Successive Synthesis of Substructure Modes

1991 ◽  
Vol 58 (3) ◽  
pp. 759-765 ◽  
Author(s):  
Luis E. Suarez ◽  
Mahendra P. Singh
Keyword(s):  
Closed Form ◽  
Characteristic Equation ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Closed Form Expression ◽  
Element Discretization ◽  
Root Finding ◽  
Complete Set ◽  
Conventional Solution ◽  
Second Eigenvalue

A mode synthesis approach is presented to calculate the eigenproperties of a structure from the eigenproperties of its substructures. The approach consists of synthesizing the substructures sequentially, one degree-of-freedom at a time. At each coupling stage, the eigenvalue is obtained as the solution of a characteristic equation, defined in closed form in terms of the eigenproperties obtained in the preceding coupling stage. The roots of the characteristic equation can be obtained by a simple Newton-Raphson root finding scheme. For each calculated eigenvalue, the eigenvector is defined by a simple closed-form expression. The eigenproperties obtained in the final coupling stage provide the desired eigenproperties of the coupled system. Thus, the approach avoids a conventional solution of the second eigenvalue problem. The approach can be implemented with the complete set or a truncated number of substructure modes; if the complete set of modes is used, the calculated eigenproperties would be exact. The approach can be used with any finite element discretization of structures. It requires only the free interface eigenproperties of the substructures. Successful application of the approach to a moderate size problem (255 degrees-of-freedom) on a microcomputer is also demonstrated.

scholarly journals NDF of the near-field of a strip source in orthogonal directions

2020 ◽  
Author(s):  
Rocco Pierri ◽  
Raffaele Moretta
Keyword(s):  
Inverse Problems ◽  
Closed Form ◽  
Degrees Of Freedom ◽  
Near Field ◽  
Closed Form Expression ◽  
Main Difficulty ◽  
Form Expression ◽  
Independent Functions ◽  
Near Zone

<div><div>In the manuscript, we address the problem of evaluating the</div><div>number of degrees of freedom (NDF) of the field radiated by a strip source along all the directions orthogonal to it. </div><div>The NDF represents at the same time the number of independent functions required to represent the data with a given degree of accuracy, and the dimension of the unknowns subspace that can be stably reconstructed. For such reason, the knowledge of the NDF gives insight on the forward and on the inverse problems and it represents one of the metrics to evaluate the achievable performance in the inversion.</div></div><div>The main difficulty arises since in near-zone the eigenvalue</div><div>problem that must be solved for the computation of the NDF,</div><div>involves a non-convolution and non-bandlimited kernel. In the paper, we show how to overcome this drawback and how to obtain a closed-form expression of the NDF which highlights the role played by the configuration parameters.</div>

Kinematics of Continuum Robots With Constant Curvature Bending and Extension Capabilities

2018 ◽  
Vol 11 (1) ◽  
Author(s):  
Arnau Garriga-Casanovas ◽  
Ferdinando Rodriguez y Baena
Keyword(s):  
Closed Form ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Rapid Development ◽  
Closed Form Expression ◽  
Great Promise ◽  
Continuum Robots ◽  
Cluttered Environments ◽  
Closed Form Solutions ◽  
Significant Attention

Continuum robots are becoming increasingly popular due to the capabilities they offer, especially when operating in cluttered environments, where their dexterity, maneuverability, and compliance represent a significant advantage. The subset of continuum robots that also belong to the soft robots category has seen rapid development in recent years, showing great promise. However, despite the significant attention received by these devices, various aspects of their kinematics remain unresolved, limiting their adoption and obscuring their potential. In this paper, the kinematics of continuum robots with the ability to bend and extend are studied, and analytical, closed-form solutions to both the direct and inverse kinematics are presented. The results obtained expose the redundancies of these devices, which are subsequently explored. The solution to the inverse kinematics derived here is shown to provide an analytical, closed-form expression describing the curve associated with these redundancies, which is also presented and analyzed. A condition on the reachable end-effector poses for robots with six actuation degrees-of-freedom (DOFs) is then distilled. The kinematics of robot layouts with over six actuation DOFs are subsequently considered. Finally, simulated results of the inverse kinematics are provided, verifying the study.

scholarly journals NDF of the near-field of a strip source in orthogonal directions

2020 ◽  
Author(s):  
Rocco Pierri ◽  
Raffaele Moretta
Keyword(s):  
Inverse Problems ◽  
Closed Form ◽  
Degrees Of Freedom ◽  
Near Field ◽  
Closed Form Expression ◽  
Main Difficulty ◽  
Form Expression ◽  
Independent Functions ◽  
Near Zone

<div><div>In the manuscript, we address the problem of evaluating the</div><div>number of degrees of freedom (NDF) of the field radiated by a strip source along all the directions orthogonal to it. </div><div>The NDF represents at the same time the number of independent functions required to represent the data with a given degree of accuracy, and the dimension of the unknowns subspace that can be stably reconstructed. For such reason, the knowledge of the NDF gives insight on the forward and on the inverse problems and it represents one of the metrics to evaluate the achievable performance in the inversion.</div></div><div>The main difficulty arises since in near-zone the eigenvalue</div><div>problem that must be solved for the computation of the NDF,</div><div>involves a non-convolution and non-bandlimited kernel. In the paper, we show how to overcome this drawback and how to obtain a closed-form expression of the NDF which highlights the role played by the configuration parameters.</div>

scholarly journals A Design Method of Sparse Array with High Degrees of Freedom Based on Fourth-Order Cumulants

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Fengtong Mei ◽  
Daming Wang ◽  
Chunxiao Jian ◽  
Yinsheng Wang ◽  
Weijia Cui
Keyword(s):  
Closed Form ◽  
Fourth Order ◽  
Degrees Of Freedom ◽  
Linear Array ◽  
Design Method ◽  
Doa Estimation ◽  
Closed Form Expression ◽  
Form Expression ◽  
Sparse Arrays ◽  
Non Gaussian

Recently, the design of sparse linear array for direction of arrival (DOA) estimation of non-Gaussian signals has attracted considerable interest due to the fact that the fourth-order difference coarray offered by non-Gaussian significantly increases the aperture of a virtual linear array, which improves the performance of DOA estimation. In this paper, a super four-level nested array (S-FL-NA) configuration based on fourth-order cumulants (FOC) is proposed. The S-FL-NA consists of uniform linear arrays which have different interelement spacing. The proposed array configuration is designed based on interelement spacing, which, for a given number of sensors, is uniquely determined by a closed-form expression. We also derive the closed-form expression for the degrees of freedom (DOFs) of the proposed array. The optimal distribution of the number of sensors in each uniform linear array of the proposed array is given for an arbitrary number of sensors. Compared with the existing sparse arrays, the proposed array can provide a higher number of degrees of freedom and a larger physical array aperture. In addition, to improve the calculation speed of the fourth-order cumulant matrix, we simplify the FOC matrix by removing some redundancy. Numerical simulations are conducted to verify the superiority of the S-FL-NA over other sparse arrays.

scholarly journals A Finite Element Approach to the Analysis of Rotating Bladed-Disk Assemblies Coupled With Flexible Shaft

1994 ◽  
Author(s):  
Wen Zhang ◽  
Wenliang Wang ◽  
Hao Wang ◽  
Jiong Tang
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Equation Of Motion ◽  
Coupled Systems ◽  
The Finite Element Method ◽  
Bladed Disk ◽  
Modal Synthesis ◽  
Element Approach ◽  
Final Equation

A method for dynamic analysis of flexible bladed-disk/shaft coupled systems is presented in this paper. Being independant substructures first, the rigid-disk/shaft and each of the bladed-disk assemblies are analyzed separately in a centrifugal force field by means of the finite element method. Then through a modal synthesis approach the equation of motion for the integral system is derived. In the vibration analysis of the rotating bladed-disk substructure, the geometrically nonlinear deformation is taken into account and the rotationally periodic symmetry is utilized to condense the degrees of freedom into one sector. The final equation of motion for the coupled system involves the degrees of freedom of the shaft and those of only one sector of each of the bladed-disks, thereby reducing the computer storage. Some computational and experimental results are given.

scholarly journals Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Yassine Zouaoui ◽  
Larbi Talbi ◽  
Khelifa Hettak ◽  
Naresh K. Darimireddy
Keyword(s):  
Closed Form ◽  
Closed Form Expression ◽  
Form Expression

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

scholarly journals Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi
Keyword(s):  
Closed Form ◽  
Topological Strings ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Infinite Family ◽  
Closed Form Expression ◽  
Two Dimensional ◽  
Laurent Polynomials ◽  
Form Expression ◽  
Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

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