Transcript ENGI 1313 Mechanics I - Memorial University of Newfoundland

ENGI 1313 Mechanics I
Lecture 37:
Analysis of Equilibrium Problems
with Dry Friction
Shawn Kenny, Ph.D., P.Eng.
Assistant Professor
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
[email protected]
Lecture 37 Objective

2
to illustrate the equilibrium analysis of rigid
bodies subjected to dry friction force by
example
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Equilibrium and Frictional Forces

Analysis Steps

FBD
• Assume frictional force to be an unknown

Do not assume Fs = sN unless impending motion is
stated
• Determine the Number of Unknowns


If more unknowns than equations, assume friction force
at some or all contact points
Apply Equilibrium Equations
• Impending motion or tipping
3
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Class of Friction Problems

1. Equilibrium
Geometry and dimensions
are known
 Draw FBD
 # Unknowns = # Equilibrium
Equations
 Solve for reaction forces
 No motion, if

W
FA  0.3NA
FC  0.5NC
4
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
W
Class of Friction Problems (cont.)

2. Impending Motion at All Contact Points

# Unknowns =

Impending Motion
# Equilibrium Equations +
# Friction Equations
Fs  sN

Motion
Fk  k N
5
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Class of Friction Problems (cont.)

2. Impending Motion at All Contact Points

Find the minimum angle () for a 100 N bar to
be placed against the wall.
• FBD
• Unknowns?

5
• Equations?
3 Equilibrium
Equations
 2 Friction Equations

FA  0.3NA
6
© 2007 S. Kenny, Ph.D., P.Eng.
FB  0.4NB
ENGI 1313 Statics I – Lecture 37
Class of Friction Problems (cont.)

3. Impending Motion at Some Contact
Points

# Unknowns <
# Equilibrium Equations +
# Friction Equations
or
 # Unknowns <
# Equilibrium Equations +
# Equations for Tipping
 May have to evaluate both scenarios
• If so, governing case has minimum requirements
7
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Class of Friction Problems (cont.)

3. Impending Motion at Some Contact
Points

Find horizontal force (P) to cause movement.
• FBD
• # Unknowns?

7
• Equations

Find minimum P
FA  0.3NA & FC  0.5NC
or
FA  0.3NA & FC  0.5NC
8
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Example 37-01

9
A uniform ladder weighs
20 lb. The vertical wall is
smooth (no friction). The
floor is rough with s =
0.8. Find the minimum
force P needed to move
(tip or slide) the ladder.
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Example 37-01 (cont.)
FBD
 # Unknowns?



4
Equilibrium Equations?


NB
W
3
FA
Assumptions?

NA
Tipping occurs
 NB  0
10
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Example 37-01 (cont.)

Analysis
F

0
y
NA  W  20lb
M
A
0
20lb3ft   P4ft   0
W
FA
P  15 lb


F
x
NA
0
FA  P  15lb
11
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Example 37-01 (cont.)

Check Tipping
Assumption
FA  15lb  sNA  0.820lb  16lb

 Tipping occurs
W
FA
NA
12
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Example 37-02

13
Drum weight is 100
lb, s = 0.5, a = 3 ft
and b = 4 ft. Find the
smallest magnitude
of P that will cause
impending motion
(tipping or slipping) of
the drum.
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
Example 37-02 (cont.)
FBD
 Assume Slipping
Occurs

P
3 ft
3
4
4 ft
W = 100lb
Fs
x
14
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
N
Example 37-02 (cont.)

For Slipping
Fs  sN  0.5N
P
3 ft
3
4
 Fx  0
4
P  0 .5 N  0
5

F
y
 100 
0
4 ft
W = 100 lb
3
P N 0
5
Fs
x
 P  100lb N  160lb
15
© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37
N
Example 37-02 (cont.)

P
Check x
 P  100lb N  160lb
M
O
3 ft
3
4
0
160lbx    3 100lb1.5 ft   
5

4

100
lb

4ft   0
5

x  1.438ft  1.5 ft

16
4 ft
W = 100 lb
O
x
 Slipping
© 2007 S. Kenny, Ph.D., P.Eng.
Fs
ENGI 1313 Statics I – Lecture 37
N
Example 37-02 (cont.)

Assume Tipping Occurs 3
MN  0
P
3 ft
4
100lb1.5ft    3 P 3ft    4 P 4ft   0
5
P  107 lb


Fs 
17

4 ft
F
0
y
3
N  P  W  164.3 lb
5

5
F
x
W = 100 lb
Fs
0
4
P  85.6 lb
5
© 2007 S. Kenny, Ph.D., P.Eng.
3 ft
x
 1.5 ft
2
ENGI 1313 Statics I – Lecture 37
N
Example 37-02 (cont.)

Check Fs
N
3
P  W  164.3 lb
5
Fs 
4
P  85.6 lb
5
P
3 ft
3
4
4 ft
Fs  85.6lb  sN  0.5164.3lb  82.2lb


18
 Slipping
Calculate minimum P based on
slipping condition
© 2007 S. Kenny, Ph.D., P.Eng.
W = 100 lb
Fs
3 ft
x
 1.5 ft
2
ENGI 1313 Statics I – Lecture 37
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References
Hibbeler (2007)
 http://wps.prenhall.com/esm_hibbeler_eng
mech_1

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© 2007 S. Kenny, Ph.D., P.Eng.
ENGI 1313 Statics I – Lecture 37