Closed-form solution for motion of a spinning rigid body.

1968 ◽  
Vol 5 (5) ◽  
pp. 606-607
Author(s):  
BRADFORD W. PARKINSON
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution

On Saint-Venant's Problem for Helicoidal Beams

2015 ◽  
Vol 83 (2) ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Fem Analysis ◽  
Original System ◽  
Form Solution ◽  
Hamiltonian Matrix ◽  
Jordan Structure ◽  
Rigid Body Motions

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

A Symbolic-Numerical Approach for the Synthesis of Planar Rigid-Body Guidance

1994 ◽  
Author(s):  
Yong Chen ◽  
Jinyuan Fu
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Fourth Order ◽  
Closed Form Solution ◽  
Numerical Approach ◽  
Form Solution ◽  
System Of Nonlinear Equations ◽  
Single Variable ◽  
All Solutions ◽  
Order Algebraic

Abstract Based on the pole method and the computer symbolic manipulating technique, a fourth order algebraic equation with single variable has been derived for the closed form solution of planar rigid-body guidance. The newly developed method eliminates the requirement of sloving system of nonlinear equations. All solutions of the problem can be obtained directly and easily without the need of initial values. A numerical example is given demonstrating the efficiency and advantages of the method.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

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