Lagrangian Formulation Research Articles

Cosmological analogies for geophysical flows, Lagrangians, and new analogue gravity systems

Formal analogies between the ordinary differential equations describing geophysical flows and Friedmann cosmology are developed. As a result, one obtains Lagrangian and Hamiltonian formulations of these equations, while laboratory experiments aimed at testing geophysical flows are shown to constitute analogue gravity systems for cosmology.

Quasi-Coordinates-Based Closed-Form Dynamic Modeling and Analysis for a 2R1T PKM with a Rigid–Flexible Structure

This work derives a closed-form dynamic model for a two rotational and one translational degrees-of-freedom (2R1T) parallel kinematic mechanism (PKM) with a hybrid rigid–flexible structure for force-control applications. Based on the three-prismatic-prismatic-spherical (3PPS) kinematic configuration of the 2R1T PKM and its zero-torsion motion characteristics, a symbolic formulation approach is proposed to establish closed-form kinematic models for both forward and inverse kinematics analysis. As the moving platform pose of the 2R1T 3PPS PKM can be readily determined by the three active prismatic joint variables and the three passive prismatic joint variables, these six joint variables are selected as the quasi-coordinates so as to systematically develop the closed-form dynamic model with a Lagrangian formulation, in which the stiffness and deformation of the three flexure-based passive prismatic joints are uniformly taken into consideration. Through eliminating the three passive prismatic joint variables based on the principle of virtual work and the relationships between the active and passive prismatic joint variables, a closed-form dynamic model for the 2R1T 3PPS PKM with a rigid–flexible structure is finally obtained. The correctness of the closed-form dynamic model was validated with the commercial dynamic simulation software. Utilizing the closed-form dynamic model, the effects of different flexure stiffness in driving directions on the required active joint force were investigated, which indicated that little flexure stiffness in driving directions is desired.

Разработка математической модели робототехнического комплекса для неразрушающего контроля стальных тросов

In order to improve the safety of lifting equipment, the task was set to create a new robotic complex for automated research of the technical condition and bearing capacity of steel cables in places that are difficult to access and dangerous for personnel. A new mathematical model of a robotic complex has been developed that describes the controlled movement of a magnetic flaw detector for performing technological procedures for non-destructive testing of steel cables. A mathematical model of the dynamics of the system is obtained using the Lagrange formalism. An algorithm for evaluating the control moments necessary for the implementation of programmed motion has been developed. Numerical simulation of the motion corresponding to the mode of approaching the magnetic flaw detector heads to the steel rope has been carried out. The graphs of the traction-speed characteristics and load diagrams of the drives providing the programmed movement of the links of the pantograph mechanism are presented. The results presented in this paper can be used to calculate the parameters of the power section of the electro-hydraulic drives included in the robotic device. The accuracy of performing flaw detection tasks is studied, taking into account the dynamics of the executive and measuring systems of the complex.

Nonlinear analysis of planar, H-shaped and U-shaped thin reinforced concrete shear walls

This paper presents and discusses the results of nonlinear finite element analysis (NL-FEA) of thin reinforced concrete shear walls. For this, an in-home developed shell structures NL-FEA program called CANECA was used. The materials modeling use the basis of the Modified Compression Field Theory and the geometric nonlinearity is evaluated using a Total Lagrangian Formulation. The influence of geometric nonlinearity and cyclic loading condition on an H-shaped shear wall behavior is discussed. From the obtained results, a higher influence of the cyclic loading condition is observed. Other two shear walls (simple planar and U-shaped) are also analyzed. Their geometry and reinforcement are modified in order to observe the influence of these parameters on the structural behavior, including the case of slender shear walls. The performed analysis on the two shear walls showed good agreement with the numerical response obtained in the NL-FEA program VECTOR4. The obtained numerical ultimate loads were much higher than the original structure design load and the slenderness ratio was insufficient to characterize the ultimate structural behavior. The results from this study can contribute for a better understanding on the structural behavior of thin reinforced concrete shear walls, and serve as guidance for future studies and normative developments.

Системный подход в оценке динамических состояний технических объектов: элементы концепции динамических инвариантов

The scientific and methodological base is being developed within the framework of systems theory and system analysis in application to the tasks of assessment, control and formation of dynamic states of technical objects, the design schemes of which are mechanical oscillatory systems with concentrated parameters. In the context of the fundamental problems of the theory of system studies of forced movements under deterministic disturbances there are considered the problems, whose solutions are aimed at developing a methodological approach to assessing the dynamic states of technical objects operating under coherent vibration loads. The purpose of the study is to develop a method for evaluating, controlling and forming dynamic states of the objects under harmonic force loads. It is assumed that mechanical oscillatory systems formed by a solid body performing small steady-state oscillations relative to the position of the static equilibrium can be used as calculation schemes of the technical objects. To construct mathematical models, Lagrange formalism, Laplace integral transformations taking into account zero conditions and structural methods are used, which make it possible to consider mechanical oscillatory systems as dynamically equivalent structural schemes of automatic control systems. The dynamic state of an object is assessed by using a transfer function, which in the physical sense represents the interpretation of a dynamic compliance. The amplitude-frequency characteristics used to assess the dynamic state of the object present a parametric family, for which the first parameter is the coefficient of connectivity of external force disturbances, and the second parameter is the coordinate of the point for which the dynamic accuracy is defined. An infinite family of amplitude-frequency characteristics is regularized due to the introduction of dynamic invariants, which can be considered as generalized dynamic states of mechanical oscillatory systems. To determine the dynamic invariants of a family of amplitude-frequency characteristics, taking into account two parameters, a map of dynamic invariants is constructed, which divides the plane of parameters into disjoint sets using boundaries. A theorem is proved to determine the explicit analytic expressions of these boundaries.

Constructing time integration with controllable errors for constrained mechanical systems

This paper investigates the numerical features of equivalent Lagrangian formulations for general mechanical systems and gives new insight into the error reduction of the corresponding numerical integrations. To this end, the modified Lagrange’s equations are derived, which give a unified form of equivalent Lagrangian formulations, featuring different generalized constraint forces. By transforming the modified Lagrange’s equations into their discrete versions, the errors of generalized constraint forces are estimated in strong and weak forms. An error analysis is employed to clarify the error generation of different formulations. This helps to identify whether the generalized constraint force contributes considerable errors compared to other terms in the system. Finally, a mechanism of error reduction is explained for the discrete system, providing a way to reduce total errors by adjusting the error generated from the generalized constraint forces. Inspired by this mechanism, a methodology for constructing synthetic integrations is developed. The synthetic integration scheme consists of the main scheme and the counter scheme. In particular, the counter scheme, discretized from the generalized constraint forces, plays an error conditioner that offsets the total error of the integration. Numerical examples demonstrate the significantly higher numerical accuracy and some other possible computational advantages of the proposed method.

Fuzzy Logic Control for Balancing a Two-Armed Inverted Pendulum

The unicycle robot is an activated model with only one wheel, which ensures its safety. Researchers were particularly interested in the unicycle robot because of its great robustness, which allows it to travel around without colliding with the ground. The inverted pendulum having two arms is modelled using a mathematical representation based on the Lagrangian formulation in this work, which embodies our concept of the unicycle robot. The fuzzy logic control algorithm will then be used to produce a high level of solidity for this system.

Preliminary Design Tools for Hydrodynamic Aspects of Submerged Impermeable Breakwaters

The boundary element method (BEM) with Lagrangian formulation is a conceptually simple and efficient method for the simulation of nonlinear wave shoaling, with or without impermeable coastal structures, up to the wave breaking. However, in post-breaking flows, the domain is no longer simply connected, and the BEM is not efficient for the generation of a new free surface. Volumes of fluid (VOF) methods are made to track the fluid-free surfaces after breaking, but they are more numerically complex and less efficient relative to the BEM before breaking. This study presents a numerical model, named BELWF—boundary elements Lagrangian wave flume—for the mathematical simulation of two-dimensional wave flumes. The BELWF can simulate the hydrodynamics of wave shoaling over a coast profile, with submerged impermeable coastal structures of any geometry. The developed model is applied to simulate and study Geotube structures. The BELWF is validated by comparisons with OpenFOAM simulations. Both the BELWF and OpenFOAM simulations show that the most critical state, regarding the sliding stability of the Geotube, occurs, typically just before breaking, where the BELWF reasonably assesses the wave loads and the sliding stability. Hence, the BELWF is a valid and efficient method for the preliminary design of impermeable coastal structures. Finally, the BELWF is applied to simulate a practical design example of a complete shoaling process along a sloped shore with a Geotube structure at the shallow water, which develops a plunging breaker. The simulation well captures the critical event considering the sliding stability of the structure.

A novel Lagrangian formulation to construct relativistic rotating stars: towards its application to their evolution calculations

ABSTRACT We present a new formulation to construct numerically equilibrium configurations of rotating stars in general relativity. Having in mind the application to their quasi-static evolutions on a secular time-scale, we adopt a Lagrangian formulation of our own devising, in which we solve force-balance equations to seek for the positions of fluid elements corresponding to the grid points, instead of the ordinary Eulerian formulation. Unlike previous works in the literature, we do not employ the first integral of the Euler equation, which is not obtained analytically in general. We assign a mass, specific angular momentum and entropy to each fluid element in contrast to the previous Eulerian methods, in which the spatial distribution of the angular velocity or angular momentum is specified. These distributions are determined after the positions of all fluid elements (or grid points) are derived in our formulation. We solve the large system of algebraic non-linear equations that are obtained by discretizing the time-independent Euler and Einstein equations in the finite-element method by using our new multidimensional root-finding scheme, named the W4 method. To demonstrate the capability of our new formulation, we construct some rotational configurations, both barotropic and baroclinic. As toy models, we also solve three evolutionary sequences that mimic the cooling, mass-loss, and mass-accretion.

Large-deformation consolidation models of saturated–unsaturated materials under self-weight and evaporation

This paper develops the one-dimensional large-deformation consolidation theory for the porous medium under alternating saturated–unsaturated conditions, through a former generalization of the well-established Gibson’s nonlinear theory from saturated to unsaturated regime. With the differential relations between the coordinate systems, the equations governing the momentum\\mass balances in the unsaturated porous stratum are transported from the moving current to the initial Lagrangian coordinate system. Without loss of generality, the collective forms of various incremental constitutive laws are then incorporated for unsaturated materials, indicating that any particular constitutive law has been embraced. Two forms of final governing equations are derived depending on the primary independents: σv-uw (total vertical stress - pore fluid pressure) and w (gravimetric water content); which are solved by two distinct finite-element codes on the initial fixed configuration. Numerical examples demonstrate that the two resulting Total Lagrangian formulations are equivalent, yield virtually identical predictions and agrees well with an existing finite-differential code that solves the equations in the convective configuration. The dynamical consolidation process evolving from a weight-driven saturated stage to an evaporation-driven unsaturated stage is quantitatively captured very well by the predictions. The impacts of evaporation, material’s compressibility, and permeability in these two distinct stages are independently explored through several parametric analyses on the consolidation/dewatering rates and the penetration of drying front. Through one factor must affect, in concert with the others, such a hydro-mechanical coupling process, these results can provide a profound understanding of the entire consolidation process of a porous stratum under large deformations.

Virtually Euler–Lagrange form for feedback passivation of a wide range of non-Euler–Lagrange systems: Adaptive passivity-based control of hepatitis B disease

In this article, a novel technique of feedback passivation is developed for a wide range of non-Euler–Lagrange systems. The reason for this naming “Virtually Euler-Lagrange” is that it is adopted from the error dynamics of Euler–Lagrange systems. An adaptive passivity-based control strategy is proposed in virtually Euler–Lagrange form to treat hepatitis B virus infection. There are several remarkable differences between classic form of passivation and the proposed method, for example, specific and predefined template of storage function in virtually Euler–Lagrange form, which has positive definite auto-extracted form. So, it will be appropriate and time-saving for designing passivity-based control. On the contrary, the process of passivation and control design in conventional formulation is based on time-invariant input-affine systems; however, in virtually Euler–Lagrange form, the system can belong to a wider category. In this regard, two theorems have been proposed to expand the scope of passivity-based control design. Some control design conditions will be released as a result of using this new method. Furthermore, for the first time, passivity-based control has been implemented in the biological area. Control profile design, simulation results, and sensitivity analysis are carried out to evaluate capability of this control approach. Some main biological limitations are also considered, which make the proposed control more realistic.

Modeling dynamic behavior of two magnetic microbubbles in ultrasonic and magnetic fields

Magnetic microbubble (microbubble coated with magnetic nanoparticles) has the advantages of hybrid encapsulated shell, non-invasive fields manipulation and drug/gene delivery, which promotes a broad application prospects in both scientific research and clinic communities. In this work, the encapsulated shell coated with magnetic nanoparticles has been translated into the equivalent problem about bubble in the magnetic fluid. Behavior of two bubbles in the magnetic fluid are theoretically studied by using Lagrange formalism. We evaluate the manipulation of acoustic and magnetic fields on two bubbles dynamics. Magnetic pressure generated by magnetic field increases the pulsating amplitude of each bubble, affects bubbles translation and enhances the gas pressure and temperature inside bubble. Besides, bubbles attraction is caused by the competition between viscous drag and attractive secondary Bjerknes force related to bubble pulsation. In addition, the attractive force caused by magnetic field is useful at very close distance between bubbles, and its effect is quite small compared to the attraction driven by ultrasound. These findings may have implications for ultrasound guided drug delivery using microbubbles coated with magnetic nanoparticles, especial their manipulation in magnetic field.

A weak form quadrature element formulation of geometrically exact beams with strain gradient elasticity

In this article, a numerical scheme for geometrically nonlinear analysis of elastic strain gradient beams is presented based on the weak form quadrature element formulation in combination with the geometrically exact beam model. Compared to standard geometrically exact beam elements, the present formulation incorporates the derivatives of displacements and rotations as additional degrees of freedom at element boundary nodes due to the C1 continuity requirements brought about by the introduction of strain gradients. A generalized differential quadrature scheme based on the Hermite interpolation functions is employed to approximate the derivatives of displacements and rotations. The quaternion-based scheme for spatial rotations and rotation derivatives is introduced to realize a total Lagrange formulation without singularities. This formulation is shown to be feasible to analyze strain gradient beam structures undergoing large displacements and rotations with the advantages of retaining strain objectivity and circumventing shear locking problems. Six typical examples are given to demonstrate the validity of the formulation.

Canonical Quantization of Yang-Mills Theory in Curved Space-time

The framework of canonical quantization of quantum field is given by Lagrangian (not Lagrangian density) formulation to avoid the problem of transfer physical quantities from 4–dimensional space–time to 3–dimensional position space (3–dimensional hypersurface with normal vector dt).

Real-Time Fuzzy Logic Control of Two-Link Flexible Manipulators

This study investigates the development of a Fuzzy Logic Controller (FLC) for tracking a sinusoidal wave trajectory and suppressing the vibration of a Two Link Flexible Manipulator (TLFM). The TLFM was modeled using Lagrange's formalism and the Assumed Mode Method (AMM). A three-part apparatus consisting of a TLFM mathematical model, a real-world TLFM, and control software was designed and implemented. The FLC was applied to both the simulated and real-world TLFM. The robustness of the FLC was investigated by considering variable payload mass and link angular velocity in both constructive and destructive link interference trajectory cases. Simulation and experimental results show the effectiveness and robustness of the proposed FLC.

Generalized Hertz Vector in the Dissipative Electrodynamics

In the electromagnetic theory, the Hertz vector reduces the number of potentials in the free fields. The further advantage of this potential is that it is much easier to solve particular radiation processes. It indicates that the related method is sometimes more effective than the scalar and vector potential-based relations. Finally, the measurable field variables, the electric and magnetic fields, can be deduced by direct calculation from the Hertz vector. However, right now, the introduction of the Hertz vector operates if the conductive current density j = 𝜎E is neglected. We suggest a generalization by the conductive currents, i.e., when the electromagnetic field dissipates irreversibly to Joule heat. The presented procedure enables us to introduce also the Lagrangian formulation of the discussed dissipated electromagnetic waves. It paves a new way involving damping physical fields in a thermodynamic frame.

Lagrangian Formulation of Fractional Nonholonomic Constrained Damping Systems

Lagrangian Formulation of Fractional Nonholonomic Constrained Damping Systems

On energy gap phenomena of the Whitney sphere and related problems

In this paper, we study Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c=0\ or \ 1)$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. We obtain some Simons' type integral inequalities and rigidity theorems for such Lagrangian submanifolds. Moreover we study Lagrangian submanifolds in $\mathbb{C}^n$ satisfying $\nabla^*\nabla^*T=0$ and introduce a flow method related to them.

Method of Solving Geometrically Nonlinear Bending Problems of Thin Shallow Shells of Complex Shape

A new numerical analytical method for solving geometrically nonlinear bending problems of thin shallow shells and plates of complex shape is given in the paper. The problem statement is performed within the framework of the classic geometrically nonlinear formulation. The parameter continuation method was used to linearize the nonlinear bending problem of shallow shells and plates. An increasing parameter t related to the external load, which characterizes the shell loading process, is introduced. For the variational formulation of the linearized problem, a functional in the Lagrange form, defined on the kinematically possible movement speeds, is constructed. To find the main unknowns of the problem of nonlinear bending of the shell (displacement, deformation, stress), the Cauchy problem was formulated by the parameter t for the system of ordinary differential equations, which was solved by the fourth order Runge-Kutta-Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometric linear deformation. The right-hand sides of the differential equations at fixed values of the parameter t, corresponding to the Runge-Kutta-Merson scheme, were found from the solution of the variational problem for the functional in the Lagrange form. Variational problems were solved using the Ritz method combined with the R-function method, which allows to accurately take into account the geometric information about the boundary value problem and provide an approximate solution in the form of a formula - a solution structure that exactly satisfies all (general structure) or part (partial structure) of boundary conditions. The test problem for the nonlinear bending of a square clamped plate under the action of a uniformly distributed load of different intensity is solved. The results for deflections and stresses obtained using the developed method are compared with the analytical solution and the solution obtained by the finite element method. The problem of bending of a clamped plate of complex shape is solved. The effect of the geometric shape on the stress-strain state is studied

Hamiltonian model for water waves in a triangular domain

We present a Hamiltonian model for free surface water waves in a 2-D symmetric triangular domain with sides that have a 45∘ slope. The construction uses a new amplitude expansion for the free surface, and a Lagrangian formulation of the water problem that is based on a formal mechanical/geometrical analogy of the fluid motion to geodesic motion in a potential force field. We define an approximate Lagrangian by choosing a physically plausible generalized coordinate and obtain the Hamiltonian by the Legendre transform. Hamilton’s equations are presented in spectral form using a known explicit construction of the linear normal modes for the domain considered. A key step is the approximation of a Dirichlet–Neumann operator appearing in the kinetic energy.