Disturbance Equations Research Articles

Disturbance Interaction Analysis and Suppression Strategy of MMC-HVDC Systems Considering Sub-Module Capacitor Voltage Ripples

This paper presents the interaction analysis of external disturbances and internal dynamics of MMC-HVDC (Modular Multilevel Converter High Voltage Direct Current) systems considering sub-module (SM) capacitor voltage ripples, and proposes an enhanced control strategy for disturbance suppression. The disturbance equation indicating the interaction of the external MMC output currents and internal SM capacitor voltage ripples are firstly established. Then based on the disturbance equation, the dynamic analytical expressions of the differential-mode capacitor ripple voltages and the converter currents are derived in the d-q reference frame, which explains that the resultant current term i0 can be regarded as an equivalent disturbance that will lead to transients in the capacitor voltages during external perturbations. Next, the disturbance observer is introduced to track the resultant disturbance current term and an enhanced control for the MMC based on disturbance suppression strategy is proposed to improve the system dynamic response during external transients. Furthermore, system stability characteristics under the enhanced control with different parameters are explored through theoretical analysis. Finally, a benchmark single-terminal MMC-HVDC simulation model is established in PSCAD/EMTDC to verify the proposed control.

Study on Vibration of Flexible Beams with Interior Fluid

In this paper, the basic governing equations of nonlinear dynamics of solid-liquid coupling system are established. Without considering the free vibration, according to the characteristic polynomial of the Jacobi matrix of the disturbance equation required by the Hopf bifurcation, the algebraic equations that the critical velocity must meet are derived. The numerical results show that when the velocity exceeds the critical value, the vibration amplitude of the beam jumps obviously.

Acoustics and dynamic materials

The governing equations for small-amplitude acoustic disturbances are derived by combining an admissible background flow or medium, balance laws, an equation of state and perturbation theory. This standard approach is used here when the background medium can vary in both space and time, thus defining a dynamic material. When the background velocity is zero, the background density cannot vary with time, assuming that the usual balance laws (such as conservation of mass) are obeyed. When mass conservation is discarded, perhaps replaced by a growth model, a time-dependent background density is permitted, leading to new equations governing acoustic disturbances in certain dynamic materials.

Analysis of Gravity Disturbance Compensation for Initial Alignment of INS

To address the accuracy requirements of initial alignment of high-precision inertial navigation systems (INSs), gravity disturbance compensation for INSs based on a spherical harmonic model is investigated herein. First, the horizontal component of gravity disturbance at an alignment point is calculated using the high-resolution Earth Gravity Model EIGEN-6C4 and then compensated to the initial alignment. Subsequently, the self-alignment algorithm of solidified coordinate frame is used to derive the misalignment angle equation of gravity disturbance affecting the initial alignment. Meanwhile, the coupling relationship between the measurement error of an inertial unit and the gravity disturbance is simulated and analyzed. Finally, a laser strapdown inertial navigation system experiment is performed. The simulation result shows that the pitch angle, roll angle, and heading angle errors reduced by $27.41^{\prime \prime }$ , $- 0.37^{\prime \prime }$ , and $6.72^{\prime \prime }$ , respectively, after the gravity disturbance compensation. Experiment result shows that the alignment performance after compensation has been improved and the heading angle error is reduced by $6.76^{\prime \prime }$ . The simulations and experiments results validate the theoretical analysis.

Nonlinear Dynamo in Obliquely Rotating Stratified Electroconductive Fluid in an Uniformly Magnetic Field

In this paper, we investigated a new large-scale instability that arises in an obliquely rotating convective electrically conducting fluid in an external uniform magnetic field with a small-scale external force with zero helicity. This force excites small-scale velocity oscillations with a small Reynolds number. Using the method of multiscale asymptotic expansions, we obtain the nonlinear equations for vortex and magnetic disturbances in the third order of the Reynolds number. It is shown that the combined effects of the Coriolis force and the small external forces in a rotating conducting fluid possible large-scale instability. The linear stage of the magneto-vortex dynamo arising as a result of instabilities of -effect type is investigated. The mechanism of amplification of large-scale vortex disturbances due to the development of the hydrodynamic - effect taking into account the temperature stratification of the medium is studied. It was shown that a «weak» external magnetic field contributes to the generation of large-scale vortex and magnetic perturbations, while a «strong» external magnetic field suppresses the generation of magnetic-vortex perturbations. Numerical methods have been used to find stationary solutions of the equations of a nonlinear magneto-vortex dynamo in the form of localized chaotic structures in two cases when there is no external uniform magnetic field and when it is present.

Network-based deployment of the second-order multi agents: a PDE approach

Deployment of a second-order nonlinear multi agent system over a desired open smooth curve in 2D or 3D space is considered. We assume that the agents have access to their velocities and to the local information of the desired curve and their displacements with respect to their closest neighbors, whereas in addition a leader is able to measure his absolute position. We assume that a small number of leaders transmit their measurements to other agents through a communication network. We take into account the following network imperfections: the variable sampling, transmission delay and quantization. We propose a static output-feedback controller and model the resulting closed-loop system as a disturbed (due to quantization) nonlinear damped wave equation with delayed point state measurements, where the state is the relative position of the agents with respect to the desired curve. To manage with the open curve we consider Neumann boundary conditions. We derive linear matrix inequalities (LMIs) that guarantee the input-to-state stability (ISS) of the system. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical example illustrates the efficiency of the method.

An Analytical Method for Coaxial Helicopter Ground Resonance

A time-frequency analytical method is presented to analyze physical mechanism of coaxial helicopter ground resonance. Eigenvalue calculation and numerical integration of disturbance equations of motions are used to obtain modal characters and time-domain response characters of coaxial helicopter ground resonance, and the interaction between rotors and body is revealed according to response of various DOFs. The analysis results show that regressive lag mode with upper rotor character is the most instability mode. In dynamic instability region, coaxial helicopter ground resonance is mainly due to energy transferred between periodic lag motion of upper rotor and body roll rotation. For this instability mode, energy transferred between periodic lag motion of lower rotor and body roll rotation is also existed, and it can enhance ground resonance instability of coaxial helicopter.

Perturbation solution for a solitary wave of the nonlinear higher dimensional disturbed Klein-Gordon equation

Perturbation solution for a solitary wave of the nonlinear higher dimensional disturbed Klein-Gordon equation

Linear Stability Analysis of Liquid Metal Flow in an Insulating Rectangular Duct under External Uniform Magnetic Field

Linear stability analysis of liquid metal flow driven by a constant pressure gradient in an insulating rectangular duct under an external uniform magnetic field was carried out. In the present analysis, since the Joule heating and induced magnetic field were neglected, the governing equations consisted of the continuity of mass, momentum equation, Ohm’s law, and conservation of electric charge. A set of linearized disturbance equations for the complex amplitude was decomposed into real and imaginary parts and solved numerically with a finite difference method using the highly simplified marker and cell (HSMAC) algorithm on a two-dimensional staggered mesh system. The difficulty of the complex eigenvalue problem was circumvented with a Newton—Raphson method during which its corresponding eigenfunction was simultaneously obtained by using an iterative procedure. The relation among the Reynolds number, the wavenumber, the growth rate, and the angular frequency was successfully obtained for a given value of the Hartmann number as well as for a direction of external uniform magnetic field.

Stability of a temporally evolving natural convection boundary layer on an isothermal wall

The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-steady assumption, whereby the unsteady base flow is frozen in time. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearized disturbance equations using a ‘frozen’ base flow as an initial-value problem at various $Gr_{\unicode[STIX]{x1D6FF}}$, where $Gr_{\unicode[STIX]{x1D6FF}}$ is the Grashof number based on the velocity integral boundary layer thickness $\unicode[STIX]{x1D6FF}$. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three-dimensional direct numerical simulations (DNS) of the temporally evolving flow. The amplification rate predicted by the linear theory compares well with the result of direct numerical simulation up to a transition point. The extent of the linear regime where the perturbations linearly interact with the base flow is thus identified. The value of the transition $Gr_{\unicode[STIX]{x1D6FF}}$, according to the three-dimensional DNS results, is dependent on the initial perturbation amplitude. Beyond the transition point, the DNS results diverge from the linear stability predictions as nonlinear mechanisms become important.

The evolution of non-linear disturbances in magnetohydrodynamic flows

In this article the stability loss of the Hartmann flow are investigated by applying the equations for disturbances. The velocity and electric potential quasi-static MHD model is used. The equations allow us to calculate time-dependent disturbance fields using a base flow and an initial disturbance. Two type of initial perturbations are considered: the eigenfunction of the linearized MHD equations and a fluid injection into the flow. These two approaches lead to identical stability results. However, we found a significant difference in the practical implementation of the two approaches. Dealing with the eigenproblem of the linearized MHD system is a laborious task. In terms of calculation costs it is equal to a series of nonlinear perturbation simulations, and if the Hartmann number is increased, the proportion becomes worse. The non-linear stability analysis produced by these two methods shows that the injection technique can also be used in numerical analysis, and that this method is less expensive in terms of calculation costs.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Stability of an Axisymmetric Liquid Metal Flow Driven by a Multi-Pole Rotating Magnetic Field

The stability of an electrically conducting fluid flow in a cylinder driven by a multi-pole rotating magnetic field is numerically studied. A time-averaged Lorentz force term including the electric potential is derived on the condition that the skin effect can be neglected and then it is incorporated into the Navier-Stokes equation as a body force term. The axisymmetric velocity profile of the basic flow for the case of an infinitely long cylinder depends on the number of pole-pairs and the Hartmann number. A set of linearized disturbance equations to obtain a neutral state was successfully solved using the highly simplified marker and cell (HSMAC) method together with a Newton–Raphson method. For various cases of the basic flow, depending on both the number of pole-pairs and the Hartmann number, the corresponding critical rotational Reynolds numbers for the onset of secondary flow were obtained instead of using the conventional magnetic Taylor number. The linear stability analyses reveal that the critical Reynolds number takes its minimum at a certain value of the Hartmann number. On the other hand, the velocity profile for cases of a finite length cylinder having a no-slip condition at the flat walls generates the Bödewadt boundary layers and such flows need to be computed including the non-linear terms of the Navier-Stokes equation.

Transient growth in the near wake region of the flow past a finite span wing

We investigate optimal perturbation in the flow past a finite aspect ratio ($AR$) wing. The optimization is carried out in the regime where the fully developed flow is steady. Parametric study over time horizon ($T$), Reynolds number ($Re$), $AR$, angle of attack and geometry of the wing cross-section (flat plate and NACA0012 airfoil) shows that the general shape of linear optimal perturbation remains the same over the explored parameter space. Optimal perturbation is located near the surface of the wing in the form of chord-wise periodic structures whose strength decreases from the root towards the tip. Direct time integration of the disturbance equations, with and without nonlinear terms, is carried out with linear optimal perturbation as initial condition. In both cases, the optimal perturbation evolves as a downstream travelling wavepacket whose speed is nearly the same as that of the free stream. The energy of the wavepacket increases in the near wake region, and is found to remain nearly constant beyond the vortex roll-up distance in nonlinear simulations. The nonlinear wavepacket results in displacement of the tip vortex. In this situation, the motion of the tip vortex resembles that observed during vortex meandering/wandering in wind tunnel experiments. Results from computation carried out at higher $Re$ suggest that, even beyond the steady flow regime, a perturbation wavepacket originating near the wing might cause meandering of tip vortices.

Dependence of instability to induce a bathtub vortex in a rectangular vessel on the aspect ratio of the horizontal cross section

Dependence of the linear instability to induce a bathtub vortex in a rectangular vessel on the horizontal aspect ratio of the vessel is investigated in the present paper. Water comes in through inlet sections on the top of two facing side walls, and is drained out through a pipe attached to the drain hole at the center on the bottom of the vessel. Water surface is assumed to be always flat without any dip even after instability yielded a large-scale vortex above the drain hole. The flow is steady and symmetric with respect to both vertical center planes along the streamwise direction and perpendicular to it at small Reynolds numbers, but becomes unstable to infinitesimal disturbances inevitably included in the flow at Reynolds numbers larger than a critical value. The steady symmetric flow is numerically obtained, and its stability is analyzed by numerically solving the linear disturbance equation. The instability is confirmed to lead to induction of a so-called bathtub vortex. It is found that there exists a threshold aspect ratio of the rectangular cross section for the instability, and the threshold value is determined.

Design procedure of first-order perturbations for collisions trajectories with a perturbing body

The article is devoted to the determination of firstand secondorder perturbations in rectangular coordinates and velocity components of body motion. Special differential equation system of perturbed motion is constructed. The right-hand sides of this system are finitesimal polynomials in powers of an independent regularizing variate. This allows constructing a single algorithm to determine firstand second-order perturbations in the form of finitesimal polynomials in powers of regularizing variates that are chosen at each approximation step. Following the calculations results with the developed method use, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the numerical results of the disturbed motion equations shows their close agreement. The developed method make it possible to calculate any visa point of body motion by the approximating polynomials.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Noether symmetrical perturbation and adiabatic invariants for disturbed non-material volumes

This paper is concerned with adiabatic invariants, applied to Noether symmetrical perturbation of disturbed non-material volumes. The disturbed equation of non-material volumes is proposed, the identity of Noether symmetry for the disturbed system is described, and the corresponding Noether exact invariant is derived. The perturbation to the Noether symmetry and adiabatic invariant for non-material volumes is employed by introducing the concept of adiabatic invariant. The Noether identity of symmetrical perturbation for the disturbed non-material volumes is reported, and Noether adiabatic invariant is obtained. Two examples are given to illustrate the application of the method, and the corresponding exact invariant and adiabatic invariant are obtained under the Noether symmetrical transformations.

The Inclined Wooding Problem

A constant-thickness thermal boundary layer is formed by the presence of uniform suction into a constant temperature hot surface bounding a porous medium—the Wooding problem. When the hot surface lies below the porous medium, the system is susceptible to thermoconvective instability. The present paper is concerned with how the classical linear stability analysis is modified by inclining the heated surface. Analysis is confined to disturbances in the form of transverse rolls because the equivalent analysis for longitudinal rolls may be described analytically in terms of that for the horizontal layer. The linear stability analysis is made difficult by the absence of a second bounding surface, and the method of multiple shooting is needed in order to obviate the consequence of having a stiff system of disturbance equations. Therefore, the computational domain is split into 5 or 10 subdomains. It is found that all modes of instability travel up the heated surface unless the surface is horizontal. The system is found to be linearly stable for all inclinations above 31.85473^circ , a value which is remarkably close to that for the inclined Darcy-Bénard problem.

Semi-analytical solutions of pulsating flow in a helically rough-walled microtube

The influence of a helically rough wall surface on fully developed pulsating laminar flows in microtube is investigated through a semi-analytical method. A two-dimensional simple harmonic function is chosen to model helically wavy structure. The velocity and pressure are decomposed into space-averaged and disturbance terms and the corresponding governing equations are established. Pulsating flow is split into a steady term and an oscillatory term based on a given oscillatory pressure drop, and the governing equations for the oscillatory terms are presented together with their steady counterparts. Analytical solutions are obtained for the space-averaged equations and a spectral collocation method is used to solve the disturbance equations numerically. An iterative approach is adopted for the coupled equations with respect to space-averaged velocities and disturbance velocities. The computational results show that a Reynolds stress layer (RSL) and a secondary swirling velocity are present in the rough-walled microtube. The relative amplitude of the waviness of the wall, its wavenumber, and the Reynolds number are found to be important parameters influencing the Reynolds stress and RSL, as well as the space-averaged velocities and pressure drop. The direction of rotation of the swirling velocity is found to depend on the sign of the azimuthal wavenumber. In addition to the parameters of the rough wall and the Reynolds number, both the oscillatory pressure drop and frequency are important factors influencing the pulsating flow and the phase shift of the space-averaged velocity.