Euler-Lagrange Research Articles

A Physical Approach to Lagrangian Equations with Bundle Structure for Minkowski 3-Space

Minkowski 3-space is important because it is the spatial structure in which physical events occur. Therefore, in this study, we examined the time-dependent Lagrangian energy equations in Minkowski 3-space. To facilitate the examination of the necessary mechanical structure and solutions with respect to time, we employed the jet bundle structure. This approach has made the creation of the necessary geometric structures more comprehensible in terms of coordinate basis. To interpret the obtained Lagrangian energy equations and to understand the significance of the time parameter, an example is also provided in our study.

Large amplitude vibration of rotating graphene reinforced titanium alloy dovetailed blades with general boundary conditions

Engine blades are generally installed on disks via dovetail joints, which can alleviate stress concentrations at the blade root and prevent heat transfer. As a result, the connection stiffness is strongly affected by the clearance between the blade and disk as well as the rotation speed of the blade. This paper first proposes a functionally graded graphene-reinforced titanium alloy trapezoid plate with elastic boundary conditions to investigate the large amplitude vibration of rotating dovetailed blades. The elastic boundaries are modelled by applying a series of artificial springs, and the related admissible displacement functions are constructed by orthogonal polynomials through the Gram–Schmidt process. The governing equations of the functionally graded graphene platelet reinforced composite (FG-GPLRC) trapezoid plate are derived by the first-order shear deformation theory (FSDT) and Lagrange equation. The numerical results of the large amplitude vibration of the plate are obtained by the weighted residual method combined with a direct iterative algorithm and verified through analytical solutions of the harmonic balance method. The effects of the stiffness of the constraint springs, rotation speed and structural parameters on the linear and nonlinear free vibrations of the FG-GPLRC trapezoid plate are thoroughly studied.

Nonlinear pitch oscillation and control of a gravitational stabilized sailcraft perturbed by solar pressure torque

The sailcraft based space missions is deeply influenced by its attitude dynamics since the solar radiation pressure force for propellantless orbital maneuver is dependent on its orientation with respect to the sun line. Therefore, it is essential to explore the attitude motion evolution of a sailcraft and understand its attitude dynamic characteristics. To address this, nonlinear pitch dynamics of a sailcraft in an Earth orbit with a sliding mass as an attitude control actuator is focused on. Firstly, the Lagrange equation method is adopted to establish the pitch dynamics of a sailcraft subjected to the gravitational gradient torque, solar radiation pressure torque considering the Earth's occlusion of sunlight, damping torque, and the attitude control torque (this torque is generated by positioning the sliding mass at proper location). Secondly, the possible occurrence of chaotic pitch motion free of control torque is analytically predicted for the sailcraft in the circular and elliptical orbits using the Melnikov method respectively. The effectiveness of the Melnikov method is verified using various numerical methods such as phase plane, Poincaré sections, and power spectrum density. Furthermore, the influence of various parameters on the chaotic evolution is analyzed in detail. Lastly, to control the chaotic pitch motion onto the selected unstable periodic orbit obtained using the close return pairs, a sliding mode controller based on control input anti-saturation considering the position restriction of the sliding mass is developed. The effectiveness of the developed pitch chaos stabilization controller is verified by two numerical simulation cases.

Dynamic Model and PD Control with Forces Compensation of Dual-Stage Gough-Stewart Platform

This paper investigates a dual-stage Gough-Stewart platform. The lower platform is responsible for simulating the oscillations of moving vehicles such as cars, ships, and airplanes. The upper platform is connected to devices that require either balance stabilization or motion stabilization according to specific requirements. The dynamic model of the robot system is derived in a general form based on the Lagrange equations of motion with Lagrange multipliers. Using these equations in a compact form, a PD controller with forces compensation in task space is designed for the robot system. Oscillation generation and balance stabilization are computed and simulated using the kinematic and dynamic parameters of two Bosch Rexroth robots. The computation and simulation results demonstrate the dynamic model's accuracy and the controller's effectiveness.

Towards an understanding of uncertainties in the Lagrangian analysis of moisture sources for tropical cyclone precipitation through a study case

Despite the increasing number of atmospheric moisture tracking tools, their validation is challenging due to the lack of observations. This work contributes to a better understanding of uncertainties in the moisture sources analysis for the precipitation of tropical cyclones (TCs) by assessing eight combinations of threshold values in tracking methods based on the Lagrangian water budget equation. We selected as a study case Hurricane Ida that formed in the North Atlantic basin in late August 2021 and extracted the air parcel trajectories from the global outputs of the Lagrangian FLEXPART model. Results indicate that the choice of relative humidity (RH) threshold for filtering precipitating parcels has a noticeable impact on the Lagrangian precipitation estimates. In addition, methods applying the atmospheric boundary layer restriction produce a weaker moisture source pattern than those accounting for moisture uptakes in the whole atmospheric column. In particular, methods imposing an RH restriction along the air parcel trajectories to filter out noise in moisture losses outperform the others, providing more reliable moisture source contributions. We also introduced a simple bias correction approach that further improves the reliability of moisture source representation.

An Axial-Vector Photon in a Mirror World

We discuss a theory in which the left- and right-handed axial-vector photons refer to long- and short-lived bosons of true neutrality, respectively. Such a difference in lifetimes expresses the unidenticality of masses, energies, and momenta of axial-vector photons of the different components, generalizing the classical Klein-Gordon equation to the case of C-odd types of particles with a nonzero spin. Together with a new Dirac equation for truly neutral particles with the half-integral spin, the latter admits the existence of the second type of the local axial-vector gauge transformation responsible for origination in a Lagrangian of an interaction Newton component, which gives an axial-vector mass to all the interacting particles and fields. The quantum axial-vector mass, energy, and momentum operators constitute herewith a new Schrödinger equation, confirming that each of them can individually influence on the matter field. They define at the new level, namely, at the level of the mass-charge structure of gauge invariance another Euler-Lagrange equation such that it has an axial-vector nature.

Static and structural dynamic analysis of thick panel kirigami deployable structures

Thick panel origami and kirigami concepts have been wildly used to design novel deployable structures in various engineering applications. However, these novel folding methods usually involve complex connected topologies, which may lead to unclear and intricate characterized relationships between system properties and structural parameters, e.g., the position of cutting creases, design parameters and hinge stiffness arrangement, etc. In this paper, we propose theoretical models to describe the static and dynamic properties of thick panel kirigami structure in the fully deployed configuration. Firstly, the connected topology of the origami and kirigami structure is analysed, and the internal coupling topology of the structure is obtained. Based on the compliant matrix method, the static model of the structure is presented, and the different crease cutting modes of origami and kirigami arrays are discussed. Then, the motion modes of slight oscillation of structure are discussed and the structural dynamic model is obtained based on the Lagrange equation and validated by simulation. On this basis, the sensitivity analysis of the parameters is carried out, and the optimization model is given based on the comprehensive performance evaluation function. A physical prototype is optimized and tested, which indicates that our model is valid. This paper provides models for the structural static and dynamic properties of thick panel kirigami structures with complex connected topology, and the findings have a potential to be developed in other thick panel structures with origami and kirigami folding concepts.

Quantitative profile decomposition and stability for a nonlocal Sobolev inequality

In this paper, we focus on studying the quantitative stability of the nonlocal Sobolev inequality given bySHL(∫RN(|x|−μ⁎|u|2μ⁎)|u|2μ⁎dx)12μ⁎≤∫RN|∇u|2dx,∀u∈D1,2(RN), where ⁎ denotes the convolution of functions, 2μ⁎:=2N−μN−2 and SHL are positive constants that depends solely on N and μ. For N≥3 and 0<μ<N, it is well-known that, up to translation and scaling, the nonlocal Sobolev inequality possesses a unique extremal function W[ξ,λ] that is positive and radially symmetric.Our research consists of three main parts. Firstly, we prove a result that provides quantitative stability of the nonlocal Sobolev inequality with the level of gradients. Secondly, we establish the stability of profile decomposition in relation to the Euler-Lagrange equation of the aforementioned inequality for nonnegative functions. Lastly, we investigate the quantitative stability of the nonlocal Sobolev inequality in the following form:‖∇u−∑i=1κ∇W[ξi,λi]‖L2≤C‖Δu+(1|x|μ⁎|u|2μ⁎)|u|2μ⁎−2u‖(D1,2(RN))−1, where the parameter region satisfies κ≥2, 3≤N<6−μ, μ∈(0,N) with 0<μ≤4, or in the case of dimension N≥3 and κ=1, μ∈(0,N) with 0<μ≤4.

An Approach to Growth Mechanics Based on the Analytical Mechanics of Nonholonomic Systems

Motivated by the convenience, in some biomechanical problems, of interpreting the mass balance law of a growing medium as a nonholonomic constraint on the time rate of a structural descriptor known as growth tensor, we employ some results of analytical mechanics to show that such constraint can be studied variationally. Our purpose is to move a step forward in the formulation of a field theory of the mechanics of volumetric growth by defining a Lagrangian function that incorporates the nonholonomic constraint of the mass balance. The knowledge of such Lagrangian function permits, on the one hand, to deduce the dynamic equations of a growing medium as the result of a variational procedure known as Hamilton–Suslov Principle (clearly, up to non-potential generalized forces that are accounted for by extending this procedure), and, on the other hand, to study the symmetries and conservation laws that pertain to a given growth problem. While this second issue is not investigated in this work, we focus on the first one, and we show that the Euler–Lagrange equations of the considered growing medium, which describe both its motion and the evolution of the growth tensor, can be obtained by reformulating a variational method developed by other authors. We discuss the main features of this method in the context of growth mechanics, and we show how our procedure is able to improve them.

In-orbit system identification of a flexible satellite with variable mass using dual Unscented Kalman filters

Modern space mission concepts are increasingly dependent on the robust and reliable deployment of spacecraft with large appendages, such as antennas, booms or solar panels. Such deployment requires the ability to properly capture and control the coupled system dynamics, which requires accurate in-orbit system identification of the mass and structural properties. This paper utilises dual Unscented Kalman filters (DUKF) to develop an online system identification strategy that captures both the structural and mass properties, and the attitude and orbit dynamics. The dynamics of the flexible multibody problem are derived from the Lagrangian equations, with the flexible body characteristics modelled with finite element software. A genetic algorithm is used to optimise the accelerometer placement, and hence improve the DUKF performance. We demonstrate that this approach can accurately capture the coupled attitude, orbit, and structural dynamics, as well as being able to provide in-orbit updates for mass properties such as the moment of inertia. The methodology is explored for two illustrative cases: one in which the initial moment of inertia is incorrectly characterised, one in which the moment of inertia changes with time. In both cases, the DUKF approach captures both the system dynamics and the mass properties, which are captured with an error of less than 1%.

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.

Active boundary control of spinning beams with elastic constraints using piezoelectric stack actuator

A spinning beam structure is usually subjected to severe vibration. To suppress this vibration, piezoelectric stacking actuators are installed on the boundary of the spinning beam. Here, an active boundary control model for the vibration of spinning beam structures with elastic constraints is established. The dynamic equation of motion of the coupled system is derived by combining the Lagrange equation with the modified Fourier series method. An active control strategy is used to suppress the vibration of the spinning beam with elastic constraints. The vibration suppression effect of the proposed method under different spinning speeds and different elastic boundary conditions is discussed. It is proven that the current active control scheme can quickly suppress the vibration of the spinning beam structure with elastic constraints. The present control method can provide a new way to suppress the vibrations of spinning beam structures with nonideal boundaries in practical engineering.

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.

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.

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.

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.

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.