Numerical Solutions of Polynomial Equations for the Eight-Position Synthesis of the Cylindrical-Spherical Dyad

2012 ◽  
Author(s):  
Chintien Huang ◽  
Chenning Hung ◽  
Kuenming Tien
Keyword(s):  
Rigid Body ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Quadratic Equation ◽  
Geometric Constraints ◽  
Numerical Continuation ◽  
Polynomial Equations ◽  
Quartic Polynomial ◽  
Solutions Of Equations ◽  
Position Synthesis

This paper investigates the numerical solutions of equations for the eight-position rigid-body guidance of the cylindrical-spherical (C-S) dyad. We seek to determine the number of finite solutions by using the numerical continuation method. We derive the design equations using the geometric constraints of the C-S dyad and obtain seven quartic polynomial equations and one quadratic equation. We then solve the system of equations by using the software package Bertini. After examining various specifications, including those with random complex numbers, we conclude that there are 804 finite solutions of the C-S dyad for guiding a body through eight prescribed positions. When designing spatial dyads for rigid-body guidance, the C-S dyad is one of the four dyads that result in systems of equal numbers of equations and unknowns if the maximum number of allowable positions is specified. The numbers of finite solutions in the syntheses of the other three dyads have been obtained previously, and this paper provides the computational kinematic result of the last unsolved problem, the eight-position synthesis of the C-S dyad.

NUMERICAL CONTINUATION FOR NONLINEAR SCHRÖDINGER EQUATIONS

2007 ◽  
Vol 17 (02) ◽  
pp. 641-656 ◽  
Author(s):  
S.-L. CHANG ◽  
C.-S. CHIEN
Keyword(s):  
Coupling Coefficient ◽  
Chemical Potential ◽  
Numerical Solutions ◽  
Continuation Method ◽  
Lanczos Method ◽  
Iterative Refinement ◽  
Numerical Continuation ◽  
Chemical Potentials ◽  
Predictor Corrector ◽  
Centered Difference

We discuss numerical methods for studying numerical solutions of N-coupled nonlinear Schrödin-ger equations (NCNLS), N = 2, 3. First, we discretize the equations by centered difference approximations. The chemical potentials and the coupling coefficient are treated as continuation parameters. We show how the predictor–corrector continuation method can be exploited to trace solution curves and surfaces of the NCNLS, where the preconditioned Lanczos method with iterative refinement is used as the linear solver. When the chemical potential is large enough, we obtain peak solutions of the NCNLS for certain values of the coupling coefficient. The contours of the peak solutions resemble those of the experimental results of Anglin and Ketterle [2002], and Anderson et al. [1995].

On the Seven Position Synthesis of a 5-SS Platform Linkage

1997 ◽  
Vol 123 (1) ◽  
pp. 74-79 ◽  
Author(s):  
Qizheng Liao ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Reference Point ◽  
Polynomial Solution ◽  
Degree Of Freedom ◽  
Prismatic Joint ◽  
Movement Characteristics ◽  
Position Synthesis ◽  
Freedom Movement

This paper builds on Innocenti’s polynomial solution for the 5-SS platform that generates a one-degree of freedom movement through seven specified spatial positions of a rigid body. We show that his 60×60 resultant can be reduced to one that is 10×10. We then actuate the linkage using a prismatic joint on the sixth leg and determine the trajectory of the reference point through the specified positions. The singularity submanifold of this associated 6-SS platform provides information about the movement characteristics of the 5-SS linkage.

Designing Planar Mechanisms Using a Bi-Invariant Metric in the Image Space of SO (3)

1994 ◽  
Author(s):  
Pierre Larochelle ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Numerical Results ◽  
Image Space ◽  
Dimensional Synthesis ◽  
Invariant Metric ◽  
Planar Mechanisms ◽  
Position And Orientation ◽  
Position Synthesis

Abstract In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of SO(3). The optimal linkage is obtained by minimizing this distance over all of the n goal positions. The paper proceeds as follows. First, we approximate planar rigid body displacements with spherical displacements and show that the error induced by such an approximation is of order 1/R2, where R is the radius of the approximating sphere. Second, we use a bi-invariant metric in the image space of spherical displacements to synthesize an optimal spherical 4R mechanism. Finally, we identify the planar 4R mechanism associated with the optimal spherical solution. The result is a planar 4R mechanism that has been optimized for n position rigid body guidance using an approximate bi-invariant metric with an error dependent only upon the radius of the approximating sphere. Numerical results for ten position synthesis of a planar 4R mechanism are presented.

Five Position Synthesis of Spatial CC Dyads

1994 ◽  
Author(s):  
Andrew P. Murray ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Kinematic Chain ◽  
New Technique ◽  
Screw Axis ◽  
Fixed Axis ◽  
A New Technique ◽  
Position Synthesis

Abstract This paper presents a new technique for determining the fixed axes of spatial CC dyads for rigid body guidance through five finitely separated positions. A CC dyad is a kinematic chain consisting of a floating link connected by a cylindric joint to a crank which in turn is connected to ground by a second cylindric joint. The lines that can be axes of the fixed joint are shown to be obtained from a “compatibility platform” constructed from selected relative screw axes associated with the five specified displacements. We show that the screw axis of the displacement of this platform is a fixed axis of a CC dyad compatible with the five positions. Roth’s original example is presented to verify the calculations. The specialization of this procedure to planar and spherical five position synthesis is also presented.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

Numerical solutions of equations describing nonisothermal flows of a real gas in tubes

1967 ◽  
Vol 13 (4) ◽  
pp. 290-293
Author(s):  
B. L. Krivoshein ◽  
E. M. Minskii ◽  
V. P. Radchenko ◽  
I. E. Khodanovich ◽  
M. G. Khublaryan
Keyword(s):  
Numerical Solutions ◽  
Real Gas ◽  
Nonisothermal Flows ◽  
Solutions Of Equations
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