Euler-Lagrange Research Articles

Euler–Lagrange equation for gradient-type Lagrangian and related conservation laws

AbstractVariational calculus with gradient-type variations has often been neglected, although it proves to be suitable for certain concrete problems governed by several evolution variables. These kinds of variations lead to Euler–Lagrange partial differential equations controlled by the right-hand member. In this context, we also introduce anti-trace Euler–Lagrange partial differential equations that are suitable for some innovative ideas. Also, some applications are provided for the theoretical results derived in the paper.

Adaptive Neural Delay Feedback Control of a Typical Robot on a Slope

ABSTRACTIn this article, an adaptive neural delay feedback control strategy is proposed for the motion control of a two‐wheel self‐balancing robot (TWSBR) on a slope with an input delay. Based on the Lagrange equation, the dynamic model of the robot is derived. Given a target trajectory vector, the corresponding error system is obtained. An input delay and uncertainties are considered. First, the system with an input delay is transformed into a delay‐free one. Second, a quadratic performance optimization criterion is defined, and a linear quadratic regulator (LQR) controller is designed to minimize the error between the system's state vector and the target vector. Thirdly, an integral sliding mode controller is added to the LQR controller to enhance the system's robustness. To reduce the “chattering” phenomenon, an adaptive neural delay feedback control scheme is developed based on the adaptive learning of the uncertainties' upper bound by a radial basis function neural network (RBFNN). It is demonstrated that a relaxation process exists at the beginning of the upslope motion. As the input delay increases, the swing range of the center of gravity of the pendulum expands. The angle can converge to the steady state more quickly, but the relaxation time is still 0.031 s which has nothing to do with the input delay. An example of the back‐and‐forth motion of a TWSBR on a slope can verify an excellent agreement between theoretical analysis and numerical results.

Stability and bifurcation analysis of a 2DOF dynamical system with piezoelectric device and feedback control

This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring’s elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system’s efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.

Dynamic Modeling and Configuration Transformation of Origami with Soft Creases

Origami has attracted much attention from scientists and engineers in recent years. Classic origami only permits rotations around creases, while origami-like structures in nature can also undergo a certain degree of extension at creases. In this paper, an earwig-inspired origami with rotational symmetry is proposed and analyzed by introducing soft creases. Such a soft crease is equivalent to a combination of a rotational spring and an extensional spring. The motion of the earwig-inspired origami is described using spherical triangles. Based on Lagrange's Equation, the nonlinear dynamic equation of the system is formulated. It is then solved by using the fourth-order Runge-Kutta method. The results of theoretical calculations are consistent with those of simulations in ADAMS. With the established framework, the bifurcation behaviors of the equilibria of the proposed system, including supercritical pitchfork and saddle-node bifurcations, are investigated. Such origami can realize both mono-stable and bi-stable mechanisms, while the corresponding design parameters are demonstrated in the design map. In particular, the properties of the bi-stable origami can vary with different equilibria. The configuration transformation could be achieved using a continuous excitation. However, it is sometimes sensitive to the initial conditions. Inspired by the working mechanism of earwig wings, a simple control approach for origami configuration transformation is proposed. The key is to stop exciting the origami when its deformation crosses an energy barrier. This work lays a foundation for the design and study of novel multi-stable and morphing structures and provides an efficient approach for their configuration transformation.

Traveling wave vibration characteristics of rotating FGM conical shell

Abstract Conical shell is common structural component in various rotating machinery. The study of traveling wave vibration performance of conical shell is of great significant for the design of rotating machinery. This paper addresses the traveling wave vibration performance of rotating functionally graded material (FGM) conical shells. Artificial springs, uniformly distributed, are employed to simulate the boundary constraints of rotating FGM conical shell. Accounting for the effects of Coriolis and centrifugal forces caused by rotation, the energy equation of rotating FGM conical shell is calculated, and then the dynamical model of the rotating FGM conical shell under elastic supported constraints is established using the Lagrange equation. The influence of key factors on traveling wave frequency and response characteristics of rotating FGM conical shell are investigated including rotational speed, half-cone angle and material components.

Thermally induced vibration of photovoltaic honeycomb-based-thermoelectric hybrid device

This paper presents a comprehensive investigation of thermally induced vibration (TIV) of elastically constrained photovoltaic honeycomb-based-thermoelectric hybrid device under space thermal environment. Based on four-variable refined quasi-3D higher-order theory as well as virtual boundary spring technique, the equation of motion of the hybrid device is derived by using Lagrange equation. The nonlinear temperature profile is obtained via finite difference scheme and the TIV responses are solved by employing Newmark approach. The effects of boundary condition, size of honeycomb cell, damping effect, thickness of insulation layer and radius of curvature on TIV of the hybrid device are studied. Results show that the TIV response of the hybrid device can be greatly weakened by replacing the dense thermoelectric leg with honeycomb-based thermoelectric leg. Designing the honeycomb cell as re-entrant architecture possesses a better suppression effect on TIV response. The amplitude of TIV reduces with increasing the thickness-length ratio and decreasing the height-length ratio of honeycomb cell. A larger thickness of insulation layer can weaken the TIV response. The oscillation of TIV response can be eliminated by fabricating hybrid device as circular cylindrical and hyperbolic parabolic shallow shells. The stiffer the boundary conditions, the less prone to TIV behavior of the hybrid device. The introduction of damping effect can suppress TIV response, but it simultaneously causes the occurrence of thermal snap. In general, the research provides several design references to weaken the TIV response of the hybrid device under space heat flux.

Study on vibration control performance of pendulum TMD with additional stoppers and its application on high-rise buildings

Although pendulum tuned mass damper (PTMD) is one of the most classic and commonly used vibration control devices, it has clear limitation due to the natural feature of linear TMD. Additional stoppers for the pendulum string (PTMD-AS) were proposed and introduced to improve the PTMD's performance by triggering its nonlinearity. A 2-DOF model was established to analyze the dynamic response of the system subjected to harmonic excitation, and the governing equations were formulated using the Lagrange equation. The extended incremental harmonic balance (EIHB) method and the Runge-Kutta (R-K) method were utilized to calculate the frequency response and time history of the system. Nonlinear dynamic characteristics of the pendulum with stiffness hardening were explored in detail. Sensitivity analyses were performed to investigate the effect of stopper position. It was found that aperiodic responses or multiple solutions could be induced when the pendulum underwent significant stiffness hardening upon passing the additional stoppers. Finally, the effectiveness and robustness of PTMD-AS are demonstrated in a numerical simulation of a high-rise building subjected to random wind excitation based on wind tunnel experiments.

Analysis of satellite attitude motion in a three-body tethered system during deployment via integral manifolds

This paper aims to analyze the impact of unideal end-body configurations on attitude motion during the tether deployment process in a linear three-body tethered system (LTBTS). A main challenge in this process is the resonance between the nutation and spin angles caused by the unideal end-body configurations, leading to severe nutation angle oscillations. These oscillations can result in tether entanglement with the end-bodies or even tether rupture. To address this issue, the deployment model for the LTBTS is first established using the Lagrangian equations, with the end-body attitude described by Eulerian angles. Secondly, the system involves the separation of two subsatellites from the central main satellite in opposite directions. The dynamics response of the deployment process under unideal configuration of end-bodies is investigated. Resonance phenomena in nutation and spin angles are observed due to errors in initial angles/angular velocities, offset errors of tether connection points, and unideal structural characteristics of the satellites. Thirdly, to further understand this resonance, analytical solutions for resonance are derived by transforming the nutation angle equations, and satellite attitude equations are systematically solved via integral manifold methods. Potential resonance issues are mitigated by reducing system asymmetry and minimizing initial disturbances. Finally, the effectiveness of the dynamic analysis is validated through simulations.

Research on circulation control mechanisms based on Lagrangian dynamics

Circulation control technology greatly improves an aircraft's aerodynamic efficiency by employing high-speed jets at the wing's trailing edge, which generates the Coanda effect. This paper analyzes and reveals the mechanism of circulation control from the perspective of Lagrangian dynamics. First, simulate the circulation control airfoil using the shear stress transport turbulence model to obtain instantaneous flow field data under various jet momentum coefficients (Cμ) and angles of attack. Subsequently, from the perspective of Lagrangian dynamics, the mixing, material transport, and momentum transfer processes between the jet and the trailing-edge separation zone are analyzed using attracting Lagrangian Coherent Structures (aLCSs) under different jet momentum coefficients. Next, through the perspective of particle motion in the Lagrangian framework, the study reveals the accelerating effect of the Coanda jet on the low-speed fluid over the upper wing surface. Additionally, repelling Lagrangian Coherent Structures are used to characterize the downward movement of the leading-edge stagnation point after applying circulation control, and the impact of this movement on the flow state is analyzed. Finally, the analysis using aLCSs assesses the flow control effectiveness of Coanda jets as “virtual aerodynamic flaps.” This analysis helps to further understand the advantages of the “virtual rudder surface” formed by Coanda jet.

The isoperimetric problem for convex hulls and the large deviations rate functionals of random walks

We study the asymptotic behaviour of the most likely trajectories of a planar random walk that result in large deviations of the area of their convex hull. If the Laplace transform of the increments is finite, such a scaled limit trajectory h solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length of h is replaced by the large deviations rate functional ∫01I(h′(t))dt and I is the rate function of the increments. Assuming that the distribution of increments is not supported on a half-plane, we show that the optimal trajectories are convex and satisfy the Euler–Lagrange equation, which we solve explicitly for every I. The shape of these trajectories resembles the optimizers in the isoperimetric inequality for the Minkowski plane, found by Busemann (1947).

Design and application of inertia switch with the zero-time trigger sensor for projectile overload impulse

The paper introduces a novel inertial trigger sensor featuring a multi-layer stacked single-arm brass beam pendulum, aimed at applications in automotive and aerospace fields, etc. It employs the Lagrangian equation for dynamic modeling and finite element analysis to achieve precise inertia force measurements. The design incorporates advanced fabrication techniques and a high-precision latching mechanism that combines circuit self-locking with mechanical triggering. Validated through simulations and testing, The results of the live ammunition tests indicate that, apart from failures of the Inertia Switch due to errors in the manufacturing process, all other inertial switches functioned normally. This demonstrates that the inertial sensor provides an adjustable trigger time of 0–5 ms and reliably withstands extreme overloads of up to 15000 g. This work significantly enhances the performance of inertial sensors in high-stress environments.

Revisiting Sommerfeld's atomic model using Euler–Lagrange dynamics

The purpose of this work is to present the atomic model proposed by Sommerfeld. We outline the classical calculation of the elliptical orbit and then apply the Wilson–Sommerfeld quantization rules to obtain expressions for the quantization of energy and orbit semi-axes. We then apply the relativistic theory to solve the problem of degenerate orbits. Thus, we see that the Sommerfeld atom is a very rich topic, involving the integration of different subjects ranging from Lagrangian mechanics to relativity, as well as plane geometry and differential equations.

Why active Willis metamaterials? A controllability and observability perspective

Recently, active Willis metamaterials (AWM) have been the focus of extensive investigations because of their unique electro-elastic coupling characteristics. However, the treatments of this class of materials have been carried out exclusively, in all the available literature, by approaches that do not rely on solid control theory basis. In this paper, the emphasis is placed on revealing very important control features that are inherent to this class of materials because of their Willis coupling characteristics. These features lie in the enhanced controllability and observability properties of the AWM as compared to non-Willis active materials. Such control properties enable the AWM to possess broad sensing and actuation capabilities that can lend this material to be an effective means for monitoring and controlling the behavior of numerous critical applications, such as acoustic cloaking, particularly when integrated with appropriate robust control strategies. A simple example of a piezoelectric-based AWM is presented to demonstrate its effective control capabilities and distinguish this class of materials from conventional materials. In the selected example, the AWM is structured from two dissimilar masses connected by a piezoelectric spring. Lagrange dynamics formulation is utilized to generate the equations governing the Willis coupling, the piezoelectric coupling, and reveal the inherent control features. With this developed controlled-based structure of the AWM, it is shown that the AWM can simultaneously monitor and control both the strain and velocity whereas the conventional active material, which is formed from two similar masses connected by a piezoelectric spring, can only measure and control the strain alone. It is envisioned that the revealed control metrics for the simple one-dimensional AMW example can serve as means for investigating the potential of AMW's of higher dimensionality.

Novel PCA-Based Lower-Dimensional Remapping of the Solution Space for a Genetic Algorithm Optimization: Estimating the Director Distribution in LC-Based SLM Devices

This work introduces a novel computational approach based on Principal Component Analysis (PCA) for dimensionality reduction of the solution space in optimisation problems with known linear interdependencies among solution variables. By creating synthetic datasets with deliberately engineered properties and applying PCA, the solution space’s remapping significantly reduces its dimensionality, leading to faster computation and more robust convergence in optimisation processes. We demonstrate this method by integrating it with a Genetic Algorithm (GA) for solving the optimal director distribution in liquid crystal (LC) devices, specifically addressing 2D and complex 3D spatial light modulator (SLM) structures such as twisted nematic liquid crystals (TN-LC) and parallel-aligned liquid crystal on silicon (PA-LCoS), respectively. The phase profiles obtained from the director vector distributions for horizontal and vertical high-frequency binary phase gratings closely match the theoretical values derived from minimising the traditional elastic Frank–Oseen functional via Euler–Lagrange equations. Beyond this specific application, our method offers a general framework for reducing computational complexity in optimisation problems by directly reducing the dimensionality of the solution space. This approach is applicable across various optimisation scenarios with well-known linear interdependencies among solution variables, enabling significant reductions in computational costs and improvements in robustness and convergence.

Output performance analysis of a piezoelectric micro nuclear battery powered by radioisotopes

Abstract This paper investigates the output performance (output voltage, electrical energy output, power output) of a piezoelectric micro nuclear battery powered by radioisotopes. The theoretical formulations are base on Euler-Bernoulli beam theory and include the effects of load nonlinearity due to radioactive source. By employing extended Hamilton’s principle, the nonlinear electromechanical Lagrange equations are derived then solved by using Runge-Kutta method to obtain the dynamic output response. The results based on present electromechanical dynamic model are validated through direct comparisons with the results in the open literatures. The effects of initial gap, length of the piezoelectric layer, thickness of the collector and load impedance on the output voltage, energy output and power output of the piezoelectric micro nuclear battery are discussed in detail.

Uncovering the Secrets of Motion: A Literature Review of Lagrange and Newtonian Mechanics

Physics is a branch of natural science. Since ancient times, humans have been interested in the movement of objects around them and motion is an area studied in mechanics. This research aims to understand the principles of mechanics in Lagrange and Newtonian mechanics for solving motion problems and their applications. The method used is the literature review method. Literature is a comprehensive overview of previous research on a particular topic. Literature review is a systematic process for identifying, evaluating, and synthesizing previously published scientific works on a particular topic. There are 20 sources used, consisting of books and scientific journals. The research results show that motion is an object that experiences movement from one place to another. Problems of motion of objects can be solved using the laws and principles of Newtonian and Lagrange mechanics. Lagrange mechanics is used to solve complex problems that cannot be solved with Newtonian mechanics. Lagrange mechanics focuses on the forces that cause objects to move, while Newtonian mechanics focuses on the energy that causes objects to move.

Development of a Lagrange Mechanics E-Module Based on a Flipbook Pendulum Spring System

This research is a type of Research and Development (R D) research that uses the 4D development model, namely Define, Design, Develop, Disseminate. This research produces a Lagrange Mechanics E-Module based on a flipbook pendulum spring system. The aim of this research is to produce a Lagrange Mechanics E-Module that is suitable for use as a learning medium for Analytical Mechanics. The data collection instrument used in this research is a questionnaire as a validation assessment for material experts and students, data analysis uses percentages with predetermined categories. The research sample was 14 Physics Education students. The validation results by material experts were 73% in the feasible category. The student response results were 85% in the very appropriate category. From these results, the Lagrange Mechanics E-Module is suitable for use with revisions as an alternative source of learning.

Research on anti-swing control system of slewing crane based on fuzzy PID.

The slewing crane is easily affected by the wind and the manipulation level of the operator when it is working, which in turn impacts its swing angle, affects the working efficiency and safety of the crane. Toimprove the operation safety andreduce the swing angle, this study design and research the anti-swing control system for the slewing crane. Firstly, based on the Lagrange equation, the dynamic model of the crane hoisting system is established. Then, the mechanical anti-swing mechanism driven by a four-motor rope is established. Finally, based on fuzzy PID control, an anti-swing platform is established, and the performance of anti-swing control is verified. Compared to no anti-swing control, the in-plane swing angle θ1 and out-of-plane swing angle θ2 of the lifting arm are reduced by 88% and 75% respectively. The results show that the four-motor rope-driven anti-swing mechanism can achieve better anti-swing effect, which provides an effective swing suppression method for the slewing crane in the actual operation.

Real Case Study of 600 m3 Bubble Column Fermentations: Spatially Resolved Simulations Unveil Optimization Potentials for l-Phenylalanine Production With Escherichia coli.

Large-scale fermentations (»100 m³) often encounter concentration gradients which may significantly affect microbial activities and production performance. Reliably investigating such scenarios in silico would allow to optimize bioproduction. But related simulations are very rare in particular for large bubble columns. Here, we pioneer the spatially resolved investigation of a 600 m³ bubble column operating for Escherichia coli based l-phenylalanine fed-batch production. Microbial kinetics are derived from experimental data. Advanced Euler-Lagrange (EL) computational fluid dynamics (CFD) simulations are applied to track individual bubble dynamics that result from a recently developed bubble breakage model. Thereon, the complex nonlinear characteristics of hydrodynamics, mass transfer, and microbial activities are simulated for large scale and compared with real data. As a key characteristic, zones for upriser, downcomer, and circulation cells were identified that dominate mixing and mass transfer. This results in complex gradients of glucose, dissolved oxygen, and microbial rates dividing the bioreactor into sections. Consequently, alternate feed designs are evaluated splitting real feed rates in two feeds at different locations. The opposite reversed installation of feed spots and spargers improved the product synthesis by 6.24% while alternate scenarios increased the growth rate by 11.05%. The results demonstrate how sophisticated, spatially resolved simulations of hydrodynamics, mass transfer, and microbial kinetics help to optimize bioreactors in silico.

Dynamics of a knuckle crane carrying a load subjected to wind pressure

AbstractThe paper presents the authors’ model of a knuckle crane offshore type that takes into account the flexibility of boom and drives, which is used to analyze the effect of constant wind pressure or cyclical gusts on the dynamics of the crane and the load. The mathematical model of the crane is derived using a formalism of joint coordinates, homogeneous transformation matrices, and Lagrange equations. Using this model, the influence of wind pressure on the torques and forces in the crane drives and on the accuracy of load positioning is investigated. The results show that the influence of wind on the accuracy of load positioning should not be assessed by limiting the observation to the movement of its gravity center but by observing the movement of characteristic points, the corners of the load. Root mean squared (RMS) indicators are used to quantitatively assess this impact on the driving torque and forces, and appropriate indicators are proposed to assess the accuracy of load positioning.