每月彙整: 2018 年 4 月

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【五‧二‧ VI 】

左傳‧昭公十四年

冬，十二月，蒲餘侯茲夫殺莒公子意恢，郊公奔齊，公子鐸逆庚與於齊，齊隰黨，公子鉏，送之，有賂田。

晉邢侯與雍子爭鄐田，久而無成，士景伯如楚，叔魚攝理，韓宣子命斷舊獄，罪在雍子，雍子納其女於叔魚，叔魚蔽罪邢侯，邢侯怒，殺叔魚，與雍子於朝，宣子問其罪於叔向，叔向曰，三人同罪，施生戮死，可也，雍子自知其罪，而賂以買直，鮒也鬻獄 ，邢侯專殺，其罪一也。已惡而掠美為昏，貪以敗官為墨，殺人不忌為賊。夏書曰，昏墨賊殺，皋陶之刑也，請從之，乃施邢侯 ，而尸雍子，與叔魚於市，仲尼曰，叔向，古之遺直也，治國制刑，不隱於親，三數叔魚之惡，不為末減，曰，義也夫，可謂直矣，平丘之會，數其賄也，以寬衛國，晉不為暴，歸魯季孫，稱其詐也，以寬魯國，晉不為虐，邢侯之獄，言其貪也，以正刑書 ，晉不為頗，三言而除，三惡加三利，殺親益榮，猶義也夫。

 

有言，恐已生︰勿掠人之美成語。

原想假借『廣義力』

Generalized forces

Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, i=1,…, n, acting on a system that has its configuration defined in terms of generalized coordinates. In the formulation of virtual work, each generalized force is the coefficient of the variation of a generalized coordinate.

Virtual work

Generalized forces can be obtained from the computation of the virtual work, δW, of the applied forces.[1]:265

The virtual work of the forces, Fi, acting on the particles Pi, i=1,…, n, is given by

\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}

where δri is the virtual displacement of the particle Pi.

Generalized coordinates

Let the position vectors of each of the particles, ri, be a function of the generalized coordinates, qj, j=1,…,m. Then the virtual displacements δri are given by

\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j},\quad i=1,\ldots ,n,

where δqj is the virtual displacement of the generalized coordinate qj.

The virtual work for the system of particles becomes

\displaystyle \delta W=\mathbf {F} _{1}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{1}}{\partial q_{j}}}\delta q_{j}+\ldots +\mathbf {F} _{n}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{n}}{\partial q_{j}}}\delta q_{j}.

Collect the coefficients of δqj so that
\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{1}}}\delta q_{1}+\ldots +\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{m}}}\delta q_{m}.

Generalized forces

The virtual work of a system of particles can be written in the form

\displaystyle \delta W=Q_{1}\delta q_{1}+\ldots +Q_{m}\delta q_{m},

where
\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}},\quad j=1,\ldots ,m,
are called the generalized forces associated with the generalized coordinates qj, j=1,…,m.

Velocity formulation

In the application of the principle of virtual work it is often convenient to obtain virtual displacements from the velocities of the system. For the n particle system, let the velocity of each particle Pi be Vi, then the virtual displacement δri can also be written in the form[2]

\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j},\quad i=1,\ldots ,n.

This means that the generalized force, Qj, can also be determined as
\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.

D’Alembert’s principle

D’Alembert formulated the dynamics of a particle as the equilibrium of the applied forces with an inertia force (apparent force), called D’Alembert’s principle. The inertia force of a particle, Pi, of mass mi is

\displaystyle \mathbf {F} _{i}^{*}=-m_{i}\mathbf {A} _{i},\quad i=1,\ldots ,n,

where Ai is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates qj, j=1,…,m, then the generalized inertia force is given by

\displaystyle Q_{j}^{*}=\sum _{i=1}^{n}\mathbf {F} _{i}^{*}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.

D’Alembert’s form of the principle of virtual work yields
\displaystyle \delta W=(Q_{1}+Q_{1}^{*})\delta q_{1}+\ldots +(Q_{m}+Q_{m}^{*})\delta q_{m}.

 

說說 Kane 法與拉格朗日之法的淵源異同，終究觀止於介紹

Welcome to the web page for AA244A.
The Matlab files for Finite-Element Analysis are available here.

 

『關鍵理念』比較文本也︰

KeyIdeas05.pdf

 

理所當然之事！宜乎多所宣講耶？

http://www.iosrjournals.org/iosr-jmce/papers/vol6-issue4/B0640713.pdf

 

承前且補對照範例爾。

 

 

 

 

 

 

 

 

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【五‧二‧ V 】

『借例使例』誰說不是學習法耶！雖尚未談及

Kane’s Method in Physics/Mechanics

mechanics provides functionality for deriving equations of motion using Kane’s method [Kane1985]. This document will describe Kane’s method as used in this module, but not how the equations are actually derived.

Structure of Equations

In mechanics we are assuming there are 5 basic sets of equations needed to describe a system. They are: holonomic constraints, non-holonomic constraints, kinematic differential equations, dynamic equations, and differentiated non-holonomic equations.

\mathbf{f_h}(q, t) = 0
\mathbf{k_{nh}}(q, t) u + \mathbf{f_{nh}}(q, t) = 0
\mathbf{k_{k\dot{q}}}(q, t) \dot{q} + \mathbf{k_{ku}}(q, t) u + \mathbf{f_k}(q, t) = 0
\mathbf{k_d}(q, t) \dot{u} + \mathbf{f_d}(q, \dot{q}, u, t) = 0
\mathbf{k_{dnh}}(q, t) \dot{u} + \mathbf{f_{dnh}}(q, \dot{q}, u, t) = 0

In mechanics holonomic constraints are only used for the linearization process; it is assumed that they will be too complicated to solve for the dependent coordinate(s). If you are able to easily solve a holonomic constraint, you should consider redefining your problem in terms of a smaller set of coordinates. Alternatively, the time-differentiated holonomic constraints can be supplied.

Kane’s method forms two expressions, F_r and F_r^{*} , whose sum is zero. In this module, these expressions are rearranged into the following form:

\mathbf{M}(q, t) \dot{u} = \mathbf{f}(q, \dot{q}, u, t)

For a non-holonomic system with oo total speeds and mm motion constraints, we will get o – m equations. The mass-matrix/forcing equations are then augmented in the following fashion:

\mathbf{M}(q, t) = \begin{bmatrix} \mathbf{k_d}(q, t) \\ \mathbf{k_{dnh}}(q, t) \end{bmatrix}

\mathbf{_{(forcing)}}(q, \dot{q}, u, t) = \begin{bmatrix} - \mathbf{f_d}(q, \dot{q}, u, t) \\ - \mathbf{f_{dnh}}(q, \dot{q}, u, t) \end{bmatrix}

 

之與『拉格朗日方法』的差異之處，何妨先讀 SymPy mechanics 範例乎？

A rolling disc

The disc is assumed to be infinitely thin, in contact with the ground at only 1 point, and it is rolling without slip on the ground. See the image below.

../../../../_images/rollingdisc.svg

We model the rolling disc in three different ways, to show more of the functionality of this module.

 

※ 謹慎

 

…★

kanes_equations(bodies, loads=None)

Method to form Kane’s equations, Fr + Fr* = 0.

Returns (Fr, Fr*). In the case where auxiliary generalized speeds are present (say, s auxiliary speeds, o generalized speeds, and m motion constraints) the length of the returned vectors will be o – m + s in length. The first o – m equations will be the constrained Kane’s equations, then the s auxiliary Kane’s equations. These auxiliary equations can be accessed with the auxiliary_eqs().

Parameters:

bodies : iterable

An iterable of all RigidBody’s and Particle’s in the system. A system must have at least one body.

loads : iterable

Takes in an iterable of (Particle, Vector) or (ReferenceFrame, Vector) tuples which represent the force at a point or torque on a frame. Must be either a non-empty iterable of tuples or None which corresponds to a system with no constraints.

 

……☆

 

甚或還可更上一層樓呦◎

Steady Motion of a Rigid Disk of Finite Thickness on a Horizontal Plane

Milan Batista
(Submitted on 2 Sep 2005 (v1), last revised 5 Sep 2005 (this version, v2))

The article discusses the steady motion of a rigid disk of finite thickness rolling on its edge on a horizontal plane under the influence of gravity. The governing equations are presented and two cases allowing for a steady state solution are considered: rolling on consistently rough ground and rolling on perfectly smooth ground. The conditions of steady motion are derived for both kinds of ground and it is shown that the possible steady motion of a disk is either on a straight line in a circle. Also oscillations about steady state are discussed and conditions for stable motion are established.

Comments: 28 pages, 7 figures
Subjects: Classical Physics (physics.class-ph)
Journal reference: International Journal of Non-Linear Mechanics, Volume 41, Issue 4, May 2006, Pages 605-621
DOI: 10.1016/j.ijnonlinmec.2006.02.005
Cite as: arXiv:physics/0509021 [physics.class-ph]
  (or arXiv:physics/0509021v2 [physics.class-ph] for this version)

 

───

An Analytical Solution of the Equations of a Rolling Disk of Finite Thickness on a Rough Plane

Milan Batista
(Submitted on 5 Sep 2005)

In this article an analytical solution of equations of motion of a rigid disk of finite thickness rolling on its edge on a perfectly rough horizontal plane under the action of gravity is given. The solution is given in terms of Gauss hypergeometrical functions.

Comments: 12 pages, 2 figures
Subjects: Classical Physics (physics.class-ph)
Journal reference: International Journal of Non-Linear Mechanics, Volume 41, Issues 6-7, July-September 2006, Pages 850-859
DOI: 10.1016/j.ijnonlinmec.2006.06.002
Cite as: arXiv:physics/0509034 [physics.class-ph]
  (or arXiv:physics/0509034v1 [physics.class-ph] for this version)

 

 

 

 

 

 

 

 

 

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【五‧二‧ IV 】

話說物有『牽一髮 而動全身』者，用之於『剛體』適切矣！若問為什麼『角速度』不如『線速度』容易想像乎？蓋僅知『歐拉角 』，尚且不知將如何運用之也！！

特介紹『歐拉盤』物理文本今說以明之呦︰

The Rolling Motion of a Disk on a Horizontal Plane

Alexander J. McDonald, Kirk T. McDonald
(Submitted on 28 Aug 2000 (v1), last revised 5 Apr 2002 (this version, v3))

Recent interest in the old problem of the motion of a coin spinning on a tabletop has focused on mechanisms of dissipation of energy as the angle alpha of the coin to the table decreases, while the angular velocity Omega of the point of contact increases. Following a review of the general equations of motion of a thin disk rolling without slipping on a horizontal surface, we present results of simple experiment on the time dependence of the motion that indicate the dominant dissipative power loss to be proportional to the Omega^2 up to and including the last observable cycle.

Comments: v2 adds experimental data on the role of friction at small angles, and adds several references. Thanks to A. Chatterjee, C. Gray and A. Ruina v3 adds discussion of the case of zero friction; thanks to Martin Olsson
Subjects: Classical Physics (physics.class-ph)
Cite as: arXiv:physics/0008227 [physics.class-ph]
  (or arXiv:physics/0008227v3 [physics.class-ph] for this version)

 

倘若深入『運動學』名目所說事，察其與觀察者參考系之關係︰

 

自能悠遊

The Advanced Part of A Treatise on the Dynamics of a System of Rigid Bodies …

Publication date 1884
Publisher Macmillan
Collection americana
Digitizing sponsor Google
Book from the collections of Harvard University
Language English
 
Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Notes

Part I. published in 1882 and entitled: The elementary part of a treatise on the dynamics of a system of rigid bodies : being Part I. of a treatise on the whole subject.

Copyright-region US
Identifier advancedpartatr03routgoog
Identifier-ark ark:/13960/t3cz38g4b
Pages 365
Possible copyright status NOT_IN_COPYRIGHT
Scandate 20070712
Scanner google
Worldcat 77435331
Year 1884

 

古代習題中哩。

 

 

 

 

 

 

 

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【五‧二‧ III 】

俗諺說︰以經驗為師。

科學始於『現象觀測』，親身的接觸取得第一手資料，能不重要乎？此所以寓教於樂之『玩具』豈只是玩具耶？？

Euler’s Disk

Euler’s Disk is a scientific educational toy, used to illustrate and study the dynamic system of a spinning disk on a flat surface (such as a spinning coin), and has been the subject of a number of scientific papers.[1] The apparatus is known for a seemingly paradoxical dramatic speed-up in spin rate as the disk loses energy and approaches a stopped condition. This phenomenon is named for Leonhard Euler, who studied it in the 18th century.

Computer rendering of Euler’s Disk on a slightly concave base

Components and use

The commercially available toy consists of a heavy, thick chrome-plated steel disk and a rigid, slightly concave, mirrored base. Included holographic magnetic stickers can be attached to the disk, to enhance the visual effect of “sprolling” or “spolling” (spinning/rolling), but these attachments are strictly decorative. The disk, when spun on a flat surface, exhibits a spinning/rolling motion, slowly progressing through different rates and types of motion before coming to rest—most notably, the precession rate of the disk’s axis of symmetry accelerates as the disk spins down. The rigid mirror is used to provide a suitable low-friction surface, with a slight concavity which keeps the spinning disk from “wandering” off a support surface.

An ordinary coin spun on a table, as with any disk spun on a relatively flat surface, exhibits essentially the same type of motion, but is normally more limited in the length of time before stopping. The commercially available Euler’s Disk toy provides a more effective demonstration of the phenomenon than more commonly found items, having an optimized aspect ratio and a precision polished, slightly rounded edge to maximize the spinning/rolling time.

Physics

A spinning/rolling disk ultimately comes to rest, and it does so quite abruptly, the final stage of motion being accompanied by a whirring sound of rapidly increasing frequency. As the disk rolls, the point of rolling contact describes a circle that oscillates with a constant angular velocity \displaystyle \omega . If the motion is non-dissipative (frictionless), \displaystyle \omega is constant, and the motion persists forever; this is contrary to observation, since \displaystyle \omega is not constant in real life situations. In fact, the precession rate of the axis of symmetry approaches a finite-time singularity modeled by a power law with exponent approximately −1/3 (depending on specific conditions).

There are two conspicuous dissipative effects: rolling friction when the coin slips along the surface, and air drag from the resistance of air. Experiments show that rolling friction is mainly responsible for the dissipation and behavior[2]—experiments in a vacuum show that the absence of air affects behavior only slightly, while the behavior (precession rate) depends systematically on coefficient of friction. In the limit of small angle (i.e. immediately before the disk stops spinning), air drag (specifically, viscous dissipation) is the dominant factor, but prior to this end stage, rolling friction is the dominant effect.

 

故耳沐浴於相同的陽光裡、臥躺在一樣的青草間的人，畢竟好奇啊！想探索東西古今經驗之異同也！！

History of research

Moffatt

In the early 2000s, research was sparked by an article in the April 20, 2000 edition of Nature,[3] where Keith Moffatt showed that viscous dissipation in the thin layer of air between the disk and the table would be sufficient to account for the observed abruptness of the settling process. He also showed that the motion concluded in a finite-time singularity. His first theoretical hypothesis was contradicted by subsequent research, which showed that rolling friction is actually the dominant factor.

Moffatt showed that, as time \displaystyle t approaches a particular time \displaystyle t_{0} (which is mathematically a constant of integration), the viscous dissipation approaches infinity. The singularity that this implies is not realized in practice, because the magnitude of the vertical acceleration cannot exceed the acceleration due to gravity (the disk loses contact with its support surface). Moffatt goes on to show that the theory breaks down at a time \displaystyle \tau before the final settling time \displaystyle t_{0} , given by:

\displaystyle \tau \simeq \left[\left({\frac {2a}{9g}}\right)^{3}{\frac {2\pi \mu a}{M}}\right]^{\frac {1}{5}}

where \displaystyle a is the radius of the disk, \displaystyle g is the acceleration due to Earth’s gravity, \displaystyle \mu the dynamic viscosity of air, and \displaystyle M the mass of the disk. For the commercially available Euler’s Disk toy (see link in “External links” below), \displaystyle \tau is about \displaystyle 10^{-2} seconds, at which time the angle between the coin and the surface, \displaystyle \alpha , is approximately 0.005 radians and the rolling angular velocity, \displaystyle \Omega , is about 500 Hz.

Using the above notation, the total spinning/rolling time is:

\displaystyle t_{0}={\frac {\alpha _{0}^{3}M}{2\pi \mu a}}

where \displaystyle \alpha _{0} is the initial inclination of the disk, measured in radians. Moffatt also showed that, if \displaystyle t_{0}-t>\tau , the finite-time singularity in \displaystyle \Omega is given by
\displaystyle \Omega \sim (t_{0}-t)^{-{\frac {1}{6}}}

Experimental results

Moffatt’s theoretical work inspired several other workers to experimentally investigate the dissipative mechanism of a spinning/rolling disk, with results that partially contradicted his explanation. These experiments used spinning objects and surfaces of various geometries (disks and rings), with varying coefficients of friction, both in air and in a vacuum, and used instrumentation such as high speed photography to quantify the phenomenon.

In the 30 November 2000 issue of Nature, physicists Van den Engh, Nelson and Roach discuss experiments in which disks were spun in a vacuum.[4] Van den Engh used a rijksdaalder, a Dutch coin, whose magnetic properties allowed it to be spun at a precisely determined rate. They found that slippage between the disk and the surface could account for observations, and the presence or absence of air only slightly affected the disk’s behavior. They pointed out that Moffatt’s theoretical analysis would predict a very long spin time for a disk in a vacuum, which was not observed.

Moffatt responded with a generalized theory that should allow experimental determination of which dissipation mechanism is dominant, and pointed out that the dominant dissipation mechanism would always be viscous dissipation in the limit of small \displaystyle \alpha (i.e., just before the disk settles).[5]

Later work at the University of Guelph by Petrie, Hunt and Gray[6] showed that carrying out the experiments in a vacuum (pressure 0.1 pascal) did not significantly affect the energy dissipation rate. Petrie et al. also showed that the rates were largely unaffected by replacing the disk with a ring shape, and that the no-slip condition was satisfied for angles greater than 10°.

On several occasions during the 2007–2008 Writers Guild of America strike, talk show host Conan O’Brien would spin his wedding ring on his desk, trying to spin the ring for as long as possible. The quest to achieve longer and longer spin times led him to invite MITprofessor Peter Fisher onto the show to experiment with the problem. Spinning the ring in a vacuum had no identifiable effect, while a Teflon spinning support surface gave a record time of 51 seconds, corroborating the claim that rolling friction is the primary mechanism for kinetic energy dissipation.[citation needed] Various kinds of rolling friction as primary mechanism for energy dissipation have been studied by Leine [7] who confirmed experimentally that the frictional resistance of the movement of the contact point over the rim of the disk is most likely the primary dissipation mechanism on a time-scale of seconds.

 

※ 參考

Finite-time singularity

 The reciprocal function, exhibitinghyperbolic growth.

A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and PDEs (Partial Differential Equations) – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically the simplest finite-time singularities are power laws for various exponents, \displaystyle x^{-\alpha }, of which the simplest is hyperbolic growth, where the exponent is (negative) \displaystyle x^{-1}. More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses \displaystyle (t_{0}-t)^{-\alpha } (using t for time, reversing direction to \displaystyle -t so time increases to infinity, and shifting the singularity forward from 0 to a fixed time \displaystyle t_{0} ).

An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finite-time singularities include the Painlevé paradox in various forms (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite, before abruptly stopping (as studied using the Euler’s Disk toy).

Hypothetical examples include Heinz von Foerster‘s facetious “Doomsday’s equation” (simplistic models yield infinite human population in finite time).

───

 

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歐拉盤

一窺銅板滾動的奧秘
※器材：銅板、鐵環、反光板
※操作步驟與現象：
相信大家都玩過旋轉銅板的遊戲，銅板旋轉後到平躺靜止，還會發生很清晰的聲響。此種看似熟悉的現象，其實蘊含了相當為複雜的運動過程。

銅板一開始運動時，如圖一；銅板保持直立並且只有「自轉」（spin）：自轉軸通過銅板直徑並垂直於桌面，銅板與桌面的接觸點是固定的。直立自轉的速度變慢後，如圖二；銅板開始傾斜並包含了二種運動：(1)「滾動」（roll）運動：銅板邊緣沿著桌面滾動（如同車輪的滾動），且繞著圈圈滾動。(2)自轉運動，但是自轉軸與銅板表面垂直，並通過銅板中心（圖二綠色線段）。最後如圖三，銅板傾斜度增加，銅板與桌面撞擊產生的聲音頻率越來越快（但是自轉速度變慢），最後停止運動。

由於運動過程包括了自轉與滾動，因此常被稱為「sproll」運動；sproll一字取自：自轉spin的sp以及滾動roll的組合。

 

所以『創造的火種』何曾熄滅過？！

 

※ 題解

 

 

 

 

 

 

 

 

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【五‧二‧ II 】

什麼是『角速度』呢？僅答以理解『轉動』概念之關鍵呦，所謂

Angular velocity

In physics, the angular velocity of a particle is the time rate of change of its angular displacement relative to the origin. The SI unit of angular velocity is radians per second. Angular velocity is usually represented by the symbol omega (ω, sometimes Ω).

In three dimensions, the angular velocity of a particle can be represented as a vector, with its direction pointing perpendicular to both the position and velocity vectors, in a fashion specified (conventionally) by the right-hand rule.[1]

 

之所指也。如是恐怕難明乎？

假使再輔之以案例

Angular velocity of a particle

 

解說︰

Particle in two dimensions

 The angular velocity of the particle at P with respect to the origin O is determined by the perpendicular component of the velocity vector v.

 The angular velocity describes the rate of change of the angular position, and the orientation of the instantaneous plane of angular displacement. If the particle is revolving about the origin, then the direction of the angular velocity pseudovector will be along the axis of rotation; in this case (counter-clockwise rotation) the vector points up.

 

The angular velocity of a particle is measured relative to a point, called the origin. As shown in the diagram (with angles ɸ and θ in radians), if a line is drawn from the origin (O) to the particle (P), then the velocity (v) of the particle has a component along the radius (radial component, v‖) and a component perpendicular to the radius (cross-radial component, v⊥). If there is no radial component, then the particle moves tangent to a circle centered about the origin. On the other hand, if there is no cross-radial component, then the particle moves tangent to a straight line that passes through the origin.

A radial motion produces no change in the direction of the particle relative to the origin, so, for the purpose of finding the angular velocity, the radial component can be ignored. Therefore, only the cross-radial (tangential) component of the velocity contributes to the angular velocity.

In two dimensions, the angular velocity ω is given by

\displaystyle \omega ={\frac {d\phi }{dt}}

This is related to the cross-radial velocity by:[1]
\displaystyle \mathrm {v} _{\perp }=r\,{\frac {d\phi }{dt}}
An explicit formula for v⊥ in terms of v and θ is:
\displaystyle \mathrm {v} _{\perp }=|\mathrm {\mathbf {v} } |\,\sin(\theta )
Combining the above equations gives a formula for ω:
\displaystyle \omega ={\frac {|\mathrm {\mathbf {v} } |\sin(\theta )}{|\mathrm {\mathbf {r} } |}}
In two dimensions, the angular velocity is a single number with an orientation but no vectorial direction. In other words, in two dimensions, it is a pseudoscalar, a quantity that changes its sign under a parity inversion (for example if one of the axes is inverted or if axes are swapped). The direction of angular velocity is taken, by convention, to be positive if the position vector turns counterclockwise, and negative if the position vector turns clockwise. If the parity is inverted, but the orientation of the angular displacement is not, then the sign of the angular velocity changes.

Particle in three dimensions

In three-dimensional space angular velocity is a pseudovector quantity that specifies the rate at which the position vector “sweeps out” angle, as well as the orientation of the plane of angular displacement. Angular velocity in three dimensions has a magnitude, and a direction. The right-hand rule indicates the positive direction of the angular velocity pseudovector.

Let \displaystyle {\mathbf {u}} be a unitary vector perpendicular to the plane of angular displacement, so that from the top of the vector the angular displacement is counter-clock-wise. The angular velocity vector \displaystyle {\vec {\omega }} is then defined as:

\displaystyle {\boldsymbol {\omega }}={\frac {d\phi }{dt}}{\mathbf {u} }

Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a plane of angular displacement in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The direction normal to this plane is defined to be the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2-dimensional case. Using the unit vector \displaystyle {\mathbf {u}} defined before, the angular velocity vector may be written in a manner similar to that for two dimensions:
\displaystyle {\boldsymbol {\omega }}={\frac {|{\mathbf {v} }|\sin(\theta )}{|{\mathbf {r} }|}}{\mathbf {u} }
which, by the definition of the cross product, can be written:
\displaystyle {\boldsymbol {\omega }}={\frac {{\mathbf {r} }\times {\mathbf {v} }}{|{\mathbf {r} }|^{2}}}
and using this vector, it can be seen that the formula for the tangential velocity of the particle is:
\displaystyle {\boldsymbol {v}}_{\perp }={\boldsymbol {\omega }}\times {\boldsymbol {r}}

 

是否足夠哉！或許『pseudo-』帶來贗、偽，將作怪耶？？

忘卻論其『性質』嗄！！

Addition of angular velocity vectors

Angular velocity can be defined as angular displacement per unit time. If a point rotates with \displaystyle \omega _{2} in a frame \displaystyle F_{2} that itself rotates with an angular velocity \displaystyle \omega _{1} with respect to an external frame \displaystyle F_{1} , we can define the addition of \displaystyle \omega _{1}+\omega _{2} as the angular velocity vector of the point with respect to \displaystyle F_{1} .

With this operation defined like this, angular velocity, which is a pseudovector, also becomes a real vector because it has two operations:

  • An internal operation (addition), which is associative, commutative, distributive and with zero and unity elements
  • An external operation (external product), with the normal properties for an external product.

This is the definition of a vector space. The only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W (see below) is skew-symmetric. Therefore, \displaystyle R=e^{Wt} is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore, it can be expanded as \displaystyle R=I+W\cdot dt+{1 \over 2}(W\cdot dt)^{2}+\ldots

The composition of rotations is not commutative; but, when the rotations are infinitesimal, the first order approximation of the previous series can be taken and \displaystyle (I+W_{1}\cdot dt)(I+W_{2}\cdot dt)=(I+W_{2}\cdot dt)(I+W_{1}\cdot dt) and therefore \displaystyle \omega _{1}+\omega _{2}=\omega _{2}+\omega _{1} .

 

故而不及見真相矣！！？？

Angular velocity vector for a frame

 Schematic construction for addition of angular velocity vectors for rotating frames

Given a rotating frame composed by three unitary vectors, all the three must have the same angular speed in any instant. In such a frame, each vector is a particular case of the previous case (moving particle), in which the module of the vector is constant.

Though it just a particular case of a moving particle, this is a very important one for its relationship with the rigid body study, and special tools have been developed for this case. There are two possible ways to describe the angular velocity of a rotating frame: the angular velocity vector and the angular velocity tensor. Both entities are related and they can be calculated from each other.

In a consistent way with the general definition, the angular velocity of a frame is defined as the angular velocity of each of the three vectors of the frame (it will be the same for any of them). The addition of angular velocity vectors for frames is also defined by movement composition, and can be useful to decompose the movement as in a gimbal. Components of the vector can be calculated as derivatives of the parameters defining the moving frames (Euler angles or rotation matrices). As in the general case, addition is commutative: \displaystyle \omega _{1}+\omega _{2}=\omega _{2}+\omega _{1}

It is known by Euler’s rotation theorem that, for any rotating frame, there always exists an instantaneous axis of rotation in any instant. In the case of a frame, the angular velocity vector is over the instantaneous axis of rotation. Any transversal section of a plane perpendicular to this axis has to behave as a two dimensional rotation. Thus, the magnitude of the angular velocity vector at a given time t is consistent with the two dimensions case.

Components from the vectors of the frame

Substituting in the expression

\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{|\mathrm {\mathbf {r} } |^{2}}}

any unitary vector e of the frame we obtain \displaystyle {\boldsymbol {\omega }}={\frac {{\boldsymbol {e}}\times {\dot {\boldsymbol {e}}}}{|{\mathbf {e} }|^{2}}} , and therefore \displaystyle {\boldsymbol {\omega }}=\mathbf {e} _{1}\times {\dot {\mathbf {e} }}_{1}=\mathbf {e} _{2}\times {\dot {\mathbf {e} }}_{2}=\mathbf {e} _{3}\times {\dot {\mathbf {e} }}_{3} .

As the columns of the matrix of the frame are the components of its vectors, this allows also the calculation of \displaystyle \omega from the matrix of the frame and its derivative.

Components from Euler angles

 Diagram showing Euler frame in green

The components of the angular velocity pseudovector were first calculated by Leonhard Euler using his Euler angles and an intermediate frame made out of the intermediate frames of the construction:

  • One axis of the reference frame (the precession axis)
  • The line of nodes of the moving frame respect the reference frame (nutation axis)
  • One axis of the moving frame (the intrinsic rotation axis)

Euler proved that the projections of the angular velocity pseudovector over these three axes was the derivative of its associated angle (which is equivalent to decomposing the instant rotation in three instantaneous Euler rotations). Therefore:[2]

\displaystyle {\boldsymbol {\omega }}={\dot {\alpha }}{\mathbf {u} }_{1}+{\dot {\beta }}{\mathbf {u} }_{2}+{\dot {\gamma }}{\mathbf {u} }_{3}

This basis is not orthonormal and it is difficult to use, but now the velocity vector can be changed to the fixed frame or to the moving frame with just a change of bases. For example, changing to the mobile frame:
\displaystyle {\boldsymbol {\omega }}=({\dot {\alpha }}\sin \beta \sin \gamma +{\dot {\beta }}\cos \gamma ){\mathbf {I} }+({\dot {\alpha }}\sin \beta \cos \gamma -{\dot {\beta }}\sin \gamma ){\mathbf {J} }+({\dot {\alpha }}\cos \beta +{\dot {\gamma }}){\mathbf {K} }
where \displaystyle {\mathbf {I} },{\mathbf {J} },{\mathbf {K} } are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles.[3]

Angular velocity tensor

See also: Skew-symmetric matrix

A similar way to describe the angular speed for a rotating frame is the angular velocity tensor. It can be introduced from rotation matrices. Any vector \displaystyle {\vec {r}} that rotates around an axis with an angular speed vector \displaystyle {\vec {\omega }} (as defined before) satisfies:

\displaystyle {\frac {d{\vec {r}}(t)}{dt}}={\vec {\omega }}\times {\vec {r}}

We can introduce here the angular velocity tensor associated to the angular speed \displaystyle \omega :
\displaystyle W(t)={\begin{pmatrix}0&-\omega _{z}(t)&\omega _{y}(t)\\\omega _{z}(t)&0&-\omega _{x}(t)\\-\omega _{y}(t)&\omega _{x}(t)&0\\\end{pmatrix}}
Notice that this is an infinitesimal angular displacement divided by an infinitesimal time. This tensor W(t) will act as if it were a \displaystyle ({\vec {\omega }}\times ) operator :
\displaystyle {\vec {\omega }}(t)\times {\vec {r}}(t)=W(t){\vec {r}}(t)

Calculation from the orientation matrix

Given the orientation matrix A(t) of a frame, defined for all t and derivable, we can obtain its instant angular velocity tensor W as follows. We know that:

\displaystyle {\frac {d{\vec {r}}(t)}{dt}}=W\cdot {\vec {r}}

As angular speed must be the same for the three vectors of a rotating frame, if we have a matrix A(t) whose columns are the vectors of the frame, we can write for the three vectors as a whole:
\displaystyle {\frac {dA(t)}{dt}}=W\cdot A(t)
And therefore the angular velocity tensor we are looking for is:
\displaystyle W={\frac {dA(t)}{dt}}\cdot A^{-1}(t)

 

蜻蜓掠過歐拉的天空的呀？？！！

於是問到『剛體』如何『轉動』ㄚ？

Rigid body considerations

See also: axes conventions

 Position of point P located in the rigid body (shown in blue). Ri is the position with respect to the lab frame, centered at O and ri is the position with respect to the rigid body frame, centered at O′. The origin of the rigid body frame is at vector position R from the lab frame.

The same equations for the angular speed can be obtained reasoning over a rotating rigid body. Here is not assumed that the rigid body rotates around the origin. Instead, it can be supposed rotating around an arbitrary point that is moving with a linear velocity V(t) in each instant.

To obtain the equations, it is convenient to imagine a rigid body attached to the frames and consider a coordinate system that is fixed with respect to the rigid body. Then we will study the coordinate transformations between this coordinate and the fixed “laboratory” system.

As shown in the figure on the right, the lab system’s origin is at point O, the rigid body system origin is at O′ and the vector from O to O′ is R. A particle (i) in the rigid body is located at point P and the vector position of this particle is Ri in the lab frame, and at position ri in the body frame. It is seen that the position of the particle can be written:

\displaystyle \mathbf {R} _{i}=\mathbf {R} +\mathbf {r} _{i}

The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector \displaystyle \mathbf {r} _{i} is unchanging. By Euler’s rotation theorem, we may replace the vector \displaystyle \mathbf {r} _{i} with \displaystyle {\mathcal {R}}\mathbf {r} _{io} where \displaystyle {\mathcal {R}} is a 3×3 rotation matrix and \displaystyle \mathbf {r} _{io} is the position of the particle at some fixed point in time, say t = 0. This replacement is useful, because now it is only the rotation matrix \displaystyle {\mathcal {R}} that is changing in time and not the reference vector \displaystyle \mathbf {r} _{io} , as the rigid body rotates about point O′. Also, since the three columns of the rotation matrix represent the three versors of a reference frame rotating together with the rigid body, any rotation about any axis becomes now visible, while the vector \displaystyle \mathbf {r} _{i} would not rotate if the rotation axis were parallel to it, and hence it would only describe a rotation about an axis perpendicular to it (i.e., it would not see the component of the angular velocity pseudovector parallel to it, and would only allow the computation of the component perpendicular to it). The position of the particle is now written as:
\displaystyle \mathbf {R} _{i}=\mathbf {R} +{\mathcal {R}}\mathbf {r} _{io}
Taking the time derivative yields the velocity of the particle:
\displaystyle \mathbf {V} _{i}=\mathbf {V} +{\frac {d{\mathcal {R}}}{dt}}\mathbf {r} _{io}
where Vi is the velocity of the particle (in the lab frame) and V is the velocity of O′ (the origin of the rigid body frame). Since \displaystyle {\mathcal {R}} is a rotation matrix its inverse is its transpose. So we substitute \displaystyle {\mathcal {I}}={\mathcal {R}}^{\text{T}}{\mathcal {R}} :
\displaystyle \mathbf {V} _{i}=\mathbf {V} +{\frac {d{\mathcal {R}}}{dt}}{\mathcal {I}}\mathbf {r} _{io}
\displaystyle \mathbf {V} _{i}=\mathbf {V} +{\frac {d{\mathcal {R}}}{dt}}{\mathcal {R}}^{\text{T}}{\mathcal {R}}\mathbf {r} _{io}
\displaystyle \mathbf {V} _{i}=\mathbf {V} +{\frac {d{\mathcal {R}}}{dt}}{\mathcal {R}}^{\text{T}}\mathbf {r} _{i}
or
\displaystyle \mathbf {V} _{i}=\mathbf {V} +W\mathbf {r} _{i}
where \displaystyle W={\frac {d{\mathcal {R}}}{dt}}{\mathcal {R}}^{\text{T}} is the previous angular velocity tensor.

It can be proved that this is a skew symmetric matrix, so we can take its dual to get a 3 dimensional pseudovector that is precisely the previous angular velocity vector \displaystyle {\vec {\omega }} :

\displaystyle {\boldsymbol {\omega }}=[\omega _{x},\omega _{y},\omega _{z}]

Substituting ω for W into the above velocity expression, and replacing matrix multiplication by an equivalent cross product:
\displaystyle \mathbf {V} _{i}=\mathbf {V} +{\boldsymbol {\omega }}\times \mathbf {r} _{i}
It can be seen that the velocity of a point in a rigid body can be divided into two terms – the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the “spin” angular velocity of the rigid body as opposed to the angular velocity of the reference point O′ about the origin O.

Consistency

We have supposed that the rigid body rotates around an arbitrary point. We should prove that the angular velocity previously defined is independent from the choice of origin, which means that the angular velocity is an intrinsic property of the spinning rigid body.

 Proving the independence of angular velocity from choice of origin

See the graph to the right: The origin of lab frame is O, while O1 and O2 are two fixed points on the rigid body, whose velocity is \displaystyle \mathbf {v} _{1} and \displaystyle \mathbf {v} _{2} respectively. Suppose the angular velocity with respect to O1 and O2 is \displaystyle {\boldsymbol {\omega }}_{1} and \displaystyle {\boldsymbol {\omega }}_{2} respectively. Since point P and O2 have only one velocity,

\displaystyle \mathbf {v} _{1}+{\boldsymbol {\omega }}_{1}\times \mathbf {r} _{1}=\mathbf {v} _{2}+{\boldsymbol {\omega }}_{2}\times \mathbf {r} _{2}

\displaystyle \mathbf {v} _{2}=\mathbf {v} _{1}+{\boldsymbol {\omega }}_{1}\times \mathbf {r} =\mathbf {v} _{1}+{\boldsymbol {\omega }}_{1}\times (\mathbf {r} _{1}-\mathbf {r} _{2})

The above two yields that
\displaystyle ({\boldsymbol {\omega }}_{1}-{\boldsymbol {\omega }}_{2})\times \mathbf {r} _{2}=0
Since the point P (and thus \displaystyle \mathbf {r} _{2} is arbitrary, it follows that
\displaystyle {\boldsymbol {\omega }}_{1}={\boldsymbol {\omega }}_{2}
If the reference point is the instantaneous axis of rotation the expression of velocity of a point in the rigid body will have just the angular velocity term. This is because the velocity of instantaneous axis of rotation is zero. An example of instantaneous axis of rotation is the hinge of a door. Another example is the point of contact of a purely rolling spherical (or, more generally, convex) rigid body.

 

理所當然爾◎