Euler Dynamics Research Articles

Alignments of triad phases in extreme one-dimensional Burgers flows.

We analyze the fine structure of nonlinear modal interactions in inviscid and viscous Burgers flows in 1D, which serve as toy models for the Euler and Navier-Stokes dynamics. This analysis is focused on preferential alignments characterizing the phases of Fourier modes participating in triadic interactions, which are key to determining the nature of energy fluxes between different scales. We develop diagnostic tools designed to probe the level of coherence among triadic interactions and apply them to Burgers flows corresponding to different initial conditions, including unimodal, extreme (in the sense of maximizing the growth of enstrophy in finite time), and generic. We find that in all cases triads involving energy-containing Fourier modes align their phases so as to maximize the energy flux toward small scales, and most of this flux is realized by only a handful of triads revealing a universal statistical distribution. We then identify individual triads making the largest contributions to the flux at different wave numbers and show that they represent a mixture of local and nonlocal interactions, with the latter becoming dominant at later times. These results point to the possibility of constructing a strongly reduced modal representation of Burgers flows that would require a much smaller number of degrees of freedom. Another interesting observation is that removing the spatial coherence from the extreme initial data (by randomizing the phases while retaining the magnitudes of the Fourier coefficients) does not profoundly change the nature of triadic interactions and synchronization as well as the resulting fluxes in these flows.

Computational Dynamics of Multi-Rigid-Body System in Screw Coordinate

This paper investigates the kinematics and dynamics of multi-rigid-body systems in screw form. The Newton–Euler dynamics equations are established in screw coordinates. All forces and torques of the multi-rigid-body system can be solved straightforwardly since they are explicit in the form of screw coordinates. The displacement and acceleration are unified in matrix form, which associates the kinematics and dynamics with variable of velocity. A one-step numerical algorithm only is needed to solve the displacements and accelerations. As a result, all absolute displacements, velocities, and accelerations are directly obtained by one kinematic equation. The kinematics and dynamics of Gough–Stewart platform validate this the method. In this paper, the kinematics and dynamics are carried out with the example of a Gough–Stewart platform, which represents the most complex multi-rigid-body system, to verify the computational dynamics method. The proposed algorithm is also fit for the kinematics and dynamics modeling of other multi-rigid-body systems.

Interaction of vortex stretching with wind power fluctuations

The transfer of turbulence kinetic energy from large to small scales occurs through vortex stretching. Also, statistical properties of the subgrid-scale energy fluxes depend on the alignment of the vorticity vector with the principal strain axis. A heuristic analysis of the present study indicates that vortex-stretching and the second invariant of the velocity gradient tensor provide a scale-adaptive parameterization of the subgrid-scale stresses and the local energy fluxes in the wakes of wind turbines. The scale-adaptivity underlies the restricted Euler dynamics of the filtered motion where vortex-stretching plays in the growth of the second invariant of filtered velocity gradient and the local energy transfer. We have analyzed wind power fluctuations in a utility-scale wind farm with 41 actuator disks. The numerical results show that the spectrum of the wind power fluctuations follows a power law with a logarithmic slope of −5/3. Furthermore, a brief analysis with the proper orthogonal decomposition method indicates that the maximum variability of wind power fluctuations depends on the incoming turbulence and its modulation by the wake interactions in wind farms.

Vectorized Formulation of Newton-Euler Dynamics for Efficiently Computing Three-Dimensional Folding Chains

Abstract Within the wide field of self-assembly, the self-folding chain has unique potential for reliable and repeatable assembly of three-dimensional structures as demonstrated by protein biosynthesis. This potential could be translated to self-reconfiguring robots by utilizing magnetic forces between the chain components as a driving force for the folding process. Due to the constraints introduced by the joints between the chain components, simulation of the dynamics of longer chains is computationally intensive and challenging. This article presents a novel analytical approach to formulate the Newton–Euler dynamics of a self-reconfiguring chain in a single vectorized differential equation. The vectorized differential equation allows for a convenient implementation of a parallel processing architecture using single instruction multiple data (SIMD) or graphical processing unit (GPU) computation and as a result can improve simulation time of rigid body chains. Properties of existing interpretations of the Newton–Euler and Euler–Lagrange algorithms are discussed in their efficiency to compute the dynamics of rigid body chains. Finally, GPU and SIMD-supported simulation, based on the vectorized Newton–Euler equations described in this article, are compared, showing a significant improvement in computation time using GPU architecture for long chains with certain chain geometry.

Mathematical modelling and virtual decomposition control of heavy-duty parallel–serial hydraulic manipulators

This paper proposes a novel modelling approach for a heavy-duty manipulator with parallel–serial structures connected in series. Each considered parallel–serial structure contains a revolute segment with rigid links connected by a passive revolute joint and actuated by a linear hydraulic actuator, thus forming a closed kinematic loop. In addition, prismatic segments, consisting of prismatic joints driven by hydraulic linear actuators, also are considered. Expressions for actuator forces are derived using the Newton–Euler (N–E) dynamics formulation. The derivation process does not assume massless actuators decoupled from manipulator links, which is common in the Lagrange dynamics formulation. Actuator pressure dynamics are included in the analysis, leading in total to a third-order system of ordinary differential equations (ODEs). With fewer parameters than its predecessors, the proposed model in the N–E framework inspires revision of the virtual decomposition control (VDC) systematic process to formulate a control law based on the new model. The virtual stability of each generic manipulator revolute and prismatic segment is obtained, leading to the Lyapunov stability of the entire robot.

Resonance-Averaged Solar Torque Dynamics for Tumbling Satellites

Observations indicate that the spin states of retired and otherwise defunct satellites are diverse and can change significantly over time. Understanding defunct satellite spin state evolution is important for solar radiation pressure modeling, active debris removal, satellite servicing, and space situational awareness. Research has shown that many defunct satellites in geosynchronous earth orbit (GEO) are primarily driven by solar radiation torques via the Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect. Recent exploration of YORP-driven nonprincipal axis tumbling has uncovered rich dynamic behavior, including tumbling cycles, spin–orbit coupling, and tumbling period resonances. Radar and optical observations of the defunct Geostationary Operational Environmental Satellite (GOES) 8 strongly suggest that it has been captured in a tumbling resonance at least once since 2018. Motivated by these findings, we develop a numerical resonance-averaged dynamic model to understand YORP-driven resonance capture, building on our earlier nonresonant averaging framework. This resonance-averaged model illuminates resonance capture mechanisms. Also, by averaging over the satellite’s rotation, this model is roughly 20 times faster to propagate than the full (Euler) dynamics. Overall, this allows for rapid, broad exploration to determine resonance capture durations, spin states changes, and the influence of initial spin rate and resonance order on resonance strength. This resonance-averaged model can be easily applied to other environmental perturbations.

Confined vortex surface and irreversibility. 1. Properties of exact solution

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution’s stability breaks the Euler dynamics’ time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a space-time region partitioned by time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into account by a new flux term, and can be integrated by method-of-line Runge-Kutta (RK) methods. The ELDG framework is designed as a generalization of the semi-Lagrangian (SL) DG method and classical Eulerian RK DG method for linear advection problems. It takes advantages of both formulations. In the EL DG framework, characteristics are approximated by a linear function in time, thus shapes of upstream cells are quadrilaterals in general two-dimensional problems. No quadratic-curved quadrilaterals are needed to design higher than second order schemes as in the SL DG scheme. On the other hand, the time step constraint from a classical Eulerian RK DG method is greatly mitigated, as it is evident from our theoretical and numerical investigations. Connection of the proposed EL DG method with the arbitrary Lagrangian-Eulerian (ALE) DG is observed. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.

Control Performance Evaluation of Serial Urology Manipulator by Virtual Prototyping

Prostatic hyperplasia and tumor are common diseases, and the minimally invasive surgery inserting the instruments through the urethra into the prostate is commonly conducted. Taking the robotic manipulator for such surgery into consideration, this paper analyses the workspace of the end effector, and proposes the distribution error of the fixed point and the tracking error of manipulator end effector on the cone bottom surface of the workspace as the basis for control implementation of the manipulator. The D-H coordinate system of the manipulator is established and the trajectory planning of the end effector in the Cartesian space is carried out. The digital model was established, and dynamics simulation was performed in Solidworks and Matlab/Simulink environment to guide the manipulator design. Trajectory mapping and synchronization control between virtual model and the actual manipulator are realized based on digital twin technique. The virtual manipulator can reflect the real-time state of the manipulator with data interaction by comparing the dynamics simulation results with the motor current values obtained by experiment. Experiment was carried out with PD feedback control and Newton–Euler dynamics based feedforward control to get the trajectory tracking characteristic of each motor, errors of the fixed point and tracking performance of the end effector of the manipulator. The results show that compared with PD feedback control, feed forward control implementation can achieve a reduction of 30.0% in the average error of the fixed point of the manipulator and a reduction of 33.3% in the maximum error.

Skidding research of a high-speed cylindrical roller bearing with beveled cage pockets

PurposeThis study aims to uncover the influencing mechanism of the tilt angles of the cage pocket walls of the high-speed cylindrical roller bearing on the bearing skidding.Design/methodology/approachA novel cylindrical roller bearing with the beveled cage pockets was proposed. Using the Hertz contact theory and the elastohydrodynamic and hydrodynamic lubrication formulas, the contact models of the bearing were built. Using the multibody kinematics and the Newton–Euler dynamics theory, a dynamics model of the bearing was established. Using the Runge–Kutta integration method, the dynamics simulations and analysis of the bearing were performed.FindingsThe simulation results show that the effects of the tilt angles of the front and rear walls of the pocket on the bearing skidding are remarkable. Under a 5° tilt angle of the front wall of the pocket and a 10° tilt angle of the rear wall, the bearing skidding can be effectively decreased in the rotational speed range of 10,000-70,000 r/min.Originality/valueIn this paper, a novel cylindrical roller bearing with the beveled cage pockets was proposed; a dynamics model of the bearing was established; the influence mechanism of the tilt angles of the front and rear walls of the pocket on the bearing skidding was investigated, which can provide fundamental theory basis for optimizing the pocket.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-01-2020-0035/

On planar self-folding magnetic chains: Comparison of Newton–Euler dynamics and internal energy optimisation

Within the wide field of self-assembly, the self-folding chain has the unique capability to pass through narrow openings, too small for the assembled structure, yet consists in one connected body. This paper presents a novel analytical framework and corresponding experimental setup to quantify the results of a self-folding process using magnetic forces at the centimetre-scale, with the aim to put experimental results and prediction methods in the context of surgical anchoring and therapy. Two possibilities to predict the folding of a chain of magnetic components in 2D are compared and investigated in an experimental setup. Folding prediction by system Coulomb energy, neglecting folding dynamics, is compared with a simulation of the system dynamics using a novel approach for 2D folding chains, derived from the Newton–Euler equations. The presented algorithm is designed for the parallel computation architecture of modern computer systems to be easily applicable and to achieve an improved simulation speed. The experimental setup for the self-folding chain used to validate the simulation results consists of a chain of magnetic components where movement is limited to one plane and the chain is agitated by the magnetic forces between the chain components. The folding process of the experimental setup is validated for its stability and predictability under different deployment modes. Finally, the results are discussed in light of the folding prediction of longer chains. The implications of the presented findings for a 3D folding chain are discussed together with the challenges to apply the novel dynamics simulation algorithm to the 3D case. The work clearly demonstrates the potential for this novel approach for complex self-folding applications such as magnetic compression anastomosis and anchoring in minimally invasive surgery.

Modeling and Stability Analysis of Coarse–Fine Composite Mechatronic System in UAV Multi-Gimbal Electro-Optical Pod

Coarse–fine composite mechatronic systems face numerous challenges due to the structural complexity and diversification of multi-gimbals. The core goal of this manuscript is to address the issue of the coarse-fine composite mechatronic system stability of a UAV (unmanned aerial vehicle) multi-gimbal electro-optical pod using USM-VCM (ultrasonic motor and voice coil motor) mechatronic design, Euler dynamics modeling, and stability DOB (disturbance observer) control. In response to this problem, a Hall effect electromagnetic circuit and USM-VCM drive acquisition circuit are designed. A Euler dynamics model in the Cartesian coordinate system is built to derive the kinematics coupling compensation matrix and mechanical parameter optimization method between the gimbals. Finally, the model is substituted into the DOB suppression control, which can monitor and compensate the motion coupling between the coarse–fine composite mechatronic systems in real time. Results show that the disturbance suppression impact of the DOB method with the Euler optimization model and USM-VCM mechatronic design is increased by up to 90% compared to the PID (proportion integration differentiation) method and 20% better than the traditional DOB method.

A Particle Filter for Stochastic Advection by Lie Transport: A Case Study for the Damped and Forced Incompressible Two-Dimensional Euler Equation

In this work, we combine a stochastic model reduction with a particle filter augmented with tempering and jittering, and apply the combined algorithm to a damped and forced incompressible two-dimensional Euler dynamics defined on a simply connected bounded domain. We show that using the combined algorithm, we are able to assimilate data from a reference system state (the “truth'') modeled by a highly resolved numerical solution of the flow that has roughly $3.1 \times 10^6$ degrees of freedom, into a stochastic system having two orders of magnitude less degrees of freedom, which is able to approximate the true state reasonably accurately for five large-scale eddy turnover times, using modest computational hardware. The model reduction is performed through the introduction of a stochastic advection by Lie transport (SALT) model as the signal on a coarser resolution. The SALT approach was introduced as a general theory using a geometric mechanics framework from Holm [Proc. A, 471 (2015)]. This work follows on the numerical implementation for SALT presented by Cotter et al. [SIAM Multiscale Model. Simul., 17 (2019), pp. 192--232] for the flow in consideration. The model reduction is substantial: the reduced SALT model has $4.9 \times 10^4$ degrees of freedom. Results from reliability tests on the assimilated system are also presented.

Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts

In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or $\alpha$-models. These models describe both nonlocal and local dynamics, with one example of the latter given by the Surface Quasi Geostrophy (SQG) model for which the existence of singularities is still under discussion. Furthermore, SQG is relevant both for atmosphere and ocean dynamics, and in particular, it is proposed to understand the oceanic submesoscale structures, front and filaments, associated with horizontal gradients of buoyancy. We discuss under which conditions the statistical theory suggests a principle of selective decay for the whole family of turbulent models, and then we explore the selective decay principle numerically, the transition to equilibrium and the formation of singularities, starting from initial conditions (i.c.s) corresponding to a hyperbolic saddle. We study the topological transitions in the flow configuration induced by filaments breaking. Furthermore we compare the theoretical equilibrium states, and the functional relation between the generalized vorticity $q$ and its correspondent stream function $\psi$, with the results of the simulations. For the particular i.c.s investigated, and domain used for the simulations, we have not noticed transition from tanh-like to sinh-like $\psi-q$ functional relation, which would be expected by the emergence of coherent structures that are, however, filtered by the time averages used to compute the mean fields.

A method of improving the anti-rudder beating of Si micromechanical gyro driven by rotating carrier

The Si micromechanical gyro driven by rotating carrier is a new kind of MEMS gyro. Since it has no a driving structure itself, it differs from the conventional Coriolis vibratory gyro. The gyro is installed on a rotating aircraft and utilizes the spinning of the carrier as a driving. Its sensing output is a amplitude-modulated signal. The amplitude is proportional to the transverse input angular velocity of the carrier, while frequency tunes the spin frequency of the carrier automatically. So the gyro is used for the damping of the single-channel control system of the rotating carrier. The control system uses a pair of the rudder surfaces to control the flight attitude of the carrier. The polar changing of the rudder surfaces will conduct a beating to the body of the carrier. In application, we found that the rudder beating distorts the gyro output signal. Thus the gyro will not work properly and even cause the channel jam of the control system. Aiming at the problem, in this paper we firstly introduce the structural principle and working mechanism of the gyro, and obtain its angular vibrating equation and the solution of the equation by using Euler dynamics equation of the rigid rotating around a fixed point. Then, we analyzed interference of the rudder beating on the gyro output signal in details. The analyses have indicated that the first harmonic of the rudder beating bring the interfering moment since the force arm between the rudder beating and the sensing axis of the gyro exists. Moreover, we tested the resonant frequency of the gyro. The tested result has shown that the second harmonic of the rudder beating is close to the resonant frequency. The situation will cause the resonance of the gyro. In order to improving the ability of anti-rudder beating of the gyro, we putted forward the installed method of 90 ° to eliminate the interfering force arm of the first harmonic. In addition, based on the model of the elastic torsion beams, we increased the thickness of the beams and enlarged the resonant frequency to avoid the resonance of the gyro. Finally, we carried out the experimental validation. The experimental results have demonstrated that the method can effectively suppress the influence of rudder beating on the output signal of the Si micromechanical gyro.

Singular behavior of a vortex layer in the zero thickness limit

The aim of this paper is to study the Euler dynamics of a 2D periodic layer of non uniform vorticity. We consider the zero thickness limit and we compare the Euler solution with the vortex sheet evolution predicted by the Birkhoff–Rott equation. The well-known process of singularity formation in shape of the vortex sheet correlates with the appearance of several complex singularities in the Euler solution with the vortex layer datum. These singularities approach the real axis and are responsible for the roll-up process in the layer motion.

Optimal capture occasion determination and trajectory generation for space robots grasping tumbling objects

This paper presents an optimal trajectory planning scheme for robotic capturing of a tumbling object. Motion planning of a space robot is much more complex than that of a fixed-based robot, due to the dynamic coupling between the manipulator and its base. In this work, the Path Independent Workspace (PIW), in which no dynamic singularity occurs, and Path Dependent Workspace (PDW) of the space robot are first calculated by the proposed algorithm. The motion equations of the tumbling object are formulated based on the Euler dynamics equations and the quaternion, which are used to predict the long-term motion of a grasping point on the tumbling object. Subsequently, the obtained PIW workspace and predicted motion trajectories are used to plan the trajectory of the end-effector to intercept the grasping point with zero relative velocity (to avoid impact) in an optimal way. In order to avoid dynamic singularity occurring at the capture moment, the optimal capture occasion is first determined by three proposed criterions guaranteeing the capture can be safely, reliably and rapidly performed, then the optimal trajectory of the end-effector is generated minimizing a cost function which acts as a constraint on acceleration magnitude. Simulations are presented to demonstrate the trajectory planning scheme for a space robot with a 3-degree of freedom (DOF) manipulator grasping a tumbling satellite, the results show that the manipulator end-effector can smoothly intercept the grasping point on the tumbling satellite with zero relative velocity.

Restricted Euler dynamics along trajectories of small inertial particles in turbulence

The fate of small particles in turbulent flows depends strongly on the velocity gradient properties of the surrounding fluid, such as rotation and strain rates. For non-inertial (fluid) particles, the restricted Euler model provides a simple low-dimensional dynamical system representation of Lagrangian evolution of velocity gradients in fluid turbulence, at least for short times. Here, we derive a new restricted Euler dynamical system for the velocity gradient evolution of inertial particles, such as solid particles in a gas, or droplets and bubbles in turbulent liquid flows. The model is derived in the limit of small (sub-Kolmogorov-scale) particles and low Stokes number. The system exhibits interesting fixed points, stability and invariant properties. Comparisons with data from direct numerical simulations show that the model predicts realistic trends such as the tendency of increased straining over rotation along heavy particle trajectories and, for light particles such as bubbles, the tendency of reduced self-stretching of the strain rate.

ESTIMATING MUSCLE FORCES AND KNEE JOINT TORQUE USING SURFACE ELECTROMYOGRAPHY: A MUSCULOSKELETAL BIOMECHANICAL MODEL

Surface electromyography (sEMG) is a useful tool for revealing the underlying musculoskeletal dynamic properties in the human body movement. In this paper, a musculoskeletal biomechanical model which relates the sEMG and knee joint torque is proposed. First, the dynamic model relating sEMG to skeletal muscle activation considering frequency and amplitude is built. Second, a muscle contraction model based on sliding-filament theory is developed to reflect the physiological structure and micro mechanical properties of the muscle. The muscle force and displacement vectors are determined and the transformation from muscle force to knee joint moment is realized, and finally a genetic algorithm-based calibration method for the Newton–Euler dynamics and overall musculoskeletal biomechanical model is put forward. Following the model calibration, the flexion/extension (FE) knee joint torque of eight subjects under different walking speeds was predicted. Results show that the forward biomechanical model can capture the general shape and timing of the joint torque, with normalized mean residual error (NMRE) of [Formula: see text]10.01%, normalized root mean square error (NRMSE) of [Formula: see text]12.39% and cross-correlation coefficient of [Formula: see text]0.926. The musculoskeletal biomechanical model proposed and validated in this work could facilitate the study of neural control and how muscle forces generate and contribute to the knee joint torque during human movement.

Approximate expressions of mean eddy current torque acted on space debris

Rotational state of space debris will be influenced by eddy current torque which is produced by the conducting body rotating within the geomagnetic field. Former expressions of instantaneous torque established in body-fixed coordinate system will change in space during rotation due to the variation of the coordinate system. In order to further investigate the evolution of the rotation of space debris subjected to the eddy current torque, approximate expressions of mean eddy current torque in inertial coordinate system are obtained from the average of the Euler dynamics equations under the assumption that two of the principal moments of inertia of the space debris are similar. Then the expressions are verified through numerical simulation, in which the orientation of the averaged variation of angular momentum is in agreement with the torque from the expressions, which is on an identical plane with magnetic field and the angular momentum. The torque and the averaged variation of the angular momentum have the same evolution trend during rotation in spite of minor deviations of their values.