Lagrangian Formulation Research Articles

Combined algorithms for analytical inverse kinematics solving and control of the Q-PRR aerial manipulator*

This article presents the design and modeling of a new aerial manipulator system, called Q-PRR, composed of three joints with a fixed base in the center of mass of the multirotor considered as a whole system. This structure has a prismatic joint as a first joint which allows to keep the center of gravity of the Q-PRR as close as possible to the center of gravity of the multirotor. This will also allow to reduce the influence of arm motion on the multirotor roll thus to ensure the stability of the system on trajectory tracking with dynamic changes in the multirotor’s center of gravity. Furthermore, the configuration of the manipulator arm for the desired position of the end-effector given by the inverse kinematics model is kept without any change in the position and attitude of the multirotor. This article develops both forward and inverse kinematics models for a nonlinear underactuated system using the Denavit–Hartenberg notation. When a new algorithm is presented for the inverse kinematics based on Levenberg–Marquardt algorithm. Then, the dynamic model in the joint spaces is developed with the Lagrangian formalism. The Q-PRR is controlled using a model-free control with a comparison of two states, a free fly and disturbance forces applied to the whole system with manipulator arm movement.

Nonlinear analysis of frame structural systems based on closed form solutions of thin beam-columns with multiple strong axial and rotation embedded discontinuities

A new thin beam-column element with capability to model multiple embedded discontinuities is developed, in which the mechanical failure process is simulated, leading to nonlinear analysis of frame structural systems. The physical and mathematical principles are based on a Lagrangian variational formulation of the mechanical behavior of thin beam-columns with multiple strong axial and rotation embedded discontinuities adapted to represent strain localization. These strong embedded discontinuities represent the material failure process as hinges for beams and cracking or crushing for bars. The developed variational formulation is derived from a decomposed continuous incremental plasticity model with hardening, softening and elastoplastic behavior, from which the strong embedded discontinuities, that represent the strain concentration at non-predetermined localization zone, are naturally obtained. The boundary value problem, which is valid throughout the domain of the structural member, is obtained. Closed form solutions for bars and thin beams with multiple strong embedded discontinuities for any boundary condition are obtained by solving the proposed variational formulation. Closed form solutions are particularized to a basic system without rigid-body displacement modes, which naturally leads to a symmetric flexibility matrix and thus a symmetric stiffness matrix of the thin beam-column element with capability to model multiple embedded discontinuities with arbitrary locations. This matrix is more efficient than those developed with the finite element method, since it is naturally condensed, its coefficients are defined values, not in terms of integrals, and any type of loads are modelled. Representative examples of beams and frames validate the capability of the formulated thin beam-column element for modelling damage. In these examples, the load-displacement curves are in agreement with those reported in the literature.

Frequency-dependent AVO inversion applied to physically based models for seismic attenuation

SUMMARYSeismic inversion of amplitude versus offset (AVO) data in viscoelastic media can potentially provide high-resolution subsurface models of seismic velocities and attenuation from offset/angle seismic gathers. P- and S-wave quality factors (Q), whose inverse represent a measure of attenuation, depend on reservoir rock and pore fluid properties, in particular, saturation, permeability, porosity, fluid viscosity and lithology; however, these quality factors are rarely taken into account in seismic AVO inversion. For this reason, in this work, we aim to integrate quality factors derived from physically based models in AVO inversion by proposing a gradient descent optimization-based inversion technique to predict the unknown model properties (P- and S-wave velocities, the related quality factors and density). The proposed inversion minimizes the non-linear least-squares misfit with the observed data. The optimal solution is iteratively obtained by optimizing the data misfit using a second-order limited-memory quasi-Newton technique. The forward model is performed in the frequency–frequency-angle domain based on a convolution of broad-band signals and a linearized viscoelastic frequency-dependent AVO (FAVO) equation. The optimization includes the adjoint-state-based gradients with the Lagrangian formulation to improve the efficiency of the non-linear seismic FAVO inversion process. The inversion is tested on synthetic seismic data, in 1-D and 2-D, with and without noise. The sensitivity for seismic quality factors is evaluated using various rock physics models for seismic attenuation and quality factors. The results demonstrate that the proposed inversion method reliably retrieves the unknown elastic and an-elastic properties with good convergence and accuracy. The stability of the inverse solution especially seismic quality factors estimation relies on the noise level of the seismic data. We further investigate the uncertainty of the solution as a function of the variability of the initial models.

Comparison of the CEL and ALE approaches for the simulation of orthogonal cutting of 15-5PH and 42CrMo4 materials

The modeling of cutting processes by the finite element method represents a great challenge for scientific researchers. Indeed, the plastic strain and strain rate in cutting induce a high risk of mesh distortion, especially for models based on a Lagrangian formulation which were the first to be developed. To overcome this problem other formulations, such as the Arbitrary Eulerian-Lagrangian (ALE), which is a combination of the classical formulations (Eulerian and Lagrangian), have been developed. Despite very interesting results in the simulation of orthogonal cutting, the implementation of this type of model remains delicate, and in particular the choice of initial geometrical conditions which can lead to a sudden stop of the calculation induced by a mesh distortion. To overcome these limitations, the Coupled Eulerian-Lagrangian (CEL) approach has been developed to simulate the orthogonal cutting process. In this paper, the CEL and ALE approaches were compared to simulate the orthogonal cutting process. The first objective was to compare the quality of the predicted physical quantities (forces, chips) over a wide range of cutting speeds and cutting thicknesses for two different materials. The second objective was to evaluate the time required for a complete simulation, integrating the time to set up the initial conditions (chip geometry) and finally the computation time of the machining operation. This work has shown that ALE and CEL formulations lead to very similar quantitative results that are pretty close to the experimental results. The ALE formulation has the advantage of having a reduced computation time compared to the CEL formulation. However, the preparation of the simulation is easy and robust for the CEL, whereas the selection of initial geometrical conditions for the ALE calculation requires several trials and errors before finding stable conditions. The CEL formulation is therefore a solution to be favored when launching several simulations in an automated way thanks to its reliability.

PGD reduced-order modeling for structural dynamics applications

We propose in this paper a Proper Generalized Decomposition (PGD) approach for the solution of problems in linear elastodynamics. The novelty of the work lies in the development of weak formulations of the PGD problems based on the Lagrangian and Hamiltonian Mechanics, the main objective being to devise numerical methods that are numerically stable and energy conservative. We show that the methodology allows one to consider the Galerkin-based version of the PGD and numerically demonstrate that the PGD solver based on the Hamiltonian formulation offers better stability and energy conservation properties than the Lagrangian formulation. The performance of the two formulations is illustrated and compared on several numerical examples describing the dynamical behavior of a one-dimensional bar.

Robust Control of Linear Systems and Shortcut to Adiabaticity Based on Superoscillations

With the advent of quantum technologies, control issues are becoming increasingly important. In this article, we address the control in phase space under a global constraint provided by a minimal energy-like cost function and a local (in Fourier space) constraint resulting from a robustness criterion. The inverse engineering Lagrangian formalism developed here generalizes the one commonly used to describe the superoscillation phenomenon. It is applied to both non-dissipative and dissipative quantum mechanics, and extended to stochastic thermodynamics. Interestingly, our approach also allows to improve the sensing capabilities by an appropriate control of the system.

Improvement of two-dimensional dynamic modeling of cable based on strain modification

<p indent="0mm">In this paper, based on the nonlinear continuum mechanics and curve theory in differential geometry, we comprehensively consider the nonlinear strain of Perkins’ model and the dynamic curvature of Howell’s model for model establishment. Under the updated Lagrangian formulation, we establish the plane dynamic equations of the cable structure, including dynamic curvature and large strain based on the current configuration. The expression of shear force is simplified using dimensionless parameter analysis, and the nonlinear geometric equation and the relative rotation angle equation are supplemented in the plane dynamic equations to form the modified Howell model. We illustrate the difference between the original and modified Howell model by analyzing the tension and shear coefficients. Moreover, we illustrate the superiority of the modified Howell model in terms of computational accuracy and efficiency through examples.

Dynamic instability and internal resonance of rotating pretwisted composite airfoil blades

A rotating pretwisted airfoil blade model reinforced with graphene platelets combined with first-order shear deformation theory is developed to study dynamic instability and internal resonance of the rotating blade, which subjected to the transverse load and in-plane load. Based on the mode shapes derived by using Rayleigh-Ritz method, Lagrange's formulation is taken into account to acquire nonlinear ordinary differential equations. The critical buckling loads and instability regions are obtained and then compared with the opening results. Then, the method of multiple scales is taken into account to solve the modulation equations of the rotating airfoil blade in the presence of 1:2 internal resonance. The detailed discussions related to frequency-response and force-response curves under different parameters are carried out.

Poisson gauge models and Seiberg-Witten map

The semiclassical limit of full non-commutative gauge theory is known as Poisson gauge theory. In this work we revise the construction of Poisson gauge theory paying attention to the geometric meaning of the structures involved and advance in the direction of a further development of the proposed formalism, including the derivation of Noether identities and conservation of currents. For any linear non-commutativity, Θab(x) = {f}_c^{ab} xc, with {f}_c^{ab} being structure constants of a Lie algebra, an explicit form of the gauge Lagrangian is proposed. In particular a universal solution for the matrix ρ defining the field strength and the covariant derivative is found. The previously known examples of κ-Minkowski, λ-Minkowski and rotationally invariant non-commutativity are recovered from the general formula. The arbitrariness in the construction of Poisson gauge models is addressed in terms of Seiberg-Witten maps, i.e., invertible field redefinitions mapping gauge orbits onto gauge orbits.

Performance enhanced piezoelectric rotational energy harvester using reversed exponentially tapered multi-mode structure for autonomous sensor systems

Piezoelectric energy harvesting from rotational motion has recently been found to be a promising way to power wireless autonomous sensor systems. In the reported studies, a piezoelectric energy harvester (PEH) structure with a high strain concentration, an inward configuration, and varied rotation axis alignments is shown to play an important role in enhancing the harvester's vibration amplitude and performance. In this paper, a novel rotation-based PEH is proposed to harvest energy from rotational motion using a multi-mode structure (mounted in an inward configuration) consisting of a reversed exponentially tapered beam (primary beam element) and six branched beams (secondary beam elements) attached with a flange to the free end of the primary beam. The beam tip mass axis does not coincide with the rotation axis along its length to improve harvester vibration amplitude. When the harvester spins at a constant speed, the gravitational force acting on the primary and branched beams causes continuous oscillations in the transverse direction. As a result, the primary beam with a piezoelectric patch continually deforms and generates electrical energy. The harvester is theoretically modeled using the Euler-Bernoulli beam theory, and its dynamic equations are derived using the Lagrangian formulation. The proposed harvester is fabricated, and its performance is evaluated through experimentation at a rotating frequency ranging from 1.5–9.5 Hz (90–570 rpm). The harvester offers greater design adaptability in tuning structural parameters to achieve the desired frequencies. An energy management system was designed after investigating the charging behavior of the capacitor with the harvester, and it was found that the proposed harvester was suitable to source wireless autonomous sensor systems.

Nonsmooth dynamic analysis of rigid-flexible interaction collision

This paper aims to explore the deformation of the collided bodies in multi-body systems and to effectively simulate the motion path of colliding bodies. First, we describe the geometrically nonlinear problems of materials by the total Lagrangian formulation. Second, a first-order integration scheme is used to solve the dynamics equations. An algorithm combining the bi-potential method with the node-to-point contact identification is proposed to solve the interface problems of rigid-flexible interaction collision. To observe the collision process more intuitively, the internal software FER/VIEW is introduced to visualize the results. The accuracy is proved by comparing the proposed method with the analytical solution or another numerical solution. Moreover, the proposed method has more numerical robustness, such as occupying less computer storage, saving the computational cost, and broadening the application range of the bi-potential method.

Numerical analysis of fully non-linear sloshing waves in an arbitrary shape tank by meshless method

This paper is devoted to develop a truly meshless numerical procedure for the fully non-linear analysis of sloshing phenomenon in an arbitrary-shape tank. For this purpose, the potential theory is considered as a liquid sloshing model, and the Lagrangian form of kinematic and dynamic boundary conditions in a moving coordinate system fixed to the tank is used to accurately capture the free surface. The multi-quadric radial basis function (MQ-RBF) augmented with the polynomial terms is employed to determine RBF-FD weights on neighboring nodes surrounding the center point for the spatial derivatives. These weight coefficients are used to solve both boundary value problems for velocity and acceleration potential. Compared with mash-based methods, the present method has the advantage of being easy to construct and flexible in dealing with the moving boundary problems. The free surface elevations, hydrodynamic pressure, and first-order natural sloshing frequencies have been reported for various geometries tanks under horizontal excitations at resonance and non-resonance cases. Fairly satisfactory agreement is observed in the numerical results and exiting results, especially for free surface elevation and first-order natural sloshing frequency.

D*/D ratio in heavy ion collisions

In this work we study the $D^*$ and $D$ multiplicities and how they change during the hadron gas phase of heavy ion collisions. With the help of an effective Lagrangian formalism, we calculate the production and absorption cross sections of the $D^*$ and $D$ mesons in a hadronic medium. We compute the time evolution of the abundances and the ratio $D^* /D$. They are approximately constant in time. Also, assuming a Bjorken type cooling and using an empirical relation between the freeze-out temperature and the central multiplicity density, we estimate $D^* /D$ as a function of $ dN /d \eta (\eta =0)$, which represents the system size. We find that, while the number of $D^*$'s and $D$'s grows significantly with the system size, their ratio remains approximately constant. This prediction can be compared with future experimental data. Our results suggest that the charm meson interactions in the hadron gas do not change their multiplicities and consequently these mesons are close to chemical equilibrium.

Bimetallic Thin-Walled Box Beam Thermal Buckling Response.

A beam model for thermal buckling analysis of a bimetallic box beam is presented. The Euler-Bernoulli-Vlasov beam theory is employed considering large rotations but small strains. The nonlinear stability analysis is performed using an updated Lagrangian formulation. In order to account for the thermal effects of temperature-dependent (TD) and temperature-independent (TID) materials, a uniform temperature rise through beam wall thickness is considered. The numerical results for thin-walled box beams are presented to investigate the effects of different boundary conditions, beam lengths and material thickness ratios on the critical buckling temperature and post-buckling responses. The effectiveness and accuracy of the proposed model are verified by means of comparison with a shell model. It is revealed that all of the abovementioned effects are invaluable for buckling analysis of thin-walled beams under thermal load. Moreover, it is shown that the TD solutions give lower values than the TID one, emphasizing the importance of TD materials in beams.

Analysis of the elasto-magnetic wave propagation in a smart magnetostrictive actuator using a mesh-free method

This work aims to propose an implicit algorithm for solving the magnetoelastic wave propagation in a magnetostrictive actuator. This algorithm is built by combining meshless methods with the Newmark implicit time scheme. The mathematical formulation of vibrations of the actuator is performed by modeling the experimental actuator device by a monodimensional magnetostrictive cylindrical rod attached to an elastic spring, subjected to an axial magnetic field. The formulation of the problem is carried out using the rod model. The study is carried out taking into account mechanical and magnetic effects described by a linear model of strain-stress magnetoelastic relations, whose coefficients depend on the square of the magnetic field, elastic, magnetostrictive, and magnetoelastic characteristics of the elastomagnetic material of which the rod is made. The dynamic equation which describes the magnetoelastic wave propagation in the actuator is obtained by applying the Lagrangian formalism based on the least action principle. The wave propagation is governed by a spatio-temporal partial differential equation whose coefficients are a function of on elastic, magnetostrictive, and magnetic characteristics and implicitly depends on time. The resolution of the dynamic equation is made by adopting two numerical approaches developed by combining the mesh-free collocation methods, the Moving Least Squares (MLS), and the Radial Point Interpolation Method (RPIM) with the Newmark implicit scheme temporal integration. By performing the spatial discretization of the displacement, we obtain a system of linear second-order ordinary differential equations (ODEs), then transform it into a system of first-order ordinary differential equations after integrating Newmark’s implicit time scheme. A comparison of results computed by the proposed numerical approaches RPIM and MLS with experimental data available in the literature is presented. The stability and the solutions qualities of two meshless methods, RPIM and MLS, are also studied. The influence of magnetical and mechanical parameters as well as the coupling coefficients on the performance of the magnetostrictive actuator is also made.

Numerical simulation of nucleating flow and shock capturing in steam turbines by a simple low-dissipation upwind scheme using an Eulerian-Lagrangian model

The rapid expansion of steam leads to non-equilibrium condensation of steam in the supersonic region which the performance of the turbines is reduced. The achievement of an accurate numerical method in a wide range of regimes to capture the aerodynamic and condensation shocks with minimum computation and numerical errors has always been of interest to researchers. Therefore, in this research, an in-house code is developed with a version of Advection Upstream Splitting Methods (AUSM), named Simple Low-dissipation AUSM (SLAU). The results of the SLAU scheme for condensing flow in three convergent-diverging nozzles are presented and compared with experimental data to validate the scheme. Then the SLAU scheme's performance in condensation and aerodynamic shock capturing for two-dimensional turbine blades is evaluated using an Eulerian-Lagrangian approach under different outlet conditions. The governing equations of flow are solved by the Eulerian approach using the SLAU scheme, and to accurately calculate wet parameters of the solution field, the corresponding equations are solved in Lagrangian form. Comparing the results of the SLAU scheme with experimental data, it was found that, despite the simplicity of the SLAU scheme, in the absence of tuning variables, it provided suitable predictions, even within shock zones. The results of the SLAU scheme for modeling steam nucleating flows and the prediction of the droplet radius are in better agreement with the experimental data and the results are improved considerably for the supersonic outlet cascade.

Stability of a plane Couette flow over inhomogeneously stressed solids

In this paper, we investigate the stability of a plane Couette flow over non-linear solids undergoing homogeneous or inhomogeneous initial stresses. Almost all biological and engineering structures are subjected to complex initial residual stresses. We probe here the stability characteristics of a plane Couette flow over these stressed solid surfaces. Since initial stress is commonly inhomogeneous in Nature, our main focus is to analyze the stability of flow over inhomogeneously stressed solids at all Reynolds number regimes, which was not investigated earlier. We apply an initially stressed neo-Hookean solid model which precludes the necessity of using the stress-free reference. A linear perturbation analysis based on an Eulerian–Lagrangian formulation determines the stability associated with short and finite wave modes for zero, low, and high Reynolds number flows. In presence of a uniform initial stress field, the short wave modes are stabilized by tensile stresses and destabilized by compressive stresses in the limit of creeping flow. A similar influence of pre-stress is observed in buckling of columns. On the other hand, the case of spatially varying initial stress is far more intricate due to the inhomogeneous shearing of the solid in the unperturbed state. A linear perturbation introduced to this complicated unperturbed state presents several interesting observations. We find multiple unstable modes for low Reynolds number (Re≪1) creeping flows which were not reported earlier to the best of our knowledge. While literature demonstrates only downstream unstable modes for a plane Couette flow, both upstream and downstream unstable modes are observed for inhomogeneously stressed solids in various Reynolds number regimes. In the creeping flow limit (Re=0), the stability depends mainly on the tensile or compressive initial stress at the solid–fluid interface. When this initial stress is compressive at the interface, only the finite wave modes determine the stability for all thicknesses instead of the short-wave modes. At higher Reynolds number, we report many more interesting results for spatially varying stress-fields which includes an unusual scaling of wall-mode validated through eigenmode analysis.

Controlling a quadrotor UAV by means of a fractional nested saturation control

In this paper, a fractional nonlinear control strategy based on the nested saturations technique and the Caputo-Fabrizio derivative for a quadrotor aircraft, to maintain desired position and orientation in stationary flight and trajectory tracking tasks, is proposed and developed. In addition, the dynamic model is obtained by using the Euler–Lagrange formulation, and then it is generalized to an arbitrary order through the Caputo-Fabrizio derivative. The traditional nonlinear control based on the nested saturation technique is also developed and implemented on the quadrotor for comparison purposes between both control schemes. Numerical simulations demonstrated the superiority of the proposed control strategy for stabilizing the quadrotor at hover and reaching the desired position and orientation, over its integer-order counterpart. In addition, three additional desired trajectories are considered, keeping the same gains and orders, to demonstrate the robustness of the proposed control scheme under different operating conditions. In this sense, our controller presents significant performance improvements, which were validated through system simulations. Our proposed control system and its traditional counterpart were tuned experimentally using the response of the system to a desired trajectory.

Real‐time control design and implementation of ball balancer system based on machine learning and machine vision

SummaryThis article addresses a hybrid pseudo‐PD/machine learning controller algorithm to improve the stabilization and trajectory tracking of the ball on the plate system (BOPS). The proposed controller design depends on a machine learning algorithm that detect the angle of the servo motor required to correct the position of the ball on the plate also the parameters of the PD controller is changed online using the fuzzy logic to enhance the performance of the trajectory and point tracking of the system. This article offers three different machine learning techniques for predicting servo motor angle that obtain higher accuracies of 99.95%, 99.908%, and 99.998% for support vector regression, decision tree regression, and random forest regression, respectively. The proposed scheme has greatly improved the system's settling time and overshoot, according to simulation and practical results. The Lagrangian formulation may be used to obtain the mathematical formulation, and a practical identification experiment can be used to determine the servo motor parameter. Simulation and practical results on a BOPS dynamic model are both demonstrated to justify the validation of the proposed design scheme. The experimental practical results justify the feasibility and effectiveness of the proposed controller design strategy.

Isochronicity Conditions and Lagrangian Formulations of the Hirota Type Oscillator Equations

Isochronicity Conditions and Lagrangian Formulations of the Hirota Type Oscillator Equations