Transcript ENGR-36_Lec-06_Particle-Equilibrium_H13e
Engineering 36
Chp 3: Particle Equilibrium
Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]
Engineering-36: Engineering Mechanics - Statics
1 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Learning Goals
Determine WHEN “Particle” Analysis can be Applied (even to Large Systems) • Determine if a Point of Concurrency Exists – Body and NO Tendency to “Twist” Draw
F
ree
B
ody
D
iagrams for Particles • Isolate particle and show Forces acting on the particle Use Particle-Equilibrium Criteria to solve for Unknown Forces
Engineering-36: Engineering Mechanics - Statics
2 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Rigid Bodies
Most Bodies In Elementary Mechanics Are Assumed To Be RIGID • i.e., Actual Deformations Are Small And Do Not Affect The Force and/or Moment analysis of the System DEFORMABLE Body Mechanics are the Subject of Later Courses • Intro to this in ENGR45 3 • More Full Treatment in a 3 rd Year Mechanics of Materials Course Bruce Mayer, PE
Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Full Mechanical Equilibrium
A Rigid Body in Static Mechanical Equilibrium is Characterized by • Balanced External Forces & Torques A Body/Force/Moment System will have no Tendency to Toward TRANSLATIONAL (forces) or ROTATIONAL (torques) Motion of the Body
Engineering-36: Engineering Mechanics - Statics
4 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Special Case
Particles
In Mechanics even very Large Bodies can be regarded as “Particles” if the Body meets Certain Criteria A 3D (or 2D) Rigid Body may be regarded as a Particle If: • There are No APPLIED Torques • ALL Forces acting on the Body are CONCURRENT – That is, all the Force LoA’s Pass Thru a COMMON Point
Concurrent Forces Engineering-36: Engineering Mechanics - Statics
5 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Special Case: Particles
The Common Point can be Called the Point of Concurrency (PoC) Use The PoC as the Point that represents the Entire Body.
• That is, the action of all forces act on a PARTICLE located at the PoC Note that Concurrent forces Generate NO Tendency to Twist the Body 6 • Thus the Body is NOT Subjected to any Torques
Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Particle Analysis → Need PoC
Particle Analysis is MUCH easier than non-Particle Analysis However Improper Application of the Particle methods produce
Incorrect
results The Particle Idealization Applies ONLY when the
LoA’s of ALL Forces
applied to the Body Pass thru
ONE
Point • This Pt is called the Point of Concurrency
Engineering-36: Engineering Mechanics - Statics
7 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Particle Equilibrium
Recall Newton’s First Law A Similar Law applies to Twisting Actions
F
T
m
a
I
α
Bodies with a Point of Concurrency are NOT subject to Torques so Only the Force Equation Applies For NonMoving (static) or Constant Velocity systems a = dv/dt = 0 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Engineering-36: Engineering Mechanics - Statics
8
Particle Equilibrium
For Static or Constant-Velocity “Particles” the Condition of Equilibrium
F
m
0
F
0 By Component DeComposition: 2D :
F x
0
F y
0 3D :
F x
0
F y
0
F z
0 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Engineering-36: Engineering Mechanics - Statics
9
Particle Equilibrium Summary
The 2D Case The 3D Case • Note the PoC
Engineering-36: Engineering Mechanics - Statics
10 • Note the PoC Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Particle Example
• The Loads at
B
&
D
are known at 500 lb & 1200 lb • Assume weights of the members and Gusset plates are negligible 11 The Gusset Plate above is used to connect 4 members of a planar truss that is in equlibrium
Engineering-36: Engineering Mechanics - Statics
Find the loads
F
C
and
F
A
acting on the Gusset Plate Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Particle Example
Thus PARTICLE ANALYSIS applies in this situation Start with ΣF
x
= 0 Note that All the Force LoA’s have a Point of Concurrency (PoC)
F A F x
F C
0 or
F A
cos 60 1200
lbs
F C
cos 60 0 1200
lbs
Engineering-36: Engineering Mechanics - Statics
12 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Particle Example
Now by ΣF
y F y
0 500
lbs
F C
sin 60 = 0 0 or
F C
500
lbs
sin 60
Engineering-36: Engineering Mechanics - Statics
13 Thus
F C
577 .
4
lbs
Sub
F C
into previous eqn for
F A F A
F C
cos 60 1200
lb F A
500 sin
lb
60 cos 60 1200
lb F A
sin 500
lb
60 cos 60 1200
lb F A F A
or 500 tan
lb
60 288 .
7
lb
1200
lb
1200
lb F A
911 .
3
lb
Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Graphical Solution (1)
Use Known Mag & Dir to Draw scaled versions of
F
B
&
F
D
• Scaling Factor = 150 lb/inch Draw “X-lines” for the know LoA’s for
F
A
•
F
C
&
F
C
LoA is 60 ° off the Horizontal
Engineering-36: Engineering Mechanics - Statics
14 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Graphical Solution (2)
Connect the intersecting LoA’s to Define the Scaled-Magnitudes for
F
A
&
F
C
Then Measure with inch-Ruler Scale-Up using 150 lbs/inch
Engineering-36: Engineering Mechanics - Statics
15 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Scaling Compared to Calculation for Gusset Plate 585 Scale Calc 577.4
915 911.3
0 100 200 300
Engineering-36: Engineering Mechanics - Statics
16 400 500
Load (lbs)
600 700 800 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
900 1,000
Special Case: Frictionless Pulley
A FrictionLess Pulley is Typically used to change the Direction of a Cable or Rope in Tension Pulley with PERFECT Axel (FrictionLess)
Engineering-36: Engineering Mechanics - Statics
17 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Special Case: Frictionless Pulley
FrictionLess Pulleys (Atwoods Machines) will Change the DIRECTION of a Tension-Force, but NOT its MAGNITUDE The Direction is determined by the TANGENT-Point of the Cord as it passes over the Pulley Circumference
Engineering-36: Engineering Mechanics - Statics
18 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Special Case: Frictionless Pulley
For a frictionless pulley in static equilibrium, the
tension
in the cable is the
same
on
both sides
of the pulley
T
2
T
1
T
2
T
1
Engineering-36: Engineering Mechanics - Statics
19 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
FrictionLess Pulley – Special Case
Since the Cables/Ropes passing over a FrictionLess Pulley generate NO Moment About the Pulley Axel, then for this case the ΣM axel = 0 by Definition.
Thus in this case, as in 20 the Particle Case:
F Engineering-36: Engineering Mechanics - Statics
0 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Example: FrictionLess Pulley
Consider the Multiple Pulley System at Right • Assume the Pulleys are Frictionless & Massless For this System Determine the Weight of the Block, W
Engineering-36: Engineering Mechanics - Statics
21 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Example: FrictionLess Pulley
Using T 1 = T 2 Draw the FBD for Pulley-C
50 lb 50 lb
By the ΣF y T C = 0 find = 100 lbs Pulley-B FBD
100 lb 100 lb T C Engineering-36: Engineering Mechanics - Statics
22
T B
Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Example: FrictionLess Pulley
By the ΣF y T B = 0 find = 200 lbs Pulley-A FBD
200 lb 200 lb
By the ΣF y = 0 find W = 400 lbs
W Engineering-36: Engineering Mechanics - Statics
23
400 lbs
Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Special Cases Summarized
2D 3D Particle:
F x
0
F x
0
F y
0
F y
0
F z
0 FrictionLess Pulley:
T
1
T
2
Engineering-36: Engineering Mechanics - Statics
24 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
WhiteBoard Work
Lets Work a Pulley Problem
25
Both Pulleys may be Regarded as Free-Wheeling (FrictionLess) Engineering-36: Engineering Mechanics - Statics
25 Find for EQUILIBRIUM • ||
P
|| • Angle α Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
25
Engineering-36: Engineering Mechanics - Statics
26 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Engineering-36: Engineering Mechanics - Statics
27 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Engineering-36: Engineering Mechanics - Statics
28 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Engineering-36: Engineering Mechanics - Statics
29 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Engineering 36
Appendix
Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected]
Engineering-36: Engineering Mechanics - Statics
30 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx