Ch6 - Dry friction

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6 Friction
2
Objectives Students must be able to

• Utilize theory of dry friction
• Describe theory of dry friction
• Describe physical meanings of frictional effects
• Describe and differentiate between static and
  kinetic coefficients of friction
• Describe the angles of frictions
• Add friction into the analyses of objects and
  structures in equilibrium

3
Objectives Students must be able to

• Describe and analyze machines with frictions
• Wedges
• Threads, screws
• Belts
• Disks and clutches
• Collar, pivot, thrust and journal bearings
• Outline rolling resistance
• Describe the physical meanings of rolling
  resistance
• Differentiate between frictions and rolling
  resistance

4
Topic in textbook
We will study this Part first.

• Section A Frictional Phenomena
• Characteristics, theory, coefficient of friction,
  angle of friction

• Section B Applications
• Wedges
• Screws
• Journal Bearings
• Thrust Bearings Disk friction
• Flexible Belts
• Rolling Resistance

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Dry Friction
Force of resistance acting on a body which
prevents or retards slipping of the body relative
to a surface with which it is in contact.
Friction exists?
roughnesses of the contacting surfaces.
Magnitude frictions magnitude limitation
will be discussed later
Direction tangent to the contacting surface
and opposed to the relative
motion or tendency for motion
Line of Action (Point of application) contact
surface
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Friction Model
Equilibrium
W
In equilibrium
a/2
a/2
FBD is correct?
modeling
P
h
F
The DN at right side is supporting force more
than its left side.
x
N
N
The object is toppling (not in equilibrium)
If x gt a/2 ?

• Frictional force F

Slipping and/or Tipping Effect
increases with force P
Slipping
toppling

• The application point (x) of N

x-limit
F-limit
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Motion
a/2
a/2
P
h
F
x
N
N

• Slipping / Sliding
• Relative sliding (translation motion) between two
  surfaces
• Toppling / Tippling
• Fall over (rotation) about the edge
• Topple, tipping, rolling, tumble, trip

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Experiment for determining Friction
mg
P
P
m
FBD
F
N
impending motion (on the verge of motion)

• Object at rest (no motion)
• coefficient of static friction

Fk
?kN

• Object with motion (steady state)
• coefficient of kinetic friction

Fsmax
?sN
constant on 2 certain contacting surfaces
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Angle of Friction
Not depend on N
not depend on P,v,a

• ?k arctan(?k) angle of kinetic friction

(object in motion)
mg
P
F
?
N
R

• ?s arctan(?s) angle of (max) static
  friction

(object at rest)
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Dry Friction Characteristics
Dry Friction

• Frictional force acts tangentially to the
  contacting surfaces, opposing the relative or
  tendency for motion.
• Fs is independent of the area of contact,
  provided that the normal pressure is not very low
  nor great enough for deformation of the surfaces.
• In equilibrium
• Impending slipping f fs
• Slipping f fk
• Very low velocity fk fs

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Static Friction Typical Values
Dry Friction
Impending Motion

• Contact Materials µs
• Metal / ice 0.03 0.05
• Wood / wood 0.30 0.70
• Leather / wood 0.20 0.50
• Leather / metal 0.30 0.60
• Aluminum / Aluminum 1.10 1.70

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Sample 6/1 Determine the maximum angle ? which
the adjustable incline may have before the block
of mass m begins to slip. The coefficient of
static friction between the block and the
inclined surface is ?s.
Impending Slip
H/2
x
Ans
(for slipping)
Three force member
Possibility of toppling?
Wmg
3 eq. , 3 unknowns
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6/125 A uniform block of mass m is at rest on an
incline z. Determine the maximum force Pgt0 that
can be applied to the block in the direction
shown before slipping begins. The coefficient of
static friction between the block and the incline
is
z
y
x
q
mg
y
P
N
q
At this max P, object is about to move at which
direction?
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Example Friction 2 1
Dry Friction

• Will this crate slide or topple over?

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1st Eq.
4 Unknowns P, F, N, x
2nd Eq.
Two possibilities
3rd Eq.
1) about to slip
2) about to tip
4th Eq.
4th Eq.
Its time consuming, Better to know it exactly
Check condition
Check condition
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The block of mass m is homogeneous, moving at a
constant velocity. The coefficient of kinetic
friction is

• Determine
• a) the greatest value that h may have so that the
  block will not tip over

b) The location of point C on the bottom face of
the block through with the resultant of the
friction and normal force act if h H/2.
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• The block of mass m is homogeneous, moving at a
  constant velocity. The coefficient of kinetic
  friction is
• Determine
• a) the greatest value that h may have so that the
  block will not tip over

Moving
With no acceleration
Solution 1
?
on the verge of tipping
Moving
The box is in Equilibrium (no acceleration)
Three force member in Statics
P
h?
Concurrent at one point OR parallel
mg
F
F
N
N
Ans
on the verge of tipping
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• The block of mass m is homogeneous.
• The block is moving at a constant velocity.
• Determine
• b) The location of point C on the bottom face of
  the block through with the resultant of the
  friction and normal force act if h H/2.

Moving
with constant velocity
Solution 1
H/2
C ?
N
Moving
The box is in Equilibrium (no acceleration)
Three force member
P
Concurrent at one point OR parallel
hH/2
mg
F
x
Ans
N
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Inequality hard to deal
Dry Frictions Problem
static friction

or
Equilibrium eq.
kinetic friction
fricitional eq.
Friction

• Kinetic motion is known.

Moving at constant vel.

Kinetic friction
Equilibrium eq.
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Static friction
not assuming impending motion
In static equilibrium
friction can support equilibrium
Static equilibrium?
No. of unknown must No. of
Equilibrium eq.
Assume static equilibrium.
Equilibrium eq.
Solve for F (and also N) using only
equilibrium eq.
Only equilibrium eq. can be used to determine
unknown values.
Assumption Checking
Equilibrium eq.
Friction is determined by equilibrium eqs.
Static friction
If
O.K.
However, F must lt mN
Assumption is not true!
Motion occurs. F ?kN
Static friction
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total unknown lt equilibrium frictional
eq.

Equilibrium eq.
Static friction
Find min q to hold equilibrium
Find min P to initiate the motion
Impending motions occur at both point in the same
time.
impending motions do not occur in both point at
the same time.
Both can be used together to determine unknown
values.
several possibilities to slip
total unknown equilibrium frictional
eq.
Assumption needs to be make and that assumption
should be checked later.
Equilibrium eq.
static friction
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Sample 6/3 Determine the friction force acting on
the block shown if P 500N and P 100N. The
block is initially at rest.
500N and 100N
Equilibrium?
Assume Body in equilibrium
Equilibrium state is unknown
(not assuming the impending motion)
2 Eq , 2 unknown
impending motion
OK !
Friction must be enough to maintain equilibrium
P500N
Assumption Checking
Correct?
P100N
Contradict!
Object not in Equilibrium
Not valid!
still valid!
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Example Hibbeler Ex 8-1 1
Dry Friction

• The uniform crate has a mass of 20 kg. If a
  force P 80 N is applied to the crate, determine
  whether it remains in equilibrium. The
  coefficient of static friction 0.3.

Equilibrium State is not known
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Example Hibbeler Ex 8-1 2
Assume Equilibrium
x lt 0.4 m (physical boundary of toppling)
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Example Bedford Ex 9.5 1
Dry Friction

• Suppose that a 10 and the coefficient of
  friction between the surface of the wedge and the
  log are ms 0.22 and mk 0.20. Neglect the
  weight of the wedge.

• If the wedge is driven into the log at a constant
  rate by vertical force F, what are the magnitude
  of the normal forces exerted on the log by the
  wedge?
• Will the wedge remain in place in the log when
  the force is removed?

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• 10 ms 0.22 and mk 0.20.
• Neglect the weight of the wedge.

Moving down at constant rate
symmetric/mirror concept
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• 10 ms 0.22 and mk 0.20.
• Neglect the weight of the wedge.

Will the wedge remain in place in the log when
the force is removed?
Think of minimum friction coefficient
that still self-lock the wedge
Impending Motion (On the verge of slipping)
symmetric/mirror concept
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Example Bedford 9.20 1
Dry Friction

• The coefficient of static friction between the
  two boxes and between the lower box and the
  inclined surface is ms. What is the largest angle
  a for which the lower box will not slip.

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The coefficient of static friction between the
two boxes and between the lower box and the
inclined surface is ms. What is the largest angle
a for which the lower box will not slip.
Impending Motion
at same time ?
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Bedford 9.30 (ex)

• The cylinder has weight W. The coefficient of
  static friction between the cylinder and the
  floor and between the cylinder and the wall is
  ms. What is the largest couple M that can be
  applied to the stationary cylinder without
  causing it to rotate.

Impending Motion at same time
Impending rotation
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Bedford 9.33 (Ex)

• The disk of weight W and radius R is held in
  equilibrium on the circular surface by a couple
  M. The coefficient of static friction between the
  disk and the surface is ms. Find that the largest
  value M can have without causing the disk to slip.

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Example Bedford 9.166 1
Dry Friction

• Each of the uniform 1-m bars has a mass of 4 kg.
  The coefficient of static friction between the
  bar and the surface at B is 0.2. If the system is
  in equilibrium, what is the magnitude of the
  friction force exerted on the bar at B.

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Example Bedford 9.166 2
Dry Friction
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Example Bedford 9.166 3
Dry Friction
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Example Bedford 9.166 4
Dry Friction
symmetry?
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Example Hibbeler Ex 8-3 1
Dry Friction

• The rod with weight W is about to slip on rough
  surfaces at A and B. Find coefficient of static
  friction.

Direction of N?
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Example Hibbeler Ex 8-3 2
Dry Friction
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Example Hibbeler Ex 8-3 3
Dry Friction
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6/9 For a 20 jaw opening, what is the minimum
coefficient of static friction between the jaws
and the tube which will enable tongs to grip the
tube with out slipping
Solution 1
y
x
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6/9 For a 20 jaw opening, what is the minimum
coefficient of static friction between the jaws
and the tube which will enable tongs to grip the
tube with out slipping
Solution 2
y
The object is in equilibrium
Two force systems
x
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Sample 6/5 Find the maximum value which P may
have before any slipping takes place.
?
Possible ways to slip
1) Middle object is going to move lonely.
2) Middle buttom object is going to move
together.
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Check the Assumption
Friction F3 can support the equilibrium of 40-kg
object
Assume
1) Middle object is going to move lonely.
Impending motion
5 unknowns
3 equilibrium eq
2 friction eq
Impossible, the assumption is wrong
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F2 can support the 50-kg and 40-kg as one body
Assume
2) Middle bottom object is going to move
together.
Impending motion
OK
Ans
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Assume
?
3) 40-kg object moves lonely -- possible?
Equation without unknown (little chance to be
true)
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8-61 The uniform 50-N slender rod rests on the
top center of the 20-N block. Determine the
largest couple moment M which can be applied to
the rod without causing motion of the rod.
Motion Possibility?
1. Slip at C, B
2. Slip at C, A
3. Slip at A, B
?
4. other
x
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8-61 The uniform 50-N slender rod rests on the
top center of the 20-N block. Determine the
largest couple moment M which can be applied to
the rod without causing motion of the rod.
Assume
2. Slip at C, A
8 unknowns, 62 equation.
Dont solve 8 eq paralleling!
Solve only 6 eq. excluding
Assumption Checking
x
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8-56 The uniform 50-N slender rod rests on the
top center of the 20-N block. Determine the
largest couple moment M which can be applied to
the rod without causing motion of the rod.
Assume
2. Slip at C, A
Block is tipping
0.12 m
62.5 N
9.5 N-m
42.5 N
25 N
Assumption Checking
7.5 N
Correct?
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8-61 The uniform 50-N slender rod rests on the
top center of the 20-N block. Determine the
largest couple moment M which can be applied to
the rod without causing motion of the rod.
Possibility?
1. Slip at C, B
2. Slip at C, A
3. Slip at A, B
?
4. C-Slip, x-limit
4. other
5. B-Slip, x-limit
x
6. A-Slip, x-limit
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8-61 The uniform 50-N slender rod rests on the
top center of the 20-N block. Determine the
largest couple moment M which can be applied to
the rod without causing motion of the rod.
Assume
4. C-Slip, x-limit
8 unknowns, 62 equation.
Dont solve 8 eq paralleling!
Solve only 6 eq. excluding
Assumption Checking
x
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8-61 The uniform 50-N slender rod rests on the
top center of the 20-N block. Determine the
largest couple moment M which can be applied to
the rod without causing motion of the rod.
Assume
4. C-Slip, x-limit
63.64 N
5.78 N-m
21.21 N
Ans
21.21 lt0.643.6426.18
43.64 N
Assumption Checking
21.21 lt0.463.6425.46
6.36 N
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6/28 Determine the minimum angle at which
slipping does not occur at either contact point A
or C. The coefficient of static friction at both
A and C is 0.05. (Note you should assume the
mass of each bar why?)
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6/28 Determine the minimum angle at which
slipping does not occur at either contact point A
or C. The coefficient of static friction at both
A and C is 0.05
2 force
A or C is going to slip first?
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8-61 The end C of the two-bar linkage rests on
the top center of the 50-kg cylinder. Determine
the largest vertical force P which can be applied
at B without causing motion. Neglect the mass of
the bars.
P applies at pin B, AB or BC?
P
P applies at pin B
2 force
From solving equilibrium eqs.
2 force
However value P are, BC will not slip at C.
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8-61 The end C of the two-bar linkage rests on
the top center of the 50-kg cylinder. Determine
the largest vertical force P which can be applied
at B without causing motion. Neglect the mass of
the bars.
Max P without causing motion
Slipping or tipping?
x
Slipping?
Tipping (x0.05)?
Ans
minimum value
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6/23 The 10-kg cylinder is initially placed at
V-Block. If , find
(1) friction force F acting at each side when P0
(2) The P to start sliding the cylinder up
y
Assume body in equilibrium
(incomplete)
y
z
Assumption Checking
OK
No tendency to move in z-axis direction !!
(2) Find P min to move
impending motion
N
N 601
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Determine the minimum coefficient of static
friction at each point of contact so that the
pile does not collapse.
B
A
weight W, radius r
D
C
Think of Equilibrium to find out the relation of
F and N
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6/108 Describe what should happen
Posibility?
1) A B and C dont move
2) A move, B and C dont move
3) B move (so do A), C dont move
Many possibilities?
4) C move (so do A and B)
Yes C not moving
No, C is moving.
C may or may not moving depends on friction at A
and B.
C not moving
B may or may not moving depends on friction at A.
A may or may not moving depends on friction at A.
y
x
56.85
75.81
99.50
189.51
379.03
Always in equilibrium, however A B C are moving
or not.
284.27
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6/108 Describe what should happen
1) A and B dont move, C dont move
Assume
75.806
379.03
y
58.403
x
125.507
Assumption Checking
125.507
Assumption is wrong !
So, A and B move and C dont move
y
x
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Recommend Problem

• 6/16 6/125 6/23 6/41 6/40 6/37 6/108

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