On the Stability of Waves in a Thin Orthotropic Spinning Disk

1982 ◽  
Vol 49 (3) ◽  
pp. 570-572 ◽  
Author(s):  
J. L. Nowinski
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Elementary Functions ◽  
Wave Stability ◽  
Spinning Disk ◽  
Perturbation Parameters ◽  
The Stability ◽  
Temporal Perturbation ◽  
Stability Of Waves

A system of two ordinary coupled differential equations with periodic coefficients of the Mathieu type for two temporal perturbation parameters is derived. A closed-form solution of the system in terms of elementary functions is found and discussed. A condition for the wave stability involving the coefficients of anisotropy is established. Illustration involves a specific range of these coefficients.

Exact closed-form solutions for Lamb’s problem—III: the case for buried source and receiver

2020 ◽  
Vol 224 (1) ◽  
pp. 517-532
Author(s):  
Xi Feng ◽  
Haiming Zhang
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Closed Form Solution ◽  
Time History ◽  
Natural Generalization ◽  
Elastic Half Space ◽  
Form Solution ◽  
Elementary Functions ◽  
Heaviside Step Function ◽  
Lamb’S Problem

SUMMARY In this paper, we derive the exact closed-form solution for the displacement in the interior of an elastic half-space due to a buried point force with Heaviside step function time history. It is referred to as the tensor Green’s function for the elastic wave equation in a uniform half-space, also a natural generalization of the classical 3-D Lamb’s problem, for which previous solutions have been restricted to the cases of either the source or the receiver or both are located on the free surface. Starting from the complex integral solutions of Johnson, we follow the similar procedures presented by Feng and Zhang to obtain the closed-form expressions in terms of elementary functions as well as elliptic integrals. Numerical results obtained from our closed-form expressions agree perfectly with those of Johnson, which validates our explicit formulae conclusively.

An Inverse Force Analysis of a Tetrahedral Three-Spring System

1995 ◽  
Vol 117 (2A) ◽  
pp. 286-291 ◽  
Author(s):  
P. Dietmaier
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Force Analysis ◽  
Equilibrium Configurations ◽  
Spring System ◽  
The Common ◽  
The Stability ◽  
Inverse Force

A tetrahedral three-spring system under a single load has been analyzed and a closed-form solution for the equilibrium positions is given. Each of the three springs is attached at one end to a fixed pivot in space while the other three ends are linked by a common pivot. The springs are assumed to behave in a linearly elastic way. The aim of the paper at hand was to find out what the maximum number of equilibrium positions of such a system might be, and how to compute all possible equilibrium configurations if a given force is applied to the common pivot. First a symmetric and unloaded system was studied. For such a system it was shown that there may exist a maximum of 22 equilibrium configurations which may all be real. Second the general, loaded system was analyzed, revealing again a maximum of 22 real equilibrium configurations. Finally, the stability of this three-spring system was investigated. A numerical example illustrates the theoretical findings.

scholarly journals Closed form solution to some mixed boundary value problems for a charged sphere

1987 ◽  
Vol 28 (3) ◽  
pp. 296-309 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Closed Form ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Closed Form Solution ◽  
Form Solution ◽  
Spherical Segment ◽  
Charged Sphere ◽  
Elementary Functions ◽  
Arbitrary Potential ◽  
Arbitrary Charge

AbstractA new method is described which allows an exact solution in a closed form to the following non-axisymmetric mixed boundary-value problem for a charged sphere: arbitrary potential values are given at the surface of a spherical segment while an arbitrary charge distribution is prescribed on the rest of the sphere. The method is founded on a new integral representation of the kernel of the governing integral equation. Several examples are considered. All the results are expressed in elementary functions. Some further applications of the method are discussed. No similar result seems to have been published previously.

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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