Euler-Lagrange Research Articles

Dynamic Modelling and Feed-forward Control of an Overhead Crane

The article presents: modelling, simulation, and controller design of a laboratory overhead crane system. A nonlinear model was developed using Lagrange equation and linearization process is performed for the purpose of analysis and controller design. Accuracy of the linear model is assessed by comparing simulation results of nonlinear and linear dynamic equations. A feed-forward control based on a low-pass filter was designed based on the crane dynamics. The controller performance is discussed in terms of trolley position response and payload sway reduction. The proposed control approach is able to suppress the payload sway, but at the expense of a slight delay in the trolley motion

On the coupling of Hamilton's principle and thermodynamic extremal principles

Extremal principles can generally be divided into two rather distinct classes. There are, on the one hand side, formulations based on the Lagrangian or Hamiltonian mechanics, respectively, dealing with time dependent problems, but essentially resting on conservation of energy and thus being not applicable to dissipative systems in a consistent way. On the other hand, there are formulations based essentially on maximizing the dissipation, working efficiently for the description of dissipative systems, but being not suitable for including inertia effects. Many attempts can be found in the literature to overcome this split into incompatible principles. However, essentially all of them possess an unnatural appearance. In this work, we suggest a solution to this dilemma resting on an additional assumption based on the thermodynamic driving forces involved. Applications to a simple dissipative structure and a material with varying mass demonstrate the capability of the proposed approach.

Drag force and heavy quark potential in a rotating background

We explored the gravity dual of a rotating quark-gluon plasma by transforming the boundary coordinates of the large black hole limit of Schwarchild-AdS5 metric. The Euler-Lagrange equation of the Nambu-Goto action and its solution become more complex than those without rotation. For small angular velocity, we obtained an analytical form of the drag force acting on a quark moving in the direction of the rotation axis and found it stronger than that without rotation. We also calculated the heavy quark potential under the same approximation. For the quarkonium symmetric with respect to the rotation axis, the depth of the potential is reduced by the rotation. For the quarkonium oriented in parallel to the rotation axis, the binding force is weakened and the force range becomes longer. We also compared our holographic formulation with others in the literature.

Emergent symmetries of relativistic fluid dynamics from local ergodicity

We show that volume-preserving diffeomorphisms and the chemical shift symmetry defining relativistic Lagrangian ideal fluid dynamics can be derived as an emerging symmetry when ergodicity is assumed to apply locally in a way that is invariant under smooth spacetime foliations. This can be used as a way to derive the ideal hydrodynamic limit in a strongly coupled but strongly fluctuating medium. We comment on the connection with thermalization in small systems, the eigenstate thermalization hypothesis and deviations from the ideal limit. Published by the American Physical Society 2024

Synchronization characteristics of two induction motors on a floating raft system

The synchronization characteristics of two exciters driven by induction motors independently on a typical double-layer vibration isolation system, that is, the floating raft system are studied in the paper to enhance its vibration attenuation effect. Firstly, a new dynamical model for a floating raft system driven by two exciters is established by the Lagrange equation under considering the coupling effect among floating raft system and exciters. Then, the vibration responses in all directions are obtained by the Laplace transform. Secondly, the synchronization and stability solution are obtained by using the average small parameter method. Thirdly, the results of the theoretical analysis are verified by simulations. Finally, the parameter scope of a floating raft system to enhance its vibration attenuation effect is given. Moreover, compared with the traditional model, the proposed model considering the coupling effect among floating raft system and exciters is more feasible. Additionally, the effects of the mass ratio of the eccentric block, the symmetrical installation distance, parameter differences of two motors, and other dynamic parameters such as system mass and stiffness on their synchronization and the system vibration are discussed. This paper can provide a theoretical reference for the design, modeling, and application of such double-layer vibration isolation systems including the floating raft systems to enhance its vibration attenuation effect.

Synchronous control of tri-motor vibration system considering an adaptive back-stepping control strategy

In oil and gas drilling engineering, the anti-phase synchronization of two co-rotating rotors in a tri-motor vibration system is easy to implement owing to its inherent coupling characteristics, which can lead to the exciting force generated by the vibrating system being very small. Thus, an adaptive back-stepping control strategy is proposed to apply in a tri-motor vibrating system and improve the processing capacity of drilling fluid shaker screens. The algorithm developed in this paper is that a complex higher-order nonlinear system is simplified into lower-order subsystems by the adaptive back-stepping control theory, and then the adaptive law for varying load torque is derived by combining with the adaptive theory to regulate the velocity of motors in a closed loop. Firstly, a dynamical model to describe the tri-motor vibration system is established, and the dynamical equations of the vibrating system are derived based on the Lagrange equation. Then, the speed error controller and the phase difference error controller are designed by combining the adaptive back-stepping control algorithm with the master–slave control (MSC) structure. According to Barbalat’s lemma, the stability of the control system is analyzed to verify the feasibility of the control method. Finally, the dynamical equations of the vibrating system are used to establish an electromechanical coupling control model of a tri-motor driven planar system through the Runge–Kutta method, and it verified the reliability of the control strategy, the robustness, and the stability of the control system, respectively, and intuitively revealed the synchronization characteristics of tri-motor vibration control system. The results demonstrate the phase difference among the three motors can be accurately stabilized at zero by a phase synchronization control strategy based on the adaptive back-stepping control method, and the method is simple and fast-speed and the control system has better robustness.

Numerical simulation of the dynamic problem of a generalized system with distributed parameters

Introduction. Questions of mathematical modeling of the dynamic problem in the form of a generalized system with one degree of freedom are discussed. Such systems include high-rise tower-type structures. Seismic stability of unique tower-type objects represents a relevant research problem.Aim. To determine the stress-strain state of a studied object under the action of external factors in the form of an earthquake accelerogram.Materials and methods. The methods of structural mechanics, dynamics of structures, and numerical simulation were used. The Lagrange equation was used as a basis for obtaining the motion equation of a generalized system with distributed parameters. The research methodology also included mathematical modeling of the considered systems, their numerical analysis, comparison of the obtained results with literature data.Results. A mathematical model was developed to investigate the stress-strain state of engineering structures under various external, including seismic, effects. The differential equation of the generalized system is solved directly using the method of successive approximations and the Duhamel integral at each time step. The developed algorithm was used to compile a software application in the FORTRAN language followed by obtaining the kinematic and static data of the investigated object. Using the example of a tower-type structure, free vibrations from the action of an instantaneous impulse were investigated. The results from a given earthquake accelerogram are presented.Conclusions. The results obtained on the free vibrations of the object under study agree well with those obtained by numerical simulation. The results obtained by numerical differentiation are effectively identical with those obtained by numerical integration, under the action of various effects. The validity of the results is confirmed by comparing the results obtained by the two methods. The developed software applications can be used for monitoring the state of unique tower-type objects.

Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase

Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase

Physics informed contour selection for rapid image segmentation

Effective training of deep image segmentation models is challenging due to the need for abundant, high-quality annotations. To facilitate image annotation, we introduce Physics Informed Contour Selection (PICS)—an interpretable, physics-informed algorithm for rapid image segmentation without relying on labeled data. PICS draws inspiration from physics-informed neural networks (PINNs) and an active contour model called snake. It is fast and computationally lightweight because it employs cubic splines instead of a deep neural network as a basis function. Its training parameters are physically interpretable because they directly represent control knots of the segmentation curve. Traditional snakes involve minimization of the edge-based loss functionals by deriving the Euler–Lagrange equation followed by its numerical solution. However, PICS directly minimizes the loss functional, bypassing the Euler Lagrange equations. It is the first snake variant to minimize a region-based loss function instead of traditional edge-based loss functions. PICS uniquely models the three-dimensional (3D) segmentation process with an unsteady partial differential equation (PDE), which allows accelerated segmentation via transfer learning. To demonstrate its effectiveness, we apply PICS for 3D segmentation of the left ventricle on a publicly available cardiac dataset. We also demonstrate PICS’s capacity to encode the prior shape information as a loss term by proposing a new convexity-preserving loss term for left ventricle. Overall, PICS presents several novelties in network architecture, transfer learning, and physics-inspired losses for image segmentation, thereby showing promising outcomes and potential for further refinement.

Nonlinear dynamic buckling analysis of the FG-GPLRC cylindrical and sinusoid panels with porous core under time-dependent axial compression

The nonlinear dynamic buckling responses of functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and sinusoid panels with porous core are presented in this paper. The governing formulations are established by applying the nonlinear higher-order shear deformation theory (HSDT). an approximation technique is used to determine the stress function for complexly curved panels. Euler-Lagrange equations can be used to achieve the nonlinear motion equation. Numerical investigations are considered using the Runge-Kutta method for dynamic responses, and using the Budiansky-Roth criterion for critical dynamic buckling loads. Some discussions on the dynamic buckling responses of panels with porous core can be achieved from the numerical examples.

Vibration Mitigation Using 3D-PSTMD for the Offshore Wind Turbine Considering Self-Excitation Behaviors

The self-excitation behaviors can be triggered by the blade rotation of the wind turbine. To mitigate the excessive vibration of offshore wind turbine (OWT) tower under this self-excitation addition with wind–wave loadings, a three-dimensional prestressed TMD (3D-PSTMD) is extended in this study. First, based on the Lagrange equation, the analytical formulations of 1P (rotor rotational speed) and 3P (blade passing frequency) loads from self-excitations are deduced, and more factors under real operational conditions of the OWT are further analyzed using theoretical derivation and numerical simulation. Then, structural aero-hydrodynamic loadings are modelled, and self-resonance phenomenon of the uncontrolled OWT and resonance dissipation capability of 3D-PSTMD are discussed at startup–operation–shutdown state. Finally, the energy harvesting competence of 3D-PSTMD is quantitatively calculated via the classical Wilson-[Formula: see text] algorithm under self-excitations with wind–wave loadings. Results indicate that this dashpot is remarkably able to mitigate the structural resonance responses over 61.3% and 40.5% respectively when under the 1P and 3P loadings, and it can also effectively reduce the bi-directionally horizontal vibrations of OWT under aero-hydrodynamic loadings.

Lagrangian approach to nonlinear waves in non-dispersive and dispersive rotating shallow water magnetohydrodynamics

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

Theoretical modeling and dynamics analysis of a rotating piezoelectric laminated beam with different setting angles

Dynamics modeling and analysis of a rotating piezoelectric laminated beam (RPLB) considering the effect of the setting angle is presented in this paper. The rotating piezoelectric energy harvester can be simplified as a RPLB which is fixed at a rotating hub. It is easy to know that a steady transversal deformation would be produced due to the transversal component of centrifugal force when the beam is not along the radial direction (the setting angle is not equal to 0° or 180°). However, the variation of steady transversal deformation and its effect were rarely considered by existing theoretical models about the rotating harvesters. To this end, a novel nonlinear dynamics model of the RPLB is developed by the combination of floating frame reference (FFR) and absolute nodal coordinate formulation. In the model, the position coordinates and gradients on the FFR are used as the nodal coordinates to describe deformation field. The enthalpy, elastic and kinetic energies of the RPLB are derived with consideration of the piezoelectric constitutive equation and the rotation of the FFR. Thus, the dynamics equations of the RPLB can be developed by using the Lagrange's equations of second kind. The results obtained from the model show agreement with those obtained from the previous literatures, which verifies the correctness of the model. Several examples are presented to analyze the effects of rotating speed, setting angle and tip mass on steady transversal deformation, output voltage and modal characteristics.

Impact-Rubbing Dynamics of Rotor with Hollow Shaft and Offset Discs Based on MDOF Timoshenko Beam Theory

The impact-rubbing dynamic characteristics of the power turbine rotor with the hollow shaft and offset discs for aircraft engine are investigated, and the impact-rubbing analytical method for the complex rotor based on MDOP Timoshenko beam theory is proposed in this paper. Compared with the traditional approach, the novel method can obtain more data to satisfy the need of engineering. The Lagrange equation is adopted to derive the equations of motion for the rotor system, and the Newmark-β method is applied to solve the equations. The diagrams such as the bifurcation, axis trajectory, spectrum, and Poincaré map are obtained to research on the effect of the rotating speed, gap, and eccentricity on the vibration response. The finite element analysis was carried out to validate the correctness of the theoretical modeling method. The research results indicate that the power turbine rotor with the hollow shaft on operation shows the various nonlinear dynamic behaviors including the multiperiod, quasi-period, jumping phenomenon, and chaotic motions; there exists an optimal gap between the rotor and the stator from the perspective of the efficiency and the dynamics; the optimal gap should make system avoid the resulting chaos or the quasi-period motion for the stability and safety of the machinery.

The influence of imbalances on the dynamic characteristics of the laboratory centrifuge HERMLE Z306

Laboratory centrifuges are used in various industries. During operation, vibrations occur that lead to resonant frequencies, which in turn impair functionality. This paper presents an overview of the computational model of the HERMLE Z306 laboratory centrifuge used in medical laboratories to separate mixtures of different fractions to determine the dynamic characteristics. The zones of stable operation of the centrifuge and the influence of the rotation speed on the natural frequencies are analytically determined. Experimental results are presented with the influence of imbalances on the dynamic characteristics of the HERMLE Z306 centrifuge. As a result of the modeling, the amplitude-frequency characteristics are determined and a Campbell diagram is constructed. Objective: Modeling of dynamic processes in laboratory centrifuges by studying the influence of imbalances on the quality of mixture separation. Determination of zones of stable operation of the centrifuge.Purpose: Construction of amplitude-frequency characteristics of the centrifuge, determination of zones of stable operation of the laboratory centrifuge during separation and Campbell diagram showing the dependence of natural frequencies on the rotation speed of the HERMLE Z306 centrifuge. This diagram makes it possible to determine the resonance zones. Methods of implementation: Based on the use of the Lagrange equation of the second kind, a model is obtained that makes it possible to determine the zones of stable operation of the centrifuge. Using experimental equipment, determine the frequency response of the centrifuge and analyze the Campbell diagram to determine the resonance zones.Results: The zones of stable operation of a laboratory centrifuge were analytically determined. The amplitude-frequency characteristics of the HERMLE Z306 centrifuge were constructed, the trajectories of the free end of the shaft were built taking into account the corresponding imbalances, and the resonance zones were experimentally determined. Conclusions: Experimental studies have shown that the free end of a laboratory centrifuge shaft moves along a surface whose shape and, accordingly, the path of movement depend on both the angles of rotation and translational movement that arise as a result of deformations of elastic supports. The analytical and experimental studies made it possible to identify unstable modes and thereby determine the areas of the centrifuge's operating modes. The separation process will be stable if the roots of the equation have a negative real part, and moreover, the motion will be asymptotically stable in the presence of resistance forces. An experimental technique for determining the dynamic parameters of the centrifuge has been developed. On its basis, the effect of the rotation speed on the natural frequencies was determined.

A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: justification as a shallow water approximation

We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order $O(\delta _1^{4N+2}+\delta _2^{4N+2})$ in the sense of consistency, where $\delta _1$ and $\delta _2$ are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and $N$ is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.

Dynamics of the Deformable Fluid Interface Interacting with an Approaching Solid under the Electrostatic Field.

A theoretical model was developed to describe the dynamics of a deformable fluid interface interacting with an approaching solid without contact by both the attractive electrostatic and van der Waals (i.e., vdW) interaction, analogous to the situation in the experiments by electric force microscopy (i.e., EFM) or electric-surface force apparatus (i.e., E-SFA) involved in the soft fluid interface. On the basis of this model, a numerical study of the deformation of the fluid interface, the force-vs-separation behavior, and the critical limiting conditions of contact has systematically been carried out. Our results show that the surface pressure induced by the electrostatic interaction plays a more prominent role in the deformation of the fluid interface than the vdW interaction does, and there exists a principal length scale associated with the relative strength of the electrostatic field to the surface tension, affecting the fluid interface shape under the electrostatic field. It was also shown that both the force-distance curves and the corresponding curves of fluid interface deformation peak versus distance for various electrostatic fields satisfy the universal scaling power law. Moreover, an analytical solution to the Euler-Lagrange differential equation governing the deformation of the fluid interface under the external electric field is obtained, and two extended formulas for explicitly describing the principal length scales that respectively characterize the lateral and longitudinal deformations of the fluid interface were determined.

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

On fluid-saturated poro-hyperelastic rotating cylinder: A centrifugal filtration

One of the applications of porous materials is centrifugal filtration. This study develops the Lagrangian equations that govern a steady-state, fluid-saturated, poro-hyperelastic, axisymmetric rotating cylinder. The formulation assumes incompressible solid and Newtonian fluid constituents to address the filtration process. The differential scheme takes into account the impact of porosity on the reduction of the drained shear and bulk moduli of the solid skeleton. Additionally, the normalized Kozeny-Carman formula considers the effect of porosity changes due to deformation on permeability. The Standard Galerkin Finite Element Method solves the derived equations using FlexPDE. ANSYS is unable to solve the current problem due to the lack of centrifugal and Coriolis accelerations in Darcy's law and its inability to account for changes in initial permeability and porosity resulting from the deformation of the solid skeleton. Therefore, the validation of the governing equations and the applied solution is investigated in three limited cases.It is shown that a rise in the angular velocity causes the expansion in the Von-Mises stress of solid constituent, radial displacement, negative pore pressure, and radial filtration velocity because of the increase in the centrifugal force. Also, the permeability and radial displacement grow with the initial porosity rise. The radial filtration velocity rises with the increased permeability resulting from the deformation. Moreover, drained shear and bulk moduli of the solid skeleton fall with increased porosity, resulting in a rise in the radial displacement. Besides, the radial displacement and filtration velocity expand with the increase of solid and fluid densities.

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler–Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler–Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.