Lagrangian System Research Articles

Analysis of a Dry Friction Force Law for the Covariant Optimal Control of Mechanical Systems with Revolute Joints

This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh’s dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh’s potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb’s law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin’s principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control.

Enhanced high-frequency continualization scheme for inertial beam-lattice metamaterials

A multifield continualization technique is introduced that offers a thermodynamically consistent description of the constitutive and dispersive properties of beam-lattice inertial metamaterials with periodic microstructures. The balance equations governing the mechanics of the discrete Lagrangian system are appropriately handled using an innovative continualization scheme to derive an equivalent integral-type continuum model. Based on formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions incorporating shift operators and appropriate pseudo-differential downscaling laws, the proposed multifield enhanced continualization scheme allows the derivation of a gradient-type continuum model of given rank and equivalent to lattices. Two different resolution techniques are proposed. Firstly, the corresponding infinite-order average differential equations are tackled using a perturbative approach to describe the forced Bloch wave propagation in the metamaterial. Secondly, higher-order continuum models are employed through proper differential equation truncation to characterize the dispersive properties of the metamaterial in both high- and low-frequency regimes. Moreover, an energetically consistent generalized equivalent Micropolar continua, with non-local inertial terms, are here identified. The multifield continualization procedure is applied to two-dimensional periodic microstructures with tetrachiral, hexachiral, and hexa-tetrachiral topologies. Illustrative examples highlight the ability of the equivalent continuum model to accurately describe the effective constitutive properties of inertial metamaterials with periodic microstructures and to define a dynamic response consistent with the discrete Lagrangian model, validated and tested through virtual experimental verification under free and forced wave conditions.

Measurement report: Source attribution and estimation of black carbon levels in an urban hotspot of the central Po Valley – an integrated approach combining high-resolution dispersion modelling and micro-aethalometers

Abstract. Understanding black carbon (BC) levels and its sources in urban environments is of paramount importance due to the far-reaching health, climate, and air quality implications. While several recent studies have assessed BC concentrations at specific fixed urban locations, there is a notable lack of knowledge in the existing literature on spatially resolved data alongside source estimation methods. This study aims to fill this gap by conducting a comprehensive investigation of BC levels and sources in Modena (Po Valley, Italy), which serves as a representative example of a medium-sized urban area in Europe. Using a combination of multi-wavelength micro-aethalometer measurements and a hybrid Eulerian–Lagrangian modelling system, we studied two consecutive winter seasons (February–March 2020 and December 2020–January 2021). Leveraging the multi-wavelength absorption analyser (MWAA) model, we differentiate sources (fossil fuel combustion, FF, and biomass burning, BB) and components (BC vs. brown carbon, BrC) from micro-aethalometer measurements. The analysis reveals consistent, minimal diurnal variability in BrC absorption, in contrast to FF-related sources that exhibit distinctive diurnal peaks during rush hours, while BB sources show less diurnal variation. The city itself contributes significantly to BC concentrations (52 ± 16 %), with BB and FF playing a prominent role (35 ± 15 % and 9 ± 4 %, respectively). Long-distance transport also influences BC concentrations, especially in the case of BB and FF emissions, with 28 ± 1 % and 15 ± 2 %, respectively. When analysing the traffic-related concentrations, Euro 4 diesel passenger cars considerably contribute to the exhaust emissions. These results provide valuable insights for policy makers and urban planners to manage BC levels in medium-sized urban areas, taking into account local and long-distance sources.

Fundamental Laws and Quantum Dynamics

Background: Recently, we obtained a unitary theory of quantum mechanics and general relativity, where a quantum particle is a continuous distribution of matter in the two conjugate spaces of the coordinates and momentum, quantized by the equality of the mass parameter describing the relativistic dynamics of the matter, with the mass as an integral of the matter density. However, in this framework, we have not explicitly revealed the connection between our new theory and the fundamental laws of quantum mechanics. Methods: We analyzed in detail the three fundamental laws of quantum mechanics, explicitly describing experimental data: 1) Planck’s law of the blackbody electromagnetic radiation of a system of electrically charged harmonic oscillators, 2) Einstein’s law of the photon energy proportionality with the photon frequency, and 3) de Broglie’s law of the quantum particle as an oscillator in space. Results: We reobtained the two dynamical equations, in the conjugate spaces of the coordinates and momentum, as functions of the Lagrangian system, unlike the Schrödinger equation, depending on the Hamiltonian. Conclusion: According to the fundamental laws of quantum mechanics, a quantum particle is a continuous distribution of matter with an intrinsic mass, unlike the conventional quantum mechanics for the state occupation probabilities of punctual entities moving with the light velocity and getting an apparent mass only by collisions with some bosons pervading the whole universe. According to these laws, we obtained a quantum theory in agreement with common sense, classical logic, and general relativity.

Dynamic multifield continualization of multilayered lattice-like metamaterials

This work focuses on dynamic continualization of multifield multilayered metamaterials in order to obtain energetically-consistent models able to provide an accurate description of the dispersive behavior of the corresponding discrete system. Continuum models, characterized by constitutive and inertial non-localities, have been identified through a recently proposed enhanced continualization scheme. They are identified by governing equations both of the integro-differential and higher-order gradient-type, whose regularization kernel or pseudo-differential functions accounting for shift operators are formally expanded in Taylor series. The adopted regularization kernel exhibits polar singularities at the edge of the first Brillouin zone, thus assuring the convergence of the frequency spectrum to the one of the Lagrangian system in the entire wave vector domain. The validity of the proposed approach is assessed through the investigation of multilayered discrete lattices with an antitetrachiral topology, where local resonators act as rigid links among the layers. The convergence of dispersion curves of the continuum model to the ones of the Lagrangian model is proved in the whole first Brillouin zone as the adopted continualization order increases, both considering the propagation and the spatial attenuation of Bloch waves inside the metamaterial. A low frequency continualization is also provided, leading to the identification of a first-order medium.

Response of MDOF nonlinear system with fractional derivative damping and driven by fractional Gaussian noise

The integration of fractional calculus into stochastic dynamics has introduced two innovative concepts, i.e., fractional derivative damping (FDD) and fractional Gaussian noise (fGn). To date, most studies on fractional stochastic dynamics have exclusively focused on either FDD or fGn, rather than concurrently exploring both. It is the first attempt for this paper to investigate a multi-degree-of-freedom (MDOF) nonlinear system with FDD and driven by fGn at the same time. Employing the generalized harmonic balance technique, FDD is equivalently decomposed into a quasi-linear restoring force and a quasi-linear damping force. Consequently, the original random Lagrangian system with FDD is transformed into the stochastically excited and dissipated Hamiltonian system. Utilizing the characteristic of the relatively flat power spectral density of fGn, it is approximated as wideband noise. This enables the application of the stochastic averaging method for quasi-integrable Hamiltonian system under wideband noise. Two detailed examples are carried out to provide the technical procedures and numerical verifications.

Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints

This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided.

Unified Sufficient Conditions for Predefined-Time Stability of Non-Linear Systems and Its Standard Controller Design

This paper presents a unified Lyapunov-based predefined-time stability theorem that includes three sufficient conditions. The standard theoretical analysis method for achieving predefined-time stability of non-linear systems using this theorem is provided within the framework of Lyapunov theory. The developed Lyapunov-based theorem facilitates the establishment of equivalence between the existing Lyapunov theorems concerning predefined-time stability. Furthermore, when the presented sufficient conditions are relaxed, the predefined-time stability conclusion for non-linear systems degenerates into a finite-time one. Consequently, a standard non-singular sliding mode control framework based on the unified Lyapunov-based theorem is developed for a Lagrangian system to ensure its predefined-time stability. Exemplary numerical simulation results are subsequently given, in order to illustrate the convergence behavior of the system states and confirm that the controlled systems are predefined-time stable.

A new hybrid local radial basis function collocation method for 2.5D thermo-mechanical modelling of continuous casting of steel

This study presents a new strong-form meshless method to solve the thermo-mechanical problem of the solidification process in the continuous casting of steel. A two-dimensional slice that travels in the casting direction is modelled in the Lagrangian system. The newly developed mechanical model is one-way coupled to the thermal model, where the heat flux due to the mould, sprays, rolls and radiation are imposed to solve heat transfer in the strand. The resulting temperature and metallostatic pressure govern the Norton-Hoff visco-plastic model used for computing shrinkage of the solid shell and induced residual stresses. The results are used to estimate critical areas susceptible to hot-tearing formation. The mechanical model uses a generalised plane strain assumption that includes linear strains perpendicular to the slice and enables the computation of the straightening of the strand. The thermo-mechanical model is spatially discretised with a local radial basis function collocation method (LRBFCM). The mechanical part includes a new hybrid method that combines LRBFCM with finite differences for increased stability. The presented work shows how the developed model is used to assess the impact of casting velocity on the solid shell shrinkage and the probability of hot-tearing occurrence in the continuous casting of square billets.

Periodic Solutions of Generalized Lagrangian Systems with Small Perturbations

In this paper we study the generalized Lagrangian system with a small perturbation. We assume the main term in the system to have a maximum, but do not suppose any condition for perturbation term. Then we prove the existence of a periodic solution via Ekeland’s principle. Moreover, we prove a convergence theorem for periodic solutions of perturbed systems.

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.

Three-dimensional wireless measurement of float–sink object motions in a gas–solid fluidized bed

We developed a Lagrangian sensor system capable of wirelessly measuring the 3D motions of objects and applied it to examine the floating and sinking behaviors in a gas–solid fluidized bed. The system consisted of a sensor particle, comprising a 9-axis sensor and a wireless module integrated into a spherical outer shell, and three pairs of magnetic coils. By utilizing this system, we can non-invasively measure the 3D motions of objects, which cannot be observed from the outside. In this study, we examined the motions of objects with various densities in the gas–solid fluidized bed. We found the similarity and dissimilarity between the float–sink motions of objects in fluidized beds and those in liquids. The experimental results revealed that the vertical motion of an object in the fluidized bed depends on its density and can be categorized into two distinct states: floating near the surface or settling down at the bottom. However, within a specific range of object densities close to the apparent density of the fluidized bed, the object exhibited complicated motions in the bed. In such cases, slight density variations induced unpredictable changes between the floating and sinking states. The results also suggested that the settling velocity of the objects in the fluidized bed varies with their vertical positions in the bed.

Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability

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Stein–Weiss type inequality on the upper half space and its applications

In this paper, we establish some Stein–Weiss type inequalities with general kernels on the upper half space and study the extremal functions of the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of positive solutions of corresponding Euler–Lagrange system. As an application, we derive some Liouville type results for the Hartree type equations on the half space.

RFWBS Model: Multilevel Hybrid Mapping Solution Framework for Designing Neurorehabilitation Physiotherapy Devices

Healthcare services and rehabilitation equipment have entered a phase of rapid development driven by user requirements. However, the development of corresponding design models for rehabilitation equipment is lacking. A general framework and development process are urgently needed for neurorehabilitation physiotherapy equipment. To address problems such as inadequate knowledge representation in the design process and difficulties in modeling the functional structure of the product, we combined a decomposition topology model of neurorehabilitation physiotherapy equipment with the modular design method. We proposed a general model for the design of neurorehabilitation physiotherapy equipment comprising the following modules: requirements (R), function (F), principle workspace (W), behavior (B), and structure (S), i.e., the RFWBS model. Using the neurorehabilitation robotic glove as an example for design practice, in terms of kinematics, the mechanism is analyzed by establishing a Lagrangian coordinate system for resolution. The mechanism has three degrees of freedom and can achieve the natural flexion and extension angles of each finger joint. In terms of dynamics, during the entire finger extension, the angular acceleration is almost zero, and the average angular velocity is approximately 30~50°/s. This indicates that the mechanism is suitable for wearable use, validating the scientific and effective nature of the RFWBS expanded model.

Prescribed‐time adaptive bipartite time‐varying formation tracking for multiple Lagrangian systems with arbitrary disturbance

SummaryThis article focuses on the problem of bipartite time‐varying formation tracking for multiple Lagrangian systems with prescribed‐time adaptive sliding mode under the signed digraph. First, for each Lagrangian system, the local prescribed‐time disturbance estimator is proposed to estimate arbitrary disturbances in the prescribed time with the condition of bounded disturbances. Then the distributed prescribed‐time observers are designed to observe the position and velocity of the leader, and the consensus problem is transformed into bipartite formation tracking via introducing above observers. Based on designing two types of observers and disturbance estimators, a new prescribed‐time adaptive sliding mode controller is designed to ensure the tracking errors of the sliding mode surface converge to origin in a prescribed time. Ultimately, the feasibility of the provided results is further demonstrated through numerical simulation.

Lagrangian steady-state discrete Boltzmann model for non-equilibrium flows at micro–nanoscale

In micro- and nanoscale channels, where the characteristic scale approaches or falls below the mean free path between gas molecules, flow characteristics exhibit pronounced discontinuity. In such scenarios, continuum-based models are inadequate, and adopting kinetic models based on statistical mechanics becomes imperative for accurately describing the gas transport phenomenon. This paper presents a novel steady-state discrete Boltzmann model (DBM) that is specifically tailored for non-equilibrium flows at the micro–nanoscale. Unlike the conventional DBM, the new model focuses on the evolution equation of the non-equilibrium component of the molecular velocity distribution function in the Lagrangian coordinate system, which follows the fluid micro-element. Through numerous numerical simulations, we demonstrate the effectiveness of our model in capturing gas flow characteristics across a wide spectrum of rarefaction parameters, ranging from slip flow to free molecular flow. Utilizing this new model, we initially examine the Onsager reciprocal relationship between heat flux generated by pressure gradients and mass flux resulting from temperature gradients in micro/nanochannels. Subsequently, we calculate higher-order non-equilibrium quantities up to the 10th order and make a comparison of their characteristics. Finally, we present and discuss the features of the non-equilibrium component of the molecular velocity distribution function.

High-order Lagrange multiplier method for the coupled Klein-Gordon-Schrödinger system

In this work, a novel class of high-order energy-preserving algorithms are developed for simulating the coupled Klein-Gordon-Schrödinger equations. We introduce a Lagrange multiplier approach and derive a new system which warrants exactly the original total energy, instead of the modified quadratic energy in the previous energy quadratization approach. The Lagrange system is then reformulated equivalently into an optimization problem subject to PDE constraints. We discretize the optimization system by applying the Gauss collocation method together with the prediction-correction technique in time and the sine pseudo-spectral method in space, which leads to a family of fully discrete high-order energy-preserving schemes. In the numerical experiments, we compare this method with other previous methods and verify the high accuracy as well as its ability to maintain the original energy at each time layer. The feasibility and validity of new method are also witnessed for the KGS model in the nonrelativistic limit regime. Additionally, enlightened by this seminal idea, another type of high-order numerical algorithms conserving two invariants of mass and energy are also presented and relevant experimental results are listed to verify the power of the suggested method.

A dynamical systems approach for most probable escape paths over periodic boundaries

Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler–Lagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The Onsager–Machlup (OM) functional, which is treated as a perturbation of the Friedlin–Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the paper is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.

Pulse width modulation control for stochastic Lagrange systems

The pulse width modulation (PWM) control for Lagrange systems driven by servo motors in stochastic environment is researched in this paper. Principles of PWM for stochastic nonlinear systems are proved, a continuous control bounded in probability of stochastic Lagrange system is designed, and the stochastic closed-loop nonlinear programming is modeled in the same form as that in the deterministic case. Sure boundedness of the control is required as a hard constraint, and the stochastic quadratic programming problem is presented for which an analytic solution is constructed and the expression of the optimal duty ratio is obtained. Finally, the main procedures of design and implementation of the stochastic PWM are summarized, and the effectiveness of the proposed scheme is demonstrated by a simulation of articulated manipulator.