scholarly journals Moving and fixed axoids of frenet thrihedral of directing curve on example of cylindrical line

2020 ◽  
Vol 11 (3) ◽  
pp. 41-48
Author(s):  
T. A. Kresan ◽  
◽  
S. F. Pylypaka ◽  
V. M. Babka ◽  
Ya. S. Kremets ◽  
...  
Keyword(s):  
Angular Velocity ◽  
Solid Body ◽  
Rotational Motion ◽  
Linear Velocity ◽  
The Body ◽  
Helical Line ◽  
Spatial Motion ◽  
Instantaneous Axis Of Rotation ◽  
Axis Of Rotation ◽  
Curvature And Torsion

If the solid body makes a spatial motion, then at any point in time this motion can be decomposed into rotational at angular velocity and translational at linear velocity. The direction of the axis of rotation and the magnitude of the angular velocity, that is the vector of rotational motion at a given time does not change regardless of the point of the solid body (pole), relative to which the decomposition of velocities. For linear velocity translational motion is the opposite: the magnitude and direction of the vector depend on the choice of the pole. In a solid body, you can find a point, that is, a pole with respect to which both vectors of rotational and translational motions have the same direction. The common line given by these two vectors is called the instantaneous axis of rotation and sliding, or the kinematic screw. It is characterized by the direction and parameter - the ratio of linear and angular velocity. If the linear velocity is zero and the angular velocity is not, then at this point in time the body performs only rotational motion. If it is the other way around, then the body moves in translational manner without rotating motion. The accompanying trihedral moves along the directing curve, it makes a spatial motion, that is, at any given time it is possible to find the position of the axis of the kinematic screw. Its location in the trihedral, as in a solid body, is well defined and depends entirely on the differential characteristics of the curve at the point of location of the trihedral – its curvature and torsion. Since, in the general case, the curvature and torsion change as the trihedral moves along the curve, then the position of the axis of the kinematic screw will also change. Multitude of these positions form a linear surface - an axoid. At the same time distinguish the fixed axoid relative to the fixed coordinate system, and the moving - which is formed in the system of the trihedral and moves with it. The shape of the moving and fixed axoids depends on the curve. The curve itself can be reproduced by rolling a moving axoid over a fixed one, while sliding along a common touch line at a linear velocity, which is also determined by the curvature and torsion of the curve at a particular point. For flat curves, there is no sliding, that is, the movable axoid is rolling over a stationary one without sliding. There is a set of curves for which the angular velocity of the rotation of the trihedral is constant. These include the helical line too. The article deals with axoids of cylindrical lines and some of them are constructed.

scholarly journals III. Researches in physical astronomy

1831 ◽  
Vol 121 ◽  
pp. 17-66
Keyword(s):  
Angular Velocity ◽  
Major Axis ◽  
The Body ◽  
Fixed Plane ◽  
Resisting Medium ◽  
Axis Of Rotation ◽  
Geographical Latitude ◽  
The Mean ◽  
The Stability ◽  
Revolving Body

In last April I had the honour of presenting to the Society a paper containing expressions for the variations of the elliptic constants in the theory of the motions of the planets. The stability of the solar system is established by means of these expressions, if the planets move in a space absolutely devoid of any resistance*, for it results from their form that however far the ap­proximation be carried, the eccentricity, the major axis, and the tangent of the inclination of the orbit to a fixed plane, contain only periodic inequalities, each of the three other constants, namely, the longitude of the node, the longitude of the perihelion, and the longitude of the epoch, contains a term which varies with the time, and hence the line of apsides and the line of nodes revolve continually in space. The stability of the system may therefore be inferred, which would not be the case if the eccentricity, the major axis, or the tangent of the inclination of the orbit to a fixed plane contained a term varying with the time, however slowly. The problem of the precession of the equinoxes admits of a similar solution; of the six constants which determine the position of the revolving body, and the axis of instantaneous rotation at any moment, three have only periodic inequalities, while each of the other three has a term which varies with the time. From the manner in which these constants enter into the results, the equilibrium of the system may be inferred to be stable, as in the former case. Of the constants in the latter problem, the mean angular velocity of rotation may be considered analogous to the mean motion of a planet, or its major axis ; the geographical longitude, and the cosine of the geographical latitude of the pole of the axis of instantaneous rotation, to the longitude of the perihelion and the eccentricity; the longitude of the first point of Aries and the obliquity of the ecliptic, to the longitude of the node and the inclination of the orbit to a fixed plane; and the longitude of a given line in the body revolving, passing through its centre of gravity, to the longitude of the epoch. By the stability of the system I mean that the pole of the axis of rotation has always nearly the same geographical latitude, and that the angular velocity of rotation, and the obliquity of the ecliptic vary within small limits, and periodically. These questions are considered in the paper I now have the honour of submitting to the Society. It remains to investigate the effect which is produced by the action of a resisting medium; in this case the latitude of the pole of the axis of rotation, the obliquity of the ecliptic, and the angular velocity of rotation might vary considerably, although slowly, and the climates undergo a con­siderable change.

scholarly journals Angular velocity of rotation of extended bodies in general relativity

1996 ◽  
Vol 172 ◽  
pp. 309-320
Author(s):  
S.A. Klioner
Keyword(s):  
General Relativity ◽  
Angular Velocity ◽  
Rigid Body ◽  
Rotational Motion ◽  
The Body ◽  
Astronomical Observations ◽  
Newtonian Approximation ◽  
Accurate Definition ◽  
Principal Axes ◽  
Definition Of

We consider rotational motion of an arbitrarily composed and shaped, deformable weakly self-gravitating body being a member of a system of N arbitrarily composed and shaped, deformable weakly self-gravitating bodies in the post-Newtonian approximation of general relativity. Considering importance of the notion of angular velocity of the body (Earth, pulsar) for adequate modelling of modern astronomical observations, we are aimed at introducing a post-Newtonian-accurate definition of angular velocity. Not attempting to introduce a relativistic notion of rigid body (which is well known to be ill-defined even at the first post-Newtonian approximation) we consider bodies to be deformable and introduce the post-Newtonian generalizations of the Tisserand axes and the principal axes of inertia.

Forces on bodies moving transversely through a rotating fluid

1975 ◽  
Vol 71 (3) ◽  
pp. 577-599 ◽  
Author(s):  
P. J. Mason
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Vertical Axis ◽  
The Body ◽  
Rotating Fluid ◽  
Size And Shape ◽  
Relative Direction ◽  
Axis Of Rotation

Measurements have been made of the net force F acting on a bluff rigid body moving with velocity U (relative to a fluid rotating about a vertical axis with uniform angular velocity Ω) in a plane perpendicular to the axis of rotation. The force F is of magnitude 2ΩρVU, where ρ is the density of the fluid and V is a volume which depends on the size and shape of the body. The relative direction of F and U is found to depend on the quantity \[ {\cal S}\equiv \frac{2\Omega L}{U}\bigg(\frac{h}{D}\bigg), \] where L and h are horizontal and vertical lengths characterizing the object and D is the depth of the fluid in which the object is placed.

A study of motions in a rotating liquid

1951 ◽  
Vol 206 (1084) ◽  
pp. 108-130 ◽  
Keyword(s):  
Angular Velocity ◽  
Equations Of Motion ◽  
Vertical Axis ◽  
The Body ◽  
Forced Oscillations ◽  
Infinite Cylinder ◽  
Two Dimensional ◽  
Small Motion ◽  
Axis Of Rotation ◽  
Rotating Liquid

Starting with the equations of motion for a perfect, incompressible fluid referred to a coordinate system which rotates about a vertical axis with uniform angular velocity R , the physical condition of ‘small motion’ is determined which permits the equations to be linearized. The small motions resulting from forced oscillations of a rotating liquid are investigated. It is shown that there are three types of flow depending on the relative magnitudes of the impressed frequency β and the angular velocity R of the fluid. Two of the regimes are studied in detail. A similarity law is developed which gives the solution of a class of problems of oscillations for β > 2 R in terms of the solutions to similar irrotational problems. An attempt is made to explain how slow, two-dimensional motion can be produced by introducing a boundary condition which is three-dimensional (as observed in experiments performed by G. I. Taylor), by considering problems from the moment at which the disturbance is created from rest relative to the rotating system, with the only initial assumption that the fluid is rotating uniformly like a solid body. For the particular cases studied the results are in agreement with Taylor’s experiments, in that the flow is found to become steady and two-dimensional if the disturbance which causes it approaches a steady state. If the disturbance is due to a body which moves along the axis of rotation of the fluid, the steady two-dimensional behaviour may be expected everywhere except in the neighbourhood of the surface of an infinite cylinder which encloses the body and whose generators are parallel to the axis of rotation. To resolve an apparent disagreement between certain theoretical results by Grace on the one hand, and experimental evidence by Taylor and the author’s conclusions, on the other, arguments are advanced that the various results may be in agreement, provided Grace’s are given a new interpretation.

Instantaneous center/axis of rotation for planar and three-dimensional motion

2021 ◽  
pp. 030641902110511
Author(s):  
Donald L Kunz
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Three Dimensional ◽  
The Body ◽  
Body Motion ◽  
Planar Motion ◽  
Instantaneous Axis Of Rotation ◽  
Axis Of Rotation ◽  
Instantaneous Center ◽  
Instantaneous Axis

This article discusses a direct analytical method for calculating the instantaneous center of rotation and the instantaneous axis of rotation for the two-dimensional and three-dimensional motion, respectively, of rigid bodies. In the case of planar motion, this method produces a closed-form expression for the instantaneous center of rotation based on a single point located on the rigid body. It can also be used to derive closed-form expressions for the body and space centrodes. For three-dimensional, rigid body motion, an extension of the technique used for planar motion locates a point on the instantaneous axis of rotation, which is parallel to the body angular velocity vector. In addition, methods are demonstrated that can be used to map the body and space cones for general rigid body motion, and locate the fixed point for the body.

Direct Kinematic Analysis of a Hexapod Spider-Like Mobile Robot

2011 ◽  
Vol 403-408 ◽  
pp. 5053-5060 ◽  
Author(s):  
Mostafa Ghayour ◽  
Amir Zareei
Keyword(s):  
Angular Velocity ◽  
Mobile Robot ◽  
Time Variation ◽  
Kinematic Analysis ◽  
The Body ◽  
Specific Time ◽  
Centre Of Gravity ◽  
Joint Variable ◽  
Direct Kinematics ◽  
Robot Platform

In this paper, an appropriate mechanism for a hexapod spider-like mobile robot is introduced. Then regarding the motion of this kind of robot which is inspired from insects, direct kinematics of position and velocity of the centre of gravity (C.G.) of the body and noncontact legs are analysed. By planning and supposing a specific time variation for each joint variable, location and velocity of the C.G. of the robot platform and angular velocity of the body are obtained and the results are shown and analysed.

Cervical Plate Stiffness Affects the Distribution of Loads and the Location of the Instantaneous Axis of Rotation of the Spine In Vitro

2016 ◽  
Vol 16 (10) ◽  
pp. S260-S261 ◽  
Author(s):  
Josh Peterson ◽  
Carolyn Chlebek ◽  
Ashley Clough ◽  
Alexandra Wells ◽  
Eric H. Ledet
Keyword(s):  
Instantaneous Axis Of Rotation ◽  
Axis Of Rotation ◽  
Cervical Plate ◽  
Instantaneous Axis

Kinematics of rotation in place during defense turning in the crayfish Procambarus clarkii

2001 ◽  
Vol 204 (3) ◽  
pp. 471-486 ◽  
Author(s):  
N. Copp ◽  
M. Jamon
Keyword(s):  
Angular Velocity ◽  
Procambarus Clarkii ◽  
Angular Acceleration ◽  
The Body ◽  
Simple Variation ◽  
Freely Behaving ◽  
Leg Motion ◽  
Two Phases ◽  
Analysis System ◽  
Final Orientation

The kinematic patterns of defense turning behavior in freely behaving specimens of the crayfish Procambarus clarkii were investigated with the aid of a video-analysis system. Movements of the body and all pereiopods, except the chelipeds, were analyzed. Because this behavior approximates to a rotation in place, this analysis extends previous studies on straight and curve walking in crustaceans. Specimens of P. clarkii responded to a tactile stimulus on a walking leg by turning accurately to face the source of the stimulation. Angular velocity profiles of the movement of the animal's carapace suggest that defense turn responses are executed in two phases: an initial stereotyped phase, in which the body twists on its legs and undergoes a rapid angular acceleration, followed by a more erratic phase of generally decreasing angular velocity that leads to the final orientation. Comparisons of contralateral members of each pair of legs reveal that defense turns are affected by changes in step geometry, rather than by changes in the timing parameters of leg motion, although inner legs 3 and 4 tend to take more steps than their outer counterparts during the course of a response. During the initial phase, outer legs 3 and 4 exhibit larger stance amplitudes than their inner partners, and all the outer legs produce larger stance amplitudes than their inner counterparts during the second stage of the response. Also, the net vectors of the initial stances, particularly, are angled with respect to the body, with the power strokes of the inner legs produced during promotion and those of the outer legs produced during remotion. Unlike straight and curve walking in the crayfish, there is no discernible pattern of contralateral leg coordination during defense turns. Similarities and differences between defense turns and curve walking are discussed. It is apparent that rotation in place, as in defense turns, is not a simple variation on straight or curve walking but a distinct locomotor pattern.

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