The Kinematics of Motion Through Finitely Separated Positions

1967 ◽  
Vol 34 (3) ◽  
pp. 591-598 ◽  
Author(s):  
Bernard Roth
Keyword(s):  
General Theory ◽  
Rigid Body ◽  
Numerical Evaluation ◽  
Companion Paper ◽  
Numerical Examples ◽  
Circle Plane ◽  
Spatial Linkages ◽  
Special Points

A rigid body is studied in a series of finitely separated positions, in order to determine those points which lie on a special locus (a sphere, circle, plane, line, or cylinder). Equations governing these special points are derived and their numerical evaluation is discussed. Several numerical examples are presented. In a companion paper [21], these results are applied to the synthesis of spatial linkages, and special motions (e.g., planer and spherical) are incorporated into the general theory presented herein.

Incompletely Specified Displacements: Geometry and Spatial Linkage Synthesis

1973 ◽  
Vol 95 (2) ◽  
pp. 603-611 ◽  
Author(s):  
Lung-Wen Tsai ◽  
Bernard Roth
Keyword(s):  
Rigid Body ◽  
Screw Axis ◽  
Numerical Examples ◽  
Spatial Linkages ◽  
Free Parameters ◽  
Analytical Expressions ◽  
Linkage Synthesis

The screw axis geometry associated with displacements of points and lines is studied. Analytical expressions are developed for rigid body screw displacements which have one or more free parameters. It is shown how to apply these results to the synthesis of spatial linkages. The theory is illustrated by numerical examples in which Cylindric-Cylindric cranks are designed to guide two points in a rigid body through five and then nine specified positions.

A Unified Theory for the Finitely and Infinitesimally Separated Position Problems of Kinematic Synthesis

1969 ◽  
Vol 91 (1) ◽  
pp. 203-208 ◽  
Author(s):  
P. Chen ◽  
B. Roth
Keyword(s):  
Rigid Body ◽  
Companion Paper ◽  
Unified Theory ◽  
Kinematic Synthesis ◽  
Special Points ◽  
General Method

A rigid body is studied in a series of different positions. These positions can be finitely separated, infinitesimally separated, or a combination of the two. A general method for determining the locations of points or lines (in the rigid body) which have their different multiple positions satisfying the constraints of binary links or combined link chains is developed. In a companion paper [10] equations governing the locations of these special points and lines are derived.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces

2017 ◽  
Vol 73 (5) ◽  
pp. 387-402 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Sajdeh Sajjadi ◽  
Bernard Shiffman ◽  
Steven M. Zucker
Keyword(s):  
General Theory ◽  
Rigid Body ◽  
Three Dimensional ◽  
Molecular Replacement ◽  
Translation Group ◽  
Common Occurrence ◽  
Space Group ◽  
Space Groups ◽  
Relevant Case ◽  
The Subject

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

scholarly journals Nonlinear morphoelastic plates II: Exodus to buckled states

2011 ◽  
Vol 16 (8) ◽  
pp. 833-871 ◽  
Author(s):  
Joseph McMahon ◽  
Alain Goriely ◽  
Michael Tabor
Keyword(s):  
General Theory ◽  
Deformation Gradient ◽  
Companion Paper ◽  
Kirchhoff Plate ◽  
Geometric Description ◽  
Elastic Materials ◽  
Multiplicative Decomposition ◽  
Linear Elastic ◽  
Kirchhoff Plates ◽  
The Stability

Morphoelasticity is the theory of growing elastic materials. The theory is based on the multiplicative decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing non-linear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed.

Conditions for the planes of symmetry of an elastically supported rigid body

2009 ◽  
Vol 223 (8) ◽  
pp. 1755-1766 ◽  
Author(s):  
S J Jang ◽  
Y J Choi
Keyword(s):  
Rigid Body ◽  
Eigenvalue Problems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Mass Matrices ◽  
Out Of Plane ◽  
Modes Of Vibration ◽  
Compliance Matrices ◽  
Special Points ◽  
Elastically Supported

Introducing the planes of symmetry into an oscillating rigid body suspended by springs simplifies the complexity of the equations of motion and decouples the modes of vibration into in-plane and out-of-plane modes. There have been some research results from the investigation into the conditions for planes of symmetry in which prior conditions for the simplification of the equations of motion are required. In this article, the conditions for the planes of symmetry that do not need prior conditions for simplification are presented. The conditions are derived from direct expansions of eigenvalue problems for stiffness and mass matrices that are expressed in terms of in-plane and out-of-plane modes and the orthogonality condition with respect to the mass matrix. Two special points, the planar couple point and the perpendicular translation point are identified, where the expressions for stiffness and compliance matrices can be greatly simplified. The simplified expressions are utilized to obtain the analytical expressions for the axes of vibration of a vibration system with planes of symmetry.

The Optimization of Dimensions of an Adjustable Mechanism for a Packaging Machine

1989 ◽  
Vol 203 (1) ◽  
pp. 37-40 ◽  
Author(s):  
A Daadbin ◽  
K S H Sadek
Keyword(s):  
Rigid Body ◽  
Complex Mechanism ◽  
Uniform Velocity ◽  
Packaging Machine ◽  
Minimum Number ◽  
Adjustable Mechanism ◽  
Spatial Linkages ◽  
Mechanical Devices ◽  
Cam Systems

Mechanisms form the basic geometrical elements of many mechanical devices including automatic packaging machinery, typewriters, textile and printing machinery, and others. A mechanism typically is designed to create a desired motion of a rigid body relative to a reference member by the help of gears, cam systems or spatial linkages. In flow pack machines a tube of wrapper containing the products moves with a uniform velocity, while the reciprocating heads move forward and backwards sealing different products. In an existing machine, these motions are produced by a rather complex mechanism involving cams and adjustable links. The paper suggests replacing these cams by a suitable quick-return mechanism with a minimum number of adjustable links. The dimensions of this mechanism are optimized such that the motions produced are as near as possible to those obtained by the original cam mechanisms. The simplification can result in reduction in the mass of different components and existing forces in the mechanism.

scholarly journals Planar dynamics of a rigid body system with frictional impacts. II. Qualitative analysis and numerical simulations

2009 ◽  
Vol 465 (2107) ◽  
pp. 2267-2292 ◽  
Author(s):  
Zhen Zhao ◽  
Caishan Liu ◽  
Bernard Brogliato
Keyword(s):  
Qualitative Analysis ◽  
Rigid Body ◽  
Numerical Simulations ◽  
Contact Point ◽  
Impulsive Differential Equations ◽  
Companion Paper ◽  
Rigid Body Dynamics ◽  
Drift Motion ◽  
Contact Points ◽  
Static Coefficient Of Friction

The objective of this paper is to implement and test the theory presented in a companion paper for the non-smooth dynamics exhibited in a bouncing dimer. Our approach revolves around the use of rigid body dynamics theory combined with constraint equations from the Coulomb's frictional law and the complementarity condition to identify the contact status of each contacting point. A set of impulsive differential equations based on Darboux–Keller shock dynamics is established that can deal with the complex behaviours involved in multiple collisions, such as the frictional effects, the local dissipation of energy at each contact point, and the dispersion of energy among various contact points. The paper will revisit the experimental phenomena found in Dorbolo et al . ( Dorbolo et al . 2005 Phys. Rev. Lett. 95 , 044101), and then present a qualitative analysis based on the theory proposed in part I. The value of the static coefficient of friction between the plate and the dimer is successfully estimated, and found to be responsible for the formation of the drift motion of the bouncing dimer. Plenty of numerical simulations are carried out, and precise agreements are obtained by the comparisons with the experimental results.

scholarly journals PROJECTION METHODS FOR INTEGRAL EQUATIONS IN EPIDEMIC

2002 ◽  
Vol 7 (2) ◽  
pp. 229-240 ◽  
Author(s):  
L. Hacia
Keyword(s):  
Mathematical Modeling ◽  
Numerical Methods ◽  
General Theory ◽  
Integral Equations ◽  
Projection Methods ◽  
Volterra Integral Equations ◽  
Temporal Development ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Spatio Temporal

In this paper numerical methods for mixed integral equations are presented. Studied equations arise in the mathematical modeling of the spatio‐temporal development of an epidemic. The general theory of these equations is given and used in the projection methods. Projection methods lead to a system of algebraic equations or to a system of Volterra integral equations. The considered theory is illustrated by numerical examples.

XIV.—Studies in Practical Mathematics. VIII. On the Iterative Methods of Lin and Friedman for Factorizing Polynomials

1954 ◽  
Vol 64 (2) ◽  
pp. 190-199
Author(s):  
A. C. Aitken
Keyword(s):  
General Theory ◽  
Iterative Methods ◽  
Point Of View ◽  
Tentative Conclusion ◽  
Numerical Examples ◽  
Practical Mathematics

SynopsisThe methods of S. N. Lin (1943) and B. Friedman (1949) for approximating to the factors of a polynomial by iterated division are studied from the point of view of convergence. The general theory, hitherto lacking, is supplied. The matrices which transform the errors in coefficients from one iterate to the next are explicitly found, and the criterion of convergence derived. Numerical examples are given. The tentative conclusion is that the methods are less simple in theory and less adaptable than the method of penultimate remainder, which admits of accelerative devices.

A General Theory for Complete Balancing of Shaking Force and Shaking Moment of Spatial Linkages Using Counterweights and Inertia-Counterweights

1994 ◽  
Author(s):  
Ting-Li Yang ◽  
Ming Zhang ◽  
Jian-Qin Zhang
Keyword(s):  
General Theory ◽  
Kinematic Equation ◽  
Force Moment ◽  
Minimum Number ◽  
Spatial Linkages ◽  
Independent Loops ◽  
Shaking Moment

Abstract The links of a spatial linkage with v independent loops can be divided into two parts — a set of v chord-links and a tree-system. Add a suitable counterweight to each chord-link for satisfying the certain conditions, under which, the v chord-links can be completely shaking force-moment-balanced by the counterweights and inertia-counterweights attached to the tree-system of the linkage; The conditions for complete balance of shaking force and shaking moment of the tree system can be written directly without extracting them from the kinematic equation of the linkage; A formula which define the minimum number of inertia-counterweights needed for a complete shaking-moment-balance and the criteria for selecting the optimum chord-link set have been presented. Two examples (a Bennett linkage and a spherical six-bar linkage) are provided.

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