Transcript Lecture 7

ENGR 3340: Fundamentals of Statics and Dynamics
Fundamentals of Statics and
Dynamics - ENGR 3340
Professor: Dr. Omar E. Meza Castillo
[email protected]
http://facultad.bayamon.inter.edu/omeza
Department of Mechanical Engineering
of Statics and Dynamics
ENGR 3340: Fundamentals
Universidad Interamericana
MSP21
- Bayamón
Tentative Lectures Schedule
Topic
Lecture
Equilibrium of a Particle in 3-D
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MSP21
- Bayamón
One thing you learn in science is that there is
no perfect answer, no perfect measure.
A. O. Beckman
Topic 7: Equilibrium of a Particle in
3-D
Equilibrium
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Objectives
 To introduce the concept of the free-body
diagram for a particle
 To show how to solve particle equilibrium
problems using the equations of equilibrium
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Three-Dimensional Force System.
Equilibrium
F  0
Particle moves with constant velocity or remains at rest.
Resolve the forces into their i, j, k components
 Fxi   Fy j   Fzk   0
  Fx  0

  Fy  0

  Fz  0
Three equations  Three unknowns
Mathematical manipulation vs. Physical interpretation
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Universidad Interamericana
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Three-Dimensional Force System - Physical Interpretation
Free Body Diagram
• Establish the x, y, z axes in any suitable orientation.
• Label all known and unknown force magnitudes and directions.
• The sense of a force having an unknown magnitude can be
assumed.
Types of Forces
Weight:
Spring:
W  mg
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F  ks
of Statics and Dynamics
ENGR 3340: Fundamentals
Universidad Interamericana
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Three-Dimensional Force System - Mathematical Manipulation
F  Fu
Position Vectors
r
uAC  AC
rAC
r  rAB  rC  rA
 (xC  x A ) i  (yC  y A ) j  (zC  z A ) k
Direction Cosines
uF1  cos i  cos  j  cos  k
 cos 60 i  cos 60 j  cos135 k
Projection angles (depends on each problem)
uF2  cos 36.8 cos 30 i  cos 36.8 sin 30 j  sin 36.8 k
 4 3   41
3
 i      j    k
   
5  2  5 2
5
7
of Statics and Dynamics
ENGR 3340: Fundamentals
Universidad Interamericana
MSP21
- Bayamón
8
of Statics and Dynamics
ENGR 3340: Fundamentals
Universidad Interamericana
MSP21
- Bayamón
Example 1
Given: A 600 N load is supported
by three cords with the
geometry as shown.
Find: The tension in cords AB, AC
and AD.
Plan:
1) Draw a free body diagram of Point A. Let the unknown force
magnitudes be FB, FC, FD .
2) Represent each force in the Cartesian vector form.
3) Apply equilibrium equations to solve for the three unknowns.
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Solution 1
z
FD
FBD at A
FC
2m
1m
2m
A
FB
x
600 N
FB = FB (sin 30 i + cos 30 j) N
= {0.5 FB i + 0.866 FB j} N
F C = – FC i N
FD = FD (rAD /rAD)
= FD { (1 i – 2 j + 2 k) / (12 + 22 + 22)½ } N
= { 0.333 FD i – 0.667 FD j + 0.667 FD k } N
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30˚
y
of Statics and Dynamics
ENGR 3340: Fundamentals
Universidad Interamericana
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Solution 1
z
Now equate the respective i , j , k
components to zero.
FD
FBD at A
FC
2m
 Fx = 0.5 FB – FC + 0.333 FD = 0
1m
 Fy = 0.866 FB – 0.667 FD = 0
2m
 Fz = 0.667 FD – 600 = 0
A
FB
x
600 N
Solving the three simultaneous equations yields
FC = 646 N
FD = 900 N
FB = 693 N
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30˚
y
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ENGR 3340: Fundamentals
Universidad Interamericana
MSP21
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Homework5  http://facultad. bayamon.inter.edu/omeza/
Omar E. Meza Castillo Ph.D.
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Universidad Interamericana
MSP21
- Bayamón
¿Preguntas?
Comentarios
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ENGR 3340: Fundamentals
Universidad Interamericana
MSP21
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GRACIAS
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