不久後終於抵達北岸碼頭，空蕩蕩似無人煙。月兒升的更高了，望著眼前土丘，顯的有些淒冷。或許 Mrphs 見我不言不語，因說道︰跨過面前的小山，就到了『幽境夢鄉』。再穿越『幽境夢鄉 』，即達《卡夫卡村》。邊聽邊走，登上了山丘，『幽境夢鄉』在目，黑壓壓的一片，不知何處是盡頭。 Mrphs 又講起︰說來『幽境夢鄉』之名原該是『幽竟夢卿』的古稱。當地原鄉人本叫它『奇幻森林』 ，林內多有五十步高之參天巨木，樹蔭之大可蔽日，又為奇藤異草纏繞，彷彿是個天然暖房。故而林中香草靈芝常生，卵生溼生常居 。遂因此生生不斷，變異驚奇而得名。此林之中央有個『林中道』正是前往《卡夫卡村》的門徑。路底有塊大石，將入村道路分成了東西兩向。誰知一日有人卻意外的用『紫外線』拍着了這石，之後驚訝的發現其上儼然有圖文。北面刻的是『 』，南面上頭有『 』。故詢之 M♪o ，得其解為『幽竟夢卿』，然而此人以為是 M♪o 誤寫，當是作『幽境夢鄉』，於是延誤至今。這麼一說反而倒讓人好奇起來，故問︰難到 M♪o 沒有分辯。 Mrphs 接答︰據『小學堂』同學講，一回有人問過這事，老師說︰『幽竟夢卿』之本義是『  』幽靜將『  』盡，此景恐不再，或終『  』在夢裡相『  』向。總帶著點『傷春悲秋』之意。錯讀為『幽境夢鄉』沒什麼不好的吧！！多少可以『安定人心』的啊？？

只覺一時惘然不知所之何境何鄉的了。或許真應該振奮精神，換個調子續彈『 Sense Hat 』的『 IMU 』inertial measurement unit 之曲的乎！！何不就反其道而行，從『實驗』起頭的哩？？

Raspberry Pi Experiment: Foucault Pendulum

The Foucault Pendulum with some RPi’s Mounted to it

Purpose

Analyze the motion of a pendulum using an inertial measurement unit (IMU) with a Raspberry Pi.

……

─── 《勇闖新世界︰ W!O《卡夫卡村》變形祭︰感知自然‧幽境夢鄉》

 

不解『時物』者︰

《春夜宴諸從弟桃花園序》‧李白

夫天地者，萬物之逆旅也；光陰者，百代之過客也。而浮生若夢 ，為歡幾何?古人秉燭夜游，良有以也。況陽春加我以煙景，大塊假我以文章。會桃花之芳園，序天倫之樂事！群季俊秀，皆為惠連。吾人詠歌，獨慚康樂。幽賞未已，高談轉清。開瓊筵以坐花 ，飛羽觴而醉月，不有詠，何伸雅懷?如詩不成，罰依金谷酒數。

 

心無『桃花園』！

有人說︰機緣可遇不可求。會有『過了這個村，就沒那個店』之慨！真耶？假耶？倘若以『萬物靜觀皆自得』考之，怕是『童趣好奇之心』易失難得的吧！？

曾寫有人想用 IMU 追跡小汽車︰

從牛頓力學來講，假使我們知道一個物體的『加速度』 ，而且如果『初始條件』︰該物位置在原點，速度為零。那麼任意時刻的『速度』是 ，『位置』為 。這麼簡易的算術有什麼重要嗎？若是我們可以『追跡物體 』，舉凡相機拍照的防震、手腳運動之練習、肢體平衡復健的監督﹐…… 實有著不勝枚舉之『用途』。然而『微機電』所作的『慣性感測器』 IMU ，一有免不了的『加速度』之『度量誤差』，此誤差在長『時間』的『積累』下將越來『錯誤』越大！再者那個『量測值』只能是『加速度』的『時間序列』 ，因此 到 時刻間之事也就不得不有『假設』的了！！就像此處問答所說的一樣︰

Tracking 2D positioning with IMU Sensor

shareimprove this question

edited Jan 18 ’14 at 12:13

asked Jan 17 ’14 at 10:14 Khashayar 1013

───

shareimprove this answer

answered Jan 17 ’14 at 10:41 Vorac 1,2961830

─── 摘自《勇闖新世界︰ W!O《卡夫卡村》變形祭︰感知自然‧幽夢‧二》

 

近日閱讀

scikit-kinematics – Documentation

scikit-kinematics is a library for scientific data analysis, with a focus on 3d kinematics.

It is hosted under https://github.com/thomas-haslwanter/scikit-kinematics, and contains the following modules:

imus Analysis routines for IMU-recordings

• calculation of orientation from velocity, recorded with IMUs or space-fixed systems (four different algorithms are implemente here:
  • simple quaternion integration
  • a quaternion Kalman filter
  • Madgwick’s algorithm
  • Mahony’s algorithm
• calculation of position and orientation from IMU-signals
• The sub-directory sensors contains utility to import in data from xio, XSens, and yei system

markers Analysis routines for 3D movements from marker-based video recordings

• a function that takes recordings from video-systems (e.g. Optotrak) and calculates position and orientation
• calculation of joint movements from marker recordings

quat Functions for working with quaternions:

• quaternion multiplication, inversion, conjugate
• conversions to rotation matrices, axis angles, vectors
• a Quaternion class, including operator overloading for multiplication and division
• also work on data arrays

rotmat Functions for working with rotation matrices

• rotation matrices for rotations about the x-, y-, and z-axis
• symbolic rotation matrices
• conversions to Euler, Fick, Helmholtz angles

vector Functions for working with vectors

• angle between vectors
• Gram-Schmidt orthogonalization
• projection
• normalization
• rotation
• also work on data arrays

view Visualization of time-series data, and of 3D orientations

 

欣喜或可用來研究四軸飛行器的運動呦☆

 

 

 

 

 

 

 

 

晨起無聊寄，明兒 同樂會，亦悲亦是喜，喜大家鵬程萬里，悲今後或難再遇。須謹記，破 坎突 險，還是有賴志 友與道 朋。當懷念，同  學共 習之情誼。莫期望 ，倚靠  幸  運。《  》 文說︰

幸，所以驚人也。从大从 。一曰大聲也。凡幸之屬皆从幸。一曰讀若瓠。一曰俗語以盜不止爲幸，幸讀若籋。

運，迻徙也。从辵軍聲。

或應知， 幸，古義枷鎖也。得脫鐐銬，何其有幸！ 運，本講遷軍。若能安泰！運何能不好？皆因不祥事耶！！？？

西諺講︰幸運女神衷情準備好了的人，莫要與之擦身而過；當上帝來敲門時，記得開門。

─── 《M♪O 之學習筆記本《巳》文章︰【䷜】樽酒簋貳》

 

已入炎炎夏日，又是同樂會之時！

時宜放暑假，偏逢天候水火未濟，致使拾穗忙？

派︰《  》畫傳︰昔有『米勒』者繪作『拾穗』圖，──

拾穗一詞淵源於舊約聖經路得記-路得與波阿斯的記載，路得在波阿斯田裏撿麥穗，以供養她的婆婆拿俄米，指農民需讓貧苦人撿拾收割後遺留穗粒以求溫飽，而該畫除了描繪3名農婦在金黃色麥田撿拾麥穗情景外，其金黃陽光、彎腰等細節，另外呈現「英雄史詩般的崇高意境」。 畫面上三位年紀不同的女性，表現出勞動家庭的命運 ，且畫面遠處可看到農地監督者坐於馬背上觀看，反映出當時資本主義產生的貧富差距。 畫面因遠方地平線處的教堂而流露出一股近似宗教情操的崇高性。──

 

，觀之令人紅眼框。無巧不成書，無事難商量，想那《水滸傳》 ，真真是，官逼民反作強梁。

─ 摘自《M♪O 之學習筆記本《寅》井井︰【紅火夏】刈麥遺穗》

 

祇能盡興『或鼓或歌』耶？！

The Physics of Quadcopter Flight

by Toglefritz | Apr 29, 2014 | theory | 22 comments

I believe that when embarking on any project, especially one as complex as multirotor construction and piloting, it is useful to have an understanding of the theoretical underpinnings involved. So, when building and flying multirotors, I think it is valuable to have at least a basic understanding of the physics of quadcopter flight. While it is certainly possible to simply follow a set of directions, like the ones on this site, for building and flying a multirotor, it will be much clearer, and more meaningful, if you can explain to yourself the rationale behind each step.

Now, I just want to give you one caveat before you read further:  there is a huge amount of physics involved in multirotor flight, and I am only going to skim the surface on this page. Here I am going to focus on the physics involved in maneuvering the multirotor which, as you will understand soon, involves adjusting the balance of forces acting on the craft. I am going to avoid talking about the physics of how the props generate lift, the physics involved in the multirotor’s power system, the physics of how brushless motors work, et cetera. If you are interested in getting really in-depth with the topic of multirotor physics, you will find many resources around the Internet (although information is a bit scattered, which  is one of the reasons I wrote this page).

 

FRIDAY, NOVEMBER 23, 2012

Quadcopter Dynamics and Simulation

Introduction

A helicopter is a flying vehicle which uses rapidly spinning rotors to push air downwards, thus creating a thrust force keeping the helicopter aloft. Conventional helicopters have two rotors. These can be arranged as two coplanar rotors both providing upwards thrust, but spinning in opposite directions (in order to balance the torques exerted upon the body of the helicopter). The two rotors can also be arranged with one main rotor providing thrust and a smaller side rotor oriented laterally and counteracting the torque produced by the main rotor. However, these configurations require complicated machinery to control the direction of motion; a swashplate is used to change the angle of attack on the main rotors. In order to produce a torque the angle of attack is modulated by the location of each rotor in each stroke, such that more thrust is produced on one side of the rotor plane than the other. The complicated design of the rotor and swashplate mechanism presents some problems, increasing construction costs and design complexity.

A quadrotor helicopter (quadcopter) is a helicopter which has four equally spaced rotors, usually arranged at the corners of a square body. With four independent rotors, the need for a swashplate mechanism is alleviated. The swashplate mechanism was needed to allow the helicopter to utilize more degrees of freedom, but the same level of control can be obtained by adding two more rotors.

The development of quadcopters has stalled until very recently, because controlling four independent rotors has proven to be incredibly difficult and impossible without electronic assistance. The decreasing cost of modern microprocessors has made electronic and even completely autonomous control of quadcopters feasible for commercial, military, and even hobbyist purposes.

Quadcopter control is a fundamentally difficult and interesting problem. With six degrees of freedom (three translational and three rotational) and only four independent inputs (rotor speeds), quadcopters are severely underactuated. In order to achieve six degrees of freedom, rotational and translational motion are coupled. The resulting dynamics are highly nonlinear, especially after accounting for the complicated aerodynamic effects. Finally, unlike ground vehicles, helicopters have very little friction to prevent their motion, so they must provide their own damping in order to stop moving and remain stable. Together, these factors create a very interesting control problem. We will present a very simplified model of quadcopter dynamics and design controllers for our dynamics to follow a designated trajectory. We will then test our controllers with a numerical simulation of a quadcopter in flight.

F3-DP-2016-Gopalakrishnan-Eswarmurthi-Quadcoptor flight mechanics model and control algorithms.pdf

 

 

 

 

 

 

 

 

 

當『螺旋槳』

Propeller

A propeller is a type of fan that transmits power by converting rotational motion into thrust. A pressure difference is produced between the forward and rear surfaces of the airfoil-shaped blade, and a fluid (such as air or water) is accelerated behind the blade. Propeller dynamics, like those of aircraft wings, can be modelled by Bernoulli’s principle and Newton’s third law. Most marine propellers are screw propellers with fixed helical blades rotating around a horizontal (or nearly horizontal) axis or propeller shaft.

History

Early developments

The principle employed in using a screw propeller is used in sculling. It is part of the skill of propelling a Venetian gondola but was used in a less refined way in other parts of Europe and probably elsewhere. For example, propelling a canoe with a single paddle using a“pitch stroke” or side slipping a canoe with a “scull” involves a similar technique. In China, sculling, called “lu”, was also used by the 3rd century AD.

In sculling, a single blade is moved through an arc, from side to side taking care to keep presenting the blade to the water at the effective angle. The innovation introduced with the screw propeller was the extension of that arc through more than 360° by attaching the blade to a rotating shaft. Propellers can have a single blade, but in practice there are nearly always more than one so as to balance the forces involved.

The origin of the screw propeller starts with Archimedes, who used a screw to lift water for irrigation and bailing boats, so famously that it became known as Archimedes’ screw. It was probably an application of spiral movement in space (spirals were a special study of Archimedes) to a hollow segmented water-wheel used for irrigation by Egyptians for centuries. Leonardo da Vinci adopted the principle to drive his theoretical helicopter, sketches of which involved a large canvas screw overhead.

In 1661, Toogood and Hays proposed using screws for waterjet propulsion, though not as a propeller.^[1] Robert Hook in 1681 designed a horizontal watermill which was remarkably similar to the Kirsten-Boeing vertical axis propeller designed almost two and a half centuries later in 1928; two years later Hook modified the design to provide motive power for ships through water.^[2] In 1752, the Academie des Sciences in Paris granted Burnelli a prize for a design of a propeller-wheel. At about the same time, the French mathematician Alexis-Jean-Pierre Paucton, suggested a water propulsion system based on the Archimedean screw.^[3] In 1771, steam-engine inventor James Watt in a private letter suggested using “spiral oars” to propel boats, although he did not use them with his steam engines, or ever implement the idea.^[4]

The first practical & applied use of a propeller on a submarine dubbed the Turtle which was designed in New Haven, Connecticut, in 1775 by Yale student and inventor David Bushnell, with the help of the clock maker, engraver, and brass foundryman Isaac Doolittle, and with Bushnell’s brother Ezra Bushnell and ship’s carpenter and clock maker Phineas Pratt constructing the hull in Saybrook, Connecticut.^[5][6] On the night of September 6, 1776, Sergeant Ezra Lee piloted the Turtle in an attack on the HMS Eagle in New York Harbor.^[7][8] The Turtle also has the distinction of being the first submarine used in battle. Bushnell later described the propeller in an October 1787 letter to Thomas Jefferson: “An oar formed upon the principle of the screw was fixed in the forepart of the vessel its axis entered the vessel and being turned one way rowed the vessel forward but being turned the other way rowed it backward. It was made to be turned by the hand or foot.”^[9] The brass propeller, like all the brass and moving parts on the Turtle, was crafted by the “ingenious mechanic” Issac Doolittle of New Haven.^[10]

In 1785, Joseph Bramah in England proposed a propeller solution of a rod going through the underwater aft of a boat attached to a bladed propeller, though he never built it.^[11] In 1802, Edward Shorter proposed using a similar propeller attached to a rod angled down temporarily deployed from the deck above the waterline and thus requiring no water seal, and intended only to assist becalmed sailing vessels. He tested it on the transport ship Doncaster in Gibraltar and at Malta, achieving a speed of 1.5 mph.^[12]

The lawyer and inventor John Stevens in the USA, built a 25-foot boat with a rotary stem engine coupled to a four-bladed propeller, achieving a speed of 4 mph, but he abandoned propellers due to the inherent danger in using the high-pressure steam engines, and instead built paddle-wheeled boats.^[13]

By 1827, Czech-Austrian inventor Josef Ressel had invented a screw propeller which had multiple blades fastened around a conical base. He had tested his propeller in February 1826 on a small ship that was manually driven. He was successful in using his bronze screw propeller on an adapted steamboat (1829). His ship “Civetta” with 48 gross register tons, reached a speed of about six knots (11 km/h). This was the first ship successfully driven by an Archimedes screw-type propeller. After a new steam engine had an accident (cracked pipe weld) his experiments were banned by the Austro-Hungarian police as dangerous. Josef Ressel was at the time a forestry inspector for the Austrian Empire. But before this he received an Austro-Hungarian patent (license) for his propeller (1827). He died in 1857. This new method of propulsion was an improvement over the paddlewheel as it was not so affected by either ship motions or changes in draft as the vessel burned coal.^[14]

John Patch, a mariner in Yarmouth, Nova Scotia developed a two-bladed, fan-shaped propeller in 1832 and publicly demonstrated it in 1833, propelling a row boat across Yarmouth Harbour and a small coastal schooner at Saint John, New Brunswick, but his patent application in the United States was rejected until 1849 because he was not an American citizen.^[15] His efficient design drew praise in American scientific circles^[16] but by this time there were multiple competing versions of the marine propeller.

……

遇上

白努利定律

白努利原理（英語：Bernoulli’s principle），又稱柏努利定律、白努利定律（英語：Bernoulli’s Law）^[1]，是流體力學中的一個定律 ，由瑞士流體物理學家丹尼爾·白努利於1738年出版他的理論《Hydrodynamica》，描述流體沿著一條穩定、非黏性、不可壓縮的流線移動行為。^[2]

在流體動力學，白努利原理指出，無黏性的流體的速度增加時，流體的壓力能或位能（位能）總和將減少。

白努利原理可以應用到不同類型的流體流動，從而是可廣泛套用的白努利方程式表示式。事實上，有不同類型的流的白努利方程式的不同形式的。白努利原理的簡單形式是有效的不可壓縮流動（如最液體流動），也為移動可壓縮流體（如氣體）在低馬赫數（通常小於0.3）。更先進的形式可被應用到在某些情況 ​​下，在更高的馬赫數（見白努利方程式的推導）可壓縮流。

白努利定律可以從能量守恆定律來推演。說明如下：在一個穩定的水流，沿著直線流向的所有點上，各種形式的流體機械能總和必定相同。也就是說，動能，位能，與內能的總和保持不變。換言之，任何的流體速度增加，即代表動態壓力和單位體積動能的增加，而在同時會導致其靜態壓力，單位體積流體的位能、內能等三者總和的減少。如果液體流出水庫，在各方向的流線上，各種形式的能量的總和是相同的；因為每單位體積能量的總和（即壓力和單位體積流體的重力位能  的總和 ）在水庫內的任何位置都相同。

白努利原理，也可以直接由牛頓第二定律推演。說明如下：如果從高壓區域往低壓區域，有一小體積流體沿水平方向流動，小體積區域後方的壓力自然比前方區域的壓力更大。所以，此區域的力量總和必然是沿著流線方向向前。在此假設，前後方區域面積相等，如此便提供了一個正方向淨力施於原先設定的流體小體積區域，其加速度與力量同方向。此假想環境中，流體粒子僅受到壓力和自己質量的重力之影響。先假設如果流體沿著流線方向作水平流動，並與流體流線的截面積垂直，因為流體從高壓區域朝低壓區域移動，流體速度因此增加；如果該小體積區域的流速降低，其唯一的可能性必定是因為它從低壓區朝高壓區移動。因此 ，任一水平流動流體之內，壓力最低處有最高流速，壓力最高處有最低流速。

水流進入文丘裡計。在不計流體壓力下而增加動能，如圖中兩列管的高度差。

物理量及定律

原表達形式

其中：

 流體速度

 重力加速度（地球表面的值為 9.8 m/s²）

 流體處於的高度（從某參考點計）

 流體所受的壓力強度

 流體質量密度

 常數

定理假設

使用白努利定律必須符合以下假設，方可使用；如沒完全符合以下假設，所求的解也是近似值。

• 定常流動（或稱穩定流，Steady flow）：在流動系統中，流體在任何一點之性質不隨時間改變。
• 不可壓縮流（Incompressible flow）：密度為常數，在流體為氣體適用於馬赫數  小於0.3的情況。
• 無摩擦流（Frictionsless flow）：摩擦效應可忽略，忽略黏滯性效應。
• 流體沿著流線流動（Flow along a streamline）：流體元素（element）沿著流線而流動，流線間彼此是不相交的。

 

快慢流速激發『轉子』產生了『推進力』！

默然咀嚼阿基米德提水器其中味？

 

回旋反思，就算『數理完備』

Blade element momentum theory

Blade element momentum theory is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.

This article emphasizes application of BEM to ground-based wind turbines, but the principles apply as well to propellers. Whereas the streamtube area is reduced by a propeller, it is expanded by a wind turbine. For either application, a highly simplified but useful approximation is the Rankine–Froude “momentum” or “actuator disk” model (1865,1889). This article explains the application of the “Betz limit” to the efficiency of a ground-based wind turbine.

A development came in the form of Froude’s blade element momentum theory (1878), later refined by Glauert (1926). Betz (1921) provided an approximate correction to momentum “Rankine–Froude actuator-disk” theory to account for the sudden rotation imparted to the flow by the actuator disk (NACA TN 83, “The Theory of the Screw Propeller” and NACA TM 491, “Propeller Problems”). In blade element momentum theory, angular momentum is included in the model, meaning that the wake (the air after interaction with the rotor) has angular momentum. That is, the air begins to rotate about the z-axis immediately upon interaction with the rotor (see diagram below). Angular momentum must be taken into account since the rotor, which is the device that extracts the energy from the wind, is rotating as a result of the interaction with the wind.

……

 

如何『傳播大眾』？？普照於人呀！！

Propeller Thrust

Most general aviation or private airplanes are powered by internal combustion engines which turn propellers to generate thrust. The details of how a propeller generates thrust is very complex, but we can still learn a few of the fundamentals using the simplified momentum theory presented here.

 

 

 

 

 

 

 

 

老子第四十二章中講︰道生一，一生二，二生三，三生萬物。是說天地生萬物就像四季循環的自然而然，如果『人』或將成為『亦大』，就得知道大自然 『益』生之道，能循本溯源貴能『得一』。他固善於『觀水』，盛讚『上善若水』，卻也深知水為山堵之『蹇』、人為慾阻之『蒙』難，故於第三十九章中又講︰

昔之得一者，天得一以清，地得一以寧，神得一以靈，谷得一以盈，萬物得一以生，侯王得一以為天下貞。其致之。天無以清則恐裂，地無以寧則恐發，神無以靈則恐歇，谷無以盈則恐竭，萬物無以生則恐滅，侯王無以貞高則恐蹶。故貴以賤為本，高以下為基。是以侯王自謂孤寡不穀，此非以賤為本耶？非乎？人之所惡，唯孤寡不穀，而侯王以為稱。故致譽無譽。不欲琭琭如玉，珞珞如石。

，希望人們知道所謂『道德』之名，實在說的是『得到』── 得道── 的啊！！如果乾坤都『沒路』可走，人又該往向『何方』？？

昔時吳越之爭，越王勾踐『臥薪嘗膽』逐夢復國，此事紀載於《史記‧越王勾踐世家》，在此我們將談及一人亦載之於史記︰

范蠡‧史記‧貨殖列傳

昔者越王句踐困於會稽之上，乃用范蠡、計然。計然曰：『知斗則修備，時用則知物，二者形則萬貨之情可得而觀已。故歲在金，穰；水，毀；木，饑；火，旱。旱則資舟，水則資車，物之理也。六歲穰，六歲旱，十二歲一大饑。夫糶，二十病農，九十病末。末病則財不出，農病則草不辟矣。上不過八十，下不減三十，則農末 俱利，平糶齊物，關市不乏，治國之道也。積著之理，務完物，無息幣。以物相貿易，腐敗而食之貨勿留，無敢居貴。論其有餘不足，則知貴賤。貴上極則反賤，賤下極則反貴。貴出如糞土，賤取如珠玉。財幣欲其行如流水。』修之十年，國富，厚賂戰士，士赴矢石，如渴得飲，遂報彊吳，觀兵中國，稱號『五霸』。

范蠡既雪會稽之恥，乃喟然而嘆曰：「計然之策七，越用其五而得意。既已施於國，吾欲用之家。」乃乘扁舟浮於江湖，變名易姓，適齊為鴟夷子皮，之陶為朱公。 朱公以為陶天下之中，諸侯四通，貨物所交易也。乃治產積居。與時逐而不責於人。故善治生者，能擇人而任時。十九年之中三致千金，再分散與貧交疏昆弟。此所謂富好行其德者也。後年衰老而聽子孫，子孫修業而息之，遂至巨萬。故言富者皆稱陶朱公。

文子治國富民之策有七，越王只用其五就洋洋得意。范蠡細省已用『天時』、『地利』二者，所餘不用，不就是『人和』不用了嗎？勾踐得國後必將失人，此時不乘扁舟浮於江湖，怕我連命都不保了。陶朱公之為後世所稱的『財神』，不因他能得之於天下成大富巨萬，而因他能『三聚三散』用之於天下。此時如果再讀讀莊子的二則寓言︰

莊子‧胠篋之竊鉤者誅，竊國者為諸侯；

聖人不死，大盜不止。雖重聖人而治天下，則是重利盜跖也。為之斗斛以量之，則並與斗斛而竊之；為之權衡以稱之，則並與權衡而竊之；為之符璽以信之，則並與符璽而竊之；為之仁義以矯之，則並與仁義而竊之。何以知其然邪？彼竊鉤者誅，竊國者為諸侯，諸侯之門而仁義存焉，則是非竊仁義聖知邪？故逐於大盜，揭諸侯，竊仁義並斗斛權衡符璽之利者，雖有軒冕之賞弗能勸，斧鉞之威弗能禁。此重利盜跖而使不禁者，是乃聖人之過也。

莊子‧秋水之無用用大；

惠子謂莊子曰：魏王貽我大瓠之種，我樹之成而實五石。以盛水漿，其堅不能自舉也；剖之以為瓢，則瓠落無所容。非不呺然大也，吾為其無用而掊之。

莊子曰：夫子固拙于用大矣！宋人有善為不龜手之藥者，世世以洴澼絖為事。客聞之，請買其方百金。聚族而謀曰：『我世世為洴澼絖，不過數金，今一朝而鬻技百金，請與之。』客得之，以說吳王。越有難，吳王使之將。冬，與越人水 戰，大敗越人，裂地而封之。能不龜手一也；或以封，或不免於洴澼絖，則所用之異也。今子有五石之瓠，何不慮以為大樽，而浮於江湖，而憂其瓠落無所容？則夫子猶有蓬之心也夫！

─── 《跟隨□？築夢！！》

 

一個『玩具』

竹蜻蜓

竹蜻蜓是一種古老的兒童玩具，由軸和槳翼組成，多以竹木製做 。嚴格來說，竹蜻蜓應包括槳翼，轉軸和套在轉軸外的竹筒三個主要部份 ，雖然光是槳翼加轉軸也能玩， 但是只能充當作直升機玩。只有三個零件組成一體才能當自轉旋翼機玩。 中國晉朝葛洪所著的《抱朴子》是紀錄類似竹蜻蜓最早的動力機械《抱朴子》 ：「若能乘蹻者，可以周流天下，不拘山河。凡乘蹻道有三法：一曰龍蹻、二曰虎蹻、三曰鹿盧蹻。或服符精思，若欲行千里，則以一時思之。若晝夜十二時思之，則可以一日一夕行萬二千里 ，亦不能過此，過此當更思之，如前法。或用棗心木為飛車 ，以牛革結環劍以引其機，或存念作五蛇六龍三牛交罡而乘之，上升四十里，名為太清。《抱朴子》一書也有這樣的記述：「或用栆心木為飛車，以牛革街環劍，以引起幾。或存念作五蛇六龍三牛 、交罡而乘之，上升四十里，名為太清。太清之中，其氣甚罡，能勝人也。」

Modern Japanese taketombo bamboo-copters; wooden type with winding thread (left); plastic type (right)

A decorated Japanese taketombopropeller

 

能『有何用』？卻足以啟發 George Cayley ，令李約瑟大感驚訝︰

Bamboo-copter

The bamboo-copter, also known as the bamboo dragonfly or Chinese top (Chinese zhuqingting (竹蜻蜓), Japanese taketombo 竹蜻蛉), is a toy helicopter rotor that flies up when its shaft is rapidly spun. This helicopter-like top originated in Warring States period China around 400 BC, and was the object of early experiments by English engineer George Cayley, the inventor of modern aeronautics.^[1]

In China, the earliest known flying toys consisted of feathers at the end of a stick, which was rapidly spun between the hands and released into flight. “While the Chinese top was no more than a toy, it is perhaps the first tangible device of what we may understand as a helicopter.”^[1]

The Jin dynasty Daoist philosopher Ge Hong‘s (c. 317) book Baopuzi (抱樸子 “Master Who Embraces Simplicity”) mentioned a flying vehicle in what Joseph Needham calls “truly an astonishing passage”.^[2]

Some have made flying cars [feiche 飛車] with wood from the inner part of the jujube tree, using ox-leather (straps) fastened to returning blades so as to set the machine in motion [huan jian yi yin chiji 環劍以引其機]. Others have had the idea of making five snakes, six dragons and three oxen, to meet the “hard wind” [gangfeng 罡風] and ride on it, not stopping until they have risen to a height of forty li. That region is called [Taiqing 太清] (the purest of empty space). There the [qi] is extremely hard, so much so that it can overcome (the strength of) human beings. As the Teacher says: “The kite (bird) flies higher and higher spirally, and then only needs to stretch its two wings, beating the air no more, in order to go forward by itself. This is because it starts gliding (lit. riding) on the ‘hard wind’ [gangqi 罡炁]. Take dragons, for example; when they first rise they go up using the clouds as steps, and after they have attained a height of forty li then they rush forward effortlessly (lit. automatically) (gliding).” This account comes from the adepts [xianren 仙人], and is handed down to ordinary people, but they are not likely to understand it.^[2]

Needham concludes that Ge Hong was describing helicopter tops because “‘returning (or revolving) blades’ can hardly mean anything else, especially in close association with a belt or strap”; and suggests that “snakes”, “dragons”, and “oxen” refer to shapes of man-lifting kites.^[3] Other scholars interpret this Baopuzi passage mythologically instead of literally, based on its context’s mentioning fantastic flights through chengqiao (乘蹻 “riding on tiptoe/stilts”) and xian (仙 “immortal; adept”) techniques. For instance, “If you can ride the arches of your feet, you will be able to wander anywhere in the world without hindrance from mountains or rivers … Whoever takes the correct amulet and gives serious thought to the process may travel a thousand miles by concentrating his thoughts for one double hour.”^[4] Compare this translation.

Some build a flying vehicle from the pith of the jujube tree and have it drawn by a sword with a thong of buffalo hide at the end of its grip. Others let their thoughts dwell on the preparation of a joint rectangle from five serpents, six dragons, and three buffaloes, and mount in this for forty miles to the region known as Paradise.^[4]

This Chinese helicopter toy was introduced into Europe and “made its earliest appearances in Renaissance European paintings and in the drawings of Leonardo da Vinci.”^[5] The toy helicopter appears in a c. 1460 French picture of the Madonna and Child at the Musée de l’Ancien Évêché in Le Mans, and in a 16th-century stained glass panel at the Victoria and Albert Museum in London.^[6][7] A c. 1560 picture by Pieter Breughel the Elder at the Kunsthistorisches Museum in Vienna depicts a helicopter top with three airscrews.^[2]

“The helicopter top in China led to nothing but amusement and pleasure, but fourteen hundred years later it was to be one of the key elements in the birth of modern aeronautics in the West.”^[8] Early Western scientists developed flying machines based upon the original Chinese model. The Russian polymath Mikhail Lomonosov developed a spring-driven coaxial rotor in 1743, and the French naturalist Christian de Launoy created a bow drill device with contra-rotating feather propellers.^[1]

In 1792, George Cayley began experimenting with helicopter tops, which he later called “rotary wafts” or “elevating fliers”. His landmark (1809) article “On Aerial Navigation” pictured and described a flying model with two propellers (constructed from corks and feathers) powered by a whalebone bow drill.^[9] “In 1835 Cayley remarked that while the original toy would rise no more than about 6 or 7.5 metres, his improved models would ‘mount upward of 90 ft (27 metres) into the air’. This then was the direct ancestor of the helicopter rotor and the aircraft propeller.”^[10]

Discussing the history of Chinese inventiveness, the British scientist, sinologist, and historian Joseph Needham wrote, “Some inventions seem to have arisen merely from a whimsical curiosity, such as the ‘hot air balloons’ made from eggshells which did not lead to any aeronautical use or aerodynamic discoveries, or the zoetrope which did not lead onto the kinematograph, or the helicopter top which did not lead to the helicopter.”^[11]

直升機

直升機是一種由一個或多個水平旋轉的旋翼提供向上升力和推進力而進行飛行的航空器。直升機具有大多數固定翼航空器所不具備的垂直升降、懸停、小速度向前或向後飛行的特點。這些特點使得直升機在很多場合大顯身手。直升機與固定翼飛機相比，其缺點是速度低、耗油量較高、航程較短。

飛行原理

直升機的升力產生原理與固定翼飛機的機翼相似，機翼與空氣之間發生相對運動，進而產生升力。只不過這個升力是來自於繞固定軸旋轉的「旋翼」。旋翼不像固定翼飛機那樣依靠整個機體向前飛行來使機翼與空氣產生相對運動，而是依靠自身旋轉產生與空氣的相對運動。但是，在旋翼提供升力的同時，直升機機身也會因反扭矩（與驅動旋翼旋轉等量但方向相反的扭矩，即反作用扭矩）的作用而具有向反方向旋轉的趨勢。對於單旋翼直升機，為了平衡反扭矩，常見的做法是以另一個小型旋翼，即尾槳，在機身尾部產生抵消反向運動的力矩。對於多旋翼直升機，多採用旋翼之間反向旋轉的方法來抵消反扭矩的作用，因此在附圖的運作說明中可以見得，由上俯視一個順時針旋轉的主翼，它的尾槳會是向黃色箭頭所指方向推力的。]

直升機和旋翼機的外觀和性能相似，但旋翼機比較簡單和低價，但不如直升機的多方面的特殊性能，而是介乎於普通的固定翼飛機和直升機中間，所以用途較狹而專業化的航空機構通常擁有直升機但鮮有採用旋翼機。

直升機旋翼運作說明

 

貫通『理論』及『實務』得一者，憑借『想象之羽翼』凌空，果不能『築夢踏實』耶☆

多旋翼直升機

四軸飛行器有四個旋翼來懸停、維持姿態及平飛。它的四個旋翼大小相同，分布位置接近對稱。

 

 

 

 

 

 

 

雖然一般都說這是個『知識爆炸』的時代，然而『知識』到底是『什麼』呢？由於人類對大自然的『認識』是『逐漸』累進的，『此時』之所知也許到了『彼時』發現它並不『全然』正確。因此在『科學史』上『典範的變遷』通常伴隨著『新舊論難』，即使『新的理論』在『特定條件』下，已經可以『推導出』既有的『舊的理論』，從整體『概念體系』的『意義一致性』上，尚且需要基本的『解釋』與『釐清』。因是之故『量子力學』的『哥本哈根詮釋』一直都有『爭議』，據聞『多世界詮釋』已經變成了現今的『主流詮釋』。事實上，這樣的活動是一直持續進行中 ，這也就說明了『歷史』的重要性，某些過去不能夠『發展完成』的『觀點』未必『不是黃金』，假使站在前人的肩膀上，用著今天的眼界，未必不能得出『新的創造』，或許『無窮小算術』只是一個很好的『例子』的吧！

在《目盲與耳聾》一文中，我們談到了『帕斯卡』的一個論證，摘引如下

過去法國的哲學家 Blaise Pascal 寫過一本《沉思錄》，這本書裡頭有一個很有意思的論證︰ 無神論者之不幸

如果你信仰上帝，但是祂不存在，你沒有損失；
假使你不信仰上帝，然而祂卻存在，你會下地獄；
權衡利弊，你還是信仰上帝的好。

這也許說明了人的『寧可信其有，切莫道其無』的心理。所謂的『非理性』 irrational ，是指人除了『理性』之外，『人性』尚有許多其它的重要『面向』。然而有時它卻彷彿變成了『不理性』的『代名詞』，就像有人說『只要我喜歡，有什麼不可以』。假使這真的可以作為一個『規範』，那麼將是人人都可以，如果不巧有人喜歡你不喜歡的，但是它的拳頭卻比較大，那又將如之何的呢？或許『怕吃了虧、貪小便宜』讓『聰明人』做了『愚笨之事』，但是人的如是種種『行止』，打開電視就可『一目了然』 ，它沒有國籍而且『無遠弗屆』。正如『左傳』上雖然多次用『歷史事件』講述『易不可行險』，然而今日還有人想以為『卜筮』既然得到了『大吉』，於是『作壞事』也能『得天保佑』，即使『不努力』終將『天降餡餅』一樣，這難到『奇怪』嗎？這難到『不奇怪』的嗎？設想『人際』之間的『倫常』都『不可知其行止』，又將奈何的呢？《呂氏春秋》上有一篇論述『慎行』與『壹行』值得一讀

《呂氏春秋‧論部》第二十二卷‧第一篇 慎行‧第四篇 壹行

先王所惡，無惡於不可知，不可知則君臣、父子、兄弟、朋友、夫妻之際敗矣。十際皆敗，亂莫大焉。凡人倫以十際為安者也，釋十際則與麋鹿虎狼無以異，多勇者則為制耳矣。不可知則知無安君、無樂親矣，無榮兄、無親友、無尊夫矣。強大未必王也，而王必強大。王者之所藉以成也何？藉其威與其利。非強大則其威不威，其利不利。其威不威則不足以禁也，其利不利則不足以勸也，故賢主必使其威利無敵，故以禁則必止，以勸則必為。威利敵，而憂苦民、行可知者王；威利無敵，而以行不知者亡。小弱而不可知，則強大疑之矣。人之情不能愛其所疑，小弱而大不愛則無以存。故不可知之道，王者行之廢，強大行之危，小弱行之滅。

今行者見大樹，必解衣縣冠倚劍而寢其下。大樹非人之情親知交也，而安之若此者信也。陵上巨木，人以為期，易知故也。又況於士乎？士義可知故也，則期為必矣。又況彊大之國？彊大之國誠可知，則其王不難矣。人之所乘船者，為其能浮而不能沈也；世之所以賢君子者，為其能行義而不能行邪辟也。孔子卜，得賁。孔子曰：『不吉。』子貢曰：『夫賁亦好矣，何謂不吉乎？ 』孔子曰：『夫白而白，黑而黑，夫賁又何好乎？』故賢者所惡於物，無惡於無處。夫天下之所以惡，莫惡於不可知也。夫不可知，盜不與期，賊不與謀。盜賊大姦也，而猶所得匹偶，又況於欲成大功乎？夫欲成大功，令天下皆輕 勸而助之，必之士可知。

『卜筮』是為『決疑』，既然『無疑』又何需『卜筮』的呢？孔子既卜得『賁』卦，又為什麼以為『不好』的呢？如果看《易經》的《十翼》，或可知其一二

彖曰：賁，亨﹔柔來而文剛，故亨。分剛上而文柔，故小利有攸往。天文也﹔文明以止，人文也。觀乎天文，以察時變﹔觀乎人文，以化成天下。

象曰：山下有火，賁﹔君子以明庶政，無敢折獄。

顯然像『子貢』說的是個『吉卦』的吧！『賁』是『文采』，想『孔老夫子』周遊列國一無所成，正感嘆著『無力於』這個『黑白不分』的時代，偏偏卻得到這麼個『紋過飾非』的現實『象徵』，他老先生當然『不高興』的啦！想必是占卜『所求不得』的吧！！

─── 《水的生命！！下》

 

什麼是『四軸飛行器』呢？

走過了『派生動力學』道路，是否已駕輕就熟耶？？

如斯者，當讀

陀螺儀

陀螺儀（英文：gyroscope），是一種用來感測與維持方向的裝置 ，基於角動量守恆的理論設計出來的。陀螺儀主要是由一個位於軸心且可旋轉的轉子構成。 陀螺儀一旦開始旋轉，由於轉子的角動量，陀螺儀有抗拒方向改變的趨向。陀螺儀多用於導航、定位等系統。

定軸陀螺儀

偏軸陀螺儀

歷史

1852年法國的物理學家萊昂·傅科為了研究地球自轉，首先發現高速轉動中的轉子（rotor），由於慣性作用它的旋轉軸永遠指向一固定方向，他用希臘文gyro（旋轉）和skopein（看）兩字合為gyroscopei一字來命名這種儀錶。

1860年代，電動馬達的演進使得陀螺儀能夠無限旋轉，進而誕生了第一組航向指示器的原型，甚至是更複雜的儀器–旋轉羅盤。第一組有功能性的旋轉羅盤於1904年由德國發明家赫爾曼·安修斯·康菲申請專利^[1]，美國人艾爾默·斯派理在一年後也提出了他自己的設計。其他國家很快地便發覺到陀螺儀在軍事方面的重要性—在這個航行技術為最重要的軍事力量指標的年代—因而創立了他們自己的陀螺儀工業。斯派理陀螺儀公司快速擴張並供應飛機與船艦的穩定器，其他陀螺儀開發商也跟進。^[2]

到了20世紀末，原本只在飛機、導彈上存在的陀螺儀逐步民用化，也從機械結構邁入電子時代，使用的原理也不盡相同，然而他們的價格依舊昂貴，感應器集成度也不高，只會用在大型儀器上，但21世紀以來，因為智慧型手機產業的進步，陀螺儀的體積不斷縮小，使得原本笨重而昂貴的陀螺儀忽然變成垂手可得的零組件，也帶動了小型無人機的發展。^[3]

結構

陀螺儀的裝置，一直是航空和航海上航行姿態及速率等最方便實用的參考儀錶。

基本上陀螺儀是一種機械裝置，其主要部分是一個對旋轉軸以極高角速度旋轉的轉子，轉子裝在一支架內（見圖一a）；在通過轉子中心軸XX1上加一內環架，那麼陀螺儀就可環繞飛機兩軸作自由運動；然後，在內環架外加上一外環架；這個陀螺儀有兩個平衡環，可以環繞飛機三軸作自由運動，就是一個完整的太空陀螺儀（space gyro）。

迴轉儀的構造

特性

陀螺儀用在飛機飛行儀錶的心臟地位，是由於其兩個基本特性：一為定軸性（inertia 或 rigidity），另一是逆動性（precession），這兩種特性都是建立在角動量守恆的原則下。

定軸性

物體維持自身轉動狀態並對抗改變的能力稱為轉動慣量，其由相對於特定旋轉軸的質量分布決定，對多質點物體轉動慣量  ，概言之：質量越大、對軸距離越遠，轉動慣量越大。一方面陀螺轉子的的對軸對稱性結構使得其具備了同質量物體較大的對軸轉動慣量，意味著其在同阻力扭矩情況下能夠更長時間保持原始運動狀態；另一方面在軸的、小摩擦與無角自由度限制的支點使得外力無法籍此產生較大且有效的阻力扭矩；因此當陀螺轉子以極高速度旋轉時，其轉動得以維持並保持其軸指向一個相對固定的方向，這種物理現象稱為陀螺儀的定軸性或慣性。

在運轉中的陀螺儀，如果外界施一力在轉子上，此力對支點的力矩當可分解為順軸方向和垂直於軸方向兩個分力矩；前者使陀螺加速、減速，但不會改變轉軸方向；後者的時間積分將會逐漸改變轉動方向（通常是短時較小而隨時間逐漸積累增大），並產生相對於原軸的章動（新的旋轉軸原軸旋轉，如轉速降低時陀螺受重力作用時的非垂直旋轉）。

逆動性

在運轉中的陀螺儀，如果外界施一作用或力矩在轉子旋轉軸上，則旋轉軸並不沿施力方向運動，而是順著轉子旋轉向前90度垂直施力方向運動，此現象即是逆動性。 逆動性的大小也有三個影響的因素： 外界作用力愈大，其逆動性也愈大； 轉子的質量慣性矩（moment of inertia）愈大，逆動性愈小； 轉子的角速度愈大，逆動性愈小。 而逆動方向可根據逆動性原理取決於施力方向及轉子旋轉方向。

三軸

 

文本，應該了然於胸的吧！

且為『三天打魚、兩天晒網』者計︰

由此觀之，持守『科學精神』的『理性』實屬不易，『科技文明 』的『發達』，也很難度杜絕『無根之言』，也許應該說面對『大自然』的『神奇奧妙』，人類其實『所知甚少』。而且一些雖然說是人們『已知之事』，但由於是『抽象』的，在缺乏了『直接經驗』下，總是顯得有些『難明難了』的吧！舉個例子來說，我們都知道『地球自轉』產生了太陽的『東升西落』，也學過牛頓力學所講的『慣性系統』，可是我們並不感覺地球在自轉的啊！一八五一年二月法國物理學家『萊昂‧傅科』Jean Bernard Léon Foucault 首度次在『巴黎天文台』的子午儀室公開展示了一的『單擺』。幾星期之後，傅科他又在『巴黎先賢祠』的拱頂下 ，用一根長六十七公尺的鋼索，其下懸掛了一顆重二十八公斤的鉛錘，然後使之擺動。這個單擺的『擺動平面』它每小時順時針方向旋轉 11° 度，經三十點七小時後環繞一圈。這就是大名鼎鼎的『傅科擺』 Foucault pendulum ，它的旋轉角速度 與『緯度』 成正比，可以表示為 ，此處，『北緯 』角度為『正』，表示『順時針方向旋轉』。據聞一八五五年，這個單擺被移到了國立巴黎工藝技術學院之國立工藝博物館。然後在二零一零年四月六日，國立工藝博物館內懸掛鉛錘的鋼索不知何故斷裂，使得單擺和博物館的大理石地板都受到無法修補的損壞。或許自傅科擺第一次以簡單的實驗證明『地球自轉』以來 ，這個擺已經善盡了『告知大眾』的『義務』的吧！！

 

，『日日學習、學習日日』是好事！！

Classical Mechanics

Lecture Notes

 

Lecture L29 – 3D Rigid Body Dynamics

 

Lecture L30 – 3D Rigid Body Dynamics: Tops and Gyroscopes