Arbitrary Reference Frame Research Articles

A finite element framework for fluid–structure interaction of turbulent cavitating flows with flexible structures

We present a finite element framework for the numerical prediction of cavitating turbulent flows interacting with flexible structures. The vapor–fluid phases are captured through a homogeneous mixture model, with a scalar transport equation governing the spatio-temporal evolution of cavitation dynamics. High-density gradients in the two-phase cavitating flow motivate the use of a positivity-preserving Petrov–Galerkin stabilization method in the variational framework. A mass transfer source term introduces local compressibility effects arising as a consequence of phase change. The turbulent fluid flow is modeled through a dynamic subgrid-scale method for large eddy simulations. The flexible structure is represented by a set of eigenmodes, obtained through a modal decomposition of the linear elasticity equations. While a partitioned iterative approach is adopted to couple the structural dynamics and cavitating fluid flow, the deforming flow domain is described by an arbitrary Lagrangian–Eulerian frame of reference. We establish the fidelity of the proposed framework by comparing it against experimental and numerical studies for both rigid and flexible hydrofoils in cavitating flows. Under unstable partial cavitating conditions, we identify specific vortical structures leading to cloud cavity collapse. We further explore features of cavitating flow past a rigid body such as re-entrant jet and turbulence-cavity interactions during cloud cavity collapse. Based on the validation study conducted over a flexible NACA66 rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in governing the structural dynamics. Subsequently, we identify a broad spectrum frequency band whose central peak does not correlate to the frequency content of the cavitation dynamics or the natural frequencies of the structure, indicating the induction of unsteady flow patterns around the hydrofoil. Finally, we discuss the coupled fluid–structure dynamics during a cavitation cycle and the underlying mechanism associated with the promotion and mitigation of cavitation.

Poincaré generators at second post-Minkowskian order

We verify the global Poincaré invariance of the Hamiltonian mechanics of gravitating binary dynamics at the second post Minkowskian (2PM) order. For spinless point particles, based on the known 2PM Hamiltonian in the center of momentum frame, we compute the general 2PM Hamiltonian valid in an arbitrary reference frame. An off-shell extension of the 1PM Hamiltonian, which contributes at the 2PM order through an iteration process, plays a crucial role. We then construct the 2PM boost generator that uniquely satisfies all the conditions imposed by the Poincaré algebra.

From the EFT of spinning gravitating objects to Poincaré and gauge invariance at the 4.5PN precision frontier

We confirm the generalized actions of the complete NLO cubic-in-spin interactions for generic compact binaries which were first tackled via an extension of the EFT of spinning gravitating objects. We first reduce these generalized actions to standard actions with spins, where the interaction potentials are found to consist of 6 independent sectors, including a new unique sector that is proportional to the square of the quadrupolar deformation parameter, CES2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {C}_{{\ extrm{ES}}^2} $$\\end{document}. We derive the general Hamiltonians in an arbitrary reference frame, and for generic kinematic configurations. With these most general Hamiltonians we construct the full Poincaré algebra of all the sectors at the fourth and a half post-Newtonian (4.5PN) order, including the third subleading spin-orbit sector, recently accomplished uniquely via our framework, thus proving the Poincaré invariance of all relevant sectors. We then derive the binding energies with gauge-invariant relations useful for gravitational-wave applications. Finally, we also derive the extrapolated scattering angles in the aligned-spins configuration for the scattering problem. Yet, as made clear already as of quadratic-in-spin sectors, the aligned-spins simplification inherent to the scattering-angle observable, entails a great loss of physical information, that is only growing with higher-spin sectors. Our completion of the full Poincaré algebra at the present 4.5PN order provides strong confidence that this new precision frontier in PN theory has now been established.

Relativistic liquids: GENERIC or EIT?

We study the GENERIC hydrodynamic theory for relativistic liquids formulated by Öttinger and collaborators. We use the maximum entropy principle to derive its conditions for linear stability (in an arbitrary reference frame) and for relativistic causality. In addition, we show that, in the linear regime, its field equations can be recast into a symmetric-hyperbolic form. Once rewritten in this way, the linearised field equations turn out to be a particular realisation of the Israel–Stewart theory, where some of the Israel–Stewart free parameters are constrained. This also allows us to reinterpret the GENERIC framework in view of the principles of extended irreversible thermodynamics and to discuss its physical relevance to model (possibly viscoelastic) fluids.

A Full-Body Relative Orbital Motion of Spacecraft Using Dual Tensor Algebra and Dual Quaternions

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative motion of two spacecraft in the specific case of Keplerian confocal orbits is proposed. The result is an analytical method without secular terms and singularities. Tensors dual algebra and dual quaternions play a fundamental role, with the solution representation being the relative problem. Furthermore, the representation theorems for the rotation and translation parts of the 6-DOF relative orbital motion problems are obtained.

Modelling and indirect field‐oriented control for pole phase modulation induction motor drives

In the recent days, for the traction and electric vehicle (EV) applications, multiphase machines with pole phase modulation (PPM) technique have been proposed. The smoother operation during pole changeovers as well as steady-state operations is a significant constraint while adopting the PPM-based multiphase induction motor (PPMIM) drives for EV and traction applications. So, in this paper, the PPMIM dynamic model and associated vector control are proposed for attaining a smoother operation of the machine. The machine modelling equations and transformation matrices are implemented in an arbitrary reference frame by considering the different pole phase combinations. Based on the modelling equations, the indirect field-oriented control (IFOC) is proposed for PPMIM drives by reflecting the associated changes in parameters for different pole phase modes. In the IFOC, for regulating the d-axis and q-axis current components, single PI control loops have been implemented for all pole-phase combinations. The proposed IFOC scheme is robust and applicable for adopting any type of pulse width modulation. The experimental, as well as simulation results, are given to illustrate the potentiality of the proposed dynamic model and IFOC. The PPMIM machine performance during the steady state as well as pole changeovers in different pole phase modes are analyzed and associated. Simulation and experimental results are presented.

Dynamic Modeling and Control of Pole-Phase Modulation-Based Multiphase Induction Motor Drives

Pole-phase modulation based multiphase induction motor (PPMIM) drives possess the capability of providing an extended range of speed-torque for high power traction application with the available additional degrees of freedom in the multiphase machine. As observed in previous open-loop results of PPMIM drives presented in the literature, the transient currents often exceed twice the rated currents. To address this issue, this paper focuses on the modelling and advanced control of the PPMIM drives during all possible pole-phase combinations. The dynamic mathematical modelling of the PPMIM in actual phase variable domain is presented in detail. Since this model involves time-dependent inductances and torque, the machine model equations are transformed into dq domain using transformation matrices. The transformation matrices, modelling equations in the arbitrary reference frame and multiphase inverter are modelled by considering all parameters of possible pole-phase combinations. Based on the proposed modelling equations, an indirect field oriented control (IFOC) is implemented for PPMIM drives for smoother operation in all possible modes of operation. The modelling equations, as well as IFOC of PPMIM drive, are implemented in MATLAB to illustrate the behaviour of machine during transients as well as different load torques. The proposed concepts are also validated on laboratory setup and the hardware results are demonstrating the smoother transition as well as steady state operation of PPMIM drive in pole changeovers as well as both pole-phase combinations.

Extensions to the Navier–Stokes equations

Historically, the mass conservation and the classical Navier–Stokes equations were derived in the co-moving reference frame. It is shown that the mass conservation and Navier–Stokes equations are Galilean invariant—they are valid in any arbitrary inertial reference frame. From the mass conservation and Navier–Stokes equations, we can derive a wave equation, which contains the speed of pressure wave as its parameter. This parameter is independent of the speed of the source—the fluid element velocity. The speed of pressure wave is determined from the thermodynamic equation of state of the fluid, which is reference frame independent. It is well known that Lorentz transformation ensures wave speed invariant in all inertial frames, and the Lorentz invariance holds for different inertial observers. Based on these arguments, general Navier–Stokes equations (conservation law for the energy–momentum) can be written in any arbitrary inertial reference frame, they are transformed from one reference frame into another with the help of the Lorentz transformation. The key issue is that the Lorentz factor is parametrized by the local Mach number. In the instantaneous co-moving reference frame, these equations will degrade to the classical Navier–Stokes equations—the limit of the non-relativistic ones. These extended equations contain a square of the Lorentz factor. When the local Mach number is equal to one (the Lorentz factor approaches infinity), the extended Navier–Stokes equations will embody an intrinsic singularity, meaning that the transitions from the subsonic flow to the supersonic flow will happen. For the subsonic flow, the square of the Lorentz factor is positive, while for the supersonic flow, the square of the Lorentz factor becomes a negative number, which represents that the speed of sound cannot travel upstream faster than the flow velocity.

Three-body inertia tensor

We derive a general formula for the inertia tensor of a rigid body consisting of three particles with which students can learn basic properties of the inertia tensor without calculus. The inertia-tensor operator is constructed by employing the Dirac’s bra-ket notation to obtain the inertia tensor in an arbitrary frame of reference covariantly. The principal axes and moments of inertia are computed when the axis of rotation passes the center of mass. The formulas are expressed in terms of the relative displacements of particles that are determined by introducing Lagrange’s undetermined multipliers. This is a heuristic example analogous to the addition of a gauge-fixing term to the Lagrangian density in gauge field theories. We confirm that the principal moments satisfy the perpendicular-axis theorem of planar lamina. Two special cases are considered as pedagogical examples. One is a water-molecule-like system in which a particle is placed on the vertical bisector of two identical particles. The other is the case in which the center of mass coincides with the incenter of the triangle whose vertices are placed at the particles. The principal moment of the latter example about the normal axis is remarkably simple and proportional to the product ‘abc’ of the three relative distances. We expect that this new formula can be used in actual laboratory classes for general physics or undergraduate classical mechanics.

Creep tide model for the three-body problem

We present a tidal model for treating the rotational evolution in the general three-body problem with arbitrary viscosities, in which all the masses are considered to be extended and all the tidal interactions between pairs are taken into account. Based on the creep tide theory, we present a set of differential equations that describes the rotational evolution of each body, in a formalism that is easily extensible to the N tidally interacting body problem. We apply our model to the case of a circumbinary planet and use a Kepler-38 like binary system as a working example. We find that, in this low planetary eccentricity case, the most likely final stationary rotation state is the 1:1 spin–orbit resonance, considering an arbitrary planetary viscosity inside the estimated range for the Solar System planets. The timescales for reaching the equilibrium state are expected to be approximately millions of years for stiff bodies but can be longer than the age of the system for planets with a large gaseous component. We derive analytical expressions for the mean rotational stationary state, based on high-order power series of the ratio of the semimajor axes a1∕a2 and low-order expansions of the eccentricities. These are found to very accurately reproduce the mean behaviour of the low-eccentric numerical integrations for arbitrary planetary relaxation factors, and up to a1∕a2 ~ 0.4. Our analytical model is used to predict the stationary rotation of the Kepler circumbinary planets and we find that most of them are probably rotating in a subsynchronous state, although the synchrony shift is much less important than our previous estimations. We present a comparison of our results with those obtained with the Constant Time Lag and find that, as opposed to the assumptions in our previous works, the cross torques have a non-negligible net secular contribution, and must be taken into account when computing the tides over each body in an N-extended-body system from an arbitrary reference frame. These torques are naturally taken into account in the creep theory. In addition to this, the latter formalism considers more realistic rheology that proved to reduce to the Constant Time Lag model in the gaseous limit and also allows several additional relevant physical phenomena to be studied.

Anisotropic Kappa Distributions. I. Formulation Based on Particle Correlations

Abstract We develop the theoretical basis for the connection of the variety of anisotropic distributions with the statistical correlations among particles’ velocity components. By examining the most common anisotropic distribution function, we derive the correlation coefficient among particle energies, show how this correlation is connected to the effective dimensionality of the velocity distribution, and derive the connection between anisotropy and adiabatic polytropic index. Having established the importance of the correlation among particles in the formulation of anisotropic kappa distributions, we generalize these distributions within the framework of nonextensive statistical mechanics and based on the types of homogeneous or heterogeneous correlations among the particles’ velocity components. The formulation of the developed generalized distributions mediates the main two types of anisotropic kappa distributions that consider either (a) equal correlations, or (b) zero correlations, among different velocity components. Finally, the developed anisotropic kappa distributions are expressed in terms of the energy and pitch angle in arbitrary reference frames.

THE CRITERION FOR TRANSFERABLE SELF-CONSISTENTLY TRANSLATIONALITY OF COORDINATE TRANSFORM OPERATORS AND REFERENCE FRAMES IN UNIVERSAL KINEMATICS

From an intuitive point of view universal kinematics are collections (sets) of changing objects, which evolve, being in a certain spatial-geometric environment, and evolution of whi- ch can be observed from many different frames of reference. Moreover, the definition of uni- versal kinematics impose the existence of some (preassigned) universal coordinate transform between every two reference frames of such kinematics. Transferable self-consistently translati- onal reference frames (in vector universal kinematics) are interesting because for such reference frames it is possible to give a clear and unambiguous definition of displacement of a moving reference frame relative to a fixed one, which does not depend on the choice of a fixed point in the moving frame of reference. In the present paper it is shown that an arbitrary reference frame m is transferable self-consistently translational relatively to a reference frame l (in some vector uni- versal kinematics F) if and only if the coordinate transform operator from the reference frame m to the reference frame l is transferable self-consistently translational. Therefore transferable self-consistently translational coordinate transform operators describe the conversion of coordi- nates from the moving and transferable self-consistently translational frame of reference to the (given) fixed frame in vector universal kinematics. Also in the paper it is described the structure of transferable self-consistently translational coordinate transform operators (this is the main result of the article). Using this result it have been obtained the necessary and sufficient conditi- on for transferable self-consistently translationality of one reference frame relatively to another in vector universal kinematics.

Object-Agnostic Dexterous Manipulation of Partially Constrained Trajectories

We address the problem of controlling a partially constrained trajectory of the manipulation frame–an arbitrary frame of reference rigidly attached to the object–as the desired motion about this frame is often underdefined. This may be apparent, for example, when the task requires control only about the translational dimensions of the manipulation frame, with disregard to the rotational dimensions. This scenario complicates the computation of the grasp frame trajectory, as the mobility of the mechanism is likely limited due to the constraints imposed by the closed kinematic chain. In this letter, we address this problem by combining a learned, object-agnostic manipulation model of the gripper with Model Predictive Control (MPC). This combination facilitates an approach to simple vision-based control of robotic hands with generalized models, enabling a single manipulation model to extend to different task requirements. By tracking the hand-object configuration through vision, the proposed framework is able to accurately control the trajectory of the manipulation frame along translational, rotational, or mixed trajectories. We provide experiments quantifying the utility of this framework, analyzing its ability to control different objects over varied horizon lengths and optimization iterations, and finally, we implement the controller on a physical system.

Generalized Analysis of Brushless Doubly Fed Induction Machine Taking Magnetic Saturation and Iron Loss Into Account

Substantial advantages of brushless doubly fed induction machine (BDFIM) bring up it as an interesting alternative for conventional solutions. In order to better analyze BDFIM performance, it is required to employ a more precise model of BDFIM. In this article, an approach, which completely relies on dynamic modeling in arbitrary reference frame, is proposed to consider both iron loss and main flux saturation effects in two-axis model of BDFIM. In this regard, the dynamic and static inductances are derived by using the magnetizing flux curve. If the cross-saturation is ignored, the time derivative of magnetizing inductances will yield incorrect simulation results. Therefore, to increase the accuracy, the proposed model contains terms that describe the cross-saturation effects. Due to the close relation between torque and power, the torque expression is also achieved in the presence of iron loss and magnetic saturation using expression of electrical input power. The simulation and experimental results are finally given to show the performance of the proposed model.

Iron Loss Modeling in Dual Stator Winding Induction Machines With Unequal Pole Pairs and Squirrel Cage Rotor

This article pertains to dual stator winding induction machines (DSWIMs) modeling and proposes an equivalent electrical circuit for DSWIMs that considers the iron loss effect. It is proved that the iron loss in DSWIMs is comprised of two independent parts resulting from their winding sets. Accordingly, the proposed model is comprised of two distinct parts and it is based on the assumption of independent operation of DSWIM winding sets. This model is valid in dynamic and steady-state operation conditions. The dq0 form of the proposed model is derived in an arbitrary reference frame. In addition, based on this model the iron loss estimation in DSWIMs is presented. The proposed model is simulated in MATLAB/Simulink based on the parameters obtained from IEEE standard characterization tests performed on a 3.3-kW DSWIM and also some experimental tests are executed via a digital signal processor (DSP)-based DSWIM drive system. Comparison of the simulation and experimental results confirms the validity of the proposed model both in steady-state and transient conditions. Moreover, this comparison clarifies that the proposed iron loss estimation method can account for the iron loss with high accuracy.

Modeling and Experimental Verification of an Unconventional 9-Phase Asymmetric Winding PM Motor Dedicated to Electric Traction Applications

Multi-phase permanent magnet motors are becoming popular in various applications such as high power traction which require low cogging and torque ripple, and reduced noise and vibration. This paper presents the development process of a full order detailed mathematical model for an unconventional nine-phase permanent magnet (PM) motor with asymmetric AC windings. The phase inductances are calculated based on the individual winding functions. Using these parameters, electrical equations for each phase based on the voltages and flux linkages are generated. These equations are transformed into an arbitrary reference frame and the torque equations are developed for the 9-phase motor. A Matlab-Simulink model is then developed for the proposed nine-phase PM motor based on the developed dynamic equations. Finally, the model is validated with the finite element analysis and laboratory testing of the prototype motor. It is shown that the developed mathematical model can be used for such complex asymmetric winding multi-phase PM motors.

Coordinated Derived Current Control of DFIG’s RSC and GSC Without PLL Under Unbalanced Grid Voltage Conditions

This paper proposes a coordinated derived current control strategy of doubly fed induction generator (DFIG) system under unbalanced grid voltage conditions. The rotor-side converter (RSC) and the grid-side converter (GSC) are synchronized to the grid with a virtual phase angle of nominal frequency, which can be simply set as $\theta =\omega _{\mathrm {n}}t$ ( $\omega _{\mathrm {n}}= 100\pi$ ). Consequently, the proposed control strategy can implement in an arbitrary reference frame without phase-locked loop (PLL). The derived d-q axis currents are obtained from the DFIG stator and GSC currents with virtual angle-based Park Transformations. For a simple implementation, a reduced-order vector integrator with a voltage vector multiplier is used to directly regulate the measured powers/currents containing both the dc part and the oscillating part without sequence extractions. By this means, RSC achieves torque ripple reduction and then GSC can rebalance the currents and smooth the active/reactive powers. Finally, the experimental results on a small laboratory setup are presented to validate the effectiveness of the proposed control strategy.

Enhanced Generalized Vector Control Strategy for Torque Ripple Mitigation of IPM-Type Brushless DC Motors

Optimal currents with appropriate harmonic components injected into the motor windings can effectively mitigate torque ripple for interior permanent magnet (IPM) type brushless dc motors (BLDCMs). However, existing approaches may fail to accurately inject the optimal currents into motors due to the limited current loop bandwidth. This paper proposes a simple enhanced generalized vector control strategy to mitigate the torque ripple for IPM-type BLDCMs. With the proposed vector control strategy, the control block diagram for IPM-type BLDCMs is as simple as that with traditional vector control for sinusoidal permanent magnet synchronous motors. First, an electromechanical energy conversion voltage (EECV), considering the effects of the non-sinusoidal back-EMF and the rotor salience property, is proposed in αβ -axis. Then, a novel coordinate frame is constructed with the proposed EECV, which is based on the arbitrary reference frame theory. As a result, the currents of IPM-type BLDCMs can be decomposed into two constant components, one being linked to the torque and the other one to the flux linkage. Thus, the bandwidth of current loop with simple proportional integral regulator can be equivalently expanded in the proposed coordinate frame. Finally, comprehensive experiments are conducted in different operation conditions to validate the effectiveness of the proposed vector control strategy.

Multi-Scale Induction Machine Model in the Phase Domain With Constant Inner Impedance

An efficient and accurate multi-scale induction machine model for simulating diverse transients in power systems is developed and validated. Voltages, currents, and flux linkages are described through analytic signals that consist of real in-phase and imaginary quadrature components, covering only positive frequencies of the Fourier spectrum. The stator is modeled in the abc phase coordinates of an arbitrary reference frame whose rotating speed is adjusted by a simulation parameter called shift frequency. When the reference frame is stationary at a zero shift frequency, then the model processes instantaneous signals to yield natural waveforms. When the reference frame is set to rotate at the synchronous frequency of the electric network, then the Fourier spectra of the analytic signals are shifted by this synchronous frequency to become dynamic phasors that allow for efficient envelope tracking. The shift frequency can be adapted during simulation. For any rotor position and independent of the variation of the magnetizing inductances with saturation, the induction machine model appears as a Norton current source with constant inner admittance in the abc phase domain to support the integration with simulators that represent the electric network in the abc phase domain. The analysis of test cases covering diverse transients substantiates the claims made in terms of accuracy and efficiency across different time scales.

Accurate numerical approximation of the absolute stability of unbounded flows

The initial stage of the laminar–turbulent transition of semi-infinite flows can be characterized as either an absolute or convective instability, naturally associated with localized wave packets. A convective instability is directly linked to an absolute instability in a different reference frame. Therefore, our aim is to determine the absolute stability of a flow in a given but arbitrary reference frame, which can only be directly inferred from the absolute eigenvalue spectrum. If advective processes are present, the associated absolute eigenfunctions grow exponentially in space in the advective direction. The eigenvalue spectrum is usually computed numerically, which requires truncating the domain and prescribing artificial boundary conditions at these truncation boundaries. For separated boundary conditions, the resulting spectrum approaches the absolute spectrum as the domain length tends to infinity. Since advective processes result in spatially exponentially growing eigenfunctions, it becomes increasingly difficult to represent these functions numerically as the domain length increases. Hence, a naive numerical implementation of the eigenvalue problem may result in a computed spectrum that strongly deviates from the (mathematically correct) absolute spectrum due to numerical errors. To overcome these numerical inaccuracies, we employ a weighted method ensuring the convergence to the absolute spectrum. From a physical point of view, this method removes the advection-induced spatial exponential growth from the eigenfunctions. The resulting (absolute) spectrum allows for a direct interpretation of the character of the pertinent perturbations and the eigensolutions can be used to construct and analyse the evolution of localized wave packets in an efficient way.