NUMERICAL SOLUTIONS OF POLYNOMIAL EQUATIONS

1968 ◽  
Author(s):  
Tadeusz Leser ◽  
Henry Wisniewski
Keyword(s):  
Numerical Solutions ◽  
Polynomial Equations

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.

Forward and Inverse Displacement Analysis of an Actuated Spoke Wheel Robot With Two Spokes and a Tail Contact With the Ground

2010 ◽  
Author(s):  
Ping Ren ◽  
Dennis Hong
Keyword(s):  
Design Optimization ◽  
Ground Motion ◽  
Numerical Solutions ◽  
Convex Surface ◽  
Polynomial Equations ◽  
Surface Geometry ◽  
Displacement Analysis ◽  
Unstructured Environments ◽  
Straight Line ◽  
Rigid Shell

Intelligent Mobility Platform with Active Spoke System (IMPASS) is a unique wheel-leg hybrid robot that can walk in unstructured environments by stretching in or out three independently actuated spokes of each wheel. The latest prototype of IMPASS has two actuated spoke wheels and one passive tail. In order to maintain its stability, the tail of the robot is designed as a rigid shell with a geometrically convex surface touching the ground. IMPASS is considered as a mechanism with variable topologies (MVTs) due to its metamorphic configurations. Its motions on the ground, such as steering, straight-line walking and other combinations, can be uniformly interpreted as a series of configuration transformations. Among all cases of its topologies, the cases with two spokes and the tail in contact with the ground possess two d.o.f and contribute the most to its ground motion. To fully understand the characteristics of such topologies, the forward and inverse displacement analysis is developed for these cases, with the polynomial equations derived. Numerical solutions from simulation are present to validate their formulation. These results lay the kinematics foundation for the motion monitoring and planning of IMPASS. It also contributes to the design optimization of the tail’s surface geometry to improve its adaptability on uneven terrains.

Dual Polynomials and Complex Dual Numbers for Analysis of Spatial Mechanisms

1996 ◽  
Author(s):  
Harry H. Cheng ◽  
Sean Thompson
Keyword(s):  
Numerical Solutions ◽  
Polynomial Equation ◽  
Mathematical Tool ◽  
Polynomial Equations ◽  
Dual Numbers ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Dual Number ◽  
Spatial Mechanisms ◽  
Language Environment

Abstract Complex dual numbers w̌1=x1+iy1+εu1+iεv1 which form a commutative ring are for the first time introduced in this paper. Arithmetic operations and functions of complex dual numbers are defined. Complex dual numbers are used to solve dual polynomial equations. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism. Like the dual number, the complex dual number is a useful mathematical tool for analytical and numerical treatment of spatial mechanisms.

Singularity Analysis of Spatial Mechanisms Using Dual Polynomials and Complex Dual Numbers

1999 ◽  
Vol 121 (2) ◽  
pp. 200-205 ◽  
Author(s):  
H. H. Cheng ◽  
S. Thompson
Keyword(s):  
Numerical Solutions ◽  
Polynomial Equation ◽  
Singularity Analysis ◽  
Polynomial Equations ◽  
Dual Numbers ◽  
Output Displacement ◽  
Spatial Mechanism ◽  
Spatial Mechanisms ◽  
Physical Insight ◽  
Language Environment

Complex dual numbers wˇ = x + iy + εu + iεv which form a commutative ring are introduced in this paper to solve dual polynomial equations numerically. It is shown that the singularities of a dual input-output displacement polynomial equation of a mechanism correspond to its singularity positions. This new method of identifying singularities provides clear physical insight into the geometry of the singular configurations of a mechanism, which is illustrated through analysis of special configurations of the RCCC spatial mechanism. Numerical solutions for dual polynomial equations and complex dual numbers are conveniently implemented in the CH language environment for analysis of the RCCC spatial mechanism.

Atomic Imaging of Crystals using Large-Angle Electron Scattering in STEM

1990 ◽  
Vol 48 (1) ◽  
pp. 74-75
Author(s):  
D.E. Jesson ◽  
S. J. Pennycook
Keyword(s):  
Wave Function ◽  
Electron Scattering ◽  
Numerical Solutions ◽  
Large Angle ◽  
Real Space ◽  
High Angle ◽  
Analytical Treatment ◽  
Order Zero ◽  
Experimental Conditions ◽  
Ewald Sphere

It is well known that conventional atomic resolution electron microscopy is a coherent imaging process best interpreted in reciprocal space using contrast transfer function theory. This is because the equivalent real space interpretation involving a convolution between the exit face wave function and the instrumental response is difficult to visualize. Furthermore, the crystal wave function is not simply related to the projected crystal potential, except under a very restrictive set of experimental conditions, making image simulation an essential part of image interpretation. In this paper we present a different conceptual approach to the atomic imaging of crystals based on incoherent imaging theory. Using a real-space analysis of electron scattering to a high-angle annular detector, it is shown how the STEM imaging process can be partitioned into components parallel and perpendicular to the relevant low index zone-axis.It has become customary to describe STEM imaging using the analytical treatment developed by Cowley. However, the convenient assumption of a phase object (which neglects the curvature of the Ewald sphere) fails rapidly for large scattering angles, even in very thin crystals. Thus, to avoid unpredictive numerical solutions, it would seem more appropriate to apply pseudo-kinematic theory to the treatment of the weak high angle signal. Diffraction to medium order zero-layer reflections is most important compared with thermal diffuse scattering in very thin crystals (<5nm). The electron wave function ψ(R,z) at a depth z and transverse coordinate R due to a phase aberrated surface probe function P(R-RO) located at RO is then well described by the channeling approximation;

A Novel Approach for Thermomechanical Analysis of Stationary Rolling Tires within an ALE–Kinematic Framework

2013 ◽  
Vol 41 (3) ◽  
pp. 174-195 ◽  
Author(s):  
Anuwat Suwannachit ◽  
Udo Nackenhorst
Keyword(s):  
Constitutive Model ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Thermomechanical Analysis ◽  
Rolling Contact ◽  
Computational Technique ◽  
Heat Equations ◽  
Arbitrary Lagrangian Eulerian ◽  
Material Particle ◽  
Operator Split

ABSTRACT A new computational technique for the thermomechanical analysis of tires in stationary rolling contact is suggested. Different from the existing approaches, the proposed method uses the constitutive description of tire rubber components, such as large deformations, viscous hysteresis, dynamic stiffening, internal heating, and temperature dependency. A thermoviscoelastic constitutive model, which incorporates all the mentioned effects and their numerical aspects, is presented. An isentropic operator-split algorithm, which ensures numerical stability, was chosen for solving the coupled mechanical and energy balance equations. For the stationary rolling-contact analysis, the constitutive model presented and the operator-split algorithm are embedded into the Arbitrary Lagrangian Eulerian (ALE)–relative kinematic framework. The flow of material particles and their inelastic history within the spatially fixed mesh is described by using the recently developed numerical technique based on the Time Discontinuous Galerkin (TDG) method. For the efficient numerical solutions, a three-phase, staggered scheme is introduced. First, the nonlinear, mechanical subproblem is solved using inelastic constitutive equations. Next, deformations are transferred to the subsequent thermal phase for the solution of the heat equations concerning the internal dissipation as a source term. In the third step, the history of each material particle, i.e., each internal variable, is transported through the fixed mesh corresponding to the convective velocities. Finally, some numerical tests with an inelastic rubber wheel and a car tire model are presented.

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