Instantaneous Properties of Trajectories Generated by Planar, Spherical, and Spatial Rigid Body Motions

1982 ◽  
Vol 104 (1) ◽  
pp. 39-50 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Roth
Keyword(s):  
Rigid Body ◽  
The Body ◽  
Spatial Motion ◽  
Third Order ◽  
Local Shape ◽  
Motion Trajectories ◽  
The Third ◽  
Moving Body ◽  
Rigid Body Motions ◽  
Point Trajectories

This paper develops relationships between the instantaneous invariants of a motion and the local shape of the trajectories generated during the motion. We consider the point trajectories generated by planar and spherical motions and the line trajectories generated by spatial motion. Those points or lines which generate special trajectories are located on (and define) so-called boundary loci in the moving body. These boundary loci define regions, within the body, for which all the points or lines generate similarly shaped trajectories. The shapes of these boundaries depend directly upon the invariants of the motion. It is shown how to qualitatively determine the fundamental trajectory shapes, analyze the effect of the invariants on the boundary loci, and how to combine these results to visualize the motion trajectories to the third order.

The Curvature Theory of Line Trajectories in Spatial Kinematics

1981 ◽  
Vol 103 (4) ◽  
pp. 718-724 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Roth
Keyword(s):  
Planar Motion ◽  
Spatial Motion ◽  
Third Order ◽  
Local Shape ◽  
The Third ◽  
Moving Body ◽  
Physical Representation ◽  
Curvature Theory ◽  
Spatial Kinematics ◽  
Differential Properties

This paper develops the differential properties of ruled surfaces in a form which is applicable to spatial kinematics. Derivations are presented for the three curvature parameters which define the local shape of a ruled surface. Related parameters are also developed which allow a physical representation of this shape as generated by a cylindric-cylindric crank. These curvature parameters are then used to define all the lines in the moving body which instantaneously generate speciality shaped trajectories. Such lines may be used in the synthesis of spatial motions in the same way that the points on the inflection circle and cubic of stationary curvature are used to synthesize planar motion. As an example of this application several special sets of lines are defined: the locus of all lines which for a general spatial motion instantaneously generate helicoids to the second order and the locus of lines generating right hyperboloids to the third order.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

scholarly journals XXI. Note relating to the attraction of spheroids

1872 ◽  
Vol 20 (130-138) ◽  
pp. 507-513
Keyword(s):  
The Body ◽  
Polar Coordinates ◽  
Homogeneous Body ◽  
Third Order ◽  
The Third

In a memoir on the Attraction o f Spheroids, published in the 'Connaissance des Tems’ for 1829, Poisson showed that certain important formulæ were true up to the third order inclusive o f the standard small quantity. The object of this note is to establish the truth o f the formulæ for all orders o f the small quantity. 1. Suppose we require the value of the potential of a homogeneous body at any assigned point. Take a fixed origin inside the body; let r ', θ ', ψ ' denote the polar coordinates of any point of the body; and let r, θ, ψ , be the polar coordinates of the assigned point; and, as usual, put μ ' for cos θ '', and μ for cos θ. The density may be denoted by unity.

Mathematical aspects of molecular replacement. III. Properties of space groups preferred by proteins in the Protein Data Bank

2015 ◽  
Vol 71 (2) ◽  
pp. 186-194 ◽  
Author(s):  
G. Chirikjian ◽  
S. Sajjadi ◽  
D. Toptygin ◽  
Y. Yan
Keyword(s):  
Rigid Body ◽  
Protein Data Bank ◽  
Data Bank ◽  
Continuous Group ◽  
Molecular Replacement ◽  
Macromolecular Crystallography ◽  
Coset Space ◽  
Space Groups ◽  
The Third ◽  
Rigid Body Motions

The main goal of molecular replacement in macromolecular crystallography is to find the appropriate rigid-body transformations that situate identical copies of model proteins in the crystallographic unit cell. The search for such transformations can be thought of as taking place in the coset space Γ\Gwhere Γ is the Sohncke group of the macromolecular crystal andGis the continuous group of rigid-body motions in Euclidean space. This paper, the third in a series, is concerned with viewing nonsymmorphic Γ in a new way. These space groups, rather than symmorphic ones, are the most common ones for protein crystals. Moreover, their properties impact the structure of the space Γ\G. In particular, nonsymmorphic space groups contain both Bieberbach subgroups and symmorphic subgroups. A number of new theorems focusing on these subgroups are proven, and it is shown that these concepts are related to the preferences that proteins have for crystallizing in different space groups, as observed in the Protein Data Bank.

scholarly journals Cases of Integrability Which Correspond to the Motion of a Pendulum in the Three-dimensional Space

2021 ◽  
Vol 16 ◽  
pp. 73-84
Author(s):  
Maxim V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
The Body ◽  
Spatial Motion ◽  
Free Rigid Body

We systematize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system

Circuit and Branch Rectification of the Spatial 4C Mechanism

2000 ◽  
Author(s):  
Pierre M. Larochelle
Keyword(s):  
Rigid Body ◽  
Finite Sequence ◽  
The Body ◽  
Degree Of Freedom ◽  
Motion Generation ◽  
Moving Body ◽  
In Series ◽  
Simply Connected ◽  
Two Degree Of Freedom ◽  
Physical Dimensions

Abstract Spatial 4C mechanisms are two degree of freedom kinematic closed-chains consisting of four rigid links simply connected in series by cylindrical(C) joints. In this work we are concerned with the design of spatial 4C mechanisms which move a rigid body through a finite sequence of prescribed locations in space. This task is referred to as rigid-body guidance by Suh and Radcliffe (20) and as motion generation by Erdman and Sandor (6). When 4C mechanisms are synthesized for such a task, for example by utilizing Roth’s spatial generalization of Burmester’s planar methods (17; 18), the result is the physical dimensions which kinematically define the mechanism. However, the motion of the mechanism which takes the workpiece through the sequence of prescribed locations in space is not determined. In fact, it may be impossible for the mechanism to move the body through all of the desired locations without disassembling the mechanism. This condition is referred to as a circuit defect. Moreover, in some cases the mechanism may enter a configuration which requires an additional mechanical input to guide the moving body as desired. These are referred to as branch defects. This paper presents a methodology for analyzing spatial 4C mechanisms to eliminate circuit and branch defects in motion generation tasks.

Characterization of Instantaneous Point-Line Motions

2004 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Rigid Bodies ◽  
Dual Quaternion ◽  
Third Order ◽  
Displacement Operator ◽  
Instantaneous Point ◽  
Rigid Body Motions ◽  
Point Line ◽  
Derivatives Of

This article discusses systematically the characterization of instantaneous point-line motions, and the higher-order relationship between a point-line motion and the associated rigid body motions. The transformation of a point-line between two positions is depicted as a pure translation along the point-line followed by a screw displacement about their common normal and expressed with a unit dual quaternion referred to as the point-line displacement operator. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. Such a treatment leads to a consistent expression or unified treatment for the transformation of lines, point-lines, and rigid bodies. The relationships between point-line motions and rigid body motions are addressed in detail up to the third order.

scholarly journals On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 80
Author(s):  
Abdukomil Risbekovich Khashimov ◽  
Dana Smetanová
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
The Body ◽  
Energy Estimates ◽  
Uniqueness Theorems ◽  
Elliptic Type ◽  
Third Order ◽  
Body Form ◽  
The Third

The paper is devoted to solutions of the third order pseudo-elliptic type equations. An energy estimates for solutions of the equations considering transformation’s character of the body form were established by using of an analog of the Saint-Venant principle. In consequence of this estimate, the uniqueness theorems were obtained for solutions of the first boundary value problem for third order equations in unlimited domains. The energy estimates are illustrated on two examples.

scholarly journals CASES OF INTEGRABILITY CORRESPONDING TO THE PENDULUM MOTION IN THREE-DIMENSIONAL SPACE

2017 ◽  
Vol 22 (3-4) ◽  
pp. 75-97 ◽  
Author(s):  
M. V. Shamolin
Keyword(s):  
Rigid Body ◽  
Force Fields ◽  
Characteristic Point ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
The Body ◽  
Spatial Motion ◽  
Free Rigid Body ◽  
Three Dimensional Space

In this article, we systemize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. The obtained results are systematized and served in the invariant form. We also show the nontrivial topological and mechanical analogies.

The Spatial Motion of a Compliantly Suspended Rigid Body Constrained by Multi-Point Frictional Contact

2003 ◽  
Author(s):  
Shuguang Huang ◽  
Joseph M. Schimmels
Keyword(s):  
Rigid Body ◽  
Friction Force ◽  
Coulomb Friction ◽  
Frictional Contact ◽  
The Body ◽  
Elastic Behavior ◽  
Inertial Forces ◽  
Spatial Motion ◽  
Frictional Surfaces

This paper addresses methods for determining the motion of an elastically suspended rigid body interacting with frictional surfaces at multiple locations. The methods developed assume: 1) that the motion of the base from which the body is suspended and the elastic behavior of the suspension are known, 2) that inertial forces are negligible (motion is quasi-static), and 3) that the interaction is characterized by Coulomb friction. The derived coupled sets of spatial rigid body equations are used to determine both the unknown direction of the friction force (at each point of contact) and the unknown motion of the rigid body.

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