Projection Equations Research Articles

On the Choice of Shape Functions of Unsymmetrical Elastic Rotors

In this paper the effects of rotordynamics under the aspect of the choice of shape functions are discussed. For this purpose a rotor system which consists of a slim shaft and two rigid disks is modeled using theProjection Equation. The shaft is assumed as an elasticEuler-Bernoullibeam, supported by two bearings modeled as radial spring systems. The rotor is driven by a permanent-magnet synchronous motor whose torque is transmitted with a spur gear pair next to one of the bearings. ARitzapproach is used to separate the elastic displacements in position and time, thereby different shape functions are evaluated. Approximated eigenfunctions are computed and used as shape functions as well. For validation, the eigenfrequencies are compared with semi-analytical ones, calculated with theTransfer-Matrix-Methodand experimental results. The insights obtained from this work should make it easier to choose the appropriate shape functions for such problems.

한국의 새로운 단일원점 평면직각좌표계 설정

최근 전 세계적으로 국가별, 기관별, 목적별로 다양하게 구축된 공간정보를 단일한 위치기반으로 하는 데이터로 통합함으로써 각 자료 간 연계성 및 활용성을 높이고 있다. 본 연구에서는 새로운 국가측지기준의 도입에 따른 위치기준의 변화와 공간정보 분야의 seamless 데이터 서비스 요구에 효율적으로 대응할 수 있도록 새로운 국가 단일원점 평면직각좌표에 대한 연구를 수행하였다. 이를 위해 한반도 전역을 투영영역으로 설정하고 투영영역 내 발생하는 투영왜곡을 분석함으로써 이를 균질화 및 최소화할 수 있는 투영파라미터를 정의하였으며, 이에 대한 표준화 및 국제기구 등록방안을 제시하였다. 연구의 결과, 국가 단일원점 평면직각좌표계를 위한 최적의 투영식은 Hooijberg 투영식으로 분석되었으며, TM 투영을 위한 5가지 투영파라미터는 위도(<TEX>${\Phi}$</TEX>) <TEX>$38^{\circ}N$</TEX>, 경도(<TEX>${\lambda}$</TEX>) <TEX>$128^{\circ}E$</TEX>, 축척계수(K) 0.99924로 각각 분석되었다. 또한, 투영원점(<TEX>${\Phi}$</TEX>, <TEX>${\lambda}$</TEX>)에 대한 가수값(false Northing & Easting)은 국가지점번호제도의 도입을 고려하여 북쪽방향 700,000m, 동쪽방향 400,000m로 각각 설정하는 것이 적합하였다. As a worldwide trend, the spatial information that is established by country, institution and purpose is integrated into the data with a single spatial reference to improve the data connectivity and usability. In this study, a new national single origin plane rectangular coordinate system was studied to efficiently respond to the changes in the spatial reference according to the introduction of a new national geodetic standard and to the demand of seamless data service in the spatial information sector. For this purpose, the Korean Peninsula was set as the projection region and the projection distortion in the projection region was analyzed. The projection parameters were defined to homogenize and minimize the projection distortion, and their standardization and registration on the international organizations were conducted. The study on the required optimal projection equation resulted in the Hooijberg projection equation and projection parameters (<TEX>${\Phi}$</TEX>, <TEX>${\lambda}$</TEX>, K, N, E) resulted in <TEX>$38^{\circ}N$</TEX> and <TEX>$128^{\circ}E$</TEX> projection origin, and a scale factor of 0.99924. The proper false northing and easting were 700,000m N and 400,000m E, respectively, considering the introduction of country station index system.

Notes on the equivalence of different variable separation approaches for nonlinear evolution equations

The equivalence of multilinear variable separation approach, the extended projective Ricatti equation method and the improved tanh-function method is firstly reported when these three popular methods are used to realize variable separation for nonlinear evolution equations. We take the (2+1)-dimensional modified Broer–Kaup system for an example to illustrate this point. All solutions obtained by the extended projective Ricatti equation method and the improved tanh-function method coincide with the one obtained by the multilinear variable separation approach. Moreover, based on one of variable separation solutions, we also find that although abundant localized coherent structures can be constructed for a special component, we must pay our attention to the solution expression of the corresponding other component for the same equation lest many un-physical related structures might be obtained.

Proyección sobre energía eléctrica en México mediante la Identidad de Kaya

We analyze the energy sector, with emphasis on electricity generation, for being one of the main sources that drive the economy, as it warns of the importance of energy in the whole country’s economic activity, showing that Mexico has a strong relationship between economic growth and energy use. It is therefore extremely important to have a vision of what the power requirements in the short, medium and long term, to identify needs earlier and avoid future energy crises, as has happened in other parts of the world, to carry out the projection equation was used called “Kaya Identity” horizons allowing us to identify and respond to us to the basic question of foresight of what if ….?

The projective translation equation and rational plane flows. I

Let X=(x,y). A plane flow is a function F(X,t): R^2*R->R^2 such that F(F(X,s),t)=F(X,s+t) for (almost) all real numbers x,y,s,t (the function F might not be well-defined for certain x,y,t). In this paper we investigate rational plane flows which are of the form F(X,t)=f(Xt)/t; here f is a pair of rational functions in 2 real variables. These may be called projective flows, and for a description of such flows only the knowledge of Cremona group in dimension 1 is needed. Thus, the aim of this work is to completely describe over R all rational solutions of the two dimensional translation equation (1-z)f(X)=f(f(Xz)(1-z)/z). We show that, up to conjugation with a 1-homogenic birational plane transformation (1-BIR), all solutions are as follows: a zero flow, two singular flows, an identity flow, and one non-singular flow for each non-negative integer N, called the level of the flow. The case N=0 stands apart, while the case N=1 has special features as well. Conjugation of these canonical solutions with 1-BIR produce a variety of flows with different properties and invariants, depending on the level and on the conjugation itself. We explore many more features of these flows; for example, there are 1, 4, and 2 essentially different symmetric flows in cases N=0, N=1, and N>=2, respectively. Many more questions will be treated in the second part of this work.

Continuous time system identification: a selected critical survey. Part II. Input error methods and optimal projection equations.

Continuous time system identification: a selected critical survey. Part II. Input error methods and optimal projection equations.

Applications of extended F-expansion and projective Ricatti equation methods to (2+1)-dimensional soliton equations

The (2+1)-dimensional Maccari and nonlinear Schrödinger equations are reduced to a nonlinear ordinary differential equation (ODE) by using a simple transformation, various solutions of the nonlinear ODE are obtained by using extended F-expansion and projective Ricatti equation methods. With the aid of solutions of the nonlinear ODE more explicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions are found out. It is shown that these methods provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Critical Parameter Values and Reconstruction Properties of Discrete Tomography: Application to Experimental Fluid Dynamics

We analyze representative ill-posed scenarios of tomographic PIV particle image velocimetry with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine: i the ability of unique reconstruction in terms of the imaging geometry and the critical sparsity parameter, and ii sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing.

Modeling and Quasi‐Static Trajectory Control of a Self‐Balancing Two‐Wheeled Vehicle

AbstractThis paper introduces the kinematical and dynamical model as well as a quasi‐static trajectory control of a self‐balancing two‐wheeled vehicle. The mobile robot is about 60cm tall, autonomous, unstable and driven by two wheels. Hence, it can be used for transport purposes. Due to the nonholonomic constraints only few modeling techniques are feasible. In this case, the modeling is based on the Projection Equation, followed by the derivation of various control strategies.In order to allow a desired velocity and to stabilize the inclination angle of the robot a partial feedback linearization in combination with a LQR controller is applied. The quasi‐static trajectory controller, which is based on the kinematical model, uses a flatness based approach in order to remain on the desired path. Continuous curvature paths, composed by clothoids, enable good performance results in simulation and experiment. (© 2012 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Recursive Methods in Modeling and Control of Modular Robotic Systems

AbstractThis paper addresses the dynamical modeling and control of reconfigurable modular robots. The modular actuators (brushless DC motors with Harmonic Drive gears) for the robots under consideration are connected by rigid links. This way the robot can be assembled in different configurations by rearranging these components. For dynamical modeling the Projection Equation in Subsystem representation is used, taking advantage of its modular structure. Due to the lack of position sensors at the gearbox output shaft, deflections caused by the elasticities in the gears can not be compensated by the PD motor joint controller. Therefore, a correction of the motor trajectory is needed, which can be calculated as part of a flatness based feed‐forward control using the exact model of the robot. With the recursive approach proposed in this paper the concept of reconfigurability is retained. For validation a redundant articulated robot arm with seven joints is regarded and results are presented. (© 2012 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Circle Pose Estimation Based on Stereo Vision

This paper presents a simple method of circle pose estimation based on binocular stereo vision. It takes the projective equation of a circle as the basis, and gives the closed form solution of the pose parameters. Since there are two possible sets of pose parameters for a circle from one calibrated perspective view, the stereo vision constraints are incorporated and the accurate pose parameters are determined. Experiments using computer simulated data and real data demonstrate the robustness and accuracy of our method.

Equivalent optimal compensation problem in the delta domain for systems with white stochastic parameters

The equivalent representation in the delta domain of the digital optimal compensation problem is provided and computed in this article. This problem concerns finding digital optimal full and reduced-order output feedback controllers for linear time-varying and time-invariant systems with white stochastic parameters. It can subsequently be solved in the delta domain using the strengthened optimal projection equations that we recently formulated in this domain as well. If the sampling rate becomes high, stating and solving the problem in the delta domain becomes necessary because the conventional discrete-time problem formulation and solution become ill-conditioned. In this article, by means of several numerical examples and compensator implementations, we demonstrate this phenomenon. To compute and quantify the improved performance when the sampling rate becomes high, a new delta-domain algorithm is developed. This algorithm computes the performance of arbitrary digital compensators for linear systems with white stochastic parameters. The principle application concerns nonconservative robust digital perturbation feedback control of nonlinear systems with high sampling rates.

A bound on the degree of schemes defined by quadratic equations

Abstract. We consider complex projective schemes defined by quadratic equations and satisfying a technical hypothesis on the fibres of the rational map associated to the linear system of quadrics defining X. Our assumption is related to the syzygies of the defining equations and, in particular, it is weaker than properties N2 , and K2 . In this setting, we show that the degree d of is bounded by a function of its codimension c, whose asymptotic behavior is given by , thus improving the obvious bound . More precisely, we get the bound . Furthermore, if X satisfies property or , we obtain the better bound . Some classification results are also given when equality holds.

Linear pose estimate from corresponding conics

We propose here a new method to recover the orientation and position of a plane by matching at least three projections of a conic lying on the plane itself. The procedure is based on rearranging the conic projection equations such that the non linear terms are eliminated. It works with any kind of conic and does not require that the shape of the conic is known a-priori. The method was extensively tested using ellipses, but it can also be used for hyperbolas and parabolas. It was further applied to pairs of lines, which can be viewed as a degenerate case of hyperbola, without requiring the correspondence problem to be solved first. Critical configurations and numerical stability have been analyzed through simulations. The accuracy of the proposed algorithm was compared to that of traditional algorithms and of a trinocular vision system using a set of landmarks.

Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geometric contact with the model to a level that is optimal, in a suitable sense. In terms of the trivialisations arising from the special frames, normal solutions of classes of natural linear PDE (so-called first BGG equations) are shown to be necessarily polynomial in the generalised homogeneous coordinates; the polynomial system is the pull back of a polynomial system that solves the corresponding problem on the model. Thus questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of the corresponding polynomial systems and algebraic sets. We show that a normal solution determines a canonical manifold stratification that reflects an orbit decomposition of the model. Applications include the construction of structures that are analogues of Poincare-Einstein manifolds.

Line-scan camera calibration in close-range photogrammetry

A novel line-scan camera calibration method in close-range photogrammetry is proposed. Since the line-scan camera is only sensing in one dimension, it's hard to recognize the space points from the linear data captured in static state. To address this problem, the camera is fixed to a programmable linear stage. With the help of the linear stage, a scan image of the pattern is grabbed by the line-scan camera in uniform rectilinear motion state. Therefore, the image points are definitely matched with the space points on the pattern. A pair of projective equations is established to describe this dynamic imaging model, which is determined by six extrinsic camera parameters, five intrinsic camera parameters and three other motion parameters. All the fourteen parameters are estimated approximately by using the direct linear transformation of a reasonably simplified camera model firstly, and then the results are further refined by non-linear least square mean (LSM). Both computer simulated data and real data are used to test our calibration method. The robustness and accuracy are verified by lots of simulated experiments, and for the real data, the root mean square error of re-projected points is less than 0.3 pixels.

Chaotic behaviors of the (2+1)-dimensional generalized Breor—Kaup system

With the help of the Maple symbolic computation system and the projective equation approach, a new family of variable separation solutions with arbitrary functions for the (2+1)-dimensional generalized Breor—Kaup (GBK) system is derived. Based on the derived solitary wave solution, some chaotic behaviors of the GBK system are investigated.

Abstract P389: Cardiovascular Disease Mortality Risk Prediction in Educational Classes across European Regions, in the MORGAM Project Populations

Aims: Although socioeconomic status is a recognized independent risk factor for CVD mortality, the recommended European risk prediction equation for primary prevention does not consider it; an approach criticized by previous results in the UK and US. We aim to assess whether the SCORE project equation adequately estimates the risk in different educational classes, across several European populations. Methods: We considered 47 prospective population-based surveys from Nordic Countries (Finland, Denmark, Sweden), UK (Belfast and Scotland), Central Europe (France, Germany and Italy) and East Europe (Lithuania, Poland) and Russia. Baseline data collection and mortality follow-up (median time 10 years) adhered to standardized MONICA-like procedures. Three educational classes were derived from population-, sex- and birth year-specific tertiles of years of schooling. The individual SCORE risk was computed from age, total cholesterol, systolic blood pressure and smoking; the risk was recalibrated to the average observed risk in each population. We estimated age- and traditional risk factors-adjusted hazard ratios (HR) for 10 year CVD mortality (highest education as the reference), from Cox models. Moreover, the observed number of fatal CVD events by educational class was compared to the expected number, as estimated by the recalibrated SCORE function. Results: The cohorts summed-up 39,215 men and 29,240 women 40 to 64 years old and free from CVD event at baseline. Education was associated with CVD mortality in men (pooled age-adjusted HR for low vs high education: 1.6, 95% CI 1.4–1.9); the hazard ratios ranged from 1.3 (95%CI: 0.9–1.8) in Central Europe to 2.1 (1.6–2.7) in East Europe and Russia. The association attenuated after adjustment for SCORE risk factors and HDL-cholesterol. Among women, the association was significant in Nordic Countries only (age-adjusted HR for low vas high education: 1.7, 95% CI 1.1–2.6), but it was no more significant after adjustment for multiple risk factors. The original SCORE equation overestimated the risk at a population level, both in men and in women, except in East Europe and Russia. After recalibration, the SCORE equation overestimated the risk among the more educated men by 20% to 50% (in Central Europe, East Europe and Russia, respectively), but underestimated it in the less educated men by 7% to 23% (in Central Europe, East Europe and Russia, respectively). Conclusions: Our results, based on a well-harmonized study comprising several European populations, suggest the need to include country-specific socioeconomic status in the risk estimation equations.

Acceleration of Unbalanced Rotors

AbstractA lot of work in rotordynamics is done in the evaluation of the Laval rotor. By accelerating this type of rotor, described by a single rigid disc centred in the middle of a slim shaft modelled as spring, one can observe the sticking of the rotational speed at the first bending eigenfrequency of the system, also named critical speed. This phenomenon only appears when the rotor is unbalanced and a small external torque accelerates the rotor. Taking a look at higher critical speeds one will notice that the Laval rotor model is insufficient accurate to take this eigenfrequencies into account. Therefore the equations of motion are derived using the Projection Equation in subsystem representation leading to Partial Differential Equations (PDE). Using the Timoshenko beam theory enables one to include the discs not only as rigid body but as real elastic part of the beam system. Applying the Transfer-Matrix-Method (TMM) to the system's linearized equations delivers adequate shape functions for the nonlinear PDE. The advantage of using the TMM is the possibility to calculate multidisc rotors and systems where the border between disc and shaft related to the diameter to length ratio cannot be defined clearly. Furthermore, the excentricity which is responsible for the unbalance of the rotor, has to be taken into account. The modeling is again done with the Projection Equation. Numerical simulation solutions are presented and compared to experimental results.

Ordering and Large-Scale Relaxation Properties of Macromolecules in Ferroelectric Polymer Films

Dynamical equations for transverse and longitudinal mean projections of elastic-deformed kinetic chain units (Gaussian subchains) with anisotropic local intra- and interchain orientation interactions of dipolar type and spectra of relaxation times for large-scale motions of the chains in an isotropic and ordered state were obtained. The long-range orientation planar order in the 3D-layered systems leads to an anisotropy of large-scale relaxation properties of the chains (relative to the axis perpendicular to layers). The anisotropy increases with increase of the chain bending rigidity, the strength of interchain interactions and decrease of the scale of its motion. It disappears for most large-scale motions of chains.