1824年，在倫敦發行的《機械雜誌》內的一副刻畫。阿基米德說：「給我一個支點，我就可以撬起整個地球。」

 

手無縛雞之力的人，能舉重乎？

倘其使用『滑輪』，易如反掌呦！

此乃『工具』設計者之『目的』也☆

Block and tackle

A block and tackle is an assembly of a rope and pulleys that is used to lift loads. A number of pulleys are assembled together to form the blocks, one that is fixed and one that moves with the load. The rope is threaded through the pulleys to provide mechanical advantage that amplifies that force applied to the rope.^[4]

In order to determine the mechanical advantage of a block and tackle system consider the simple case of a gun tackle, which has a single mounted, or fixed, pulley and a single movable pulley. The rope is threaded around the fixed block and falls down to the moving block where it is threaded around the pulley and brought back up to be knotted to the fixed block.

The mechanical advantage of a block and tackle equals the number of sections of rope that support the moving block; shown here it is 2, 3, 4, 5, and 6, respectively.

      

Let S be the distance from the axle of the fixed block to the end of the rope, which is A where the input force is applied. Let R be the distance from the axle of the fixed block to the axle of the moving block, which is B where the load is applied.

The total length of the rope L can be written as

where K is the constant length of rope that passes over the pulleys and does not change as the block and tackle moves.

The velocities V_A and V_B of the points A and B are related by the constant length of the rope, that is

or

The negative sign shows that the velocity of the load is opposite to the velocity of the applied force, which means as we pull down on the rope the load moves up.

Let V_A be positive downwards and V_B be positive upwards, so this relationship can be written as the speed ratio

where 2 is the number of rope sections supporting the moving block.

Let F_A be the input force applied at A the end of the rope, and let F_B be the force at B on the moving block. Like the velocities F_A is directed downwards and F_B is directed upwards.

For an ideal block and tackle system there is no friction in the pulleys and no deflection or wear in the rope, which means the power input by the applied force F_AV_A must equal the power out acting on the load F_BV_B, that is

The ratio of the output force to the input force is the mechanical advantage of an ideal gun tackle system,

This analysis generalizes to an ideal block and tackle with a moving block supported by n rope sections,

This shows that the force exerted by an ideal block and tackle is n times the input force, where n is the number of sections of rope that support the moving block.

 

所以阿基米德豪情壯志，想移動地球哩！？

Lever

A lever (/ˈliːvər/ or US: /ˈlɛvər/) is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. A lever is a rigid body capable of rotating on a point on itself. On the basis of the location of fulcrum, load and effort, the lever is divided into three types. It is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provideleverage. The ratio of the output force to the input force is the mechanical advantage of the lever.

Etymology

The word “lever” entered English about 1300 from Old French, in which the word was levier. This sprang from the stem of the verb lever, meaning “to raise“. The verb, in turn, goes back to the Latin levare, itself from the adjective levis, meaning “light” (as in “not heavy“). The word’s primary origin is the Proto-Indo-European (PIE) stem legwh-, meaning “light”, “easy” or “nimble”, among other things. The PIE stem also gave rise to the English word “light”.^[1]

Early use

The earliest remaining writings regarding levers date from the 3rd century BCE and were provided by Archimedes. ‘Give me a place to stand, and I shall move the Earth with it’ is a remark of Archimedes who formally stated the correct mathematical principle of levers (quoted by Pappus of Alexandria).

It is assumed^{[by whom?]} that in ancient Egypt, constructors used the lever to move and uplift obelisks weighing more than 100 tons.

Force and levers

A lever in balance

 

A lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever does not dissipate or store energy, which means there is no friction in the hinge or bending in the beam. In this case, the power into the lever equals the power out, and the ratio of output to input force is given by the ratio of the distances from the fulcrum to the points of application of these forces. This is known as the law of the lever.^{[citation needed]}

The mechanical advantage of a lever can be determined by considering the balance of moments or torque, T, about the fulcrum.

where F₁ is the input force to the lever and F₂ is the output force. The distances a and b are the perpendicular distances between the forces and the fulcrum.

Since the moments of torque must be balanced,  . So,  .

The mechanical advantage of the lever is the ratio of output force to input force,

This relationship shows that the mechanical advantage can be computed from ratio of the distances from the fulcrum to where the input and output forces are applied to the lever, assuming no losses due to friction, flexibility or wear. This remains true even though thehorizontal distance (perpendicular to the pull of gravity) of both a and b change (diminish) as the lever changes to any position away from the horizontal.

 

若以今日『物理概念』思之，實需了解雖說『能量守恆』，那個『能量』的『利用方式』為何不可以不同勒？！

Virtual work and the law of the lever

A lever is modeled as a rigid bar connected to a ground frame by a hinged joint called a fulcrum. The lever is operated by applying an input force F_A at a point A located by the coordinate vector r_A on the bar. The lever then exerts an output force F_B at the point Blocated by r_B. The rotation of the lever about the fulcrum P is defined by the rotation angle θ in radians.

Let the coordinate vector of the point P that defines the fulcrum be r_P, and introduce the lengths

which are the distances from the fulcrum to the input point A and to the output point B, respectively.

Now introduce the unit vectors e_A and e_B from the fulcrum to the point A and B, so

The velocity of the points A and B are obtained as

where e_A^⊥ and e_B^⊥ are unit vectors perpendicular to e_A and e_B, respectively.

The angle θ is the generalized coordinate that defines the configuration of the lever, and the generalized force associated with this coordinate is given by

where F_A and F_B are components of the forces that are perpendicular to the radial segments PA and PB. The principle of virtual work states that at equilibrium the generalized force is zero, that is

Thus, the ratio of the output force F_B to the input force F_A is obtained as

which is the mechanical advantage of the lever.

This equation shows that if the distance a from the fulcrum to the point A where the input force is applied is greater than the distance b from fulcrum to the point B where the output force is applied, then the lever amplifies the input force. If the opposite is true that the distance from the fulcrum to the input point A is less than from the fulcrum to the output point B, then the lever reduces the magnitude of the input force.

 

故而知『凡有所得，亦有所失』夫？？

Gear train

A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage.

Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, providing a smooth transmission of rotation from one gear to the next.^[1]

The transmission of rotation between contacting toothed wheels can be traced back to the Antikythera mechanism of Greece and the south-pointing chariot of China. Illustrations by the Renaissance scientist Georgius Agricola show gear trains with cylindrical teeth. The implementation of the involute tooth yielded a standard gear design that provides a constant speed ratio.

Features of gears and gear trains include:

• The ratio of the pitch circles of mating gears defines the speed ratio and the mechanical advantage of the gear set.
• A planetary gear train provides high gear reduction in a compact package.
• It is possible to design gear teeth for gears that are non-circular, yet still transmit torque smoothly.
• The speed ratios of chain and belt drives are computed in the same way as gear ratios. See bicycle gearing.

An Agricola illustration from 1580 showing a toothed wheel that engages a slotted cylinder to form a gear train that transmits power from a human-powered treadmill to mining pump.

Mechanical advantage

Gear teeth are designed so the number of teeth on a gear is proportional to the radius of its pitch circle, and so the pitch circles of meshing gears roll on each other without slipping. The speed ratio for a pair of meshing gears can be computed from ratio of the radii of the pitch circles and the ratio of the number of teeth on each gear.

The velocity v of the point of contact on the pitch circles is the same on both gears, and is given by

where input gear A with radius r_A and angular velocity ω_A meshes with output gear B with radius r_B and angular velocity ω_B. Therefore,

where N_A is the number of teeth on the input gear and N_B is the number of teeth on the output gear.

The mechanical advantage of a pair of meshing gears for which the input gear has N_A teeth and the output gear has N_B teeth is given by

This shows that if the output gear G_B has more teeth than the input gear G_A, then the gear train amplifies the input torque. And, if the output gear has fewer teeth than the input gear, then the gear train reduces the input torque.

If the output gear of a gear train rotates more slowly than the input gear, then the gear train is called a speed reducer. In this case, because the output gear must have more teeth than the input gear, the speed reducer amplifies the input torque.

Analysis using virtual work

For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear.

The size of the gears and the sequence in which they engage define the ratio of the angular velocity ω_A of the input gear to the angular velocity ω_B of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let Rbe the speed ratio, then

The input torque T_A acting on the input gear G_A is transformed by the gear train into the output torque T_B exerted by the output gear G_B. If we assume the gears are rigid and there are no losses in the engagement of the gear teeth, then the principle of virtual workcan be used to analyze the static equilibrium of the gear train.

Let the angle θ of the input gear be the generalized coordinate of the gear train, then the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear:

The formula for the generalized force obtained from the principle of virtual work with applied torques yields:^[2]

The mechanical advantage of the gear train is the ratio of the output torque T_B to the input torque T_A, and the above equation yields:

The speed ratio of a gear train also defines its mechanical advantage. This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And if the input gear rotates slower than the output gear, the gear train reduces the input torque.

 

彷彿清風掀開面紗，一睹『功率不變』之容顏◎

Efficiency

Mechanical advantage that is computed using the assumption that no power is lost through deflection, friction and wear of a machine is the maximum performance that can be achieved. For this reason, it is often called the ideal mechanical advantage (IMA). In operation, deflection, friction and wear will reduce the mechanical advantage. The amount of this reduction from the ideal to the actual mechanical advantage (AMA) is defined by a factor called efficiency, a quantity which is determined by experimentation.

As an ideal example, using a block and tackle with six ropes and a 600 pound load, the operator would be required to pull the rope six feet and exert 100 pounds of force to lift the load one foot. Both the ratios F_out / F_in and V_in / V_out from below show that the IMA is six. For the first ratio, 100 pounds of force in results in 600 pounds of force out; in the real world, the force out would be less than 600 pounds. The second ratio also yields a MA of 6 in the ideal case but fails in real world calculations; it does not properly account for energy losses. Subtracting those losses from the IMA or using the first ratio yields the AMA. The ratio of AMA to IMA is the mechanical efficiency of the system.

Ideal mechanical advantage

The ideal mechanical advantage (IMA), or theoretical mechanical advantage, is the mechanical advantage of a device with the assumption that its components do not flex, there is no friction, and there is no wear. It is calculated using the physical dimensions of the device and defines the maximum performance the device can achieve.

The assumptions of an ideal machine are equivalent to the requirement that the machine does not store or dissipate energy; the power into the machine thus equals the power out. Therefore, the power P is constant through the machine and force times velocity into the machine equals the force times velocity out–that is,

The ideal mechanical advantage is the ratio of the force out of the machine (load) to the force into the machine (effort), or

Applying the constant power relationship yields a formula for this ideal mechanical advantage in terms of the speed ratio:

The speed ratio of a machine can be calculated from its physical dimensions. The assumption of constant power thus allows use of the speed ratio to determine the maximum value for the mechanical advantage.

Actual mechanical advantage

The actual mechanical advantage (AMA) is the mechanical advantage determined by physical measurement of the input and output forces. Actual mechanical advantage takes into account energy loss due to deflection, friction, and wear.

The AMA of a machine is calculated as the ratio of the measured force output to the measured force input,

where the input and output forces are determined experimentally.

The ratio of the experimentally determined mechanical advantage to the ideal mechanical advantage is the efficiency η of the machine,

 

 

 

 

 

 

 

如果將日常使用的個人電腦看成『計算機』 + 『輸出入界面』 + 『操作者』，那麼『自動機器』 Automata 就像無需『操作者』的獨立運行系統。既然早有『圖零測試』的議論，要是將

計算機 → 大腦

輸出入界面 → 神經網路

操作者 → ？

，『生物』能夠被當作『生化物理自動機器』嗎？？大語言學家 Noam Chomsky 認為語言是人類的一種天賦，而且為人類所獨有。這從『進化論』的角度來講，『人的語言』必然得是『生物言語』的一種『突破』變化！當真人是『有靈魂』的？還是『五蘊皆空』之『人無我』與『法無我』的呢？？要有人說︰他在腦海中，聽到 Tux 在『講話』！那他定然是『瘋』了的吧！！

 

『色聲香味觸法』進了『腦海』怕它都是『編過碼』之『符號』，連笛卡爾都只能以『我思故我在』帶過，那『桶中之腦？？』聽到無有言乎！！

在『惠施』的【泛愛萬物，天地一體也。】之論裡，作者認為︰

假使人們『體會』大自然中『陽光』與『生命』的關係， 也許能夠『感受』太陽的『久』和『大』，生起『愛．惜』萬物之心。如果人類多些『純樸』，少點『貪慾』，將更能夠認識『慎終追遠』，興起『不忍之心』。即使科學家果真找到了所謂『自私的基因』，『天下』依然會有『慈悲人』！『造物者』仍舊是將『抉擇』之『自由』交與了『生命』的吧！！

既然人類能學會『萬邦言語』，那該會有『基本 ○ □ 』的吧！大千世界中的『情意相通』生命現象也許訴說同一件事而已，『創造』以及『賦予』符號『意義』乙事，祇是『自然而然』的哩！！怎就不能有『解讀者』的存在呢？？

那些『死掉的文字』與『滅絕的語言』並非是為著『來者解謎』而存在，若問『謎題』大概將是怎『興替不斷』的啊！只不過不管『歷史』曾經多少『邦國』，隨著『地域』和『時代』講過哪些『語音』的言語，也許還是必須有『解讀者』的吧！！

怎可『 ㄒㄗ 』 = 『 ㄒㄧㄢˊ ㄗㄜˊ 』這樣論斷呢？更不要說那還是『被翻譯』過的『企鵝語』哩！又誰知 Tux 會如何『笑』的了？ ？

─── 《TUX@RPI ︰ 《咸澤碼訊》》

 

曾經有『自勝者強』之時，恐不以『勝人者有力』為尚，如

今果不合時宜乎？？

聞道或有先後，術業也可專攻，此古今之法則也。所以『鬼谷子』一書可以通達遊戲『設計者』之心思，往來縱橫無礙︰

鬼谷子姓王名禪，字詡，道號鬼谷^[1][2]。

鬼谷子被喻為縱橫家之鼻祖的原因是其下有蘇秦與張儀兩個叱吒戰國時代的傑出弟子〔見《戰國策》〕。另有孫臏與龐涓亦為其弟子之說^[3][4]。

《鬼谷子》

捭闔第一

反應第二

───

 

 也可以究觀遊戲『使用者』的想法，出入心想事成。然而天下事卻常有『所思』之不周，『所想』又不備之時，終究如之奈何耶？！

─── 摘自《W!O+ 的《小伶鼬工坊演義》︰【新春】 復古派 《九》神機鬼藏》

 

況且還存在『機心巧詐』之說，此

所以先言機械效益也！！

Mechanical advantage

Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force. The model for this is the law of the lever. Machine components designed to manage forces and movement in this way are called mechanisms.^[1] An ideal mechanism transmits power without adding to or subtracting from it. This means the ideal mechanism does not include a power source, is frictionless, and is constructed from rigid bodies that do not deflect or wear. The performance of a real system relative to this ideal is expressed in terms of efficiency factors that take into account departures from the ideal.

The law of the lever

The lever is a movable bar that pivots on a fulcrum attached to or positioned on or across a fixed point. The lever operates by applying forces at different distances from the fulcrum, or pivot. The location of the fulcrum determines a lever’s class. Where a lever rotates, continuously, it functions as a rotary 2nd-class lever. The motion of the lever’s end-point describes a fixed orbit, where mechanical energy can be exchanged. (see a hand-crank as an example.)

In modern times, this kind of rotary leverage is widely used; see a (rotary) 2nd-class lever; see gears, pulleys or friction drive, used in a mechanical power transmission scheme. It is common for mechanical advantage to be manipulated in a ‘collapsed’ form, via the use of more than one gear (a gearset). In such a gearset, gears having smaller radii and less inherent mechanical advantage are used. In order to make use of non-collapsed mechanical advantage, it is necessary to use a ‘true length’ rotary lever. See, also, the incorporation of mechanical advantage into the design of certain types of electric motors; one design is an ‘outrunner’.

As the lever pivots on the fulcrum, points farther from this pivot move faster than points closer to the pivot. The power into and out of the lever is the same, so must come out the same when calculations are being done. Power is the product of force and velocity, so forces applied to points farther from the pivot must be less than when applied to points closer in.^[1]

If a and b are distances from the fulcrum to points A and B and if force F_A applied to A is the input force and F_B exerted at B is the output, the ratio of the velocities of points A and B is given by a/b, so the ratio of the output force to the input force, or mechanical advantage, is given by

This is the law of the lever, which was proven by Archimedes using geometric reasoning.^[2] It shows that if the distance a from the fulcrum to where the input force is applied (point A) is greater than the distance bfrom fulcrum to where the output force is applied (point B), then the lever amplifies the input force. If the distance from the fulcrum to the input force is less than from the fulcrum to the output force, then the lever reduces the input force. Recognizing the profound implications and practicalities of the law of the lever, Archimedes has been famously attributed the quotation “Give me a place to stand and with a lever I will move the whole world.”^[3]

The use of velocity in the static analysis of a lever is an application of the principle of virtual work.

Speed ratio

 

The requirement for power input to an ideal mechanism to equal power output provides a simple way to compute mechanical advantage from the input-output speed ratio of the system.

The power input to a gear train with a torque T_A applied to the drive pulley which rotates at an angular velocity of ω_A is P=T_Aω_A.

Because the power flow is constant, the torque T_B and angular velocity ω_B of the output gear must satisfy the relation

which yields

This shows that for an ideal mechanism the input-output speed ratio equals the mechanical advantage of the system. This applies to all mechanical systems ranging from robots to linkages.

Gear trains

 

Gear teeth are designed so that the number of teeth on a gear is proportional to the radius of its pitch circle, and so that the pitch circles of meshing gears roll on each other without slipping. The speed ratio for a pair of meshing gears can be computed from ratio of the radii of the pitch circles and the ratio of the number of teeth on each gear, its gear ratio.

Two meshing gears transmit rotational motion.

The velocity v of the point of contact on the pitch circles is the same on both gears, and is given by

where input gear A has radius r_A and meshes with output gear B of radius r_B, therefore,

where N_A is the number of teeth on the input gear and N_B is the number of teeth on the output gear.

The mechanical advantage of a pair of meshing gears for which the input gear has N_A teeth and the output gear has N_B teeth is given by

This shows that if the output gear G_B has more teeth than the input gear G_A, then the gear train amplifies the input torque. And, if the output gear has fewer teeth than the input gear, then the gear train reduces the input torque.

If the output gear of a gear train rotates more slowly than the input gear, then the gear train is called a speed reducer (Force multiplier). In this case, because the output gear must have more teeth than the input gear, the speed reducer will amplify the input torque.

 

其實是『時』來『時』去，『應時』而已矣◎

 

 

 

 

 

 

 

 

夏日晨起天光好，微風吹花氣味佳，忽而 眼 皮直跳，不知是何兆？欲法 梅 花心易，急讀《   》觀水之法︰

……

，  眼為離 ䷝ 目，左通心，其跳吉。跳者 眉居離眼上，或是喜上眉梢之兆。剛過夏至日，兆體取為乾 ䷀ ，目離變二爻，眉梢動上爻，樹欲靜，風不止，定前後。蓋指夬 ䷪ 之革 ䷰ ，當是『己日乃革』，『君子豹變』之象。

因是速往學堂。剛上迴廊，就見課堂外擺著桌子，三五同學看到我來，忙向前導引桌前，在那『未曾有』的『簽到簿』上 畫 卯，還得『工筆』寫句『勵志』語，只覺興筆寫下

 

，卻瞧見同學搔首笑說︰這可是句 『畫』。趕緊添補上一筆

學而時習之不亦悅乎。

一進教室大吃一驚，桌子圍成了『圓』，拱著講桌而排，中央拼作『方』，上有派生碼訊，正播放著學習日子裡的點點滴滴，甚至還配著音樂呢！？一時如在夢中，以為今兒已經是『禮拜一』了，將要開『同樂會』？？正惶惑間，……學長走進了教室

派︰同學們，大家好。【彷彿一點都不驚訝】

今天的課堂佈置，嗯！天圓而地方，這可不是那『孔方』，擺明了要『化緣』，也罷！！請這兩位同學，出外跑一遭，將『下』星期一才辦的 同樂會，先行個籌備會，排演一番。在此等

『吃』的之際，正好回顧彼此相聚的短暫時光。…

沒多久，大家的桌上，擺著各樣點心、糖果、飲料，零零種種的有一整桌。祇聽學長講︰今兒，出『錢』的與出『力』的都不講課，大家都來『聽』課，聽 M♪o 講這學期的『最後一堂課』。………【熱烈鼓掌，久久不歇】

原來是學長帶頭『作怪』，看來不上『講台』，這掌聲是不會歇的了。

─── 《M♪O 之學習筆記本《巳》文章︰【䷪】有戎勿恤》

 

既已知道一顆『小馬達』的『電路模型』，正好乘著

Modular Circuits

Andras Tantos

Motor Modeling

………

 

文字

A Practical Example: the R/C Car

But just how useful this model is? To be honest not terribly, as you rarely use a motor without anything connected to it. Modeling the mechanical properties of just the motor is not that practical. Luckily, the same model can be used for a wide range of applications, for example for moving platforms.

In this example, I’ll dust of my old Tumbleweed robot. It is based on a Stampede R/C car. A first-level mechanical model is fairly simple for this kind of platform: the motor through a number of gears, drives a wheel, which than moves the whole body.

From the motors perspective, it looks as if it moves the whole weigh of the car as if it was spread around the circumference of the wheel. The gears can be simplified into a single gear-box with a ratio on ‘1:N’, and we can further assume that the inertia of the gears are negligible compared to the inertia of the other components. With this, we get to the following model:

 

Here, the motor is represented by our usual model of a lossy rotating disc (parameters with the ‘m’ suffix), and our load is represented by another lossy rotating disc of the same kind (parameters with the ‘w’ suffix).

The new component is the gear-box, so let’s talk about it first! Gears work as torque-speed-converters. They multiply the torque by ‘N’ and reduce the speed by a factor of ‘N’:

T_w = T_m*N
s_w = s_m/N

We have our previous equations for the torques ‘consumed’ by a disc and the draw from before:

T_dw= f_w and T_Jw= J_w * (ds_w/dt)

Putting s_m into these equations we get:

T_dw= f_w and T_Jw= J_w/N * (ds_m/dt)

Now, from the motors’ perspective, it only sees 1/N-th of this torque through the gear-box (remember, gear-boxes convert torque). So the torque of these components expressed on the motor-end of the gear-box are:

T_dm= 1/N *f_w and T_Jm= 1/N² * J_w * (ds_m/dt)

It looks like our components kept their properties, but their effect shrunk. So, now that we’ve eliminated the gear-box, we ended up with the following model:

 

This can be further simplified, by combining the two discs and the two frictions into one of each:

 

And now, we’re back to the previous model: a single disc and a single friction. We just have more complex equations for calculating the values. Of course this also means that the equivalent circuit will be identical as well, except that the values of the current-source and the capacitance are now a little more complex:

I_f = (f_m+f_w/N)/K_r

and

C = (J_m+J_w/N²)/(K_E*K_T)

A further simplification can be made as well: we can say that the inertia of the motor is negligible compared to the inertia of the whole body (even after the gear-conversion). That results in a further simplified capacitor model:

C = (J_w/N²)/(K_E*K_T)

 

羽翼，回到《運動學》耶？！

重新認識『小齒輪』之機械優點也！？

Gear

Two meshing gears transmitting rotational motion. Note that the smaller gear is rotating faster. Since the larger gear is rotating less quickly, its torque is proportionally greater. One subtlety of this particular arrangement is that the linear speed at the pitch diameter is the same on both gears.

 

A gear or cogwheel is a rotating machine part having cut like teeth, or cogs, which mesh with another toothed part to transmit torque. Geared devices can change the speed, torque, and direction of a power source. Gears almost always produce a change in torque, creating a mechanical advantage, through their gear ratio, and thus may be considered a simple machine. The teeth on the two meshing gears all have the same shape.^[1] Two or more meshing gears, working in a sequence, are called a gear train or a transmission. A gear can mesh with a linear toothed part, called a rack, producing translation instead of rotation.

The gears in a transmission are analogous to the wheels in a crossed, belt pulley system. An advantage of gears is that the teeth of a gear prevent slippage.

When two gears mesh, if one gear is bigger than the other, a mechanical advantage is produced, with the rotational speeds, and the torques, of the two gears differing in proportion to their diameters.

In transmissions with multiple gear ratios—such as bicycles, motorcycles, and cars—the term “gear” as in “first gear” refers to a gear ratio rather than an actual physical gear. The term describes similar devices, even when the gear ratio is continuous rather than discrete, or when the device does not actually contain gears, as in a continuously variable transmission.^[2]

 

 

 

 

 

 

 

 

 

此『事』── 山頭主義 ── 易經有『之』，但看『該不該』『隨』的吧？？

《易經》第十七卦‧澤雷隨

隨：元亨利貞，無咎。

彖曰：隨，剛來而下柔，動而說，隨。大亨貞，無咎，而天下隨時，隨之時義大矣哉！

象曰：澤中有雷，隨﹔君子以嚮晦入宴息。

初九：官有渝，貞吉。 出門交有功。
象曰：官有渝，從正吉也。 出門交有功，不失也。

六二：系小子，失丈夫。
象曰：系小子，弗兼與也。

六三：系丈夫，失小子。 隨有求得，利居貞。
象曰：系丈夫，志舍下也。

九四：隨有獲，貞凶。有孚在道，以明，何咎。
象曰：隨有獲，其義凶也。 有孚在道，明功也。

九五：孚于嘉，吉。
象曰：孚于嘉，吉﹔位正中也。

上六：拘系之，乃從維之。 王用亨于西山。
象曰：拘系之，上窮也。

 

，或許已落於『無可奈何』，方不得不說『凶』之情事罷了！！

『東方』曾如是說，『西方』後有研究︰

 

引出了『令人震驚』之『整合分析』 meta-analysis 『結論』︰

Thomas Blass ──《電醒全世界的人》的作者 ── of the University of Maryland, Baltimore County performed a meta-analysis on the results of repeated performances of the experiment. He found that the percentage of participants who are prepared to inflict fatal voltages remains remarkably constant, 61–66 percent, regardless of time or country.

The participants who refused to administer the final shocks neither insisted that the experiment itself be terminated, nor left the room to check the health of the victim without requesting permission to leave, as per Milgram’s notes and recollections, when fellow psychologist Philip Zimbardo asked him about that point.

 

假使再添上『阿希從眾實驗』的『從眾效應』所說

實驗結果︰受試者中有百分之三十七之回答是依據了『大多數』的『錯誤回答』，大概有四分之三的人至少有過一次『從眾行為』，只有大約四分之一的人維持了『獨立自主』性。

『獨立自主』之不易正如《易經‧乾卦》所講︰

初九曰：潛龍勿用。何謂也？
子曰： 龍德而隱者也。不易乎世，不成乎名﹔遯世而無悶，不見是而無悶﹔樂則行之，憂則違之﹔確乎其不可拔，潛龍也。

『尊重事實』，有著『實驗查證』的『科學精神』，從古今歷史來看，實在並不容易！『待人處事』能夠『進取』『有所不為』，不落『鄉愿』『德之戝也』窠臼，誠屬難能可貴！！

─── 《《隨》□ 起舞？！》

 

贈東林總長老‧蘇軾

溪聲便是廣長舌，山色豈非清凈身。
夜來八萬四千偈，他日如何舉似人。

 

提筆欲寫『電感』之『串‧並』現象︰

串聯與並聯電路

串聯電路

如上圖所示，  個電感器串聯在一起。現將電源連接於這串聯電路的兩端。按照電感的定義，第  個電感器兩端的電壓  等於其電感  乘以通過的電流的變率  ：

 。

按照克希荷夫電流定律，從電源（直流電或交流電）給出的電流  等於通過每一個電感器的電流  。所以，

 。

根據克希荷夫電壓定律，電源兩端的電壓等於所有電感器兩端的電壓的代數和：

 。

所以， 個電感器串聯的等效電感  為

 。

由於電感器產生的磁場會與其鄰近電感器的纏繞線圈發生耦合，很難避免緊鄰的電感器彼此互相影響。物理量互感  能夠給出對於這影響的衡量。

例如，由電感分別為  、  ，互感為  的兩個電感器構成的串聯電路，其等效互感  有兩種可能：

• 假設兩個電感器分別產生的磁場或磁通量，其方向相同，則稱為「串聯互助」，以方程式表示，

 。

• 假設兩個電感器分別產生的磁場或磁通量，其方向相反，則稱為「串聯互消」，以方程式表示，

 。

對於具有三個或三個以上電感器的串聯電路，必需考慮到每個電感器自己本身的自感和電感器與電感器之間的互感，這會使得計算更加複雜。等效電感是所有自感與互感的代數和。

例如，由三個電感器構成的串聯電路，會涉及三個自感和六個互感 。三個電感器的自感分別為  、  、  ；互感分別為  、  、  、  、  、  。等效電感為

 。

由於任意兩個電感器彼此之間的互感相等，  =   ，後面兩組互感可以合併：

 。

導引

串聯互助電路圖。

如上圖所示，兩個電感器串聯互助在一起。將電源連接於這串聯電路的兩端。應用克希荷夫電壓定律，按照點規定，可以得到

 ；

其中，  是電源兩端的電壓，  是電流。

電壓  和電流  之間的關係為

 。

所以，兩個電感器串聯互助的有效電感為

 。

類似地，可以得到兩個電感器串聯互消的有效電感。

並聯電路

如上圖所示，  個電感器並聯在一起，類似前面所述方法，可以計算出其等效電感  為

 。

由於電感器產生的磁場會與其鄰近電感器的纏繞線圈發生耦合，很難避免緊鄰的電感器彼此互相影響。物理量互感  能夠給出對於這影響的衡量。上述方程式描述  個電感器無互感並聯的理想案例。

由電感分別為  、  ，互感為  的兩個電感器構成的並聯電路，其等效互感  為^[6]：

• 假設兩個電感器分別產生的磁場或磁通量，其方向相同，則稱為「並聯互助」，以方程式表示，

 。

• 假設兩個電感器分別產生的磁場或磁通量，其方向相反，則稱為「並聯互消」，以方程式表示，

 。

對於具有三個或三個以上電感器的並聯電路，必需考慮到每個電感器自己本身的自感和電感器與電感器之間的互感，這會使得計算更加複雜。

導引

並聯互消電路圖。

如上圖所示，兩個電感器並聯互助在一起。將電源連接於這並聯電路的兩端。應用克希荷夫電壓定律，按照點規定，可以得到

 、

 ；

其中，  是電源兩端的電壓，  和  分別是通過兩個支路的電流。

所以，電流  和  之間的關係為

 。

應用克希荷夫電流定律，總電流  為

 。

電流  和  之間的關係為

 。

電壓  和電流  之間的關係為

 。

所以，兩個電感器並聯互助的有效電感為

 。

類似地，可以得到兩個電感器並聯互消的有效電感。

 

之際，忽耳思及『合縱連橫』︰

名字源起

戰國時代，秦國位於西方，六國位於其東

連橫一計，出自諸子百家中的縱橫家。《s:韓非子/五蠹》：「從者 ，合眾弱以攻一強也；而衡者，事一強以攻眾弱也。」

「縱」與「橫」的來歷，據說是因「南北向」稱為「縱」，「東西向」稱為「橫」。秦國位於西方，六國位於其東。六國結盟抗秦為南北向聯合，故稱「合縱」；六國分別與秦國結盟為東西向聯合，故稱「連橫」。

 

或因『自感』與『互感』複雜計算而起？

怎知竟陷入回憶裡！

哈佛大學 Michael Sandel 教授《正義：一場思辨之旅》開放式課程名聞遐邇座無虛席，一十二講談論『道德決定』的『困境』。

如果我們連『美德』是什麼都不知道︰

美諾篇

《美諾篇（Meno）》，是柏拉圖記載的蘇格拉底對話錄，以蘇格拉底對話體寫成。其試圖確定德行（virtue）的定義。是德行的本質定義，而非某些特定的美德（如正義與節制等）。目標在於一個普適的定義，適用於一切特定的德行。

簡介

開篇

• 美諾向蘇格拉底請教：「德行是什麼？」
• 蘇格拉底一如既往地回答：「不知道。」
• 美諾便列舉了很多類型的例子：男人德、女人德、奴隸德、兒童德等。蘇格拉底並不接受這些解說。他想知道，到底是什麼特性（quality）使得這些行為被稱做德行。

 

將要如何免於『悖論』的呢？

Meno’s paradox

Meno asks Socrates: “And how will you inquire into a thing when you are wholly ignorant of what it is? Even if you happen to bump right into it, how will you know it is the thing you didn’t know?“^[9] Socrates rephrases the question, which has come to be the canonical statement of the paradox: “[A] man cannot search either for what he knows or for what he does not know[.] He cannot search for what he knows–since he knows it, there is no need to search–nor for what he does not know, for he does not know what to look for.“^[10]

 

真相真的是『靈魂』之『遺忘』與『回憶』耶？？

Dialogue with Meno’s slave

 

Socrates responds to this sophistical paradox with a mythos (poetic story) according to which souls are immortal and have learned everything prior to transmigrating into the human body. Since the soul has had contact with real things prior to birth, we have only to ‘recollect’ them when alive. Such recollection requires Socratic questioning, which according to Socrates is not teaching. Socrates demonstrates his method of questioning and recollection by interrogating a slave who is ignorant of geometry.

Socrates begins one of the most influential dialogues of Western philosophy regarding the argument for inborn knowledge. By drawing geometric figures in the ground Socrates demonstrates that the slave is initially unaware of the length that a side must be in order to double the area of a square with two-foot sides. The slave guesses first that the original side must be doubled in length (four feet), and when this proves too much, that it must be three feet. This is still too much, and the slave is at a loss.

Socrates claims that before he got hold of him the slave (who has been picked at random from Meno’s entourage) might have thought he could speak “well and fluently” on the subject of a square double the size of a given square.^[11] Socrates comments that this “numbing” he caused in the slave has done him no harm and has even benefited him.^[12]

Socrates then draws a second square figure using the diagonal of the original square. Each diagonal cuts each two foot square in half, yielding an area of two square feet. The square composed of four of the eight interior triangular areas is eight square feet, double that of the original area. He gets the slave to agree that this is twice the size of the original square and says that he has “spontaneously recovered” knowledge he knew from a past life^[13] without having been taught. Socrates is satisfied that new beliefs were “newly aroused” in the slave.

After witnessing the example with the slave boy, Meno tells Socrates that he thinks that Socrates is correct in his theory of recollection, to which Socrates replies, “I think I am. I shouldn’t like to take my oath on the whole story, but one thing I am ready to fight for as long as I can, in word and act—that is, that we shall be better, braver, and more active men if we believe it right to look for what we don’t know…”^[14] It has been argued variously that this implies Socrates is skeptical regarding knowledge or that he is a pragmatist.^{[citation needed]} It also prepares us for the subsequent discussion of knowledge by hypothesis.

This demonstration shows the slave capable of learning a geometrical truth, because “he already has the knowledge in his soul.”^{[citation needed]} In this way, Socrates shows Meno that learning is possible through recollection, and that the learner’s paradox is false. Meno’s paradox claims that learning is impossible, but the examination of the slave shows that it is possible.

─── 摘自《萬象在說話︰美德可以教導嗎》

 

 

 

 

 

 

 

 

《鞦韆曲》清‧鮑之蕙

芳園四壁花光聞，鞦韆動處朝霞飛。
美人妝成對花立，欲上不上嬌無力。
㩳身一舉穿林梢，流鶯驚起花旛搖。
翩然反側妙容與，隱隱紅潮上眉𡧃。
藕絲裙輭罥游蜂，杏子衫輕濕香雨。
拖煙約霧東風顛，珠翠彷彿雲中斬。
琤瑽仙珮潄嗚玉，蘭香萼綠相齊肩。
紅纏雪腕綵索勁，綠鬆雲髮金釵偏。
小鬢扶下日初轉，徙倚花陰息嬌喘。
栩栩魂猶夢蝶驚，行行足訝蒼苔輭。
美人會得春難駐，不放芳華等閒度。
來日清明風雨多，落紅滿地奈愁何。

 

盪鞦韆有道乎？

秋千搖蕩千秋已，
春暖花開打韆鞦。
不管己身有無力，
想方設法出枝頭。

借力使力之術而已耶？？

圓周運動的思路，帶給我們另一種考察『受驅振子』系統行為的觀點。在此再次引用《【Sonic π】聲波之傳播原理︰振動篇》一文中的方程式

是驅動力的振幅大小。在線性微分方程式如 的『求解』裡，如過『』是 的一個解，『』是 一個『特解』，那麼『』就是該方程是的『通解』。我們已經知道 的『低阻尼振子』之解在若干個弛豫時間後數值將變得太小了，所以它對於系統長時間之後的『行為』沒有太多的貢獻。因此我們說這個系統的『穩態解』steady-state solution 是

，此處

是『響應阻抗』函數。而 是驅動力引發的相位角，可由

所決定，一般它表達著相位『遲滯』 lag 現象。

─── 《STEM 隨筆︰古典力學︰動力學【五‧一】》

 

那麼

這個系統的『穩態解』steady-state solution 是

，此處

是『響應阻抗』函數。而 是驅動力引發的相位角，可由

所決定，一般它表達著相位『遲滯』 lag 現象。

 

到底怎麼來的呢？

尚請思考哩！

 

切莫忘卻基本功呦！！

Using the Impedance Method