere to a generalized Huygens principle.

Avalanche dynamics in nonconservative water droplet

The self-organized criticality (SOC) exhibits excellent performance in describing the dynamic features of multi-body systems in nature. It is usually considered that conservative law is a necessary condition for the existence of SOC. However, mass or energy dissipation typically exists in the process from self-organization to a critical state in natural systems such as earthquakes and air pollution, namely, non-conservation. At present, few physical experiments have systematically explored the SOC behavior in a non-conservative system, compared to numerical simulations. This study is the first to design a non-conservative sandpile experiment using water droplets. Based on this experiment, the dynamic evolution process of avalanches in a non-conservative system has been analyzed. The results show that the avalanche size of water droplets follows a power-law distribution, as the water mass decays with time. This phenomenon obeys the rules of scale-free distribution, which is consistent with SOC behavior. Moreover, the waiting time of water droplets avalanche follows a stretched exponential distribution. These results suggest that SOC behavior still exists in non-conservative multi-body systems, which is affected by the decay coefficient. We then explore the mechanism by which the decay coefficient affects the transition of the system to a critical state in the non-conservative sandpile model (NBTW). We find that only when the decay coefficient is small do the inequality measures, Gini coefficient g and Kolkata index k, exhibit nearly universal values, similar to the mechanism of the system transitioning to a critical state in the traditional BTW model. When the decay coefficient is large, the NBTW model shows scaling-limited effects and the scaling exponent of avalanche size increases with the decay coefficient. Furthermore, we use the NBTW model to study how the decay coefficient influences the scaling exponent of avalanche size. We find that the NBTW model can effectively explain the variability of scaling exponents observed in different SOC systems and establish a quantitative relationship between the mass decay coefficient and the avalanche size scaling exponent. This study provides new insight into SOC behavior in non-conservative systems in nature.

A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations

A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations

Non‐Hermitian Skin Effect in Non‐Hermitian Optical Systems

AbstractThe development of non‐Hermitian (NH) physics in nonconservative systems has led to various interesting wave phenomena, including the emergence of the non‐Hermitian skin effect (NHSE). NHSE is a topological effect caused by the nontrivial winding number of the energy spectrum, which results in many bulk eigenmodes localized on low‐dimensional boundaries under open boundary conditions. Although this phenomenon is striking, the flexible and dynamic tuning of NHSE has not been reviewed in optics systems. As an ideal NH platform, an optical system can introduce artificially constructed gains and losses and flexibly design optical structures to control the transmission and regulation of light. This exploration based on NH optical systems deepens the understanding of NHSE and facilitates technological breakthroughs in high‐energy directional localized optics applications. In this review, the focus is on the latest progress and applications of NHSE in NH optical systems, including the basic theory and local property modulation methods. Finally, possible future research directions for NHSE based on NH optical systems are discussed.

High‐order in‐cell discontinuous reconstruction path‐conservative methods for nonconservative hyperbolic systems–DR.MOOD method

AbstractIn this work, we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in‐cell discontinuous reconstruction operator are the key points to develop a new family of high‐order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two‐Layer Shallow Water system.

Modal analysis of non-conservative systems with friction-induced strong nonlinear damping by using response-controlled testing

In recent years, the response-controlled stepped-sine testing (RCT) method has made important contributions to the evolution of the single nonlinear mode (SNM) method from a postulate to a well-established theory with the accurate modal identification of various nonlinear mechanical systems ranging from simple benchmark beams to real missiles. Since the RCT method resides on the SNM theory which is based on Rosenberg’s periodic motion definition of nonlinear modes, it has been regarded as limited to weakly damped mechanical systems that mainly exhibit strong conservative nonlinearities. An important contribution of this current paper is extending the RCT method to non-conservative mechanical systems with complex nonlinear modes. The extension is first numerically validated on a strongly friction-damped cantilever beam and then experimentally validated on the actuation mechanism of a real control fin that exhibits strong softening-stiffening nonlinearity (up to 50% shift of natural frequency) due to friction/backlash and very high and nonlinear damping (up to 30% modal damping ratio) due to friction. Another important contribution of this paper is to demonstrate the applicability of the superposition principle in nonlinear dissipative systems with closely spaced modes at an acceptable accuracy if the nonlinear modes are similar in shape to their corresponding linear modes and the modal coupling terms in the equation of motion can be ignored. The assumption of similar nonlinear modes is reasonable for most practical applications that exhibit dissipative nonlinearities due to mechanical joints. Although the assumption of ignorable coupling terms does not reside on a solid theoretical framework, numerical studies on a friction-damped lumped system with closely spaced modes give promising results, pointing to the significant potential of the modal superposition in nonlinear systems.

Reliability assessment of ballasted railway bridges considering soil-structure interaction using ensemble of surrogate models

ABSTRACT The increasing speeds of modern trains lead to excessive vibrations on the bridges, which have the potential to destabilize the ballast particles. The occurrence of this phenomenon not only increases the track maintenance cost, but can also disrupt the load path from the rail level to the bridge deck, posing a risk to the train running safety. The design regulations indirectly control this limit-state by restricting the vertical acceleration of the bridge deck. The assessments pertaining to this purpose often neglect the soil-structure interaction (SSI) effects considering that as a conservative assumption. Such effects can positively contribute by increasing the system damping, but they can also increase the bridge flexibility making it more susceptible to vibrations due to reduction on critical speed. Therefore, this study investigates the influence of considering/disregarding SSI effects on the ballast destabilization phenomenon using a probabilistic methodology. The results are classified based on the maximum permissible train speeds and the bridge span length. Due to the high computational costs of the reliability analyses, the associated limit-state is approximated by an ensemble of classification-based surrogate models using the stack-generalization concept. Subsequently, the upper/lower bounds of the failure probability in the presence of SSI effects are compared with those obtained for simply-supported bridges. It is pointed out that neglecting SSI effects for shorter span bridges may lead to an underestimation of system safety. For longer span bridges, however, this may lead to an overestimation of safety, which means that a non-conservative system can be designed.

The Herglotz variational principle for dissipative field theories

In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg–de Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.

Data-driven discovery of interpretable Lagrangian of stochastically excited dynamical systems

Exploring the intersection of deterministic and stochastic dynamics, this paper delves into Lagrangian discovery for conservative and non-conservative systems under stochastic excitation. Traditional Lagrangian frameworks, adept at capturing deterministic behavior, are extended to incorporate stochastic excitation. The study critically evaluates recent computational methodologies for learning Lagrangians from observed data, highlighting the limitations in interpretability and the exclusion of stochastic excitation. To address these gaps, an automated data-driven framework is proposed for the simultaneous discovery of Lagrange densities and diffusion coefficients of stochastically excited dynamical systems by leveraging sparse regression. This novel framework offers several advantages over existing approaches. Firstly, it provides an interpretable description of the underlying Lagrange density, allowing for a deeper understanding of system dynamics under stochastic excitations. Secondly, it identifies the interpretable form of the diffusion coefficient of generalized stochastic force, addressing the limitations of existing deterministic approaches. Additionally, the framework demonstrates robustness and versatility through numerical case studies encompassing both stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs), with results showing almost exact approximations to true system behavior and minimal relative error in derived equations of motion.

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given.

Symmetries and perturbations of a singular nonconservative system on time scales

Symmetries and perturbations of a singular nonconservative system on time scales

Nonlinear vibrations of fractional viscoelastic PET membranes subjected to tangential follower force

Nonlinear forced vibrations of simply supported viscoelastic Polyethylene Terephthalate (PET) membranes subjected to a harmonic excitation and tangential follower force are investigated in this paper. The consideration of vibration systems under follower force requires special attention to the modeling of non-conservative force and its influence on nonlinear analysis. Herein, a fractional Kelvin-Voigt Hamilton-based framework accounting for the viscoelastic PET membrane is developed. Based on the Hamilton principle for the non-conservative system, the mathematical modeling of viscoelastic PET membrane taking into account non-conservative force and geometric nonlinearity is formulated, and the derived equations are discretized via the Galerkin schemes. A technique of multiple scales is employed to numerically acquire the frequency–response curves with different system configurations. Numerical investigations are conducted to analyze the effects of tangential follower force, viscous coefficient, as well as fractional order on the dynamic stability of the considered structure. Further ramifications of the present study point to the paramount importance of tangential follower force and an accurate viscoelasticity model on the dynamic behaviors of the viscoelastic structures.

Stability analysis of multi-trigger nano tube sandwich structures considering thermal and magnetic effects

This research investigates the stability analysis of multi-trigger nanotube-based sandwich structures that carry fluid subjected to external stimulus. Acrylic-based material and magnetorheological elastomers (MRE) are considered multi-trigger cores in which the external stimulus alters the mechanical properties of the core and, subsequently, the sandwich structures. The system is subjected to an external magnetic field while the ambient temperature varies. The results demonstrate that incorporating a multi-trigger viscoelastic core can accelerate the divergence instability of the non-conservative system while delaying its flutter. Additionally, the impact of the external magnetic field, as a control parameter, for the frequency and stability region of the MRE is more significant than that of the Acrylic-based material. Furthermore, employing Acrylic-based material with a higher shear modulus in a constant magnetic field leads to a more stable system. Moreover, the effect of the external magnetic force on stability is affected smoothly by the ambient temperature. Finally, the influence of temperature on stability is nearly the same for both MRE and Acrylic-based material, with a more pronounced effect observed in systems with larger length-to-diameter ratios.

The concentration and cavitation in the Riemann solution for non-homogeneous logrithmic equation of state with magnetic field

In the present study, the phenomena of concentration and cavitation in the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field is analyzed. Firstly, we introduced new state variable for the velocity to modify the non-conservative system into conservative system and solved the Riemann problem for modified system constructively. Further, the Riemann solutions for the transport equations is investigated as pressure and magnetic field vanish. It is proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two shock waves converges to the delta shock wave solution of the transport equations as pressure and magnetic field vanish. It is also proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two rarefaction waves converges to the contact discontinuity solution of the transport equations as pressure and magnetic field vanish.

MUSES: A nonlinear magnetohydrodynamics discontinuous Galerkin code for fusion plasmas

The article describes initial steps toward the development of our nonlinear magnetohydrodynamics code, MUSES. Unstructured grids and general coordinates are employed so that MUSES code is expected to have a capability to represent realistic geometries of fusion devices. Discontinuous Galerkin method is chosen as a code framework because of its high spatial accuracy and numerical stability on unstructured grids. The governing equation of the ideal magnetohydrodynamics in this work is Powell 8-wave model which can mitigate the divergence-free issue. Roe scheme is used as a numerical flux since the Powell 8-wave model is a nonconservative system. A cross code comparison is carried out with a linear magnetohydrodynamics code, MINERVA. Owing to discontinuous Galerkin method, a linear theory of an ideal internal kink mode is reproduced quantitatively although the upwind method is employed for numerical stability.

K voprosu o chislennom reshenii nekonservativnykh giperbolicheskikh sistem uravneniy

Issues related to the lack of convergence in the application of formally path-conservative difference schemes for solving nonconservative hyperbolic systems of equations are numerically investigated. This problem is central in constructing well-posed difference schemes for solving this class of problems. The basic concepts of the theory of nonconservative hyperbolic systems of equations and the corresponding problems of constructing difference schemes for their solution are outlined. A variant of the HLL method is proposed that allows using an arbitrary explicitly specified path. For a model system of Burgers equations, the shock adiabates and paths corresponding to the viscous regularization of a system of a given form are explicitly calculated. The reasons for the lack of convergence of numerical solutions of exact ones in the case of incorrect application of the corresponding algorithms are analyzed. It is shown that, at least in the particular case considered, a variant of the HLL method that is formally conservative along the way gives the correct solution of the problem.

Modeling of temperature effect on electromechanical properties of dielectric elastomer minimum energy structures

Due to the characteristics of large deformation and fast response to excitation voltage, dielectric elastomer minimum energy structures (DEMES) can actively adjust its shape to adapt to the complex environments, which is a research hotspot in the field of soft robotics. The electromechanical properties of DEMES are sensitive to temperature variation. However, the analysis of the effect of temperature on the viscoelastic dynamic behavior of DEMES is incomplete in existing studies. In this paper, an analytical framework for investigating the electromechanical performances of DEMES actuator is presented. By introducing the effects of temperature on the dielectric constant and elastic modulus of DEMES actuator, the governing equations of motion for the underlying non-conservative system affected by temperature are established by combining the DE viscoelastic rheological model and the Euler-Lagrange equation. The proposed model is used for building insights into the effect of temperature on the stiffness, equilibrium state, actuation range and the evolution law of the dynamic response under the DC and AC excitations of DEMES. The results show that a significant enhancement in the equilibrium angle attained by the structure with the improving of the temperature is observed, indicating a favorable impact of ambient temperature. Phase paths and Poincaré maps are presented for assessing the temperature effect on the stability and periodicity of the nonlinear oscillations. The frequency responses of the actuator for the AC loads indicate that a suitable temperature can significantly improve the dynamic performance and stability of DEMES actuator. The proposed analytical model and results presented in this study can provide guidance for the design and development of DEMES based soft robot subjected to harsh environments.

INFLUENCE OF CONTROL CURRENT WAVEFORM ON ROTOR NUTATION OF THREE-DEGREES-OF-FREEDOM ELECTRIC MACHINE

The design of the electric machine with three degrees of freedom of the rotor with an external magnetic core, containing permanent magnets, and an internal magnetic core is considered. The both magnetic cores are rotating. The stator windings (control winding and two-phase rotation winding) are located in the air gap between the magnetic cores. The windings are powered by alternating current of corresponding phase. The positive property of rotor gyrostabilization in such structure is shown. At the same time, the negative fact consists in the nutation under the conditions of forced change in the orientation of rotation axis. There are no magnetic losses in the structure due to synchronous rotation of both external and internal magnetic cores. The immobility of the center of mass is assumed. There are no losses that give a non-conservative mechanical system. The mathematical model of the mechanic system and magnetic field is developed to compute the electromagnetic torques acting on the rotor. The expressions for electromagnetic torque components are obtained to use them in the Comsol Multiphysics interface. The effect of the sinusoidal and pulse currents in the control winding on the nutation swing is studied. The optimality of the control winding supplied by undisturbed sinusoidal current or the pulse current having square-wave length of about one-third of the period is revealed. References 15, figures 6, table 1.

Fractional Complex Euler–Lagrange Equation: Nonconservative Systems

Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system.