Euler-Lagrange Research Articles

Analysis of dynamic modeling and power flow of ABH laminated composite thin beam with elastic boundaries

To investigate the energy distribution characteristics of a laminated structure with an acoustic black hole (ABH), this study introduces the dynamic modeling and power flow analysis of an ABH laminated thin beam (ABH laminated beam) with elastic boundary conditions for the first time. ABH profile is defined by a power function in the dynamic modeling process. Utilizing the Euler-Bernoulli beam theory, the constitutive equation of ABH laminated beams is formulated by using the isogeometric method. Three sets of artificial springs are employed to replicate the elastic boundary conditions, incorporating the potential energy of the springs. Subsequently, the dynamic model of the ABH laminated beam is developed by solving the Lagrange equation concerning the total energy within the uniform region, ABH region, and boundary. Through numerical examples, the convergence and accuracy of the proposed modeling approach are validated by comparing the results with those obtained from COMSOL Multiphysics and experimental data. The analysis of power flow and structural intensity elucidates the energy propagation behavior in ABH laminated beams and the underlying mechanism of the ABH effect. The results demonstrate that the ABH laminated beam displays remarkable absorption and dissipation characteristics, introducing a new perspective for the design and application of ABH structures.

Analysis of nonlinear vibration of lateral-torsional coupling for drill string in deviated well

To clarify the motion characteristic of the drill string in deviated well, the nonlinear dynamic model of lateral-torsional coupling for drill string system is established by the Lagrange equation. This model incorporates the contact between the drill string and borehole wall, torque dissipation, and borehole trajectory effects. Additionally, the contact behavior using linear elastic contact model is simulated. Meanwhile, the torque transfer law in the drill string system is described by a discrete torque-drag model. Finally, numerical simulations are employed to determine the dynamic properties of the drill string system. The results reveal that friction losses in drill string systems are increased with higher well inclination angles. The motion of BHA along the x direction is predominantly concentrated near the wellhole center at inclination angle of 65°, while in the y direction it primarily focuses on the low side of the wellhole. An increase in inclination angle leads to a more prominent occurrence of stick-slip motion in the drill string. When inclination angle more than 25°, there is a slightly higher collision frequency observed between the BHA and the low side of the borehole wall compared to that with the upper side. When increasing the WOB (weight on bit) to 160 kN, stick-slip motion becomes more pronounced within the drill string. Through parametric dynamics analysis of the drill string system, the rotary speed can be controlled in range from 40 to 70 r/min, and the WOB should be restricted in range from 20 to 40 kN in well depth 5000 m.

Breakage characteristics of large mineral particles in pneumatic conveying

Pneumatic conveying of mineral particles is widely used in underground roadway material transportation and support. However, the breakage of large particles at bends is inevitable. Based on a coupled Computational Fluid Dynamics (CFD) and Discrete Element Method (DEM) method, the Euler-Lagrange equation was used to investigate the breakage of quartz particles (8 to 12 mm) during pneumatic conveying. By the Tavares UFRJ model, single large particle transport was simulated and the particle breakage was described as preliminary breakage and final breakage. The pneumatic conveying of multiple particles was simulated, and a detailed analysis was conducted on the effects of airflow velocity, particle size, and gravity direction on the particle breakage rate. The results indicated that the particle breakage rate increased with the rise in airflow velocity and particle size. The breakage rate was highest when the direction of gravity was in the negative x-direction and lowest in the negative y-direction.

A new variational integrator for constrained mechanical system dynamics

A new variational integrator is proposed to solve constrained mechanical systems. The main distinguishing feature of the present integrator comes from the distinct discretization of Lagrangians based on the Hamilton's principle in its most general form. Specifically, Hermite interpolation is used for discrete positions, which provides at least C1 continuity for generalized coordinates. The velocities and momentums are interpolated using quadratic polynomials for the consistency, such that the kinematic relation between velocities and positions can be exactly satisfied. Meanwhile, the Gauss-Legendre quadrature rule is employed to guarantee the accuracy of discrete Lagrange equations. To tackle constrained mechanical systems, a coordinate partition approach is used to eliminate the constraint equations. The local incremental rotation vector is exploited to get rid of rotation singularities in spatial problems. Moreover, an adaptive stepsize strategy is implemented to improve the efficiency. The strengths of the new integrator lie in the accessible large step sizes in the simulation and its global second-order accuracy for positions as well as velocities. Several examples are performed and analyzed to validate its accuracy and capabilities.

Gust response alleviation of the aircraft wing aeroelastic systems in supersonic flows using nonlinear energy sink

In order to reduce the gust response level of the aircraft wing aeroelastic systems in supersonic flows, the nonlinear energy sink (NES) is introduced into the aeroelastic system, and the effect of the nonlinear energy sink on the aeroelastic gust response is studied. The gust response dynamic model without and with NES is established by Lagrange equation, considering the cubic nonlinear characteristics of both plunge and pitch stiffness. The piston theory model as well as the '1-cos' model are employed to describe the unsteady aerodynamic load and the gust velocity, respectively. Besides, the comparison study of the gust response amplitude and energy characteristics of the wing aeroelastic systems with and without NES are analyzed. For systems with NES, the amplitude of gust dynamic response decreases rapidly, while vibration can be transmitted to the NES mechanism and causing its vibration. From an energy perspective, about 72.15 % of the original vibration energy has been transferred into NES, and further consumed through its damping. The influence of NES parameters, such as frequency ratio, mass ratio, damping ratio, and installation position on gust response, is discussed. Numerical simulations demonstrate that: as the frequency ratio, damping ratio and mass ratio increase, the dynamic response amplitude of the aeroelastic system decays faster, and the energy transferred from the wing system also increases. In addition, an optimal relative distance for the installation of NES can be found with best gust alleviation effect. The influence of nonlinear energy sink on the targeted transfer of vibration energy in aeroelastic systems are analyzed under different working conditions. The NES shows great gust alleviation effect and good robustness under different flight speeds, gust intensities, and gust lengths.

Comparisons of Inverse Dynamics Formulations in a Spatial Redundantly Actuated Parallel Mechanism Constrained by Two Point Contact Higher Kinematic Pairs.

A spatial redundantly actuated parallel mechanism (RAPM) constrained by two point contact higher kinematic pairs (HKPs) has been designed, arising from the inspiration of mastication in human beings: the end effector is the lower jaw, the six kinematic chains are the primary chewing muscles, and the constraints at HKPs are the temporomandibular joints. In this paper, firstly, the constrained motions of the mechanism are described in detail; thereafter, five models are formulated by the well-known Newton-Euler's law, the Lagrangian equations, and the principle of virtual work, to explore its rigid-body inverse dynamics. The symbolic results show that the model structures based on these approaches are quite different: the model via the Newton-Euler law well reflects the nature of the mechanism in terms of the constraint forces from HKPs with six equations and eight unknowns, and the existence of reaction forces at the spherical joints is tightly dependent on the number of kinematic chains. In comparison, from the latter two methods, the constraint forces and the reaction forces at spherical joints do not appear in the four models in which there are only four equations and six unknowns. Further, by using the dynamics model of the non-redundantly actuated counterpart as the core in both the second models from the energy and virtual work-related methods, their computational cost is only about 16.7% and 36.63% of the two first models, respectively. Finally, the comparisons between the dynamics models of the RAPM and its counterpart clarify that the HKP constraints greatly alter the model structures and raise the technical difficulties.

An immersed multi-material arbitrary Lagrangian–Eulerian finite element method for fluid–structure-interaction problems

Fluid–structure-interaction (FSI) phenomena are widely concerned in engineering practice and challenge current numerical methods. In this article, the finite element method is strongly coupled with the multi-material arbitrary Lagrangian–Eulerian (MMALE) method to develop a monolithic FSI method named the immersed multi-material arbitrary Lagrangian–Eulerian finite element method (IALEFEM). By immersing the finite elements in the MMALE computational grid, the fluid–solid interface is directly tracked by the element boundary with accurate normal directions. The fluid–structure-interaction is implicitly implemented by assembling the nodal variables and updating the Lagrangian momentum equation on the MMALE grid. Combining the advantages of both MMALE and FEM with the immersed boundary method, the IALEFEM is effective for solving complicated FSI problems with multi-material fluid flow. A slip fluid–structure-interaction method is also proposed to enhance the computational accuracy in simulating FSI problems with significantly different velocity fields. The accuracy and effectiveness of the IALEFEM are verified and validated by several benchmark numerical examples including the shock-cylinder obstacle interaction, flexible panel deformation induced by shock wave, dam break problem with large structural deformation, water entry of a wedge, fragmentation of a cylinder shell induced by blast, response of elastic plate subjected to spherical near-field explosion and structural damage of open-frame building under blast loading.

Impact of compliant electrodes on the dynamics of electromagnetoactive membranes

The dynamics of electromagnetoactive polymer (EMAP) membranes have attracted much attention recently because of their wide range of modern robotic applications. Such applications majorly centered on how the dynamics of this novel class of membranes are affected by the mechanical behavior of the compliant electrode. This article presents the dynamic modeling and analysis of EMAP membranes, examining how the inertia of the electrode, coupled with its inherent viscoelastic properties, impacts its dynamic performance. Both the compression and suspension stages of the membrane are covered here in broad terms. An Euler–Lagrange equation of motion is implemented to deduce the governing dynamic model equation of the membrane system. The findings of the model solutions provide preliminary insights to characterize the dynamic response, instability analysis, periodic behavior, and resonance properties across varying parameters such as inertia, electric field, magnetic field, and prestress. Moreover, the study also evaluates the periodicity and stability of the nonlinear oscillations using Poincaré maps and phase portraits, facilitating an assessment of quasi-periodic to periodic transitions.

A unified nonlinear dynamic model for bolted flange joint disk-drum structures under different interface states: Theory and experiment

Bolted flange joint disk-drum structures (BFJDDSs) are key components in aero-engines, however, bolt looseness inevitably occurs due to extreme service conditions. This can cause the bolted joint interface to exhibit complex contact behaviors, which significantly complicate the vibration characteristics of BFJDDSs. Existing dynamic models for BFJDDSs have not considered the effect of bolt looseness (slip and separation), so their vibration behaviors are not well understood. In this study, a unified nonlinear dynamic model for BFJDDSs considering different interface states (stick, slip, and separation) is proposed, which is effective under both bolt looseness and non-looseness (stick) conditions. The bi-linear hysteretic model combined with the piecewise linear model is used to simultaneously consider different interface states. The Kirchhoff plate theory, the Sanders’ shell theory, and the Euler-Bernoulli beam theory are used to derive the energy functions of the disk, drum, and flange, respectively. Then, the Lagrange equations are employed to derive the governing equations of BFJDDSs. Modal and nonlinear forced vibration experiments are carried out on a BFJDDS to prove the correctness of the proposed mathematical model. Research results show that the proposed mathematical model is able to achieve good predictions of nonlinear dynamic properties of BFJDDSs under both bolt looseness and non-looseness conditions. This model is also able to well predict the jumping phenomenon observed in the experiment.

Design and dynamic analysis of a spanwise morphing wing for Mars exploration

The application of spanwise morphing wings makes it possible to transport and launch large-size Mars exploration UAVs. This article proposes a modular variable spanwise morphing wing for high-aspect-ratio aircrafts, and analyses the characteristics of the morphing wing by theoretical analysis, numerical simulation and experimental verification. The novel spanwise morphing wing is based on the Sarrus-inspired deployable structure, which can increase the wing's lift by changing the spanwise length. The pre-strain torsion springs are assembled to provide the initial driving moment of the morphing mechanism. A regular triangle cross section is selected by evaluating the bending and torsional stiffness, and deployable triangular prism mechanisms are identified. Through releasing the pre-strain torsion springs to achieve expansion and implant Sarrus linkages along the straight motion paths. A lockable structure is designed, and the main factors affecting the self-locking property are analysed. A rigid origami skin is proposed, which maintains the airfoil's continuous smoothness after spanwise morphing. The kinematics of the spanwise morphing wing unit is developed using the Lagrange equation, and the theoretical models of morphing wings with different numbers of units are obtained. The unit numbers influence fully expanded time, and the time decreases gradually along the wingtip's direction. Finally, a one-way fluid-structure interaction analysis is performed to investigate the spanwise morphing mechanism and origami skin response under aerodynamic loads. Results show that the skeleton mechanism's maximum stress is below the material's yield strength, and the origami skin's maximum out-of-plane deformation is less than 0.5% of the wing chord, which provides a necessary theoretical basis for applying the spanwise morphing wing.

Exergy Flow as a Unifying Physical Quantity in Applying Dissipative Lagrangian Fluid Mechanics to Integrated Energy Systems.

Highly integrated energy systems are on the rise due to increasing global demand. To capture the underlying physics of such interdisciplinary systems, we need a modern framework that unifies all forms of energy. Here, we apply modified Lagrangian mechanics to the description of multi-energy systems. Based on the minimum entropy production principle, we revisit fluid mechanics in the presence of both mechanical and thermal dissipations and propose using exergy flow as the unifying Lagrangian across different forms of energy. We illustrate our theoretical framework by modeling a one-dimensional system with coupled electricity and heat. We map the exergy loss rate in real space and obtain the total exergy changes. Under steady-state conditions, our theory agrees with the traditional formula but incorporates more physical considerations such as viscous dissipation. The integral form of our theory also allows us to go beyond steady-state calculations and visualize the local, time-dependent exergy flow density everywhere in the system. Expandable to a wide range of applications, our theoretical framework provides the basis for developing versatile models in integrated energy systems.

Vibrational and stability analysis of planar double pendulum dynamics near resonance

AbstractThe focus of this paper is to examine the motion of a novel double pendulum (DP) system with two degrees of freedom (DOF). This system operates under specific constraints to follow a Lissajous curve, with its pivot point moving along this path in a plane. The nonlinear differential equations governing this system are derived using Lagrange's equations. Their analytical solutions (AS) are subsequently calculated using the multiple-scales method (MSM), which provides higher-order approximations. These solutions are considered new, as the traditional MSM has been applied to this novel system for the first time. Additionally, the accuracy of these solutions is validated through numerical results obtained using the fourth-order Runge–Kutta method. The solvability conditions and characteristic exponents are determined based on resonance cases. The Routh–Hurwitz criteria (RHC) are employed to assess the stability of the fixed points corresponding to the steady-state solutions. They are also used to demonstrate the frequency response curves. The nonlinear stability analysis is performed by examining the stability and instability ranges. Resonance curves and time history plots are presented to analyze the behavior of the system for specific parameter values. The investigation delves into a comprehensive analysis of bifurcation diagrams (BDs) and Lyapunov exponent spectra (LEs), aiming to uncover the various types of motion present within the system. Systematic examination of these charts reveals critical insights into transitions between stable, quasi-stable, and chaotic dynamical behaviors. This work has practical applications in various fields, such as robotics, pump compressors, rotor dynamics, and transportation devices. It can be used to study the vibrational motion of these systems.

Integration of 3-D 4-directional braided composites into dynamic modeling of plates with large overall motions

In this study, a dynamic model of a 3-D 4-directional braided composite plate with large overall motions is proposed. Based on the Kirchhoff’s plate hypothesis, the deformation of the orthotropic plate is described. The assumed mode method is employed to discretize the displacement variables, and the rigid-flexible coupled dynamic model of the plate is derived from the Lagrange equation. The equivalent stiffness matrices of 3-D 4-directional braided composites with different braided angles are obtained using the homogenization method. The effects of different velocities and braided angles on the dynamic characteristics of the plate with translational motion and fixed-axis rotation, respectively, are investigated. The present study lays the theoretical foundation for analyzing the dynamic properties of orthotropic plates in engineering components. For example, the dynamic model established in this study can be used to estimate the dynamic responses of braided composite parts in aeronautical structures.

The Euler-Lagrange equations of nabla derivatives for variational approach to optimization problems on time scales

This paper investigates the variational approach using nabla (denoted as ∇) within the framework of time scales. By employing two different methods, we derive the Euler-Lagrange equations for first-order variational approach to optimization problems involving exponential functions, as well as for those with both exponential functions and their ∇-derivatives. To establish the high-order variational approach to optimization problem, we present the Leibniz Formula for ∇-derivatives along with its proof. Additionally, we determine the high-order variational approach to optimization problem incorporating ∇-derivatives of exponential functions. Through these analyses, we aim to contribute to the understanding and application of the variational calculus on time scales, offering insights into the behavior of dynamic systems governed by exponential functions and their derivatives.

Global reliability assessment of coupled transmission tower-insulator-line systems considering soil-structure interaction subjected to multi-hazard of wind and ice

This study aims to assess the global reliability of coupled transmission tower-insulator-line systems considering soil-structure interaction (CTTILSs-SSI) subjected to multi-hazard of wind and ice. Firstly, two simplified models of CTTILSs-SSI under stochastic wind and ice loads are established, and their governing equations are derived through the Lagrange equation. On this basis, a global reliability assessment framework of CTTILSs-SSI is proposed based on the improved maximum entropy method, in which the global performance function of CTTILSs-SSI is defined via the state variable description method. Finally, a practical ultra-high voltage overhead transmission line located on the icing zone is chosen to illustrate the proposed framework. Moreover, to investigate the influence of soil-structure interaction (SSI) and soil types on the global reliability of transmission tower-insulator-line systems (TTILSs), the global reliability of CTTILSs-SSI with various soil types is compared with that of a fixed base system. The results indicate that under the in-plane wind and ice loads, the global failure probability of TTILSs is almost zero, and it is essentially not affected by the SSI effect and soil types. However, under the out-of-plane wind and ice loads, the global failure probability of CTTILSs-SSI is higher than that of the fixed base system, and the global failure probability of CTTILSs-SSI increases as the soil stiffness decreases. Accordingly, it may be more reasonable to consider the SSI effect into the global reliability analysis of TTILSs subjected to ice and out-of-plane wind loads, particularly when the soil is soft.

Nonlinear interactions of an n-layer X-shape low-frequency vibration isolator equipped with a nonlinear vibration absorber at 1:1 internal resonance: analytical and numerical investigations

This work addresses, for the first time, the nonlinear dynamics of an n− layers bioinspired X− shape low-frequency vibration isolation system equipped with a nonlinear vibration absorber. The comprehensive mathematical model governing the two-degree-of-freedom system has been derived using Lagrangian mechanics. The method of averaging was applied to obtain an accurate analytical solution for the derived mathematical model. Utilizing the established analytical solution, the dynamical characteristics of the X− shape vibration isolator have been explored both as a single-degree-of-freedom system and after coupling with the proposed vibration absorber, as a two-degree-of-freedom system. The influence of different system parameters, including the number of structure layers, the initial inclination angle, the X− shape rod length, and the absorber stiffness and damping coefficients, on the vibration isolation performance has been investigated. The main findings indicate that while the single-degree-of-freedom X− shape low-frequency vibration isolator performs well in eliminating low-frequency vibration excitations, it exhibits strong vibration levels when the base excitation frequency exceeds the structure's natural frequency. Additionally, it was found that coupling a highly damped linear vibration absorber to the X− shape isolator not only eliminates the strong oscillations of the structure when subjected to high-frequency base excitations but also enhances the structure's low-frequency vibration isolation performance. Moreover, it was reported that integrating a nonlinear vibration absorber with either soft or hard spring characteristics instead of a linear one may degrade the vibration isolation performance of the structure rather than enhance it. Finally, numerical validation of all the analytical results was performed, which showed excellent correspondence with the analytical findings.

Numerical modeling of earthquake-induced landslides using updated Lagrangian nonlocal general particle dynamics method

Developing a robust numerical method to model earthquake-induced landslides has long been a persistent challenge in the field of computational geotechnical engineering. Recently, the meshless methods based on nonlocal theory have piqued the interest of researchers. However, the application of nonlocal theory in seismic analysis is currently limited. This paper proposed a numerical framework based on the updated Lagrangian nonlocal general particle dynamics (UL-NGPD) method to analyze earthquake-induced landslide problems. The UL-NGPD method benefiting from the update support domain can capture the whole process of slope run-out induced by earthquakes. To enhance the numerical stability of the proposed method, several optimizing strategies are proposed. In the current framework, the seismic waves are input through the proposed boundary treatments of nonlocal form. Besides, a nonlocal friction model with velocity-weakening is proposed to accurately simulate the movement process of soils on a rocky sliding bed. The proposed framework is validated by the simulations of several classic problems, including the collapse of sand and the shake table test. The performance of the proposed approach is further demonstrated through simulating the landslides induced by earthquakes. The numerical results consistent with recorded data indicate that the UL-NGPD method has an excellent capacity to deal with earthquake-induced landslide problems.

A Mumford-Shah Model-based Method for Inpainting Markers in Ultrasound Images

Abstract During ultrasound diagnosis in clinics, clinicians often mark the lesion area, resulting in the presence of unnecessary objects in the collected ultrasound images. These markers potentially affect subsequent image analysis. To address this problem, we proposed an image inpainting method that combines image processing with Mumford-Shah algorithm to remove markers from ultrasound images. The proposed method consists of two parts. First, the input ultrasound images were processed by several image processing algorithms, including contrast enhancement, image binarization, edge detection, hole filling, and connected domain marking. These algorithms were combined to extract the mask of the manual marker in the ultrasound images. Second, the original ultrasound images and mask images were put into the Mumford-Shah model to implement ultrasound image inpainting by minimizing the following energy functional. The continuous iteration of Euler-Lagrange equation was employed to update the iteration item. If the difference between the two output results was less than the pre-set threshold, the output result could be regarded as the repaired ultrasound image. Experimental results show that the designed image processing method accurately extracted the binary mask, whether they are colourful or binary, number or line in ultrasound images. Meanwhile, the improved Mumford-Shah model effectively implemented the ultrasound image inpainting, with the accuracy of 96.55%. Therefore, the proposed method can be used to remove unnecessary objects in ultrasound images, which is beneficial to improve the accuracy of the artificial intelligence model.

Discrete-time optimal control problems with time delay argument: New discrete-time Euler–Lagrange equations with delay

Discrete-time optimal control problems are a crucial type of control problems that deal with a dynamic system evolving in discrete time-steps. This paper introduces a new technique for solving linear discrete-time optimal control problems with state delays, applicable to both finite and infinite time horizons. Our method employs a Riccati matrix equation, optimizing control strategies and ensuring system stability through bounded control inputs. We adopt a Bolza problem for the performance index, which guides the classification of control issues. The technique simplifies problems into manageable Riccati matrix equations using the Euler–Lagrange equations and Pontryagin maximum principle, ensuring stability and necessary condition compliance. The paper validates the approach with numerical examples from quantum mechanics and classical physics, demonstrating its practicality and potential for broader application.

Elastic ribbons in bubble columns: When elasticity, capillarity, and gravity govern equilibrium configurations.

Taking advantage of the competition between elasticity and capillarity has proven to be an efficient way to design structures by folding, bending, or assembling elastic objects in contact with liquid interfaces. Elastocapillary effects often occur at scales where gravity does not play an important role, such as in microfabrication processes. However, the influence of gravity can become significant at the desktop scale, which is relevant for numerous situations including model experiments used to provide a fundamental physics understanding, working at easily accessible scales. We focus here on the case of elastic ribbons placed in two-dimensional bubble columns: by introducing an elastic ribbon inside the central soap films of a staircase bubble structure in a square cross-sectioncolumn, the deviation from Plateau's laws (capillarity-dominated case dictating the shape of usual foams) can be quantified as a function of the rigidity of the ribbon. For long ribbons, gravity cannot be neglected. We provide a detailed theoretical analysis of the ribbon profile, taking into account capillarity, elasticity, and gravity. We compute the total energy of the system and perform energy minimization under constraints, using Lagrangian mechanics. The model is then validated via a comparison with experiments with three different ribbon thicknesses.