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摩擦力Frictional Force
Chen-Ching Ting(丁振卿)
Mechanical Engineering, National Taipei University of Technology(國立台北科技大學機械系)
Homepage: http://cct.me.ntut.edu.tw/E-mail: [email protected]
CCT Group
助教: 吳穎彥
June 9, 2011
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 1 / 45
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課程大綱
1 靜摩擦力(Static Frictional Force)
2 動摩擦力(Kinetic Frictional Force)
3 乾摩擦(Dry Friction)
4 楔(Wedge)
5 螺旋摩擦(Frictional Force on Screw)
6 平帶摩擦(Frictional Force on Flat Belt)
7 環軸承、樞軸承、圓盤摩擦(Frictional Force on Collar Bearing, PivotBearing, and Disk)
8 滑動軸承摩擦(Frictional Force on Journal Bearing)
9 滾動摩擦(Rolling Resistance)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 2 / 45
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參考文獻
R.C Hibbeler, Statics, Pearson Education Inc., 歐亞書局有限公司,ISBN=978-986-154-861-6, Chapter 8, 2009.
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 3 / 45
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摩擦力
不同物體之介面間的剪力稱為摩擦力,其力主要來自物體之介面間彼此的粗糙度所致,還有物體之介面間彼此分子的吸引力。當物體之介面間的剪力使得物體彼此間沒有產生滑動現象時,稱為靜摩擦力(Fs),當物體之介面間開始產生滑動前的剪力為最大,稱為最大靜摩擦力,靜摩擦力的大小等於施加於物體並與物體之介面平行的力(P);當物體之介面間開始產生滑動後,此時的剪力比最大靜摩擦力為小,稱為動摩擦力(Fk),動摩擦力在物體之介面間的滑動沒有產生空氣間隙層時,其力為固定常數,當物體之介面間的滑動速度加快時,會產生空氣間隙層,因而使得動摩擦力下降。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 4 / 45
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靜靜靜摩摩摩擦擦擦力力力Static Frictional Force
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 5 / 45
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靜摩擦力(Static Frictional Force)
不同物體之介面間沒有產生滑動時的剪力稱為靜摩擦力(Fs),其力主要來自物體之介面間彼此的粗糙度所致,靜摩擦力的大小等於施加於物體並與物體之介面平行的力(P)。由左圖W為物體重、P為平行介面的施力、∆Fn為摩擦力、∆Nn為介面法線方向的力。
−−→∆Rn =
−−→∆Nn +
−−→∆Fn (1)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 6 / 45
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靜摩擦力(Static Frictional Force) - Continued
當物體靜止時,達力與扭矩的平衡,也就是合力(FR = 0)及合扭矩(MR = 0)均為零。
~FR = ~W + ~P + ~N + ~F (2)
= −W~j + P~i + N~j − F~i (3)
= (P − F )~i + (N −W )~j (4)= 0 (5)
MRO = Wx − Ph = 0 (6)
x =Ph
W(7)
當物體開始滑動前(臨界傳動(Impending Motion))的靜摩擦力稱為最大靜摩擦力(Fs)
Fs = µsN (8)
其中,µs為靜摩擦係數(Coefficient of Static Friction)。
φs = tan−1
(FsN
)= tan−1
(µsNN
)= tan−1µs (9)
µs = tanφs (10)
其中,φs為合阻力(Rs)與垂直力(N)之夾角。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 7 / 45
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動動動摩摩摩擦擦擦力力力Kinetic Frictional Force
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 8 / 45
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動摩擦力(Kinetic Frictional Force)
當物體開始滑動後之摩擦力稱為動摩擦力(Fk)
Fk = µkN (11)
其中,µk為動摩擦係數(Coefficient of KineticFriction)。
φk = tan−1(FkN
)= tan−1
(µkNN
)= tan−1µk
µk = tanφk (12)
其中,φk為合阻力(Rk)與垂直力(N)之夾角。一般而言,µk小於µs約25%,φs ≥ φk。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 9 / 45
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乾乾乾摩摩摩擦擦擦Dry Friction
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 10 / 45
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乾摩擦(Dry Friction)
乾摩擦(Dry Friction)也稱為庫倫摩擦(CoulombFriction),簡單來說,乃不同物體之介面間沒有潤滑劑。當結構體靜止時,達力與扭矩的平衡,也就是合力(FR = 0)及合扭矩(MR = 0)均為零。
FA ≤ µANA (13)FC ≤ µCNC (14)
其中,µA與µC分別為A點與C點的靜摩擦係數。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 11 / 45
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乾摩擦(Dry Friction) - Continued
考慮摩擦力時,除了合力外,還有合扭矩。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 12 / 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 13 / 45
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解題技巧
首先畫出受力圖(Free-Body Diagram),接著進行合力(FR = 0)及合扭矩(MR = 0)均為零。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 14 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
合力與合扭矩均為零
系統達靜平衡
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 14 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
PN
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 14 / 45
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Example 8.1 解答
由合力( ~FR = 0)及合扭矩(−−→MRO = 0)均為零可得
~P = 80cos30◦~i − 80sin30◦~j (15)~W = −196.2~j (16)~NC = NC~j (17)~F = −F~i (18)~FR = ~P + ~W + ~NC + ~F (19)
= (80cos30◦ − F )~i + (NC − 80sin30◦ − 196.2)~j = 0 (20)
又
−→OP = −0.4~i + 0.2~j (21)−→ON = x~i (22)−−→MRO =
−→OP × ~P +
−→ON × ~NC = 0 (23)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
達開始滑動前,求靜摩擦係數
合力與合扭矩均為零
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
達開始滑動前,求梯子角度
合力與合扭矩均為零,摩擦力達最大靜摩擦力
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
flip_clock_black_12.swfMedia File (application/x-shockwave-flash)
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
達開始滑動前,求施力P
合力與合扭矩均為零,摩擦力達最大靜摩擦力
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
flip_clock_black_12.swfMedia File (application/x-shockwave-flash)
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
flip_clock_black_12.swfMedia File (application/x-shockwave-flash)
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
達開始滑動前,求施力P
合力與合扭矩均為零,摩擦力達最大靜摩擦力
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
flip_clock_black_12.swfMedia File (application/x-shockwave-flash)
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 15 / 45
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楔楔楔Wedge
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 16 / 45
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楔(Wedge)
楔(Wedge)具有傾斜角,一般用來轉換力成為相對此力大許多的力。如左圖,當系統達平衡時,合力為零。 ∑
Fx =∑
Fy = 0 (24)
F = µN (25)
當推力P不存在時,如果系統還是達平衡,此時稱為自動鎖住(Self-Locking)。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 17 / 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 18 / 45
flip_clock_black_12.swfMedia File (application/x-shockwave-flash)
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 18 / 45
flip_clock_black_12.swfMedia File (application/x-shockwave-flash)
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 18 / 45
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螺螺螺旋旋旋摩摩摩擦擦擦Frictional Force on Screw
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 19 / 45
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螺旋摩擦(Frictional Force on Screw)
螺絲(Screw)常被用來作為固定器,還有傳送力量或傳送運動使用,方螺紋螺絲(Square-ThreadedScrew)或V螺紋螺絲(V-Threaded Screw) 常被用來傳送力量或傳送運動。下圖θ為方螺紋螺絲的導角(Lead Angle)。
tanθ =2πr
l(26)
θ = tan−1(2πr
l) (27)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 20 / 45
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向上臨界傳動(Upward Impending Motion)
所謂臨界傳動(Impending Motion)指的是物體開始滑動之臨界點,也就是最大靜摩擦點。假設為向上臨界傳動行為(Upward Impending Motion),M為力偶矩(Couple Moment),W為垂直旋轉方向的力,F為摩擦力,由受力分析圖得知
φs = tan−1(
F
N) = tan−1(
µsN
N) = tan−1µs
當系統達平衡時,其合力與合扭矩均等於零∑Fx =
M
r− Rsin(φs + θ) = 0 (28)∑
Fy = Rcos(φs + θ)−W = 0 (29)
由式(28)除以式(29)得
M = rWtan(φs + θ) (30)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 21 / 45
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自動鎖上螺旋(Self-Locking Screw)
所謂自動鎖上螺旋(Self-Locking Screw)指的是當沒有外加扭矩時,螺旋自動旋上,發生自動鎖上的條件是當靜摩擦角(φs)大於或等於螺牙導角(θ)時,假設為自我鎖住螺旋(Self-LockingScrew)
φs ≥ θ (31)
自動鎖上螺旋的扭矩來自螺旋本身重量之分力所造成。當靜摩擦角(φs)等於螺牙導角(θ)時,則螺旋無法進行;當靜摩擦角(φs)大於螺牙導角(θ)時,螺旋自動進行。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 22 / 45
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向下臨界傳動(Downward Impending Motion)
向下臨界傳動(Downward Impending Motion)產生有兩種情形,一種情形是具備自動鎖上(Self-Locking)條件時,而且靜摩擦角(φs)大於螺牙導角(θ)。M’為力偶矩(CoupleMoment),W為垂直旋轉方向的力,F為摩擦力,由受力分析圖得知
M ′ = rWtan(θ − φs), φs > θ (32)
另一種情形是靜摩擦角(φs)小於螺牙導角(θ)。M”為力偶矩(Couple Moment),W為垂直旋轉方向的力,F為摩擦力,由受力分析圖得知
M ′′ = rWtan(φs − θ), φs < θ (33)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 23 / 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 24 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 24 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 24 / 45
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平平平帶帶帶摩摩摩擦擦擦Frictional Force on Flat Belt
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 25 / 45
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平帶摩擦(Frictional Force on Flat Belt)
平帶摩擦(Frictional Force on Flat Belt)為傳動及煞車常使用的元件,平帶為簡化的元件,一般常使用V形帶(V-Belt),其在靜力學上的分析相同。如左圖,當平帶貼附在半徑為r的轉軸上時,其貼附在轉軸上由轉軸之軸心的張角β,平帶與轉軸面之摩擦係數為µ,如果欲使轉軸逆時針轉動,拉力T2除了對抗T1外,還須對抗摩擦力,因此
T2 > T1 (34)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 26 / 45
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平帶摩擦(Frictional Force on Flat Belt) - Continued
考慮平帶上的摩擦力,首先取微小段ds進行平衡系統之合力分析,當系統達平衡時,合力為零。∑
Fx = 0, Tcos(dθ
2) + µdN − (T + dT )cos(dθ
2) = 0
⇒ Tcos(dθ2
) + µdN − Tcos(dθ2
)− dTcos(dθ2
) = 0
⇒ µdN = dTcos(dθ2
) = dT∑Fy = 0, dN − (T + dT )sin(
dθ
2)− Tsin(dθ
2) = 0
⇒ dN − Tsin(dθ2
)− dTsin(dθ2
)− Tsin(dθ2
) = 0
⇒ dN = 2Tsin(dθ2
) + dTsin(dθ
2) = Tdθ
其中,dθ → 0,sin( dθ2 ) ≈dθ2 ,cos(
dθ2 ) ≈ 1,
dTsin( dθ2 ) ≈ 0。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 27 / 45
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平帶摩擦(Frictional Force on Flat Belt) - Continued
由式(35)除以式(36)得
µdN = dT (35)
dN = Tdθ (36)
dT
T= µdθ (37)
ˆ T2T1
dT
T= µ
ˆ β0
dθ (38)
lnT2T1
= µβ (39)
T2 = T1eµβ (40)
由上式(40)結果得知,平帶張力與轉軸半徑無
關,而與平帶貼附在轉軸上的張角有關,因
此,式(40) 可適用在平帶貼附於任何曲線面
上。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 28 / 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 29 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 29 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 29 / 45
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環環環軸軸軸承承承、、、樞樞樞軸軸軸承承承、、、圓圓圓盤盤盤摩摩摩擦擦擦Frictional Force on Collar Bearing, Pivot
Bearing, and Disk
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 30 / 45
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環軸承摩擦(Collar Bearing)
環軸承(Collar Bearing)一般以乾摩擦或部份潤滑的方式支撐轉動軸的軸向上負載(Axial Load)。左下圖為受力分析,其中,軸向受力P,支撐軸向力的面積A,單位面積軸向承受的力(壓力)p,考慮環軸承為均勻受力,也就是壓力為均勻分佈,N為正向力,則微小面積dA受力dF為∑
Fz = 0 (41)
A = π(R22 − R21 ), dA = rdθdr (42)
p =P
π(R22 − R21 )(43)
dN = pdA (44)
dF = µsdN = µspdA (45)
=µsP
π(R22 − R21 )dA
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 31 / 45
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環軸承摩擦(Collar Bearing) - Continued
系統之正向力N與z軸同方向,因此只有摩擦力F造成環軸承之轉動扭矩。由系統達平衡時之合扭矩為零得到∑
Mz = 0 (46)
M −ˆA
rdF = 0 (47)
將dF與dA代入式(47)並進行積分可得
M =
ˆ R2R1
ˆ 2π0
r [µsP
π(R22 − R21 )]rdθdr (48)
=µsP
π(R22 − R21 )
ˆ R2R1
r2dr
ˆ 2π0
dθ (49)
=2
3µsP(
R32 − R31R22 − R21
) (50)
其中,M為環軸承靜摩擦扭矩,若環軸承等速轉動,則動摩
擦扭矩以動摩擦係數µk取代靜摩擦係數µs。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 32 / 45
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樞軸承摩擦(Pivot Bearing)
樞軸(Pivot Bearing)與環軸承(Collar Bearing)類似,一般以乾摩擦或部份潤滑的方式支撐轉動軸的軸上負載(Axial Load)。左圖為受力分析,其中,軸向受力P,支撐軸向力的面積A,單位面積承受的力(壓力)p,考慮樞軸承為均勻受力,也就是壓力為均勻分佈,由達平衡時之合力為零得到∑
Fz = 0 (51)
A = πR2 (52)
p =P
πR2(53)
N = pA (54)
F = µsN = µspA =µsP
πR2A (55)
假如樞軸承非均勻受力,則必須考慮壓力分佈情況才
能決定摩擦力大小。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 33 / 45
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圓盤摩擦(Disk)
圓盤摩擦可考慮環軸承之R2 = R且R1 = 0時,由系統達平衡時之合扭矩為零及環軸承摩擦扭矩得到 ∑
Mz = 0 (56)
M =2
3µsP(
R32 − R31R22 − R21
) (57)
=2
3µsPR (58)
其中,M為圓盤靜摩擦扭矩,若圓盤等速轉動,則動摩擦扭矩乃以動摩擦係數µk取代靜摩擦係數µs。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 34 / 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 35 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 35 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 35 / 45
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滑滑滑動動動軸軸軸承承承摩摩摩擦擦擦Frictional Force on Journal Bearing
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 36 / 45
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滑動軸承摩擦(Journal Bearing)
滑動軸承(Journal Bearing)一般以乾摩擦或部份潤滑的方式支撐轉動軸的側向負載(Lateral Load)。左下圖為受力分析,其中,A點為滑動軸承開始轉動的任一受力點,與軸向垂直之方向受力P,A點上之反作用力R,由力的平衡可得R與P之大小相等、方向相反。由系統達平衡時,以滑動軸承之軸心為轉軸之合扭矩為零得到
∑Mz = 0 (59)
M − (Rsinφk)r = 0 (60)M = Rrsinφk (61)
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 37 / 45
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滑動軸承摩擦(Journal Bearing) - Continued
由式(61)得知滑動軸承摩擦扭矩(M)為
M = Rrsinφk (62)
其中,φk為動摩擦角(Angle of Kinetic Friction),其與動摩擦係數(µk)之關係為
tanφk =F
N=µkN
N= µk (63)
rf = rsinφk (64)
(65)由左下圖之半徑(rf )所形成的圓稱為摩擦圓(FrictionCircle),A點上之反作用力R的作用方向總是緣著摩擦圓的切線方向。另外,當滑動軸承被部份潤滑時,表示動摩擦係數(µk)非常小,
sinφk ≈ tanφk ≈ φk (66)M ≈ Rrµk ≈ −Prµk
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 38 / 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 39 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 39 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 39 / 45
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滾滾滾動動動摩摩摩擦擦擦Rolling Resistance
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 40 / 45
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滾動摩擦(Rolling Resistance)
滾動摩擦因為不等同於摩擦力的定義,因此稱為滾動阻抗(Rolling Resistance)。 剛體滾輪在剛體面上的滾動摩擦力的方向為滾輪與接觸面上的點之切線方向,但是,真實環境並不存在完全的剛體(Rigid Body)。分析非剛體間的滾動摩擦問題,在此先將問題簡化成非常硬的滾輪在軟性的面上滾動,則滾輪與接觸面為一曲線,在此接觸曲線上有壓力分佈,其力的方向均指向滾輪之轉軸,施加在滾輪上的力分成在滾輪行進方向並造成接觸面形變(Deformation)的Nd及讓變形接觸面恢復(Restoration)的Nr,但是,驅動滾輪前進的力(滾動摩擦力)P不易得知,因此必須對滾動摩擦問題進行簡化。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 41 / 45
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滾動摩擦(Rolling Resistance) - Continued
將滾輪與接觸面的分佈負載之合力(~N)及施加作用點(A)求出,則以作用點A為轉軸點的合扭矩為零。
~N = ~Nd + ~Nr (67)∑MRA = Wa− Prcosθ = 0 (68)
P ≈ War
(69)
a ≈ PrW
(70)
其中,一般狀況θ ≈ 0, cosθ ≈ 1,P為滾動摩擦力,W為輪重,a稱為滾動摩擦係數(Coefficient of Rolling Resistance),滾
動摩擦係數的單位是長度,例如:mm。對於堅硬的鋼珠滾
軸a ≈0.1 mm,對於輪子在低碳鋼上滾動a ≈0.5 mm,滾動摩擦係數不易被精準預測,但是,War 一般極小於µkW,也就是
滾動摩擦力一般極小於滑動摩擦力,這也是轉動軸常使用滾珠
取代滑動的原因。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 42 / 45
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課課課堂堂堂練練練習習習
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 43 / 45
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Engineering Mechanics: Statics, Twelfth EditionRussell C. Hibbeler
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 43 / 45
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結結結論論論
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 44 / 45
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結論
不同物體之介面間的剪力稱為摩擦力,其力主要來自物體之介面間彼此的粗糙度所致,還有物體之介面間彼此分子的吸引力。當物體之介面間的剪力使得物體彼此間沒有產生滑動現象時,稱為靜摩擦力(Fs),當物體之介面間開始產生滑動前的剪力為最大,稱為最大靜摩擦力,靜摩擦力的大小等於施加於物體並與物體之介面平行的力(P);當物體之介面間開始產生滑動後,此時的剪力比最大靜摩擦力為小,稱為動摩擦力(Fk),動摩擦力在物體之介面間的滑動沒有產生空氣間隙層時,其力為固定常數,當物體之介面間的滑動速度加快時,會產生空氣間隙層,因而使得動摩擦力下降。
Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] CCT Group[2mm] 助教: 吳穎彥 ()摩擦力Frictional Force June 9, 2011 45 / 45
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靜摩擦力(Static Frictional Force)動摩擦力(Kinetic Frictional Force)乾摩擦(Dry Friction)楔(Wedge)螺旋摩擦(Frictional Force on Screw)平帶摩擦(Frictional Force on Flat Belt)環軸承、樞軸承、圓盤摩擦(Frictional Force on Collar Bearing, Pivot Bearing, and Disk)滑動軸承摩擦(Frictional Force on Journal Bearing)滾動摩擦(Rolling Resistance)