scholarly journals Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex-Plane Formulation

2000 ◽  
Vol 123 (3) ◽  
pp. 382-387 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
Numerical Solutions ◽  
Minimal Size ◽  
Input Output ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Revolute Joints ◽  
Generalized Eigenvalue ◽  
General Method

This paper presents a general method for the analysis of any planar mechanism consisting of rigid links connected by revolute joints. The method combines a complex plane formulation [1] with the Dixon determinant procedure of Nielsen and Roth [2]. The result is simple to derive and implement, so in addition to providing numerical solutions, the approach facilitates analytical explorations. The procedure leads to a generalized eigenvalue problem of minimal size. Both input/output problems and the derivation of tracing curve equations are addressed, as is the extension of the method to treat slider joints.

scholarly journals Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex-Plane Formulation

2000 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
Numerical Solutions ◽  
Generalized Eigenvalue Problem ◽  
Minimal Size ◽  
Input Output ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Generalized Eigenvalue ◽  
General Method

Abstract This paper presents a general method for the analysis of any planar mechanism consisting of rigid links connected by revolute and slider joints. The method combines the complex plane formulation of Wampler (1999) with the Dixon determinant procedure of Nielsen and Roth (1999). The result is simple to derive and implement, so in addition to providing numerical solutions, the approach facilitates analytical explorations. The procedure leads to a generalized eigenvalue problem of minimal size. Both input/output problems and the derivation of tracing curve equations are addessed.

Solving the Kinematics of Planar Mechanisms

1999 ◽  
Vol 121 (3) ◽  
pp. 387-391 ◽  
Author(s):  
C. W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
Generalized Eigenvalue Problem ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms ◽  
Generalized Eigenvalue ◽  
General Method

This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a generalized eigenvalue problem, or in some cases, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.

Solving the Kinematics of Planar Mechanisms

1998 ◽  
Author(s):  
Charles W. Wampler
Keyword(s):  
Eigenvalue Problem ◽  
Complex Plane ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms ◽  
General Method

Abstract This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a standard eigenvalue problem, or if preferred, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.

Stiffness characterization of a 3-PPR planar parallel manipulator with actuation compliance

2014 ◽  
Vol 229 (12) ◽  
pp. 2291-2302 ◽  
Author(s):  
Guanglei Wu ◽  
Shaoping Bai ◽  
Jørgen Kepler
Keyword(s):  
Eigenvalue Problem ◽  
Stiffness Matrix ◽  
Parallel Manipulator ◽  
Rotational Stiffness ◽  
Planar Mechanisms ◽  
Spatial Mechanisms ◽  
Generalized Eigenvalue ◽  
Stiffness Model ◽  
Planar Parallel Manipulator

This paper investigates the stiffness of a compliant planar parallel manipulator. Instead of establishing stiffness matrix directly for planar mechanisms, we adopt the modeling approach for spatial mechanisms, which allows us to derive two decoupled homogeneous matrices, corresponding to the translational and rotational stiffness. This is achieved by resorting to the generalized eigenvalue problem, through which the eigenscrew decomposition is implemented to yield six screw springs. The principal stiffnesses and their directions are then identified from the eigenvalue problem of the two separated submatrices. In addition, the influence of the nonlinear actuation compliance to the manipulator stiffness is investigated, and the established stiffness model is experimentally verified.

Singularities of Single-DOF Spherical Mechanisms Identified by Means of Pole Axes’ Properties

2010 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Singularity Analysis ◽  
Systems Of Equations ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Geometric Conditions ◽  
Spherical Mechanisms ◽  
Spherical Mechanism ◽  
General Method

Instantaneous pole axes (IPAs) play, in spherical-mechanism kinematics, the same role as instant centers in planar-mechanism kinematics. IPA-based techniques have not been proposed yet for the singularity analysis of spherical mechanisms, even though instant-center-based algorithms have been already presented for planar mechanisms’ singularity analysis. This paper addresses the singularity analysis of single-dof spherical mechanisms by exploiting the properties of pole axes. A general method for implementing this analysis is presented. The presented method relies on the possibility of giving geometric conditions for any type of singularity, and it is the spherical counterpart of an instant-center-based algorithm previously proposed by the author for single-dof planar mechanisms. It can be used to generate systems of equations useful either for finding the singularities of a given mechanism or to synthesize mechanisms that have to match specific requirements about the singularities.

Automatic Generation of the Input-Output Equations of Planar Mechanisms

1994 ◽  
Author(s):  
Kunter A. Kanberoglu ◽  
Resit Soylu
Keyword(s):  
Closed Form ◽  
Functional Relation ◽  
Automatic Generation ◽  
Computational Effort ◽  
Input Output ◽  
Symbolic Manipulation ◽  
Planar Mechanisms ◽  
Planar Mechanism ◽  
Transmission Angle ◽  
Output Equation

Abstract In this article, a methodology, which yields (in closed-form) the functional relation between “any” two joint variables of a one degree-of-freedom planar mechanism, is developed. For instance, the transmission angle and crank-rotatibility synthesis algorithms (Soylu, 1993; Soylu and Kanberoğlu, 1993) require such a generic input-output equation. The equation is obtained in an optimal manner which minimizes the computational effort associated with it. The tools of theory of elimination and symbolic manipulation are also used in the developed method.

Solving the Input/Output Problem for Planar Mechanisms

1998 ◽  
Author(s):  
James Nielsen ◽  
Bernard Roth
Keyword(s):  
Solution Procedure ◽  
Polynomial Equations ◽  
Input Output ◽  
Planar Mechanisms ◽  
Revolute Joints

Abstract This paper presents a method for solving the input/output problem for all planar mechanisms composed of revolute joints. The solution procedure is a modification of the Dixon resultant method, which was developed to solve sets of polynomial equations; in this paper the method is applied to sets of equations which are linear in the sines and cosines of unknown angles. A particular planar multi-circuit mechanism is analyzed to illustrate the solution procedure, and implementation details are discussed.

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

Determination of the Instantaneous Pole Axes in Single-DOF Spherical Mechanisms

2008 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
A Algorithm ◽  
Planar Mechanisms ◽  
Spherical Mechanisms ◽  
Kinematic Behavior ◽  
General Method

In spherical mechanisms, the instantaneous pole axes play the same role as the instant centers in planar mechanisms. Notwithstanding this, they are not fully exploited to study the kinematic behavior of spherical mechanisms as the instant centers are for planar mechanisms. The first step to make their use possible and friendly is the availability of efficient techniques to determine them. This paper presents a general method to determine the instantaneous pole axes in single-dof spherical mechanisms as a function of the mechanism configuration. The presented method is directly deduced from a algorithm already proposed by the author for the determination of the instant centers in single-dof planar mechanisms.

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