Dual Coordinate Systems Research Articles

Dual Attitude Representations and Kinematics for Six-Degree-of-Freedom Spacecraft Dynamics

This paper focuses on the development of three newly formed representations of dual attitude and their respective kinematics equations - Dual Classical Rodrigues Parameters, Dual Modified Rodrigues Parameters, Dual Principal Axis and Principal Angle (Dual Principal Rotation Vector). These three formulations are developed by building off of the theories and concepts used with Dual Quaternions and utilizing numerous attitude identities for quaternions to create new identities using dual quaternions and dual attitudes while detailing the process used throughout the derivation. The accompanying dual kinematic equations using these dual attitude parameters are also generated using their respective dual attitude coordinate systems. Furthermore, the new dual attitude representations with their respective, distinct kinematic equations and identities are validated against a simulated 6-DOF rigid body spacecraft control problem to verify equivalency across equations using the dual quaternion as a baseline.

Information-geometric structure for chemical thermodynamics: An explicit construction of dual affine coordinates.

We construct an information-geometric structure for chemical thermodynamics, applicable to a wide range of chemical reaction systems including nonideal and open systems. For this purpose, we explicitly construct dual affine coordinate systems, which completely designate an information-geometric structure, using the extent of reactions and the affinities of reactions as coordinates on a linearly constrained space of amounts of substances. The resulting structure induces a metric and a divergence (a function of two distributions of amounts), both expressed with chemical potentials. These quantities have been partially known for ideal-dilute solutions, but their extensions for nonideal solutions and the complete underlying structure are novel. The constructed geometry is a generalization of dual affine coordinates for stochastic thermodynamics. For example, the metric and the divergence are generalizations of the Fisher information and the Kullback-Leibler divergence. As an application, we identify the chemical-thermodynamic analog of the Hatano-Sasa excess entropy production using our divergence.

Fabrication and Characterization of Dual Coordinate Self Examined Thermal Flow Sensor Arrays Based on Longitudinal Heat Conduction

Thermal flow sensors have been widely used in wind energy utilization, dangerous chemicals detection, automobiles, intelligent robots and unmanned aerial vehicles (UAV). New applications put forward higher requirements for detecting parameters of a two-dimensional (2-D) gas-flow vector over a full range of 360°. A novel thermal flow sensor on Si/SiO2/Pt/AlN substrate is presented to optimize sensor parameters. Eight thermal elements patterned as equally divided sectors of a regular octagon compose the sensor probe. Dual coordinate systems are created based on the regular octagon distribution to realize the self examined function. The longitudinal heat conduction mode shortens the heat transfer distance from mm level to micron level. And the a-axis oriented AlN film improves the heat transfer efficiency due to its high thermal conductivity. Dual coordinate self examined method is proven through experiments for the first time. Measurement results show that the maximum measurement flow velocity is up to 37m/s, with an output voltage sensitivity of 28.1 mV/(m/s). Flow direction was measured over a full range of 360° with an accuracy of 0.5 m/s at low velocity and 5% at high velocity. We believe that the sensor presented in this work will pave the way for novel applications in calorimetric sensors and array sensors.

Publication quality 2D graphs with less manual effort due to explicit use of dual coordinate systems

Creating visually pleasing graphs in data visualization programs such as Matlab is surprisingly challenging. One common problem is that the positions and sizes of non-data elements such as textual annotations must typically be specified in either data coordinates or in absolute paper coordinates, whereas it would be more natural to specify them using a combination of these coordinate systems. I propose a framework in which it is easy to express, e.g., “this label should appear 2 mm to the right of the data point at (3, 2)” or “this arrow should point to the datum at (2, 1) and be 5 mm long.” I describe an algorithm for the correct layout of graphs of arbitrary complexity with automatic axis scaling within this framework. An implementation is provided in the form of a complete 2D plotting package that can be used to produce publication-quality graphs from within Matlab or Octave.

Twelve-dimensional Stroh-like formalism for Kirchhoff anisotropic piezoelectric thin plates

A twelve-dimensional Stroh-like formalism is developed for the coupled stretching, bending and polarization of a Kirchhoff anisotropic piezoelectric thin plate which is inhomogeneous and laminated along the thickness direction. The structure and explicit expressions of the fundamental piezoelectric plate matrix and its inverse form are established. The formalism in a rotated coordinate system is presented. Then the fundamental piezoelectric plate matrix in dual coordinate systems can be addressed. Some identities associated with the new formalism are derived. It is rigorously proved that each 3×3 partitioned matrix of the introduced three 6×6 real matrices S, H and L and the 6×6 Hermitian matrix M is a second-order tensor. Sixty-four permuted forms of the twelve-dimensional formalism are presented. Finally, to demonstrate the applications of the new formalism, we investigate the effective electroelastic properties of a microcracked anisotropic piezoelectric thin plate within the framework of non-interaction approximation (NIA) and study the asymptotic problem associated with a semi-infinite interface crack between two dissimilar piezoelectric thin plates.

Reverberation-ray analysis of continuous Timoshenko beams subject to moving loads

The method of reverberation-ray matrix is extended to investigate the transient response of a multi-span, continuous Timoshenko beam subject to a moving load. Dual local coordinate systems are introduced and the corresponding analytical solutions in the frequency domain are presented, with the particular solution derived by the dual Fourier transform. The transient responses are then obtained using the Fast Fourier Transform algorithm. To avoid the error due to the singularity (or near singularity) of the kernel of integral, the technique of Neumann series expansion is adopted for the undamped response of beams. Numerical results are compared with those obtained by the existing methods and a good agreement is observed. It is found that the present method is uniform in formulating for dynamic analysis of beam-typed structures with multi-branches and is especially suitable for calculating transient responses containing high-frequency components. Besides, no discretization error is involved as the method is completely based on continuum models.

Structure and properties of the fundamental elastic plate matrix

AbstractThis work presents further development of the octet formalism established by the authors for the classical Kirchhoff anisotropic plate theory. The structure of the fundamental elastic plate matrix is fully explored and the explicit expression is provided. The matrices N2 and N3 are proved to be positive semi‐definite. Thus, H and L are positive definite. Further studies are concerned with a rotated coordinate system. The transform relation between the eigenvectors in the original and the rotated coordinate system is given. The fundamental elastic plate matrix associated with the eigenrelation referring to the dual coordinate systems, NΘ, is studied. The major properties that hold in the Stroh sextic formalism for generalized plane strain problems are also valid in the octet formalism for thin plate bending problems. In particular, we generalize a property in Stroh's formalism for any non‐semisimple matrix NΘ. We show a new property in the octet formalism. The non‐semisimple cases of NΘ are discussed. Finally, we make it transparent that the mixed/hybrid formalism of others is precisely one of sixteen permuted forms of the octet formalism.

Exponential and mixture families in quantum statistics: Dual structure and unbiased parameter estimation

The differential-geometric formulation of statistics (the so-called information geometry) concerning the structure of a smooth manifold in the parameter space Θ of classical probabilities, S = { p(·, θ), θ ϵ Θ}, discussed by Amari, is extended to the same manifold but for quantum states (density matrices), S = { ϱ( θ); θ ϵ Θ} in N × N matrix algebras. This is done by introducing an n-tuple of tangent vectors { δ} n i = 1 in analogy to the classical ones { ∂ i } n i = 1 . On this basis, a special problem of quantum information geometry is treated; namely, the analysis of the exponential and the mixture families defined, respectively, as (e) ϱ(θ) = exp(θ i A i − ψ(θ)). θ ϵ Θ = R n. A i ϵ B s(H N) . (m) ϱ(θ) = θ iA i + θ 0 A 0. θ ϵ Θ = (0,1) n + 1. ∑ i=0 n θ i=1. A i ϵ B +(H N) Tr A i = 1 (the tensorial summation convention for repeated indices is used). We prove some of the basic theorems known in the classical information geometry by extending the formulation to a non-commutative smooth manifold. We establish the existence of a pair of dual affine coordinate systems in (e) or (m) and a projection theorem in order to ensure the Cramer-Rao inequality and an identification of the efficient estimator.

Anisotropic Elasticity: Theory and Applications

1. Matrix Algebra 2. Linear Anisotropic Elastic Materials 3. Antiplane Deformations 4. The Lekhnitskii Formalism 5. The Stroh Formalism 6. The Structures and Identities of the Elasticity Matrices 7. Transformation of the Elasticity Matrices and Dual Coordinate Systems 8. Green's Functions for Infinite Space, Half-space, and Composite Space 9. Particular Solutions, Stress Singularities, and Stress Decay 10. Anisotropic Matrials with an Elliptic Boundary 11. Anisotropic Media with a Crack or a Rigid Line Inclusion 12. Steady State Motion and Surface Waves 13. Degenerate and Near Degenerate Materials 14. Generalization of the Stroh Formulism 15. Three-Dimensionsal Deformations