Friction Term Research Articles

Multidimensional and Nonlinear Fractional Langevin Equations: Advancing the Theory in of Anomalous Diffusion

This paper presents a comprehensive theoretical analysis of fractional Langevin equations (FLEs) and their applications in modeling anomalous diffusion processes. We extend the classical FLE framework in several significant directions to address the complexities observed in various physical, biological, and social systems. First, we derive a multidimensional extension of the FLE, providing a robust tool for studying anomalous diffusion in higher-dimensional spaces. Second, we incorporate non-Gaussian Lévy α-stable noise into the FLE, enabling the description of systems characterized by heavy-tailed distributions and extreme events. Third, we develop a nonlinear variant of the FLE to model more intricate restoring forces and potential landscapes. Finally, we analyze a generalized FLE with different orders of fractional derivatives for the inertial and friction terms, offering a flexible framework capable of capturing a wider range of anomalous diffusion phenomena. For each extension, we prove the existence and uniqueness of solutions, derive the corresponding probability distributions, and analyze the scaling behavior of the mean squared displacement in the long-time limit. Our results demonstrate how these generalized FLEs can describe various regimes of anomalous diffusion, from subdiffusive to superdiffusive behavior. This work provides a unified theoretical foundation for understanding and modeling complex diffusive processes across diverse disciplines, paving the way for more accurate descriptions of systems exhibiting memory effects, non-Gaussian statistics, and multiscale dynamics.

Parametric resonance of gravitational waves in general scalar-tensor theories

Gravitational waves offer a potent mean to test the underlying theory of gravity. In general theories of gravity, such as scalar-tensor theories, one expects modifications in the friction term and the sound speed in the gravitational wave equation. In that case, rapid oscillations in such coefficients, e.g. due to an oscillating scalar field, may lead to narrow parametric resonances in the gravitational wave strain. We perform a general analysis of such possibility within DHOST theories. We use disformal transformations to find the theory space with larger resonances, within an effective field theory approach. We then apply our formalism to a non-minimally coupled ultra-light dark matter scalar field, assuming the presence of a primordial gravitational wave background, e.g., from inflation. We find that the resonant peaks in the spectral density may be detectable by forthcoming detectors such as LISA, Taiji, Einstein Telescope and Cosmic Explorer.

Trapezoidal breakwater on reducing resonant wave amplitude on a rectangular basin

In this study, we delve into the effectiveness of trapezoidal breakwaters in mitigating resonance phenomena. The challenge at hand is to identify the optimal configuration capable of halting resonance. Employing Shallow Water Equations with a friction term to encapsulate the breakwater’s roughness, we analytically solve the model to determine the natural wave period. This period serves as the instigator of an unstoppable harmonic oscillation. Numerically, we use the Finite Volume Method on a Staggered Grid to simulate the model for several cases to pinpoint the infimum value of the natural wave period required to impede resonance phenomena. This research holds significance for those involved in coastal protection design, particularly in the context of trapezoidal breakwaters. The findings contribute to the effective reduction, rather than amplification, of wave height, thereby enhancing the reliability of such coastal protection structures.

On the Interpretation of Cosmic Acceleration

In relativity, the Newtonian concepts of velocity and acceleration are observer-dependent quantities that vary with the chosen frame of reference. It is well established that in the comoving frame, cosmic expansion is currently accelerating; however, in the rest frame, this expansion is actually decelerating. In this paper, we explore the implications of this distinction. The traditional measure of cosmic acceleration, denoted by q, is derived from the comoving frame and describes the acceleration of the scale factor a for a 3D space-like homogeneous sphere. We introduce a new parameter qE representing the acceleration experienced between observers within the light cone. By comparing qE to the traditional q using observational data from Type Ia supernovae (SN) and the radial clustering of galaxies and quasars (BAO)—including the latest results from DESI2024—our analysis demonstrates that qE aligns more closely with these data. The core argument of the paper is that Λ—regardless of its origin—creates an event horizon that divides the manifold into two causally disconnected regions analogous to conditions inside a black hole’s interior, thereby allowing for a rest-frame perspective qE in which cosmic expansion appears to be decelerating and the horizon acts like a friction term. Such a horizon suggests that the universe cannot maintain homogeneity outside. The observed cosmological constant Λ can then be interpreted not as a driver of new dark energy or a modification of gravity but as a boundary term exerting an attractive force, akin to a rubber band, resisting further expansion and preventing event horizon crossings. This interpretation calls for a reconsideration of current cosmological models and the assumptions underlying them.

Limiting current in a collisional crossed-field gap

Crossed-field devices are often used in pulsed power and high-power microwave applications. Previous studies derived closed-form solutions for the limiting current of a vacuum crossed-field system, corresponding to the maximum permissible current for laminar flow, below and above the Hull cutoff BH for magnetic insulation. We extend these studies by introducing collision frequency into the electron force law as a friction term to derive the limiting current in a collisional crossed-field gap. The resulting solution recovers the vacuum crossed-field case in the limit of no collisions and the collisional space-charge limited current with general initial velocity for magnetic field B→0. In the limit of infinite collisions, we obtain a crossed-field equivalent to the Mott–Gurney law for the maximum current permissible in a collisional, nonmagnetic diode. When the collision frequency ν is less than the electron cyclotron frequency Ω, increasing initial velocity makes the critical current nonmonotonic with increasing ν with the critical current higher at B=BH for ν=Ω. As for a misaligned crossed-field gap where a component of the magnetic field was introduced parallel to the electric field across the gap, magnetic insulation is eliminated and the discontinuity at B=BH for limiting current observed in a vacuum crossed-field gap vanishes. As B→∞, the limiting current approaches a constant that depends on the initial velocity and the collision frequency.

A vertically non-uniform temperature approach for the friction term computation in depth-averaged viscoplastic lava flows

Recently, depth-averaged shallow flow models have been adapted to modelling liquefied lava flows, generally characterized by a marked temperature-dependent non-Newtonian rheology. Modelling these complex flows requires to include the effects of the depth-averaged temperature gradients in the governing equations. The most significant term to correctly predict the lava mobility is the flow resistance term, which is widely estimated using the linear viscoplastic Bingham model. This non-Newtonian model allows to relate the bed shear stress to the depth-averaged lava flow features by means of a cubic equation with analytical solution when assuming a uniform temperature distribution along the vertical profile. Nevertheless, the lava temperature is non-uniform along the vertical due to the heat transfer at the bottom and the free surface, and hence the classical cubic Bingham model is not valid anymore. In this work, a depth-averaged shallow flow model is adapted for realistic lava flows considering influence of the non-uniform vertical temperature profile in the non-Newtonian resistance. This requires to modify the rheological viscoplastic models for ensuring the coupling between flow dynamics and temperature evolution. Three non-uniform temperature vertical distributions are considered: linear, piece-wise and diffusion profiles. Synthetic tests are used to show the influence of the temperature vertical profile on the numerical results. Furthermore, laboratory experimental data are used to validate this novel viscoplastic resistance formulation and to show that the calibration of its parameters is possible.

Development of enhanced force models to analyze the nonlinear hysteresis response of miniaturized pneumatic artificial muscles

Miniaturized pneumatic artificial muscles (MPAMs), designed to replicate natural muscle actuation, offer unique attributes such as a high power-to-weight ratio, flexibility, easy integration, and compactness, making them favourable for many applications. The present paper aims at the development of an accurate semi-analytical force model considering the effect of the bladder material and friction terms to predict the nonlinear force-deformation response of MPAMs during contraction and expansion cycles. Existing force models for MPAMs exhibit limitations to accurately capturing the force-deformation behaviours due to several simplification factors. This study enhances these models by integrating correction terms to accurately address the nonlinearity and frictional effects exhibited by MPAMs. An analysis of the hysteresis loops resulting from the cyclic loading and unloading of MPAMs under specific pressures is undertaken to compare different methodologies in order to determine the most accurate correction terms. To investigate the nonlinear behaviour of MPAMs, the stress-strain relationship of the bladder material and results from force-deformation experimental tests on the entire actuator are considered and for the effect of friction term, theoretical and empirical approaches are investigated. Results suggest that the theoretical force model based on analytical and empirical friction forces, respectively, slightly overestimates and underestimates the force experienced by MPAMs during contraction while slightly underestimate and overestimates during expansion, respectively. A comparative analysis between MPAMs featuring Ecoflex-50 silicone and Ecoflex30 + PDMS mixture as bladder materials has also been conducted to further investigate the effect of bladder materials on their force and contraction outputs under inlet pressures ranging from 0 kPa to 300 kPa. It is shown that the MPAM feauting Ecoflex-50 bladder, exhibits lower dead-band pressure and an overall reduced blocked force in comparison to MPAM with bladder made of Ecoflex30 + PDMS while exhibiting a substantially enhanced free contraction capacity.

A Novel Friction Compensation Method for Machine Tool Drive Systems in Insufficient Lubrication.

Friction is the dominant factor restricting tracking accuracy and machining surface quality in mechanical systems such as machine tool feed-drive. Hence, friction modeling and compensation is an important method in accurate tracking control of CNC machine tools used for welding, 3D printing, and milling, etc. Many static and dynamic friction models have been proposed to compensate for frictional effects to reduce the tracking error in the desired trajectory and to improve the surface quality. However, most of them focus on the friction characteristics of the pre-sliding zone and low-speed sliding regions. These models do not fully describe friction in the case of insufficient lubrication or high acceleration and deceleration in machine tool systems. This paper presents a new nonlinear friction model that includes the typical Coulomb-Viscous friction, a nonlinear periodic harmonic friction term for describing the lead screw property in insufficient lubrication, and a functional component of acceleration for describing the friction lag caused by the acceleration and deceleration of the system. Experiments were conducted to compare the friction compensation performance between the proposed and the conventional friction models. Experimental results indicate that the root mean square and maximum absolute tracking error can be significantly reduced after applying the proposed friction model.

A Scale‐Dependent Analysis of the Barotropic Vorticity Budget in a Global Ocean Simulation

AbstractThe climatological mean barotropic vorticity budget is analyzed to investigate the relative importance of surface wind stress, topography, planetary vorticity advection, and nonlinear advection in dynamical balances in a global ocean simulation. In addition to a pronounced regional variability in vorticity balances, the relative magnitudes of vorticity budget terms strongly depend on the length‐scale of interest. To carry out a length‐scale dependent vorticity analysis in different ocean basins, vorticity budget terms are spatially coarse‐grained. At length‐scales greater than 1,000 km, the dynamics closely follow the Topographic‐Sverdrup balance in which bottom pressure torque, surface wind stress curl and planetary vorticity advection terms are in balance. In contrast, when including all length‐scales resolved by the model, bottom pressure torque and nonlinear advection terms dominate the vorticity budget (Topographic‐Nonlinear balance), which suggests a prominent role of oceanic eddies, which are of km in size, and the associated bottom pressure anomalies in local vorticity balances at length‐scales smaller than 1,000 km. Overall, there is a transition from the Topographic‐Nonlinear regime at scales smaller than 1,000 km to the Topographic‐Sverdrup regime at length‐scales greater than 1,000 km. These dynamical balances hold across all ocean basins; however, interpretations of the dominant vorticity balances depend on the level of spatial filtering or the effective model resolution. On the other hand, the contribution of bottom and lateral friction terms in the barotropic vorticity budget remains small and is significant only near sea‐land boundaries, where bottom stress and horizontal viscous friction generally peak.

Penetration of a spinning sphere impacting a granular medium.

We investigate experimentally the influence of rotation on the penetration depth of a spherical projectile impacting a granular medium. We show that a rotational motion significantly increases the penetration depth achieved. Moreover, we model our experimental results by modifying the frictional term of the equationdescribing the penetration dynamics of an object in a granular medium. In particular, we find that the frictional drag decreases linearly with the velocity ratio between rotational (spin motion) and translational (falling motion) velocities. The good agreement between our model and our experimental measurements offers perspectives for estimating the depth that spinning projectiles reach after impacting onto a granular ground, such as happens with seeds dropped from aircraft or with landing probes.

Modeling and simulation of pollution transport in the Mediterranean Sea using enriched finite element method

This paper presents a novel numerical method for simulating the transport and dispersion of pollutants in the Mediterranean sea. The governing mathematical equations consist of a barotropic ocean model with friction terms, bathymetric forces, Coriolis and wind stresses coupled to an advection–diffusion equation with anisotropic dispersion tensor and source terms. The proposed numerical solver uses a multilevel adaptive semi-Lagrangian finite element method that combines various techniques, including the modified method of characteristics, finite element discretization, coupled projection scheme based on a rotational pressure correction algorithm, and an adaptive L2-projection. The approach employs the gradient of the concentration as an error indicator for enrichment adaptations and increasing the number of quadrature points where needed without refining the mesh. The method is shown to provide accurate and efficient simulations for pollution transport in the Mediterranean sea. The proposed approach distinguishes itself from the well-established adaptive finite element methods for incompressible viscous flows by retaining the same structure and dimension of linear systems during the adaptation process.

A well-defined Godunov-type scheme in a Navier-Stokes model with a friction term

A well-defined Godunov-type scheme in a Navier-Stokes model with a friction term

A model-independent precision test of general relativity using bright standard sirens from ongoing and upcoming detectors

Abstract Gravitational waves (GWs) provide a new avenue to test Einstein’s General Relativity (GR) using the ongoing and upcoming GW detectors by measuring the redshift evolution of the effective Planck mass proposed by several modified theories of gravity. We propose a model-independent, data-driven approach to measure any deviation from GR in the GW propagation effect by combining multi-messenger observations of GW sources accompanied by EM counterparts, commonly known as bright sirens (Binary Neutron Star (BNS) and Neutron Star Black Hole systems (NSBH)). We show that by combining the GW luminosity distance measurements from bright sirens with the Baryon Acoustic Oscillation (BAO) measurements derived from galaxy clustering, and the sound horizon measurements from the Cosmic Microwave Background (CMB), we can make a data-driven reconstruction of deviation of the variation of the effective Planck mass (jointly with the Hubble constant) as a function of cosmic redshift. Using this technique, we achieve a precise measurement of GR with redshift (z) with a precision of approximately 7.9 % for BNSs at redshift z=0.075 and 10 % for NSBHs at redshift z=0.225 with 5 years of observation from LIGO-Virgo-KAGRA network of detectors. Employing Cosmic Explorer and Einstein Telescope for just 1 year yields the best precision of about 1.62 % for BNSs and 2 % for NSBHs at redshift z=0.5 on the evolution of the frictional term, and a similar precision up to z=1. This measurement can discover potential deviation from any kind of model that impacts GW propagation with ongoing and upcoming observations.

Alleviating both H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} and σ8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _8$$\\end{document} tensions in Tsallis cosmology

We present how Tsallis cosmology can alleviate both H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} and σ8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _8$$\\end{document} tensions simultaneously. Such a modified cosmological scenario is obtained by the application of the gravity-thermodynamics conjecture, but using the non-additive Tsallis entropy, instead of the standard Bekenstein–Hawking one. Hence, one obtains modified Friedmann equations, with extra terms that depend on the new Tsallis exponent δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} that quantifies the departure from standard entropy. We show that for particular δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} choices we can obtain a phantom effective dark energy, which is known to be one of the sufficient mechanisms that can alleviate H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} tension. Additionally, for the same parameter choice we obtain an increased friction term and an effective Newton’s constant smaller than the usual one, and thus the σ8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _8$$\\end{document} tension is also solved. These features act as a significant advantage of Tsallis modified cosmology.

A Numerical Study of Quantum Entropy and Information in the Wigner-Fokker-Planck Equation for Open Quantum Systems.

Kinetic theory provides modeling of open quantum systems subject to Markovian noise via the Wigner-Fokker-Planck equation, which is an alternate of the Lindblad master equation setting, having the advantage of great physical intuition as it is the quantum equivalent of the classical phase space description. We perform a numerical inspection of the Wehrl entropy for the benchmark problem of a harmonic potential, since the existence of a steady state and its analytical formula have been proven theoretically in this case. When there is friction in the noise terms, no theoretical results on the monotonicity of absolute entropy are available. We provide numerical results of the time evolution of the entropy in the case with friction using a stochastic (Euler-Maruyama-based Monte Carlo) numerical solver. For all the chosen initial conditions studied (all of them Gaussian states), up to the inherent numerical error of the method, one cannot disregard the possibility of monotonic behavior even in the case under study, where the noise includes friction terms.

A novel efficient and robust treatment of the friction source term in 2D shallow water inundation models

The technological advancements of the last few decades have fostered the use of two-dimensional (2D) Shallow water Equations (SWE) flood simulations not only in academic research but also in practical real-world applications and territorial planning. The evaluation of flow resistance due to friction is crucial as it plays a relevant role in a variety of flow conditions. However, problems arise in dam-break and overland flow applications involving very small water depth because the time scale connected to the friction term may be much smaller than the hydrodynamic time scale. To cope with this issue, known as source term stiffness, an implicit treatment of the friction term is frequently adopted in Finite Volume (FV) numerical schemes, but there are cases (null roughness coefficient or null flow velocity) where a division by zero may occur during calculations, leading to a crash of the algorithm. Herein, we propose a reformulation of the implicit friction term that is general, more stable and computationally efficient (with a speed-up of approximately 3 times for the routine running the friction computation) than currently available approaches. The novel implicit method allows to manage special conditions with null roughness or null discharge, without resorting to ad-hoc thresholds, and can be easily implemented in existing schemes with pointwise friction treatment. The novel friction approach is implemented in a purposely simple FV numerical scheme (hydrostatic reconstruction to account for the bed slope terms, first-order accuracy in time and space) - to better focus on the friction term treatment - and it is validated against analytical, synthetic, laboratory and real-world case studies, showing promising capabilities.

On radially symmetric oscillations of a collisional cold plasma

We study the influence of the friction term on the radially symmetric solutions of the repulsive Euler–Poisson equations with a non‐zero background, corresponding to cold plasma oscillations in multiple spatial dimensions. It is shown that for any arbitrarily small non‐negative constant friction coefficient, there exists a neighborhood of the zero equilibrium in the norm such that the solution of the Cauchy problem with initial data belonging to this neighborhood remains globally smooth in time. Moreover, this solution converges to zero as . This result contrasts with the situation of zero friction, where any small deviation from the zero equilibrium generally leads to a blow‐up. Our method allows us to estimate the lifetime of smooth solutions. Further, we prove that for any initial data, one can find such a coefficient of friction that the respective solution to the Cauchy problem keeps smoothness for all and converges to zero. We also present the results of numerical experiments for physically reasonable situations, which allows us to estimate the value of the friction coefficient, which makes it possible to suppress the formation of singularities of solutions.

Effect of Hygrothermal Conditioning on the Machining Behavior of Biocomposites

Abstract This work aims to study the cutting behavior of biocomposites under different controlled hygrothermal conditions. This investigation choice is motivated by the fact that natural plant fibers such as flax are characterized by their hydrophilicity which makes them able to absorb water from a humid environment. This absorption ability is intensified by increasing the conditioning temperature. The moisture diffusion process affects considerably the mechanical properties of the resulting composite, which causes many issues during the machining operations. In this paper, moisture diffusion, chip form, cutting and thrust forces, and scanning electron microscope (SEM) observations are considered to explore the cutting behavior of flax fiber-reinforced polylactic acid (PLA) depending on the hygrothermal conditioning time. Results reveal that moisture content in the biocomposite is significantly influenced by the conditioning temperature and the fiber orientation. Moisture content and fiber orientation affect both the curling behavior of the removed chip as well as the tool/chip interaction in terms of friction. The machinability of flax fiber-reinforced PLA biocomposites depending on hygrothermal conditioning time is then investigated using SEM analysis in addition to analytical modeling. An analysis of variance is used finally to quantify the observed results.

Empirical mathematical model for describing anisotropic dry friction forces

The paper presents an experimental and simulative study on anisotropic dry friction. Friction force components tangential and normal to sliding velocity are measured for the corrugated cardboard-felt frictional pair. Six different curves with two-fold symmetry are analysed in terms of modelling anisotropic friction. Next, the parameters of the analysed curves are identified based on the experimental results. Friction hodograph is best described by the Gielis’ curve and the superellipse, whereas the ellipse seems to be the worst choice. The sliding potential is described best by the oval. It is shown that modelling of the sliding potential can be crucial in capturing the dynamics of the system and the superelliptical sliding rule can lead to qualitatively different results.

Scaling between volume and runout of rock avalanches explained by a modified Voellmy rheology

Abstract. Rock avalanches reach considerably greater runout lengths than predicted by Coulomb friction. While it has been known for a long time that runout length increases with volume, explaining the increase qualitatively is still a challenge. In this study, the widely used Voellmy rheology is reinterpreted and modified. Instead of adding a Coulomb friction term and a velocity-dependent term, the modified rheology assigns the two terms to different regimes of velocity. While assuming a transition between Coulomb friction and flow at a given velocity is the simplest approach, a reinterpretation of an existing model for the kinetic energy of random particle motion predicts a dependence of the crossover velocity on the thickness of the rock avalanche. Analytical solutions for a lumped mass on a simple 1D topography reveal the existence of a slope-dominated and a height-dominated regime within the regime of flow. In the slope-dominated regime, the kinetic energy at the foot of the slope depends mainly on the slope angle, while the absolute height relative to the valley floor has little effect, and vice versa. Both regimes can be distinguished by the ratio of a length scale derived from the rheology and the length scale of the topography. Long runout occurs in the height-dominated regime. In combination with empirical relations between volume, thickness, and height, the approach based on the random kinetic energy model reproduces the scaling of runout length with volume observed in nature very well.