Independent Coordinates Research Articles

Free vibration analysis of rectangular plate with a hole by means of independent coordinate coupling method

This paper is concerned with the vibration analysis of a rectangular plate with a rectangular hole or a circular hole. In the previous researches, the effect of the hole is taken into account by subtracting the energies of the hole domain from the total energies of the whole plate. In doing so, the integrals of the functions consisting of the admissible functions based on the plate domain over hole domain lead to complex numerical computations. In this paper, a new methodology called Independent Coordinate Coupling Method (ICCM) was developed, in which the energies corresponding to the rectangular plate domain and the hole domain were derived independently and the two independent coordinates were coupled by imposing kinematic relations. By matching the deflection condition imposed on the expressions, the relationship between the global axes and the local axes can be easily derived and then used to construct the mass and stiffness matrices. The numerical results show the efficacy of the proposed method.

Dual-bounded generating problems: Efficient and inefficient points for discrete probability distributions and sparse boxes for multidimensional data

We show that ∣ X ∣ ≤ n ∣ Y ∣ must hold for two finite sets X , Y ⊂ R n whenever they can be separated by a nonnegative linear function such that X is above Y and the componentwise minimum of any two distinct points in X is dominated by some point in Y . As a consequence, we obtain an incremental quasi-polynomial time algorithm for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p -inefficient points of a given discrete probability distribution, and for generating all maximal hyper-rectangles which contain a specified fraction of points of a given set in R n . This provides a substantial improvement over previously known exponential time algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, implying that for discrete probability distributions with independent coordinates, both p -efficient and p -inefficient points can be separately generated in incremental polynomial time.

Subsystem synthesis method with approximate function approach for a real-time multibody vehicle model

This paper presents the subsystem synthesis method with approximate function approach for a real-time multibody vehicle dynamics model. In the subsystem synthesis method, equations of motion for the car body of a vehicle and the equations of motion for suspension subsystems are formed separately for efficient computation. Joint coordinates are used to construct suspension subsystem equations of motion. Since these joint coordinates must satisfy the loop closure constraint equations that represent suspension linkage kinematics, they are not all independent. Using the generalized coordinate partitioning method, suspension subsystem equations of motion can be represented only in terms of independent generalized coordinates. To represent dependent coordinates as a function of independent coordinates in the generalized coordinate partitioning method, expensive numerical approaches, such as the Newton–Raphson method, must be applied. For real-time computation of the multibody vehicle model, an approximate function approach is proposed to express the dependent coordinates as polynomial functions of the independent coordinates within the framework of the subsystem synthesis method. Different orders of candidate polynomial functions are investigated for solution accuracy. Efficiency of the proposed method has been studied theoretically by counting arithmetic operators. By measuring actual CPU times of the simulations with a quarter car and a full car model, efficiency of the proposed method has also been investigated.

A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.

A new locking-free shear deformable finite element based on absolute nodal coordinates

The absolute nodal coordinate formulation has been recently extended to shear deformable beam or plate elements. This has been accomplished, in practice, by parameterizing the complete volume of the elements instead of a line or surface in the element kinematics description. In the absolute nodal coordinate formulation, the position of any point of the element volume is defined employing independent slope coordinates. The use of a large number of slope coordinates leads to unusual kinematic features that must be accounted for in order to avoid the element locking. This study demonstrates that the shear deformable element based on the absolute nodal coordinate formulation suffers from curvature thickness locking and shear locking in addition to the previously reported Poisson’s locking. Due to the tendency of locking, the use of the absolute nodal coordinate formulation can lead to elements with weak performance. In order to eliminate locking problems, this study introduces a new absolute nodal coordinate-based finite element. The introduced element uses redefined polynomial expansion together with a reduced integration procedure. The performance of the introduced element is studied by means of certain dynamic problems. The element exhibits a competent convergence rate and it does not suffer from the previously mentioned locking effects.

A tibial-based coordinate system for three-dimensional data

An accurate and repeatable tibial measurement system will aid in the definition of tibial geometry and improving tibial prosthesis design. Unlike in the femur, there is no standardized method for constructing a tibial coordinate frame. Most tibial measurements are given relative to femoral axes or the coordinate frame of the CT/MRI scanner or radiograph machine. The objective of this study was to establish an independent tibial coordinate frame. Data consisted of CT scans from 34 subjects. The tibial anatomical axis was chosen as the axial axis. The anteroposterior (AP) axis was selected to be parallel to the lateral surface of the tibial shaft and orthogonal to the anatomical axis and from this the mediolateral axis could be derived. The selected AP axis was compared with the surgical tibial AP axis by measuring their variability relative to a common axis, the posterior tibial condylar line (PTCL). The mean angle between the selected AP axis and the perpendicular to the PTCL was measured as −4.07°, standard deviation of 4.28°. The mean angle between the surgical AP axis and the perpendicular to the PTCL was measured as −18.56°, standard deviation of 4.66°. There was no significant difference in the variance of the two sets of measurements (p=0.63). Variability of the selected AP axis was even smaller (standard deviation of 2.74°) when measured independently from the PTCL reference axis, by aligning virtual resection profiles. Anatomically, the selected AP axis was almost perpendicular to the posterior tibial condylar axis. This coordinate system can aid in gathering consistent and repeatable anthropometric data that can be used to improve tibial implant design and could also, in combination with CT/MR imaged-based computer assisted surgery, be used as a guideline for tibial component positioning in TKR.

Modelling and simulation of a sphere rolling on the inside of a rough vertical cylinder

A classical problem of nonholonomic system dynamics—the motion of a sphere on the inside of a rough vertical cylinder—is extended to rolling friction. The case study is modelled in independent coordinates. Due to the nonholonomic constraints imposed on the sphere, the governing equations arise as a set of differential-algebraic equations. The results of numerical simulations show the transition of the sphere from a sinusoid path on the vertical cylinder surface to a fall with slip. The physics of the ‘paradoxical’ motion is explained in detail.

A Large Deviations Approach to the Geometry of Random Polytopes

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

Functional Degrees of Freedom

Mechanical degrees of freedom (DOF) are defined as the minimum number of independent coordinates needed to describe a system's position. The human musculoskeletal system has many mechanical DOF through which countless movements are accomplished. In the motor control field, one of the aspirations is to understand how the many DOF are organized for movement execution-the so-called DOF problem. Natural movements are characterized by the coordination of the DOF such that few vary independently. The concept of functional degrees of freedom (fDOF) is introduced to describe the very limited DOF of purposeful, coordinated movements. Deterministic (i.e., constraint satisfaction) and statistical (i.e., principal component analysis) approaches are used to determine fDOF. In contrast to DOF as a mechanical descriptor, fDOF emphasizes the mechanisms of human movements and corroborates our search for the solution to the DOF problem.

Gravoturbulent Formation of Planetesimals

We explore the effect of magnetorotational turbulence on the dynamics and concentrations of boulders in local box simulations of a sub-Keplerian protoplanetary disk. The solids are treated as particles, each with an independent space coordinate and velocity. We find that the turbulence has two effects on the solids. (1) Meter and decameter bodies are strongly concentrated, locally up to a factor of 100 times the average dust density, whereas decimeter bodies only experience a moderate density increase. The concentrations are located in large-scale radial gas density enhancements that arise from a combination of turbulence and shear. (2) For meter-sized boulders, the concentrations cause the average radial drift speed to be reduced by 40%. We find that the densest clumps of solids are gravitationally unstable under physically reasonable values for the gas column density and for the dust-to-gas ratio due to sedimentation. We speculate that planetesimals can form in a dust layer that is not in itself dense enough to undergo gravitational fragmentation, and that fragmentation happens in turbulent density fluctuations in this sublayer.

Kinetic aspects of deactivation of catalysts in prolonged use

A theoretical approach to the dynamic analysis of deactivation and aging of heterogeneous catalysts is proposed — analysis of the rate derivative in a kinetic equation of a given form. Methodology was demonstrated for the simplest kinetic equations — zero-and first-order. Regardless of the form of the kinetic equation for the dynamic model of deactivation, the rate constant is a function of an independent reaction coordinate. The method can also be used for solving reactivation problems and for explaining the dynamics of cation exchange in zeolite.

Performance Optimization of the Magnetohydrodynamic Generator at the Scramjet Inlet

The performance optimization problem of a magnetohydrodynamic (MHD) generator at the inlet of a scramjet engine is studied. The generator performance is characterized in terms of both the net energy extracted from the flow and the flow characteristics through the channel before it enters the combustion chamber of the scramjet engine. The analysis assumes a steady-state one-dimensional flow in which the position along the channel is treated as an independent coordinate. The performance optimization problem can then be handled by optimal control theory, in which position now plays the role of time as in a standard control problem. In this work, an optimal neural-networks-based controller design technique developed in our previous work is used. The technique is well suited for this application, because the MHD system is highly nonlinear. Furthermore, the technique is data-based, in that experimental data obtained from the system can be used to design the optimal controller without requiring an explicit model of the system. Nomenclature A f = cross-sectional area of the channel, m 2 B f

Historical data analysis based on plots of independent and parallel coordinates and statistical control limits

Interactive multidimensional visualisation based on parallel coordinates has been studied previously as a tool for process historical data analysis. Here attention is given to improvement of the technique by the introduction of dimension reduction and upper and lower limits for separating abnormal data to the plots of coordinates. Dimension reduction using independent component analysis transforms the original variables to a smaller number of latent variables which are statistically independent to each other. This enables the visualisation technique to handle a large number of variables more effectively, particularly when the original variables have recycling and interacting correlations and dependencies. Statistical independence between the parallel coordinates also makes it possible to calculate upper and lower limits (UL and LL) for each coordinate separating abnormal data from normal. Calculation of the UL and LL limits requires each coordinate to satisfy Gaussian distribution. In this work a method called the Box–Cox transformation is proposed to transform the non-Gaussian coordinate to a Gaussian distribution before the UL and LL limits are calculated.

Special relativity: Interpretation and implications for space-time geometry

The paper commemorates the centenary of the special theory of relativity, which effectively sets the limit for the structure of space-time to that of the stationary system. The long lasting debate for definition of concepts of instantaneity and simultaneity was thus resolved by the declaration of constancy of speed of light in vacuo as a law of physics. All motions were thus bounded by the light cone and described by the properties of differential geometry, firmly anchored in the calculus of variations. The key contribution underpinning the theory was the resolution of the contradiction imposed by the Galilean transformation through physical explanation and the adoption of the Lorentzian transformation. This highlighted the relative nature of both space and time and the linkage of these to preserve the sanctity of the light cone. The resulting space-time geometry was then founded on the traditional calculus of variation with the addition of this transformation. This retains the time as an independent coordinate and its linkage to space in an explicit form. One implication of this approach has been the retention of the concept of infinitum for some physical quantities as a drawback for use of the Lorentzian transformation. The paper shows that this singular behaviour need not arise if the explicit linkage in space-time is abandoned in favour of the implicit inclusion of time as a link between the curved structure of space and the speed of light, thus restating the calculus of variation in line with special relativity. This points to a closed loop space-matter field, which may belie the fabric of the continuum. One implication of this interpretation is that a small variation in speed of light within the field would be required to dispense with the aforementioned singular nature of the Lorentzian boost, while still remaining within the spirit of special relativity.

Scalar Field at the Phase Transition Point of RNdS Space

The real scalar field equation between the outer black holehorizon and the cosmological horizon is solved in theReissner–Nordström–de Sitter space when it is at thephase transition point. We use an accurate approximation, thepolynomial approximation, to approximate the tortoisecoordinate x(r) for obtaining the inverse function r = r(x)and then for solving the wave equation. The case where the twohorizons are very close to each other is discussed in detail. Itis found that the wave function is characteristically similar tothe harmonic in the whole range with x as the independentcoordinate, while the waves pile up near the horizons with r asthe independent coordinate. Furthermore, we find that the heightof the potential increases as the cosmological constant Λdecreases.

Implicit Numerical Integration for Design Sensitivity Analysis of Rigid Multibody Systems

ABSTRACT An implicit Runge-Kutta numerical integration algorithm based on generalized coordinate partitioning is presented for solution of the differential-algebraic equations of motion and design sensitivity analysis for rigid multibody systems. The direct differentiation method for dynamic sensitivity analysis is employed to obtain differential-algebraic sensitivity equations. The implicit integration formula expresses design sensitivity of independent generalized coordinates and their first-time derivative as functions of sensitivities of independent accelerations at discrete integration times. The same integration Jacobian is used to determine generalized independent accelerations and their sensitivities, by iteratively solving the equations obtained from state-space, second-order ordinary differential equations in independent coordinates. Design sensitivities of dependent variables in the formulation, including design sensitivities of Lagrange multipliers, are recovered by satisfying the full design sensitivity system of kinematic and kinetic equations of motion. Design sensitivity analysis results are verified using finite differences, and a comparison with an explicit numerical integration algorithm is performed to validate the efficiency of the proposed algorithm for design sensitivity analysis of stiff vehicle models.

Subsystem synthesis methods with independent coordinates for real-time multibody dynamics

For real time dynamic simulation, two different subsystem synthesis methods with independent generalized coordinates have been developed and compared In each formulation, the subsystem equations of motion are generated in terms of independent generalized coordinates The first formulation is based on the relative Cartesian coordinates with respect to moving subsystem base body The second foimulation is based on the relative joint coordinates using recursive formulation Computational efficiency of the formulations has been compared theoretically by the arithmetic operational counts In order to verify real-time capability of the formulations, bump run simulations of a quarter car model with SLA suspension subsystem have been carried out to measure the actual CPU time

Linearizing the Equations of Motion for Multibody Systems Using an Orthogonal Complement Method

The equations of motion of a multibody system are linearized and reduced to independent coordinates, using an orthogonal complement method. The orthogonal complement is used to eliminate the terms that result from a variation of the constraint forces. The resulting equations contain the derivative of the constraint Jacobian with respect to the coordinate vector in the stiffness matrix. The technique is suitable for a computer implementation. Examples are used to illustrate the process.

Bridge‐Mediated Electron Transfer and Multiple Reaction Coordinates

AbstractIn the case of superexchange tunneling, sensitivity of the tunneling gap (Δ) to progress along the thermal et reaction coordinate (including molecular as well as solvent contributions) has been formulated in terms of a general linear response framework, which is used to express the correlation between linearly independent reaction coordinates for distinct et steps. When the tunneling gap is relatively small (i.e., of the order of magnitude of reorganization energies), the corresponding sensitivity of the donor (D)/acceptor (A) coupling (HDA) and its distance dependence (decay coefficient ß) may be significant. The magnitude of Δ at the thermal et transition state has been compared with other reference values obtainable from optical or thermodynamic data. The initial state at equilibrium yields an upper bound for Δ, and so also does the equilibrium free energy change for charge injection onto the bridge in cases of et processes which are only moderately exothermic. The in vacuo value of Δ may constitute either an upper or lower bound, depending on the nature of the et process (charge separation or charge shift) and the details of the relevant reorganization energies.

A procedure for outlier identification in data sets from continuous distributions

We propose a procedure, based on sums of reciprocals ofp-values, for the identification of outliers in univariate or multivariate data sets coming from continuous distributions. Using results of Csorgő (1990), we find the limiting distribution of the relevant statistic for completely specified models. By simulations, we obtain approximate quantiles for the asymptotic distribution, (which does not depend on the specific model or the dimension where the data live) and for the finite sample distribution in different dimensions of our statistic when parameters are estimated, for the multivariate Gaussian model and a multivariate double exponential model with independent coordinates. Monte Carlo evaluation shows that the procedure proposed is effective in the identification of outliers, and that it is sensitive to sample size, a feature seldom found in outlier identification methods.