Title: Ch3 - Force 3d
12-D Force Systems
3-D Force Systems
Force Moment, Couple Resultants
Force Moment,Couple Resultants
23D-Force Systems
- Rectangular Components, Moment, Couple, Resultants
3(No Transcript)
4Moment (3D)
A
moment axis
X
Moment about point P
Y
-Magnitude
d
-Direction
right-hand rule
P
-Point of application point O
O
(Unit newton-meters, N-m)
In 3D, forces (generally) are not in the same
plane.
In many cases on 3D, d (the
perpendicular distance) is hard to find. It is
usually easier to find the moment by using the
vector approach with cross product multiplication.
5Cross Product
-
-
-
Beware xyz axis must complies with right-hand
rule
6Moment (Cross Product)
Physical Meaning
Mx - Fyrz Fzry
Fz
z
Fy
A
Fx
My Fxrz - Fzrx
rz
y
rx
O
Mz -Fxry Fyrx
ry
x
7Moment About a Point 4
Moment
Resultant Moment of Forces
z
y
O
x
8Varignons Theorem (Principal of Moment)
- Two or more concurrent forces
- their moments about a point may be found in two
ways - for nonconcurrent forces see Resultants sections
(2D - 2/6, 3D- 2/9)
r
A
O
- Sum of the moments of a system of concurrent
forces about a given point equals the moment of
their sum about the same point
9 Determine the vector expression for the
moment of the 600-N force about point O. The
design specification for the bolt at O would
require this result.
Ans
10z
6m
x
0.8m
O
y
400N
P
1.2m
N-m Ans
11z
6m
x
0.8m
O
y
400N
1.2m
VD2
N-m Ans
12plus
plus
rx
rz
N-m Ans
Not-Recommended Method
13Example Hibbeler Ex 4-4 1
Moment
Determine the moment about the support at A.
14Example Hibbeler Ex 4-4 2
Moment
15Example Hibbeler Ex 4-4 3
Moment
16Example Hibbeler Ex 4-4 4
Moment
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18z
x
y
Moment about line
Moment about Point
( projection effect )
19Finding moment of force about (arbitary) axis ?
?
O
Depend on line l only, Not depend on point O
?, ?, ? are the directional cosines of the unit
vector
20?
F
F
A
Moment of about point A,B in the
direction of l
(generally)
where A, B on line l
Moment of in the direction of l Moment of
projected to line l
Moment of about line l
where A, B are any points on the line l
Moment about axis is sliding vector.
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22Finding moment of force about (arbitary) axis ?
?
O
Depend on line l only, Not depend on point O
?, ?, ? are the directional cosines of the unit
vector
23Moment about Point P
Moment about line l (Definition)
moment axis
Line l (moment axis)
X
X
Direction right-hand rule
d
P
O
d
Q
A Any point on line l
How to find Moment about line l ?
A
Hard to find
Hard to find
24Moment about line l
We will prove that
A Any point on line l
Line l (moment axis)
is equal to
X
Moment of about point A,Q projected to
line l
d
Q
A
must prove to be
A Any point on line l
24
25Moment about line l
Line l (moment axis)
is equal to
X
Point A is any point in the line l
Moment about axis is sliding vector.
d
Q
A
where A, B are any points on the line l
Moment of about line l
Moment of about point A in the
direction of l
Moment of in the direction of l
Moment of projected to line l
25
Moment of about point B in the direction
of l
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28Find of (the moment of about
z-axis passing through the base O )
y
A
15 m
T 10 kN
O
Ans
x
z
9 m
12 m
B
OK
OK
Figure must be shown
OK
not OK
292/133 A 5N vertical force is applied to the knob
of the window-opener mechanism when the crank BC
is horizontal. Determine the moment of force
about point A and about line AB.
D
r
N-mm
Ans
N-mm
N-mm
N-mm
Ans
30Example Hibbeler Ex 4-8 1
Moment
Determine the moments of this force about the x
and a axes.
31Example Hibbeler Ex 4-8 2
Moment
32Example Hibbeler Ex 4-9 1
Moment
Determine the moment MAB produced by F (600i
200j 300k) N, which tends to rotate the rod
about the AB axis.
33Example Hibbeler Ex 4-9 2
Moment
34Example Hibbeler Ex 4-9 3
Moment
Vector r is directed from any point on the AB
axis to any point on the line of action of the
force.
35(No Transcript)
36position vector from A to point of
application of the force
r
X
A
d
position vector from A to any point on line
of action of the force.
?
p
F
O
a
r
position vector from any point on line l to
any point on tline of action of the force.
r
A
X
Y
d
Z
37parallel with line l
O
P
Why?
Forces which interest or parallel with axis, do
not cause the moment about that axis
38(No Transcript)
39Couple
Couple is a summed moment produced by two force
of equal magnitude but opposite in direction.
d
B
A
O
from any point on line of the action to any
point on the other line of action
magnitude and direction Do not depend on O
Moment of a couple is the same about all point
? Couple may be represented as a free vector.
40The followings are equivalent couples
F
F
F
d/2
F
?
?
?
2F
F
2F
F
Every point has the equivalent moment.
2D representations (Couples)
couple is a free vector
M
M
M
41- Couple tends to produce a pure rotation of
the body about an axis normal to the plane of the
forces (which constitute the couple) i.e. the
axis of the couple.
- Couples obey all the usual rules that govern
vector quantities.
- Again, couples are free vector. After you add
them (vectorially), the point of application are
not needed!!!
- Compare to adding forces (i.e. finding
resultant), after you add the forces vectorially
(i.e. obtaining the magnitude and direction of
the resultant), you still need to find the line
of action of the resultant (2D - 2/6, 3D - 2/9).
4230 N
30 N
60?
- Replace the two couples with a single couple
that still produces the same external effect on
the block. - Find two forces and on two faces of
the block that parallel to the y-z plane that
will replace these four forces.
60?
0.06m
x
y
0.04m
0.05 m
25 N
25 N
0.1 m
(forces act parallel to y-z plane)
z
(25)(0.1) 2.5 N-m
60?
M
y
M
60?
z
(30)(0.06) 1.8 N-m
43Example Hibbeler Ex 4-13 1
Moment
Replace the two couples acting on the pipe column
by a resultant couple moment.
44Example Hibbeler Ex 4-13 2
Moment
45y
MO,240N-m
O
x
z
250mm
200mm
30O
240N-m
1200N
Vector Diagram
N-m Ans
46Concepts 1
Review
- Vectors can be manipulated by scalar
multiplication, addition, subtraction, dot
product, cross product and mixed triple product.
Vectors representing can be classified into free,
sliding and fixed vectors. - Position vectors describe the position of a point
relative to a reference point or the origin. - Statically, force is the action of one body on
another. In dynamics, force is an action that
tends to cause acceleration of an object. To
define a force on rigid bodies, the magnitude,
direction and line of action are required. Thus,
the principle of transmissibility is applicable
to forces on rigid bodies.
47Concepts 2
Review
- To define a moment about a point, the magnitude,
direction and the point are required. To define a
moment about an axis, the magnitude, direction
and the axes are required. To define a couple,
the magnitude and direction are required.
48Chapter Objectives Descriptions 1
- Use mathematical formulae to manipulate physical
quantities - Specify idealized vector quantities in real
worlds and vice versa - Obtain magnitude, direction and position of a
vector - Manipulate vectors by scalar multiplication,
addition, subtraction, dot product, cross product
and mixed triple product - Describe the physical meanings of vector
manipulations - Obtain position vectors with appropriate
representation.
49Chapter Objectives Descriptions 2
- Use and manipulate force vectors
- Identify and categorize force vectors
- Describe the differences between force
representation in rigid and deformable bodies - Identify and represent forces in real worlds with
sufficient data and vice versa - Manipulate force vectors
50Chapter Objectives Descriptions 3
- Use and manipulate moment vectors
- Identify and categorize moment vectors
- Describe the differences between moments about
points, moments about axes and couple - Identify and represent moments in real worlds
with sufficient data and vice versa - Manipulate moment vectors
51Review Quiz 1
Review
- Use mathematical formulae to manipulate physical
quantities - Give 4 examples of vector quantities in real
world. - In how many ways can we specify a 2D/3D vector?
Describe each of them. - How can we prove that two vectors are parallel?
- What are the differences between the vector
additions by the parallelogram and triangular
constructions? - Even though we can manipulate vectors
analytically, why do we still learn the graphical
methods?
52Review Quiz 2
Review
- Use mathematical formulae to manipulate physical
quantities - What are the mathematical definitions of dot,
cross and mixed triple products? - What are the physical meanings of addition,
subtraction, dot product, cross product and mixed
triple product? - What are the meanings of associative,
distributive and commutative properties of
products? - What are the differences between 2D and 3D vector
manipulation?
53Review Quiz 3
Review
- Obtain position vectors with appropriate
representation. - Given points A and B, what information do you
need to obtain the position vector and what name
will you give to the position vectors and
distance vector between the two points?
54Review Quiz 4
Review
- Use and manipulate force vectors
- For the following forces tension in cables,
forces in springs, weight, magnetic force, thrust
of rocket engine, what are their classification
in the following force types external/internal,
body/surface and concentrated/distributed forces? - If a surface is said to be smooth, what does that
mean? - What are the differences between force
representation in rigid and deformable bodies? - What are the additional cautions in force vector
manipulation that are not required in general
vector manipulation?
55Review Quiz 5
Review
- Use and manipulate moment vectors
- Give 5 examples of moments in real world and
approximate them into mathematical models. - What information do you need to specify a moment?
- What is the meaning of moment direction?
- If a force passes through a point P, what do you
know about the moment of the force about P? - What are the differences between physical
meanings of moments about points, moments about
axis and couples?
56Review Quiz 6
Review
- Use and manipulate moment vectors
- As couples are created from forces, why do we
write down the couple vectors instead of forces
in diagrams? - Given a couple of a point P, what do you know of
the couple about a different point Q? - If we know moments about different points or
axes, why cant we add components of moments as
in vector summation? - Why can we simply add couple components together?
57Resultant Definition
Resultant
- The force-couple systems or force systems can
be reduced to a single force and a single couple
(together called resultant) that exert the same
effects of - Net force ç Tendency to translate
- Net moment ç Tendency to rotate
- Two force-couple systems are equivalent if their
resultants are the same.
58(No Transcript)
59Force Couple Systems
A
A
A
B
B
B
No changes in the net external effects
Couple of about point B
from new location (point B) to any point on the
line of action of
calculated the same way as Moment of Point B by
the force F at the old position
(which applied at the old point)
60R
MO
Vector diagram
Move 3 forces to point O
Ans
Ans
N-m Ans
61Example Hibbeler Ex 4-15 1
Resultant
Replace the current system by an equivalent
resultant force and couple moment acting at its
base, point O.
62Example Hibbeler Ex 4-15 2
Resultant
63Example Hibbeler Ex 4-15 3
Resultant
64Recommended Problems
- 3D Moment and Couples
- 2/124 2/125 2/129 2/132
65(No Transcript)
66Wrench Resultant
O
O
O
3) Add the forces vectorially to get the
resultant force (since the forces are concurrent
now) and add the couple vectorially to get the
resultant couple
2) Replace each force with a force at point O a
couple
1) Pick a point (easy to find moment arms)
67Vector
Scalar (2D 3Plane)
68- The choice of point O is arbitrary
the resultant couple will not be the same for
each point O selected (in general), but the
resultant force will be the same.
O
Ex)
- The resultant couple cannot be cancelled by
moving the resultant force (in general).
M which // R, cannot be cancelled M which _ R,
can be cancelled.
Wrench Resultant (not very useful) - All force
systems can be represented with a wrench
resultant as shown in the figures
Positive if right-hand rule
Negative wrench
Positive wrench
69How to find Wrench Resultant
M-R plane
O
Vector approach see ex. 2/16
How to find ( knowing
)
70The simplest form of force-couple system
any forces couples system
3D
single-force single couple (which // with
each other)
wrench resultant
2D
any forces couples system
single-force system (no-couple)
O
OR single-couple system
Why 2D is different from 3D?
71Special cases Wrench Resultant
O
1) Coplanar 2D (Article 2/6)
O
2) Concurrent force the resultant will
pass through the point of concurrency. No
resultant moment at concurrent point. Pick the
point of concurrency!
z
3) Parallel forces (not in same plane)
x
O
single-force system (no-couple)
y
OR single-couple system
72Sample problem 2/13
Find the resultant
z
Move all force to point O
70 N-m
50 N
100 N-m
80 N
50 N
96 N-m
O
1.2
80 N
100 N
x
1.6
1
Ans
100 N
y
pass thru O no need to calculate couple
7350 N
Find the resultant
z
Move all force to point O
x
O
500 N
.35
y
R _ M
300 N
200 N
.35
0.5
Moving R can erase M completely
0.5
z
New point (x,y,z)
x
O
R
y
M
Which quadrant?
74(No Transcript)
75Find the wrench resultant, give coordinates on
x-y plane that the wrench resultant acts.
Solution 0 (Wrong)
Move all force to point O
Move R to point P (x,y,z), to cancel the couple
z
y
P
unable to solve!!
x
z
O
y
Generally in 3D, we can not change force-couple
system to single-force system.
x
76Find the wrench resultant, give coordinates on
x-y plane that the wrench resultant acts.
Solution 1 Direct Method
Move all force to point O
negative wrench
77Find the wrench resultant, give coordinates on
x-y plane that the wrench resultant acts.
new point P (x,y,z)
old point O (0,0,0)
line of action
Ans
78Find the wrench resultant, give coordinates on
x-y plane that the wrench resultant acts.
Solution 2 Equivalent System
Assume (x,y,0) is the point where wrench passes.
Parallel Condition
M ( or is ok)
79Find the wrench resultant, give coordinates on
x-y plane that the wrench resultant acts.
The coordinate in x-y plane, where wrench
resultant passes
Magnitude 2.4 N-m
Direction opposite with R
(negative wrench)
(negative wrench)
Ans
80Find the wrench resultant, give coordinates on
x-y plane that the wrench resultant acts.
Solution 3 wrench condition
Move forces to P (x,y,0)
z
wrench condition
y
P
x
P
O
y
x
Take it as the other unknown
81Find the wrench resultant, give coordinates on
x-y plane that the wrench resultant acts.
M ( or is ok)
The coordinate in x-y plane, where wrench
resultant passes
Magnitude 2.4 N-m
(negative wrench)
Direction opposite with R
(negative wrench)
Ans
82Hibbeler Ex 4-136
The three forces acting on the block each have a
magnitude of 10 N. Replace this system by a
wrench and specify the point where the wrench
intersects the z axis, measured from point O.
y
erasable
x
Positive wrench
Ans
83Hibbeler Ex 4-136
84Example Hibbeler Ex 4-136 1
Equivalent System
The three forces acting on the block each have a
magnitude of 10 N. Replace this system by a
wrench and specify the point where the wrench
intersects the z axis, measured from point O.
85Example Hibbeler Ex 4-136 2
Equivalent System
86Example Hibbeler Ex 4-136 3
Equivalent System
87Example Hibbeler Ex 4-136 4
Equivalent System
88- ?????????????????????????
- ??????????????? ??????????????????????????????????
?????? ?????????????? ?? ?????????????????????????
???????? - - ????????? ?????? moment/couple
???????????????????????????? ?????????????????
?????????????? xy ????????????????????????????????
???? - - ???????? ???????????????????????????????????????
?????????
89Reduction Summary
Equivalent System
Single force single couple
General force systems
2D force systems
Single force or single couple
simplest systems
3D force systems
Wrench
90A flagpole is guyed by 3 cables. If the tensions
in the cables have the same magnitude P (N),
replace the forces exerted on the pole with an
eqivalent wrench and determine the resultant
force R and the point where the axis of the
wrench intersects the x-z plane
Assume (x,0,z) is the point where wrench passes.
y
z
x
P (x,0,z)
wrench condition
91y
z
x
P (x,0,z)
92wrench condition
M ( or is ok)
y
z
x
P (x,0,z)
93M has no component in the direction of R.
We can move R to new position to eliminate
this couple completely
Force Systems
Resultants (3D)
94Ans
95z
F
a
c
F
O
y
b
x
VD1
We move R to the new location (x,y,z) where there
is no couple.
Generally in 3D, we can not change force-couple
system to single-force system.
96F
a
c
F
O
y
b
x
VD1
Ans
Note we can calculate wrench just in 1 step see
sample 2/16.
97Sample problem 2/15
Replace the two force and the negative wrench by
a single force and a couple at A
x
700 N
30mm
500 N
60
25 N-m
z
40
A
100 mm
80 mm
50mm
60mm
45
120 mm
600 N
40 mm
y
98Recommended Problems
- 3D Resultants
- 2/140 2/142 2/149 2/150
99(No Transcript)
100Equivalent System
A
A
?
B
B
P
P
C
C
force-couple System B
force-couple System A
In Statics Mechanics, we treat these two systems
are equivalent if and only if
(Pure Tendency to translate)
(Pure Tendency to rotate)
(in fact, just any one point is ok)
(just one point, and can be any point)
- Tendency to translate
- Net moment ç Tendency to rotate
- Two force-couple systems are equivalent if their
resultants are the same.
101Equivalent System
- useful for
- reducing any force-couple system
- gt simplest
resultant
- General (3D) Force System
- Concurrent Force System
- Parallel Force System
- Coplanar Force System (2D System)
102General-3D Force Systems
simplest system
z
y
x
P
O
y
x
103 Concurrent Force Systems (and no couple)
simplest system
O
O
No benefit to use, because it is satisfied by
default (moment at O)
104Coplanner System
simplest system
x
x
y
y
O
O
for most case (99.9)
(Moment at point O)
105(No Transcript)
106Example Hibbeler Ex 4-16 1
Equivalent System
Determine the magnitude, direction and location
on the beam of a resultant force which is
equivalent to the system of forces measured from
E.
107Example Hibbeler Ex 4-16 2
Equivalent System
108Example Hibbeler Ex 4-16 3
Equivalent System
109Example Hibbeler Ex 4-19 1
Equivalent System
Determine the magnitude and direction of a
resultant equivalent to the given force system
and locate its point of application P on the
cover plate.
110Example Hibbeler Ex 4-19 2
Equivalent System
111Example Hibbeler Ex 4-19 3
Equivalent System
112Reduction 3D System to a Wrench 1
Equivalent System
113Reduction 3D System to a Wrench 2
Equivalent System