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  • Babe Ruth, a legend of Major League Baseball

    貝比.魯斯 美國職棒的傳奇人物

  • He led the league in home runs during a season 12 times

    曾經奪得12次全壘打王的頭銜

  • and made a record of 60 home runs in 1927 season

    並且在1927年創下單季60支全壘打的紀錄

  • as well as 714 home runs of his life time record

    在其職業生涯總共累積了714支全壘打

  • In the dead-ball era

    在那個被稱為「死球時代」的時期

  • Ruth's home run and slugging percentage dominated the league

    貝比.魯斯的長打能力遠遠的超越同時代的打擊者

  • He creates an unchallengeable position in the heart of baseball fans

    於是 他在球迷的心中建立了不可挑戰的神格地位

  • and was respectfully named "Baseball Demigod"

    被尊稱為「棒球之神」

  • Roger Maris, an outfielder of New York Yankee

    羅傑.馬里斯 洋基隊的外野手

  • who challenged Ruth's record after 34 years-

    在30多年後挑戰貝比.魯斯的紀錄

  • - 60 home runs in a season

    單季60支全壘打

  • With his home run number getting close to 60

    然而 眼看著馬里斯的全壘打數字步步逼近

  • however, there was no joy for fans

    球迷的心中並沒有喜悅

  • because there can be only one hero in their heart

    因為英雄只能有一個

  • Roger, up here

    嗨 羅傑 看上面

  • Hey it's the voice from above

    這是來自天堂的聲音 哈哈哈...

  • Hey Maris up here

    嘿 馬里斯 看上面

  • it's the Babe

    貝比.魯斯在此

  • Hey you wnat my record? you want my record?

    你想破我的紀錄嗎? 想破嗎?

  • Catch this you ape piece of shit

    垃圾來了給我接好 猿猴

  • Stop

    暫停

  • In the game 154 in 1961

    1961年 洋基第154場比賽

  • Maris hit number 59 in the 3rd inning

    馬里斯在第三局擊出了第59號全壘打

  • Two outs, in the top of 9th

    九局上半 兩出局

  • and it's the last chance for Maris

    馬里斯最後一次打擊機會

  • Orioles was behind but made an unusual substitution

    為了防止他追平紀錄

  • The closer Hoyt Wilhelm was called on to stop him tying Ruth's record

    金鶯隊在落後的情況下做出不尋常的調度

  • I hate that knuckleball

    王牌投手 霍依.威漢登板救援

  • In this weather it's gonna be dancin' all over the place

    我痛恨蝴蝶球

  • Even if he gets wood on it it's not goin' anywhere

    今天的風會讓蝴蝶球亂飄

  • Bushlinger

    就算能碰到球 球也飛不到哪去

  • Wilhelm takes the sign

    小人

  • but we all know what's coming

    威漢收到暗號

  • Does Roger Maris have one last home rum in him

    但大家都知道會配什麼球

  • he fouled it back

    羅傑能從他手中擊出最後一支全壘打嗎

  • Roger had a good cut on that ball, but it's dancin' all over the place

    打成後方界外

  • Wilhelm's got it

    那一球的揮擊角度很好 問題是球四處亂竄

  • Out

    威漢接到球

  • The knuckleball dominated the hitter and Maris failed to tie the record on that day

    出局

  • Knuckleball is a very special pitch in baseball games

    受到蝴蝶球的壓制 馬里斯只能鎩羽而歸

  • It may suddenly change in direction during its flight

    在棒球的世界中 蝴蝶球是一種極為特殊的球路

  • and dance all over the place with erratic movement

    它能夠忽左忽右 或者突然的下墜

  • so that it can fool the batter

    以幾乎不太規則的方式運動

  • In the following we will investigate knuckleball

    也就是這樣能夠愚弄打擊者

  • base on aerodynamics

    接下來 我們將從空氣動力學的角度出發

  • Furthermore, we will then introduce various pitches of baseball

    了解蝴蝶球的原理

  • When it comes to aerodynamics, most people are familiar with Bernoulli's principle

    並且進一步的 介紹並分析其他各種變化球球路

  • But, to establish enough background knowledge

    講到流體力學 一般人最熟知的就是柏努力定律

  • we need to learn more about the fluid theories

    但是在柏努力定律之外

  • besides the Bernoulli's principle

    還有其他重要的流體理論需要了解與認識

  • First let's talk about boundary layer theory

    以便具備足夠的基礎知識

  • When a fluid flows over a obstacle, they would interact with each other

    我們先來談邊界層理論

  • Let's look into a simplest case

    當流體流過障礙物時 會與障礙物發生交互作用

  • When a fluid passing through a flat plate with velocity V

    我們先來研究最簡單的情況

  • it causes friction on the surface

    當流體以速度V流過一個平板

  • In general, fluids are viscous

    流體會跟物體表面發生摩擦

  • so the flow speeds decrease gradually when it get close to the surface

    因為流體一般都是有黏滯性的

  • and end up at zero

    所以越往平板表面靠近 流速會越來越慢

  • The region where flow speeds gradually decrease

    直到趨近於零

  • is called the "Boundary layer"

    這個流速逐步遞減的範圍

  • Viscosity dominate physical phenomenon in this thin layer

    稱之為「邊界層」

  • and the Bernoulli's principle does not apply in this region

    在這個薄層內 黏滯力主導物理現象

  • However, viscosity has no effect on the region outside the boundary layer

    有黏滯力的區域就不適用柏努力定律

  • where the fluid behaves like a inviscid ideal fluid

    然而 在邊界層外面

  • such that the Bernoulli's principle applies

    黏滯力沒有發揮什麼作用

  • We will back to Bernoulli's principle soon

    流體的行為接近於無黏滯性的理想流體

  • Now pay attention to the phenomenon inside the boundary layer

    所以可以使用柏努力定律加以描述

  • If the flow velocity V is not large

    我們稍後會再回來談柏努力定律

  • The airflow inside the boundary layer is stable

    現在先讓我們來關注邊界層內的現象

  • and the streamlines look like a Mille Crepe

    當流速V不大的情況下

  • This is named a "Laminar boundary layer"

    邊界層內的氣流是穩定的

  • The geometric shape of obstacles have effects on boundary layer

    流體的流線就像千層派一樣

  • For example, when substitute the plate by a sphere

    這樣的邊界層稱之為「層流邊界層」

  • a new physical phenomenon may occur

    障礙物的幾何形狀會影響邊界層

  • Streamlines are drew around the sphere

    例如說從平板變成球體的時候

  • and the green part represents a boundary layer

    會有新的物理現象發生

  • which is a important region

    我們把球體周圍的流線畫出來

  • Let's study the region outside boundary layer first

    綠色的部位代表邊界層

  • where the Bernoulli's principle applies

    它是重要的區域

  • Observe the motion of a small volume element

    我們先來探討邊界層外面的區域

  • Since the speed of a baseball is much lower than sound speed

    那是可以運用柏努力定律的地方

  • the compression effect can be ignore

    觀察一小塊體積單元的運動

  • and the volume of this small element keeps unchanged

    因為棒球運動遠低於音速

  • Imagine it moves in a tube that continuously changes in diameter

    不必考慮壓縮效應

  • When it moves toward A

    這一塊小體積在運動過程中保持體積不變

  • its speed decreases

    而且可以把它想像成是在口徑不斷變化的管子中移動

  • A is called the "stagnation point"

    當它靠近A點時

  • around which the fluid separates to both sides

    速度逐漸變慢

  • The speed increases from A to B

    A點又叫做「停滯點」

  • and reach the highest value at B

    流體會從停滯點開始往兩側分離

  • After passing through B the speed decreases

    當它離開A點後速度逐漸增加

  • and reach the minimum speed when it arrive at C

    到達B點時速度最高

  • Bernoulli's principle states that

    然後通過B點之後又逐漸減速

  • high speed corresponds to low pressure

    到達C點時流速又降到最低

  • and low speed corresponds to high pressure

    柏努力定律是說

  • Therefore, the pressure maximums occurs at A and C

    高流速的地方壓力小

  • and B has the minimum pressure

    低流速的地方壓力大

  • Interesting thing occurs in the downstream region

    所以

  • The pressure gradually increases from B to C

    A跟C點的壓力最大

  • Outside the boundary layer, when the external pressure variation is large enough

    B點壓力最小

  • it can force the downstream boundary layer to move in opposite direction

    有趣的地方是發生在下游的區域

  • Once the inverse flow occurs

    從B到C壓力是逐漸變大的

  • upstream boundary layer can be pushed away from the surface

    邊界層在這樣的外部壓力變化下

  • This phenomenon is called "Boundary layer separation"

    如果這個壓力差足夠大的話

  • The location of "Separation point" is near B

    會發生邊界層往上游逆流的現象

  • and the flow is very unstable behind the separation point

    一旦發生逆流

  • Round and round

    上游的邊界層就會被推擠離開球體表面

  • When various scales of vortices emerge, it is called "Turbulence"

    這種現象叫做「邊界層分離」

  • The tail-like turbulent region behind a obstacle is also named a "Wake"

    「分離點」大約會發生在B點附近

  • Since the separation points occur close to B and D

    在分離點之後的流體非常的不穩定

  • the wake width is comparable to the diameter of the sphere

    會不停打轉

  • As a whole, wake is a low pressure region

    然後生出各種大大小小的渦流

  • and this is the main reason for air resistance

    這種情況稱之為「亂流」

  • However, the location of separation points

    因為它是出現在障礙物的後方

  • are sensitive to the geometric shape of obstacles

    所以這整個亂流區域還有個特別的名子

  • For a streamlined body

    叫做「尾流」

  • the separation occur near its tail

    因為分離點就發生在B跟D點附近

  • The smaller the wake width

    所以尾流的寬度大致就和球體的寬度差不多

  • the smaller the drag

    整體來說

  • This is the reason why airfoils and solar cars

    尾流是一個低壓區域

  • are designed to be streamlined form

    這是造成空氣阻力的主要原因

  • Besides the geometric shape of obstacles

    然而分離點發生的位置

  • the surface roughness also determines the location of separation points

    很容易因為障礙物的外形而變化

  • Take a look at this interesting experiment

    像是流線形的物體

  • The left photo represents a smooth sphere in wind tunnel

    分離點會發生在尾端附近

  • and the separation points are close to the largest cross section of the sphere

    尾流變小了

  • The right photo shows that with a thin trip wire in the upstream

    阻力也就大為縮減

  • the separation points move backward and the wake shrinks

    這就是為什麼機翼

  • What's going on here?

    以及太陽能車要設計成流線型的原因

  • Why a small raised surface leads to this result?

    除了障礙物的外形

  • The reason is that the state of boundary layer has been changed

    另一個決定分離點的因素是障礙物表面的粗糙程度

  • It is originally a laminar flow in the upstream boundary layer

    先來看一組有趣的實驗

  • however, since the raised surface creats small scale vortices

    左邊的圖是光滑球體的實驗

  • the boundary layer becomes turbulent

    分離點大約發生在球體最大截面積的地方

  • This is called a "Turbulent boundary layer"

    右圖是在上游的地方加上一圈細線

  • Fluid flows faster outside the boundary layer because no viscous effect there

    結果分離點向後退

  • A turbulent boundary layer is stirred by vortices

    尾流變小了

  • which makes it mixed with the outer and faster fluid

    這是怎麼回事呢?

  • Therefore, the average velocities of boundary layer increases

    為什麼球面上的微小凸起會產生這樣的變化?

  • and get more momentum toward downstream

    原因就在於邊界層的狀態改變了

  • So that the separation points retreat and the wake shrinks

    原本應該是穩定的「層流邊界層」

  • The purpose of dimple design on a golf ball

    因為球面上的凸起部分製造出小尺度的渦流

  • is to create a turbulent boundary layer and reduce the drag

    使得邊界層演變成亂流的狀態

  • Baseballs are not smooth spheres, too

    這稱之為「亂流邊界層」

  • The seam lines on the surface of a balseball can disturb boundary layer

    我們知道邊界層外面的流體不受黏滯力的影響

  • In general, baseball seams can cause the retreat of separation points

    所以有較大的流速

  • Baseball seams are not uniformly distributed on the surface

    亂流邊界層因為受到渦流的攪動

  • and this could result in the wake deflection

    使得它與周圍較快的流體發生混合

  • Moreover, the rotations of a baseball are classified into two types

    這樣一來

  • One is four-seamer: There are 4 seams passing by in one full rotation

    邊界層的平均速度變快了

  • The other one is two-seamer: There are 2 seams passing by in one full rotation

    有更大的動量往下游衝擊過去

  • The changes in attack angle results in non-symmetric force

    如此便造成分離點向後退

  • which can be measured by wind tunnel experiments

    尾流也跟著縮小了

  • This figure shows the force versus attack angle

    高爾夫球在表面上設計了很多的小凹洞

  • for a four-seamer at zero rotation speed

    目的也是為了製造出亂流邊界層

  • As you can see, the force has four periods of variation in one full rotation

    以達到減少阻力的目的

  • The detail explanations for that figure are given in the following computer graphics

    然而

  • The air flows in steadily from the left

    棒球也不是光滑球體

  • In the beginning the attack angle is zero

    表面有約一公釐高的縫線會干擾邊界層

  • and the seams are symmetrically distributed

    通常的情況

  • therefore, the wake is right behind the ball without being deflected

    棒球的縫線也會導致分離點的位置延後

  • But the wake deflects dramatically when the attack angle changes

    縫線在棒球表面的分布是不均勻的

  • The largest upward deflection occurs when the attack angle is about 22 degree

    這會使得尾流發生偏折的現象

  • What is the reason for wake deflection ?

    然而

  • The air in upstream boundary layer becomes turbulent after passing by the seams

    棒球的旋轉又可以分為兩種方式

  • Notice the seam in the lower half part is more close to the stagnation point

    一種是旋轉一圈的時候

  • and the lower half turbulent boundary leads the upper one

    有四條縫線劃過

  • With longer path and more chance for mixing

    稱之為「四縫線球」

  • the boundary layer gets more velocity and momentum moving downward

    另一種是旋轉一圈

  • so the lower half part separation point retreats more

    有兩條縫線劃過

  • On the contrary

    稱之為「二縫線球」

  • the upper half part separation point is in advance and the average velocity is lower

    攻角的變化會使棒球受到不對稱的作用力

  • After mixing of these two sides of boundary layer flow in the wake

    這可以藉由風洞實驗加以測量

  • the average velocity deviates upward

    這張圖是代表四縫線球

  • and this is the explanation for wake deflection

    在轉速等於零時

  • To realize how the force acting on the ball

    各種角度所產生的作用力

  • one can regard the complex interaction processes as a black box

    可以看到棒球自轉一周

  • The horizontal air flow deflects upward after passing through the black box

    作用力重覆變化四個週期

  • implies that a upward force acting on the air flow

    我們現在運用電腦動畫

  • Simultaneously, there must be a reaction force in the black box exert on the ball

    更詳細的來說明這個實驗結果

  • which is equal in magnitude and opposite in direction

    空氣固定的由左邊流進來

  • as specified by Newton's third law

    一開始角度等於零的時候

  • This is the result of non-symmetric seam distribution

    縫線在兩側的分布是對稱的

  • Keep on varying the attack angle

    所以尾流沒有偏折

  • the force vanishes at about 45 degree

    位於球的正後方

  • beyond which the wake deflects to another side

    一旦改變球的攻角

  • and the force reachs its maximum at about 68 degree

    尾流很快的就發生偏折

  • Then the force vanishs again at 90 degree

    在大約22度附近向上偏折最大

  • that it has goes through a period

    要如何解釋偏折的原因呢?

  • As a result, there are 4 periods in a full rotation of 360 degree

    上游的邊界層氣流在經過縫線之後

  • In the case of two-seam rotation for a knuckleball

    會變成亂流邊界層

  • there are two main periods of force variation

    請注意下半邊的縫線比上半邊更接近停滯點

  • The maximum force is equivalent to 2/3 of baseball weight

    也就是下半邊的亂流邊界層提早發生

  • and the period is two times larger than the case of four-seam rotation

    走過更長的路徑

  • Define the coordinate system first before follow-up discussions

    有更多的機會發生混合

  • Use the rectangular coordinate system

    因此邊界層得到更大的速度與動量向下游流去

  • and let X axis pointing toward home plate

    下半邊的分離點就延後了

  • Y axis pointing toward first base

    相反的

  • Let XY plane be parallel to the ground

    上半邊的分離點較早

  • so Z axis stands vertically

    流速也較慢

  • For this coordinate system

    兩側的邊界層氣流在尾流中混合後

  • The lift force is in the positive Z direction

    平均速度偏向上方

  • the gravity is in the negative Z direction

    這就是尾流發生偏折的原因

  • the lateral force is in the positive/negative Y direction

    要理解棒球所受到的作用力

  • and the drag force is in the negative X direction

    我們可以把中間複雜的交互作用過程當成黑盒子看待

  • Now consider the rotation axis of a knuckleball is perpendicular to the ground

    原本水平流進來的空氣

  • Observe the trajectory from top view

    經過黑盒子後向上偏折

  • and assume the ball spins half a rotation during the flight

    這表示空氣受到向上的作用力

  • The wind tunnel experiment implies that

    從牛頓第三運動定律可以知道

  • there are two periods of left and right motion for a four-seam knuckleball

    在這個黑盒子中

  • In the case of two-seam rotation

    一定有大小相等

  • the knuckleball trajectory has one period of variation

    方向相反的反作用力

  • In practice

    作用在棒球上

  • a pitcher would try to throw the ball as no spin as possible

    這就是縫線的不對稱分布所造成的結果

  • The way is to push the ball with fingertip

    繼續改變攻角

  • but it makes the ball hard to be controlled

    到達45度時作用力回復為零

  • Since the spin axis may not be fixed in some specific direction

    再來尾流會向另一側偏折

  • and since the force is sensitive to attack angle

    到68度時作用力最大

  • the knuckleball trajectory becomes hard to be predicted

    90度時又回復為零

  • This is a demonstration of knuckleball's movement

    這樣子就走完一個週期

  • by Tim Wakefield in a Japan TV show

    所以轉一圈360度作用力就有四個週期的變化

  • From the research of knuckleball we learn that at extremely low spin rate

    若是改成以二縫線的方式旋轉

  • the seam locations or say attack angle

    實驗結果發現自轉一周有兩個大的週期變化

  • determines the force on the baseball

    作用力的最大值相當於棒球重量的2/3

  • In baseball games, however

    週期比四縫線大一倍

  • other kinds of pitches have spin rate over 10 times higher than knuckleball

    我們先把座標系定義好以方便之後的討論

  • and a new physical effect of force enters in-

    使用直角座標系

  • - the "Magnus force"

    X方向指向本壘板

  • To get rid of the disturbance by seams

    Y方向指向一壘側

  • let's begin with a smooth sphere

    並且令XY平面與地面平行

  • When it start to spin

    Z軸垂直於地表

  • The B-side surface and the ambient air move in the same direction

    這樣定義的座標系

  • while C-side moves reversely

    升力剛好在正Z方向

  • Since the fluid inside boundary layer is governed by viscosity

    重力則在負Z方向

  • the B-side boundary layer flows faster than that of C-side toward downstream

    側向力在正負Y方向

  • and carries more momentum

    阻力在負X方向

  • Therefore, the B-side separation point becomes relative backward

    我們先假設蝴蝶球的自轉軸垂直於地表

  • so the wake deflects downward after mixing of these two sides of air

    從俯視圖觀察

  • and the transverse force arises

    假設飛行過程中球自轉了半圈

  • This is called the "Magnus effect"

    根據前面風洞實驗的結果來推斷

  • that resulted from the rotation of moving object in the fluid

    四縫線的蝴蝶球會有兩個週期的左右搖擺

  • The magnitude of Magnus force is probably proportional to

    改成二縫線旋轉的話

  • the air flow speed V and the angular frequency ω of the ball

    蝴蝶球的軌跡會有一個週期的變化

  • So the empirical formula for Magnus force is given by

    在實際應用上

  • 1/2 times "Magnus coefficient CM"

    投手必須要盡可能的壓抑球的轉動

  • "air density ρ"

    所以投球時要利用指尖把球向前推出

  • "baseball cross-sectional area A"

    這樣的投法是相當難以控制的

  • "radius R"

    自轉軸不一定會固定在特定方向

  • "angular frequency ω" and the "flow speed V"

    再加上作用力的變化對於攻角相當敏感

  • The Magnus coefficient is a dimensionless quantity

    這使得蝴蝶球的軌跡非常難以捉摸

  • and is the only one parameter to be determined from experiments

    在一次的練習當中

  • Rotation is the motion characterize with direction

    紅襪隊的威克菲爾為觀眾展示了蝴蝶球的漂移能力

  • and can be well described by a "Vector"

    從蝴蝶球的研究我們已經了解到

  • For example, a spin vector S

    在極低轉速下

  • Point your right-hand thumb in the direction of the arrow

    縫線的位置

  • and the grip of the other four fingers represents gyration of the object

    或者說攻角

  • In addition

    完全決定棒球的受力情形

  • the length of the arrow is used to represent the spin rate

    但是在棒球運動中

  • Therefore, a vector can give a complete description of rotation

    其他種類的球路都比蝴蝶球自轉快10倍以上

  • Now we can utilize the right-hand rule

    在高速自轉的情況下還要考慮新的物理效應

  • to determine the direction of Magnus force

    那就是「馬格納斯力」

  • Point your four fingers toward the direction of ball flight

    為了要排除縫線的干擾

  • and align the thumb with spin vector S

    我們以光滑的球體來說明

  • then you got your palm faces the direction of Magnus force

    當球體開始自轉的時候

  • We've learned the Magnus effect by a spinning smooth ball

    B側的表面跟周圍空氣的運動是順向的

  • What about a spinning ball with seams?

    C側剛好是反向

  • The wake becomes fluttering

    我們知道邊界層內的流體受到黏滯力控制

  • The spin of the ball as well as the surface seams result in joint effect

    所以B側的邊界層會比C側具有較大的流速往下游流動

  • and the Magnus force varies periodically

    也就是B側的邊界層具有較大的動量

  • Although the complex behavior of boundary layer has not been well studied

    這樣

  • the average force can still be obtained from experiments

    B側的分離點自然較為延後

  • If the spin vector points toward the third base horizontally

    兩側的氣體混合後造成尾流向下方偏移

  • the Magnus force will totally contribute to lift

    球體因此受到橫向的作用力

  • This figure is the measurement results

    這種作用力是由轉動所引起的

  • The horizontal axis represents the spin parameter SP

    叫做「馬格納斯效應」

  • which is defined as the surface speedof a rotating sphere

    馬格納斯力的大小

  • divided by the flow speed V

    大致上正比於空氣的流速V

  • The vertical axis represents the lift coefficient CL

    還有球體的自轉角頻率ω

  • which is defined by Magnus coefficient times SP

    所以馬格納斯力的經驗公式為

  • The upper curve represents the lift coefficient of 4-seamer

    1/2乘以「馬格納斯係數CM」

  • and the lower one represents the lift coefficient of 2-seamer

    「空氣密度ρ」

  • Taking a 140 km/hr and 20 rev/sec fastball as an example

    「棒球截面積A」

  • The lift coefficient for a 4-seamer is about two times larger than that of a 2-seamer

    「半徑R」

  • The difference between them decreases

    「自轉角頻率ω」以及「空氣流速V」

  • with increasing the spin rate or decreasing the ball speed

    馬格納斯係數是一個無因次的量

  • In this 140kmh and 20rps case:

    也是唯一的待定係數

  • The lift for a 4-seamer is about 60% of the weight

    可以從實驗測量得知它的值

  • while it is about 30% for a 2-seamer

    任何物體的轉動是有方向性的

  • Now the trajectory can be estimated since the lift force is known

    所以適合使用「向量」來加以描述

  • Green and red curves denote the trajectories of 4-seamer and 2-seamer, respectively

    譬如說自旋向量S

  • And the dashed line is a straight line

    把拇指指向箭頭方向

  • When the ball arrives at the plate

    那麼四指握起來就代表物體迴旋的方向

  • the 4-seamer drops about 40 cm

    更進一步的

  • while the 2-seamer drops about 70 cm

    還可以用箭頭的長短來代表轉速的快慢

  • and the difference is about 30 cm

    所以向量這種東西可以完整的描述一個物體的自轉

  • If there is no lift force

    至於說馬格納斯力的方向

  • and consider only the gravitational force on the ball

    可以利用右手定則來決定

  • the ball would drop about 1 meter

    以四隻指頭指向棒球前進的方向

  • Another extreme case is a ball with vertical spin axis

    拇指指向自旋向量

  • such that Magnus force acts totally in the direction of lateral force

    這樣掌心面對的方向就代表馬格納斯力作用的方向

  • Consider the same spin rate and ball speed discussed above

    我們已經了解光滑球體轉動的馬格納斯效應

  • A 2-seamer moves about 30 cm to the left or right

    那如果把縫線的干擾加進來呢?

  • when it arrive at the plate

    尾流的方向開始變得搖擺不定

  • and the horizontal movement for a 4-seamer is about 60 cm

    球體的自轉

  • The ability of horizontal or vertical motion is called "tail-strength"

    與表面的縫線產生綜合的效應

  • in Taiwan's baseball terminology

    使得馬格納斯力變成隨時間變化

  • Owing to the surface roughness

    雖然說

  • the drag coefficient of a baseball is in between that of a smooth ball and a golf ball

    邊界層複雜的行為還沒有被完全的研究透澈

  • General speaking, people think that

    然而

  • 2-seamers have larger drag than 4-seamers

    透過實驗的測量

  • But the experiment data show that difference is not big

    還是可以得到平均的作用力

  • On the contrary

    如果球的自旋向量水平

  • 2-seamers have obviously smaller drag for some special conditions

    指向三壘的一側

  • To pitch a 4-seam fastball

    這樣馬格納斯力就會貢獻在升力上面

  • place your index and middle fingertips on the baseball seam

    這張圖是實驗測量結果

  • and place your thumb right beneath the ball

    橫軸是自轉參數SP

  • At release point, press the fingers downward

    定義為球體表面圓周運動速度Rω

  • and get the ball backspin like this

    除以流速V

  • The spin axis of a 4-seam fast ball is in general oblique

    縱軸是升力係數CL

  • which results in the inside movement for a right-handed batter

    定義為馬格納斯係數乘以SP

  • 4-seam fastball

    上面的曲線代表四縫線球的升力係數

  • In the pitcher's view angle

    下面的是二縫線球的升力係數

  • Spin vector S pointing toward lower right

    以一顆時速140公里

  • and the right-hand rule tells that

    轉速每秒20轉的快速球為例

  • Magnus force M pointing toward the upper right

    四縫線球的升力係數大約是二縫線球的兩倍

  • As for gravity, Fg pointing downward

    增加轉速

  • The resultant force is obtained by making a parallelogram

    或者降低球速

  • To pitch a cutter

    兩者的差距會明顯的縮小

  • place the index and middle fingers a little bit outside

    跟棒球的重量相比

  • and press the fingers downward at release point

    這個四縫線球的升力相當於棒球重量的60%左右

  • and get the ball spin like this

    而二縫線球則約為30%

  • Cutter (Pitcher's view angle)

    知道升力的大小就可以估算運動軌跡

  • To pitch a 2-seam fastball

    綠線與紅線分別代表四縫線與二縫線快速球的運動軌跡

  • place the index and middle fingertips on the narrow part of the seams

    虛線是一條直線

  • and place the thumb right beneath the ball

    當球底達本壘板

  • then it will rotate as a 2-seamer after delivery

    四縫線球大約會掉落40公分左右

  • 2-seam fastball (Pitcher's view angle)

    而二縫線球大約掉落70公分左右

  • If push off the index finger at release point and let the ball side spin

    兩者差距約30公分

  • the ball will get more lateral movement and sinking

    若是完全不考慮升力

  • and it becomes a "sinker"

    在只受重力作用的情況下

  • 2-seam sinker (Pitcher's view angle)

    棒球會下落接近1公尺的程度

  • To pitch a slider

    另一種極端是

  • place the index and middle fingers outside of the ball

    自轉軸垂直地面的情況

  • and rotate your palm a little bit

    這樣馬格納斯力完全作用在側向力的方向上

  • At release point, press the fingers downward

    以同樣的球速與轉速作為例子

  • and let the ball spin in this way

    當二縫線球在抵達本壘板時

  • Slider (Pitcher's view angle)

    會有大約30公分向左

  • To pitch a curveball

    或者向右的橫向移動

  • rotate your palm to the left

    四縫線球則大約有60公分的橫向移動

  • At release point, rotate the fingers forward

    在棒球的術語裡面

  • and let the ball spin in this way

    這種橫向或垂直移動的能力稱之為「尾勁」

  • Another view angle for curveball

    由於表面粗糙程度的關係

  • To pitch a forkball

    棒球的阻力係數介於光滑球體與高爾夫球之間

  • Split your index and middle fingers apart to grip the ball

    雖然說一般認為

  • and place the thumb beneath the ball

    二縫線球比四縫線球阻力大

  • It leads to low spin rate and large sink

    但是從一般的實驗資料來看

  • Forkball (Pitcher's view angle)

    兩者的差別並不大

  • To pitch a changeup

    而且

  • make an "OK" gesture

    在特殊條件下

  • then put the ball in your hand

    二縫線球的阻力反而還明顯的小於四縫線球

  • This results in low spin rate

    投四縫線快速球的時候

  • and is similar to a forkball

    把食指跟中指按在縫線上

  • Changeup (Pitcher's view angle)

    拇指放在正下方

  • Vertical slider is a special variant of slider

    投出去的時候手指往下扣

  • The Magnus force vanishes

    讓球產生這樣子的旋轉

  • since the spin axis aligns with its motion

    四縫線快速球的自轉軸通常會有些傾斜

  • Gravity and drag are the rest of forces acting on the ball

    這會產生往右打者內側移動的尾勁

  • therefore, it drop fast vertically

    從投手的視角來看

  • The spin axis of a gyroball is in between that of V-slider and cutter

    自旋向量S指向右下

  • and so does its characteristics

    利用右手定則得知

  • It can move as fast as a cut fastball

    馬格納斯力M指向右上

  • and may also sinks like a V-slider

    至於說重力則是指向下

  • Nevertheless

    做一個平行四邊形

  • since every pitch has wide range of physical characteristics

    就得到合成力

  • and since there are no standard criteria for classification

    投卡特球的時候

  • some people think that gyroball can just be classified into cutter or slider

    食指與中指稍微偏向外側

  • To pitch a screwball

    放球時

  • Turn your palm inside out to pitch the ball

    手指往下扣

  • The spin direction of a screwball for a right-handed pitcher

    讓球產生這樣子旋轉

  • is similar to that of a left hander's slider or curve

    投二縫線快速球的時候

  • so the pitch moves down and in on a right-handed batter

    食指與中指放在兩條縫線最靠近的地方

  • The screwball pitchers are rare

    拇指放在正下方

  • because it tends to damage pitcher's arms

    投出後會以二縫線方式旋轉

  • Conclusions

    如果食指稍微用力的話

  • We have gone into the detail of the physics of boundary layer

    讓球產生這樣子的旋轉

  • and have leaned various baseball pitches

    產生更大幅度的側移跟下沉

  • Hope you have fun watching and playing baseball game

    這就是所謂的伸卡球

Babe Ruth, a legend of Major League Baseball

貝比.魯斯 美國職棒的傳奇人物

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