[email protected] engr-36_lec-22_wedge-n-belt_friction.pptx 1 bruce mayer, pe engineering-36:...
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[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx1
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engineering 36
Ch08: Wedge &
Belt Friction
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx2
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Outline - Friction
The Laws of Dry Friction• Coefficient of Static Friction• Coefficient of Kinetic (Dynamic) Friction
Angles of Friction• Angle of static friction• Angle of kinetic friction• Angle of Repose
Wedge & Belt Friction• Self-Locking & Contact-Angle
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx3
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Basic Friction - Review The Static Friction Force Is The force that Resists
Lateral Motion. It reaches a Maximum Value Just Prior to movement. It is Directly Proportional to Normal Force:
NF sm After Motion Commences The Friction Force Drops
to Its “Kinetic” Value NF kk
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx4
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction Consider the
System Below
Find the Minimum Push, P, to move-in the Wedge
The Wedge is of negligible Weight
Then the FBD of the Two Blocks using Newton’s 3rd Law
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx5
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx6
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx7
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
For Equilibrium of the Heavy Block
Solve for FA,n
For Equilibrium of the Wt-Less Wedge
sincos0 ,, nAsnAy FFWF
sincos,s
nA
WF
cossin0
sincos0
,,,
,,,
nAnAsnCy
nAnAsnCsx
FFFF
FFFPF
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx8
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
In the last 2-Eqns Sub Out FA,n
Eliminating FC,n from the 2-Eqns yields an Expression for Pmin:
cossincos
sinsincos
0
sinsincos
cossincos
0
,
,
sssnCy
sssnCsx
WWFF
WWFPF
cos2sin1sincos
2min ss
s
µW
P
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx9
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
MATLAB Plots for P when W = 100 lbs
0 2 4 6 8 10 12 14 16 18 2040
45
50
55
60
65
70
75
80
85
(°)
P (
lbs)
W = 100 lbs, µ = 0.2
0 5 10 15 20 25 3010
20
30
40
50
60
70
80
90
µ (%)
P (
lbs)
W = 100 lbs, = 10°
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx10
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
MATLAB Code% Bruce Mayer, PE% ENGR36 * 22Jul12% ENGR36_Wedge_Friction_1207.m%u = 0.2W = 100a = linspace(0,20);P = W*((1-u*u)*sind(a) +2*u*cosd(a))./(cosd(a)-u*sind(a))plot(a,P, 'LineWidth',3), grid, xlabel('\alpha (°)'), ylabel('P (lbs)'), title('W = 100 lbs, µ = 0.2')disp('showing 1st plot - Hit Any Key to Continue')pause%a = 10;u = linspace(0,0.3);P = W*((1-u.*u)*sind(a) +2*u*cosd(a))./(cosd(a)-u*sind(a));plot(100*u,P, 'LineWidth',3), grid, xlabel('µ (%)'), ylabel('P (lbs)'), title('W = 100 lbs, \alpha = 10°')disp('showing 2nd plot - Hit Any Key to Continue')pause%u = linspace(0, .50);aSL =atand (2*u./(1-u.^2));plot(100*u,aSL, 'LineWidth',3), grid, xlabel('µ (%)'), ylabel('\alpha (°)'), title('Self-Locking Wedge Angle')disp('showing LAST plot')
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx11
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction Now What Happens
upon Removing P
The Wedge can• Be PUSHED OUT• STAY in Place
– SelfLocking condition
Then the FBD When P is Removed• Note that the
Direction of the Friction forces are REVERSED
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx12
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx13
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx14
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
For Equilibrium of the Heavy Block
Solve for FA,n
For Equilibrium of the Wt-Less Wedge
sincos0 ,, nAsnAy FFWF
cossin0
sincos0
,,,
,,,
nAnAsnCy
nAnAsnCsx
FFFF
FFFF
KW
Fs
nA
sincos,
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx15
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
To Save Writing sub K for FA,n
Eliminate FC,n
Now Divide Last Eqn by Kcosα
0sincos
0sincos
,
,
KKF
KKF
snC
snCs
01sincos20
0sincos
0sincos
2,
,
ss
ssnC
snCs
µKK
KKF
KKF
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx16
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
Dividing by Kcosα
Recognize sinu/cosu = tanu
01cos
sin2
cos
01sincos2
2
2
ss
ss
µ
K
µKK
222
2
1
2
1
2
1
2tan
21tan
s
s
s
s
s
s
ss
µµµ
µ
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx17
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
After all That AlgebraFind The Maximum α to Maintain the Block in the Static Location
Since Large angles Produce a Large Push-Out Forces, and a ZERO Angle Produces NO Push-Out Force, the Criteria for Self-Locking
2max 1
2arctan
s
s
21
2arctan
s
sSL
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx18
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Push-Out
SMALL PushOut Force• Likely SelfLocking
LARGE PushOut Force• Likely NOT SelfLocking
21
2arctan
s
sSL
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx19
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Friction
0 5 10 15 20 25 30 35 40 45 500
10
20
30
40
50
60
µ (%)
(
°)
Self-Locking Wedge Angle
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx20
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Belt Friction Consider The Belt Wrapped
Around a Drum with Contact angle .
The Drum is NOT Free-Wheeling, and So Friction Forces Result in DIFFERENT Values for T1 and T2
To Derive the Relationship Between T1 and T2 Examine a Differential Element of the Belt that Subtends an Angle • The Diagram At Right Shows
the Free Body Diagram
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx21
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Belt Friction cont Write the Equilibrium Eqns for
Belt Element PP’ if T2>T1
2sin
2sin0
2cos
2cos0
TTTNF
NTTTF
y
sx
Eliminate N from the Equations
2sin
2sin
2sin
2cos
2cos
2cos0
2sin
2sin
2cos
2cos0
2sin
2sin
TTTTTT
TTTTTTF
TTTNF
s
sx
y
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx22
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Belt Friction cont.1 Combining Terms
2
sin22
cos0
TTT s
Divide Both Sides by
2
2sin2
2cos0
TTT
s
Now Recall From Trig And Calculus
d
dLimLim
00
1sin
10cos
So in the Above Eqn Let: /2 →0; Which Yields
TdTTTd
dTdTT
d
dTss 2 as20
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx23
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Belt Friction cont.2 The Belt Friction Differential Eqn
Integrate the Variables-Separated Eqn within Limits• T( = 0) = T1
• T( = ) = T2
From Calculus
Now Take EXP{of the above Eqn}
ddTT
Td
dTss
1Vars Sep
12120lnlnln
12
1
TTTTddTT ss
T
T
ss eTTee TT 12ln 12
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx24
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Belt Friction Illustrated This is a VERY
POWERFUL Relationship
Condsider the Case at Right. Assume• A ship Pulls on the Taut
Side With A force of 4 kip (2 TONS!)
• The Wrap-Angle = Three Revolutions, or 6
• µs = 0.3
seT
T
1
2
The Tension, T1, Applied by the Worker
lbe
kip
e
TT
s14
463.0
21
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx25
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
WhiteBoard Work
Let’s WorkThese NiceProblems
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx26
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PERegistered Electrical & Mechanical Engineer
Engineering 36
Appendix 00
sinhT
µs
T
µx
dx
dy
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx27
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
WhiteBoard Work
Let’s WorkThis NiceProblem
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx28
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx29
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx30
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-22_Wedge-n-Belt_Friction.pptx31
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Wedge Push-Out
SMALL PushOut Force• Likely SelfLocking
LARGE PushOut Force• Likely NOT SelfLocking