scholarly journals The curl of angular velocity and its mechanical significance

2021 ◽  
Author(s):  
PU GuangYi ◽  
Pu Cheng-Xi
Keyword(s):  
Angular Velocity ◽  
Vortex Ring ◽  
Velocity Vector ◽  
Mathematical Expression ◽  
Mathematical Equation ◽  
Vector Parameter ◽  
Angular Velocity Vector ◽  
Single Vortex ◽  
Motion Characteristics ◽  
The Right

Abstract The curl of the vector field is widely used in modern field theory, fluid mechanics, mathematics, electromagnetic field, and other fields. In this paper, by introducing an auxiliary vector parameter (We called 𝑷𝑼⃗⃗⃗⃗⃗⃗ ) whose direction satisfies the right-hand thread rule the mathematical expression of angular velocity vector curl (𝛁×𝝎⃗⃗⃗ ) was obtained by analogy with the method of defining velocity vector curl (𝛁×𝒗⃗⃗ ). We also pointed out that the laminar flow of viscous fluid in a circular pipe (Hogen-Poiseuille flow) in nature is a typical real example of angular velocity vector curl (𝛁×𝝎⃗⃗⃗ ). Moreover, a concise mathematical equation (Equation(11)) was given, which could be used to describe some motion characteristics of vortex ring theoretically; Therefore, the motion of a single vortex ring has the dual characteristics of the velocity curl(𝛁×𝒗⃗⃗ ) and the angular velocity curl(𝛁×𝝎⃗⃗⃗ ) at the same time.

Sensor Placement for Angular Velocity Determination

2005 ◽  
Vol 128 (3) ◽  
pp. 543-547 ◽  
Author(s):  
Guy M. Genin ◽  
Joseph Genin
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Vector ◽  
Constraint Equation ◽  
Complete Characterization ◽  
Angular Velocity Vector ◽  
Algebraic Constraint ◽  
Transducer Placement

Velocity transducer placement to uniquely determine the angular velocity of a rigid body is investigated. The angular velocity of a rigid body can be determined with no fewer than five properly placed velocity transducers, if no other types of sensors are present and no algebraic constraint equation involving the angular velocity vector can be written. Complete characterization of the velocity of a rigid body requires six transducers. Choice of transducer placement and orientation requires care, as suboptimal transducer placement can result in data from which the determination of a unique angular velocity vector is impossible. Conditions for successful transducer placement are established, and two examples of adequate transducer placement are presented: an Earth-penetrating projectile, and a bioengineering device for the measurement of head motion.

Development of angular velocity vector measurement of a spherical rotor based on color sensors

2017 ◽  
Vol 40 (13) ◽  
pp. 3736-3743
Author(s):  
ChengGang Pan ◽  
GaoFei Zhang
Keyword(s):  
Angular Velocity ◽  
Attitude Control ◽  
Velocity Vector ◽  
Spherical Surface ◽  
Contact Method ◽  
Angular Velocity Vector ◽  
Loop Control ◽  
Direction Error ◽  
Black And White ◽  
Color Sensors

Spherical actuators have many applications in satellite attitude control. It is very important for closed-loop control to measure the angular velocity vector of the spherical rotor accurately and quickly. This paper proposes one non-contact method for measuring angular velocity vector of a spherical rotor, using four color sensors. The spherical rotor surface should be color-coded with red, blue, green, magenta, yellow, cyan, black and white, while each color covers an octant in sequence. Four color sensors are mounted about the vertexes of a tetrahedron, used to sense the color of the spherical surface and transform them into angular velocity vector. Simulations approve the correctness of this method, within a module error of approximately 5.2 rpm and a direction error of 0.88 degree within 4000 rpm. Experiments verify that the random error is smaller than 2.7 rpm and direction error smaller than 1.1 degree within 2000 rpm.

Kinematics of an Elastic Sphere Rolling on a Plane and Between Two Planes

1993 ◽  
Vol 115 (3) ◽  
pp. 476-480
Author(s):  
E. Kingsbury
Keyword(s):  
Angular Velocity ◽  
Elastic Deformation ◽  
General Expression ◽  
Velocity Vector ◽  
Elastic Sphere ◽  
Angular Velocity Vector ◽  
Spiral Path ◽  
Symmetry Argument

A sphere rolling between a stationary and a spinning plane traces out a spiral path, even under quasistatic conditions. Published theory suggests that radial creep due to pivot produces the spiral path. We show experimentally a component of the sphere’s angular velocity not considered in the published analysis, raising questions about pivot in producing the spiral. We give a general expression for the sphere angular velocity vector which accommodates a linear, circular or spiral path, pivot or no pivot, and one or two planes in contact. We show that a sphere can roll in a circle on one or between two plane without pivot, but not between a stationary and a spinning plane. We show that a circumferential component of angular velocity results in a spiral path. A symmetry argument suggests that the spiral might be due to elastic deformation in the planes rather than to pivot, but the question is still open.

Algorithm of the Time-Optimal Reorientation of an Axially Symmetric Spacecraft in the Class of Conical Motions

2018 ◽  
Vol 19 (12) ◽  
pp. 10-17587/mau.19.797-805
Author(s):  
Ya. G. Sapunkov ◽  
A. V. Molodenkov ◽  
T. V. Molodenkova
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Vector ◽  
Space Shuttle ◽  
Control Function ◽  
Axially Symmetric ◽  
Bounded Control ◽  
Angular Velocity Vector ◽  
Time Optimal ◽  
Optimal Turn

The problem of the time-optimal turn of a spacecraft as a rigid body with one axis of symmetry and bounded control function in absolute value is considered in the quaternion statement. For simplifying problem (concerning dynamic Euler equations), we change the variables reducing the original optimal turn problem of axially symmetric spacecraft to the problem of optimal turn of the rigid body with spherical mass distribution including one new scalar equation. Using the Pontryagin maximum principle, a new analytical solution of this problem in the class of conical motions is obtained. Algorithm of the optimal turn of a spacecraft is given. An explicit expression for the constant in magnitude optimal angular velocity vector of a spacecraft is found. The motion trajectory of a spacecraft is a regular precession. The conditions for the initial and terminal values of a spacecraft angular velocity vector are formulated. These conditions make it possible to solve the problem analytically in the class of conical motions. The initial and the terminal vectors of spacecraft angular velocity must be on the conical surface generated by arbitrary given constant conditions of the problem. The numerical example is presented. The example contain optimal reorientation of the Space Shuttle in the class of conical motions.

scholarly journals On the Geodetic Rotation of the Major Planets, the Moon and the Sun

2009 ◽  
Vol 44 (2) ◽  
pp. 43-52
Author(s):  
G. Eroshkin ◽  
V. Pashkevich
Keyword(s):  
Angular Velocity ◽  
Reference Frame ◽  
Velocity Vector ◽  
Previous Investigation ◽  
The Body ◽  
The Other ◽  
The Sun ◽  
The Moon ◽  
Angular Velocity Vector

On the Geodetic Rotation of the Major Planets, the Moon and the SunThe problem of the geodetic (relativistic) rotation of the major planets, the Moon and the Sun was studied in the paper by Eroshkin and Pashkevich (2007) only for the components of the angular velocity vectors of the geodetic rotation, which are orthogonal to the plane of the fixed ecliptic J2000. This research represents an extension of the previous investigation to all the other components of the angular velocity vector of the geodetic rotation, with respect to the body-centric reference frame from Seidelmann et al. (2005).

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