Euler-Lagrange Research Articles

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.

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.

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.

A new semi-analytical approach for nonlinear dynamic responses of functionally graded porous graphene platelet-reinforced circular plates and spherical shells

This paper presents the semi-analytical approach for nonlinear dynamic responses of porous graphene platelet-reinforced circular plates (CiPs) and spherical shells (SpSs) resting on the Pasternak foundation in thermal environment. The graphene platelets (GPLs) are distributed in the functionally graded (FG) distribution patterns and uniform distribution pattern through the CiP and SpS thickness direction. The governing equations of the GPL-reinforced structures subjected to time-dependent external pressure respected to the harmonic and linear functions of time are established based on the geometrically nonlinear higher-order shear deformation theory. A simple and effective semi-analytical solution for the considered problem is established using the Euler-Lagrange equations, and the damping viscosity of the foundation can be counted using the Rayleigh dispassion function. The Runge-Kutta method is employed to determine the dynamic responses of the GPL-reinforced structures, while the fundamental frequencies of free and linear vibration, and frequency-amplitude curves are obtained in explicit form. The numerical results obtained reveal that the GPL and porous distributions, geometrical and material properties can significantly affect the frequency-amplitude curves, and dynamic responses of harmonically loaded and linearly loaded structures, specially, the dynamic buckling phenomenon is clearly observable with the SpSs but not with the CiPs.

Conservation laws of the one-dimensional relativistic magnetogasdynamics equations

Abstract The present paper offers a comprehensive analysis of the one-dimensional relativistic magnetogasdynamics equations in Lagrangian coordinates. These equations, describing the behavior of a relativistic magnetized gas, are reformulated in variational form to facilitate further analysis. A complete group analysis of the resulting Euler-Lagrange equation is performed, allowing us to systematically identify the symmetries inherent in the system. Using Noether’s theorem, we derive conservation laws in Lagrangian coordinates, offering critical insights into the invariants and conserved quantities of the system. This work expands the theoretical understanding of relativistic magnetogasdynamics.

Adaptive Saturated Obstacle Avoidance Trajectory Tracking Control for Euler–Lagrange Systems With Velocity Constraints

ABSTRACTThis article pays attention to the obstacle avoidance trajectory tracking control problem for uncertain Euler–Lagrange (EL) systems subject to velocity constraints and input saturation. The existing obstacle avoidance results do not consider velocity constraints under input saturation, which means that an EL system may not be able to obtain sufficient control inputs to avoid a collision with an obstacle if it has a high speed when approaching the obstacle. Therefore, the velocity constraints in the obstacle avoidance tracking control are considered in this paper. A novel velocity constraint function that depends on the distance between the system and the obstacle is proposed. Integral‐multiplicative Lyapunov‐barrier functions (LBFs) are constructed and incorporated into the backstepping procedure to design an adaptive fuzzy obstacle avoidance tracking control scheme. Moreover, an auxiliary dynamic system is designed by constructing a bounded nonlinear vector related to an auxiliary variable to compensate for the effects of saturation. Through the Lyapunov method and boundedness analysis for the barrier function, it is shown that the protocol achieves obstacle avoidance for the EL system without violating the velocity constraints inside the obstacle detection region, while also guaranteeing the ultimate uniform boundedness of all the closed‐loop signals. Numerical simulations are presented to demonstrate the efficacy of the proposed control strategy.

The Dirichlet problem for the Lagrangian mean curvature equation

The Dirichlet problem for the Lagrangian mean curvature equation

A semi-analytical transient undisturbed velocity correction scheme for wall-bounded two-way coupled Euler-Lagrange simulations

Force closure models employed in Euler-Lagrange (EL) point-particle simulations rely on the accurate estimation of the undisturbed fluid velocity at the particle center to evaluate the fluid forces on each particle. Due to the self-induced velocity disturbance of the particle in the fluid, two-way coupled EL simulations only have access to the disturbed velocity. The undisturbed velocity can be recovered if the particle-generated disturbance is estimated. In the present paper, we model the velocity disturbance generated by a regularized forcing near a planar wall, which, along with the temporal nature of the forcing, provides an estimate of the unsteady velocity disturbance of the particle near a planar wall. We use the analytical solution for a singular in-time transient Stokeslet near a planar wall (Felderhof [1]) and derive the corresponding time-persistent Stokeslets. The velocity disturbance due to a regularized forcing is then obtained numerically via a discrete convolution with the regularization kernel. The resulting Green's functions for parallel and perpendicular regularized forcing to the wall are stored as pre-computed temporal correction maps. By storing the time-dependent particle force on the fluid as fictitious particles, we estimate the unsteady velocity disturbance generated by the particle as a scalar product between the stored forces and the pre-computed Green's functions. Since the model depends on the analytical Green's function solution of the singular Stokeslet near a planar wall, the obtained velocity disturbance exactly satisfies the no-slip condition and does not require any fitted parameters to account for the rapid decay of the disturbance near the wall. The numerical evaluation of the convolution integral makes the present method suitable for arbitrary regularization kernels. Additionally, the generation of parallel and perpendicular correction maps enables to estimate the velocity disturbance due to particle motion in arbitrary directions relative to the local flow. The convergence of the method is studied on a fixed particle near a planar wall, and verification tests are performed in the Stokes regime on a settling particle parallel to a wall and a free-falling particle perpendicular to the wall.

Nonlinear dynamic modeling and vibration analysis for early fault evolution of rolling bearings

In rotating machinery, the condition of rolling bearings is paramount, directly influencing operational integrity. However, the literature on the fault evolution of rolling bearings in their nascent stages is notably limited. Addressing this gap, our study establishes an innovative nonlinear dynamic model for early fault evolution of rolling bearings based on collision impact. Firstly, considering the fault evolution characteristics, the influence of the rolling element and fault structure, the dynamic model of early fault evolution between the rolling element and the local fault is established. Secondly, according to the Hertzian contact deformation theory, a nonlinear dynamic model of rolling bearings expressed as mass-spring is established. Thirdly, the energy contribution method is used to integrate the fault evolution model and the nonlinear dynamic model of the rolling bearing. A nonlinear dynamic model of early fault evolution of the rolling bearing is proposed by using the Lagrangian equation. Comparing the simulation results of the nonlinear dynamic with the experimental results, it can be seen that the numerical model can effectively predict the evolution process and vibration characteristics of the fault evolution of rolling bearings in the early stage.

Теоретические исследования процесса смешивания кормов центробежным смесителем

The article presents theoretical studies aimed at understanding the physical processes occurring in the production of compound feeds using a centrifugal cascade mixer, which open up new horizons in optimizing feed production technology. Special attention in these studies is paid to the analysis of the mixing process, which, as it turned out, is closely interrelated with the principles of operation of the dispenser. As a result of the analysis, it was found that the dispenser and mixer should be considered as a single system, regardless of the mixing mechanism used in the production process. This made it possible to develop a cascade mixer of feed materials of centrifugal type and continuous action, which is an innovative solution in the field of feed production. This mixer provides a homogeneous mixture of loose feed components and can be integrated into production lines for the preparation of both feed and combined mineral briquettes. The key advantage of the proposed mixer is the use of a centrifugal force field, which, unlike the gravity field, provides more efficient mixing. In this case, the process of preparing compound feeds can be represented as a sequence of three key stages: the metered supply of components, the introduction of one component into the composition of another, transportation (removal) of the finished product from the working area of the mixer. The research allowed us to obtain analytical dependences of the components of the centrifugal force on the angular velocity of the rotor and its radius. This allows you to optimize the operating parameters of the mixer to achieve maximum mixing efficiency. During the research, a mathematical model of the spatial motion of a particle in a mixer was developed. The model is presented in the form of Lagrange equations of the second kind, the generalized coordinates of which are the displacement of the particle along the cone and the polar angle of the particle position. This makes it possible to solve the optimization problems of centrifugal mixers according to a given objective function. Thanks to the conducted research, it was possible to develop an innovative mixer that provides a homogeneous mixture of components of loose compound feeds. The use of centrifugal forces in the mixer made it possible to increase the mixing efficiency and ensure that the final product meets the requirements of the formulation. The research results are of great practical importance and can be used to improve feed production technologies. Keywords: COMPOUND FEED, FEED MIXTURE, MIXER, DISPENSER, CENTRIFUGAL MIXER, PARTICLE MOTION IN THE MIXER

Modeling and Simulation of Fractional PID Controller for Under-actuated Inverted Pendulum Mechanical System

The stabilization of the non-linear inverted pendulum system requires a robust control strategy, as this system is inherently unstable and sensitive to disturbances. This research utilizes Lagrangian mechanics, a powerful technique in analytical dynamics, to derive the mathematical representation of the system. By applying the principles of Lagrangian dynamics, we can accurately model the energies involved and derive the equations of motion that govern the pendulum’s behavior. Following this, state-space feedback is employed to determine the Proportional, Integral, and Derivative (PID) values essential for effective control. This control strategy is particularly useful due to its ability to minimize error over time and ensure stability. To further enhance the control process, a comprehensive mathematical model is developed to establish the transfer function that correlates the pendulum's angle with the displacement of the cart. This relationship is crucial for understanding how changes in the cart's position affect the pendulum's stability. To validate the proposed control law, extensive simulations are conducted, allowing for comparative analysis against an Integer Order Controller. These simulations not only highlight the effectiveness of the PID controller but also provide insights into the dynamic behavior of the system under various conditions. The results demonstrate significant improvements in settling time and overshoot, showcasing enhanced performance metrics for the selected objective functions. This research contributes to the broader field of control systems engineering, suggesting that advanced control strategies can effectively manage complex, non-linear systems.

Study of first-order quantum corrections of thermodynamics to a Dyonic black hole solution surrounded by a perfect fluid

In this work, we review the metric for a dyonic global monopole with a perfect fluid. To calculate the standard temperature of a Dyonic black hole surrounded by a perfect fluid, we analyze the graphical interpretation of the Hawking temperature concerning the horizon under the effects of the black hole's surrounding field and electric-magnetic charges. For this purpose, we follow the semi-classical method, the Wentzel-Kramers-Brillouin (WKB) approximation, and the Lagrangian equation in the presence of quantum gravity as seen in the generalized uncertainty principle (GUP). We calculate the improved temperature of a Dyonic black hole using a bosonic tunneling strategy based on the Hamilton-Jacobi technique. We observe that the physical state of the Dyonic black hole is surrounded by a perfect fluid under the effects of the black hole solution and the gravity parameter. Further, we examine the improved entropy to study the influences of quantum gravity and black hole geometry on entropy. We explore the graphical behavior of entropy based on the black hole's horizon structure and analyze the influence of perfect fluid parameters, electric charge, magnetic charge, and quantum gravity on entropy. Finally, we investigate the unstable and stable conditions of a black hole via graphical results.

Fixed Switching Frequency Model Predictive Control and Passivity Based Control for DC-DC Converter

The DC-DC converter represents a crucial component in the renewable energy sources. The stability and dynamic capability enhancement of the DC/DC converter have emerged as a significant research topic in the current era. Model predictive control (MPC) is particularly prevalent due to its high dynamic response speed, simplicity of the controller design, and capacity for multi-objective optimization. However, the traditional finite control set model predictive control (FCS-MPC) method is suffer to variable switching frequency and vast computing. To improve the dynamic performance of the converter, a novel nonlinear control strategy named fixed switching frequency MPC and passivity-based control (PBC), named FSFPBMPC, is proposed, which could achieve fixed switching frequency and enhance the system's dynamic response speed. Firstly, the Euler-Lagrange (EL) model of the boost converter is established. Secondly, the relationship between duty cycle and MPC is established. Ultimately, the output voltage of PBC is incorporated into the cost function of the FCS-MPC. The characteristics of PBC power shaping and damping injection can enhance the system's immunity to interference, improve the system's dynamic response speed, and thus reinforce the system's stability. Then, depending on MATLAB, the simulation results proved that the proposed strategy has the same effect as we expected.

Unified non-hourglass formulation for total Lagrangian SPH solid dynamics

AbstractThe persistence of hourglass modes poses a significant numerical instability issue in total Lagrangian smoothed particle hydrodynamics (TLSPH) solid dynamics, especially when dealing with substantial deformations, regardless of material properties. However, existing hourglass control methods have shown effectiveness only within limited applications. Thus far, a comprehensive solution capable of addressing hourglass issues across a wide range of material models, including elasticity, plasticity, and anisotropy, remains elusive. In this study, we introduce a unified TLSPH formulation grounded in volumetric-deviatoric stress decomposition, aimed at fundamentally mitigating hourglass modes in general simulations. Different conceptually from previous approaches using stress points or extra viscous or hourglass-control stresses within the momentum equation, our formulation is based on the weighted average of a standard but hourglass-prone formulation and an essentially non-hourglass formulation for elastic materials, employing a single limiter to dynamically adjust the weighting between the two formulations. Crucially, the dimensionless characteristic of the formulation enables seamless handling of complex material models. To validate the effectiveness of our formulation, we conduct simulations across a range of benchmark cases involving elastic, plastic, and anisotropic materials. To illustrate its versatility, we apply the formulation to simulate a complex scenario involving viscous plastic Oobleck material, contacts, and very large deformation. Our work addresses a critical gap in TLSPH simulations by offering a unified approach to mitigate hourglass modes, enhancing the reliability and accuracy of simulations across diverse material models and complex scenarios.

Optimal Regularity for Lagrangian Mean Curvature Type Equations

Optimal Regularity for Lagrangian Mean Curvature Type Equations

Piezoelectric energy harvesting from walking motion of a passive biped robot model with flexible legs

Passive dynamic walking biped robots utilize the inherent dynamics of their structure to achieve walking motion without continuous actuation, enhancing energy efficiency and stability. Replacing conventional rigid legs with flexible elastic beams in the simplest passive walker will be associated with undesirable vibrations. Despite the positive effect of the damper in reducing vibrations, the current research has used the piezoelectric energy harvesting with the aim of reducing the wasted energy of the gait cycle and improving its stability range. For this purpose, the electromechanical Lagrange equations are determined by applying Hamilton's principle and the assumed mode method and then, the new definition of the step function is obtained through numerical methods, solving a boundary value problem to establish proper initial conditions for stable walking. Numerical simulations indicate that the piezoelectric energy harvester can create a stable period-one gait cycle by optimizing the length of the piezo layers attached to the undamped elastic legs despite the presence of small vibrations. Additional results are presented in bifurcation diagrams, investigating the effect of important physical and structural parameters such as slope angle, length and thickness of the piezo layer.