Upload
others
View
32
Download
1
Embed Size (px)
Citation preview
Rectangular Components in Space
• Vector F is
contained in the
plane OBAC
• Resolve F into
horizontal and
vertical
components.
yh FF sin
yy FF cos
• Resolve Fh into
rectangular
components
sinsin
sin
cossin
cos
y
hz
y
hx
F
FF
F
FF
1ME101 - Division III Kaustubh Dasgupta
Spatial Components (Direction Cosines)
2ME101 - Division III Kaustubh Dasgupta
Spatial Components (Direction Cosines)
• With the angles between F and the axes,
kji
F
kjiF
kFjFiFF
FFFFFF
zyx
zyx
zyx
zzyyxx
coscoscos
coscoscos
coscoscos
3ME101 - Division III Kaustubh Dasgupta
• is a unit vector along the line of
action of F;
are the direction cosines
zyx cos and,cos,cos
Rectangular Components in Space
• Direction of force F
– Defined by location of two points
• M (x1, y1 and z1) and N (x2, y2 and z2)
d
FdF
d
FdF
d
FdF
kdjdidd
FF
zzdyydxxd
kdjdid
NMd
zz
yy
xx
zyx
zyx
zyx
1
and joining vector
121212
4ME101 - Division III Kaustubh Dasgupta
Rectangular Components in SpaceExample: The tension in the guy
wire is 2500 N. Determine:
a) components Fx, Fy, Fz of the
force acting on the bolt at A,
b) the angles qx, qy, qz defining the
direction of the force
SOLUTION:
• Based on the relative locations of the
points A and B, determine the unit
vector pointing from A towards B.
• Apply the unit vector to determine
the components of the force acting
on A.
• Noting that the components of the
unit vector are the direction cosines
for the vector, calculate the
corresponding angles.
5ME101 - Division III Kaustubh Dasgupta
Rectangular Components in SpaceSolution
• Determine the unit vector pointing from Atowards B.
m 3.94
m30m80m40
m30m80m40
222
AB
kjiAB
• Determine the components of the force.
kji
kji
FF
N 795N 2120N1060
318.0848.0424.0N 2500
kji
kji
318.0848.0424.0
3.94
30
3.94
80
3.94
40
6ME101 - Division III Kaustubh Dasgupta
Solution
• Noting that the components of the unit
vector are the direction cosines for the
vector, calculate the corresponding angles.
kji
kji zyx
318.0848.0424.0
coscoscos
5.71
0.32
1.115
z
y
x
Rectangular Components in Space
7ME101 - Division III Kaustubh Dasgupta
Product of 2 Vectors: Dot Product
• Dot Product (Scalar product)
– A.B = AB cos θ
– Applications
• Determination of the angle between two vectors
A
B
A.B = (Axi+Ayj+Azk).(Bxi+Byj+Bzk) = AxBx+AyBy+AzBz
A.B = ABcos
Obtain
8ME101 - Division III Kaustubh Dasgupta
Product of 2 Vectors: Dot Product
• Applications
– Determination of the projection of a vector on a
given axis
PAB
A.B = AxBx+AyBy+AzBz
PAB = Acos = (A.B)/B
AB
9ME101 - Division III Kaustubh Dasgupta
Product of 2 Vectors: Cross Product
• Cross Product (Vector Product)
– C = A B
• C = AB sin θ
10ME101 - Division III Kaustubh Dasgupta
Product of 2 Vectors: Cross Product
• Cross Product
Cartesian Vector
11ME101 - Division III Kaustubh Dasgupta
Product of 2 Vectors: Cross Product
• Cross Product
– Distributive property
• C (A+B) = C A + C B
i j k
Ax AY AZ
BX BY BZ
=
12ME101 - Division III Kaustubh Dasgupta
AB = (Axi+Ayj+Azk) (Bxi+Byj+Bzk)
= (AyBz- AzBy)i + (….)j + (….)k
Moment of a Force (Torque)
• Moment of a Force (F) @ point A
– Mo = r F
13ME101 - Division III Kaustubh Dasgupta
r = position
vector
directed from
O to any
point on the
line of
action of F
Moment of a Force
• Magnitude of Moment
– tendency of F to cause rotation of the body
about an axis along MO
FdrFrFMO )sin(sin
14ME101 - Division III Kaustubh Dasgupta
Moment of a Force
• Characteristic
– Moment arm (d = r sinα) does not depend on the
particular point on the line of action of F to
which the vector r is directed
• Sliding vector
– Line of action same as the moment axis
15ME101 - Division III Kaustubh Dasgupta
Moment of a Force
• Principle of Transmissibility
– Two forces are equivalent
• Same magnitude, direction and lines of action
• Same magnitude, direction and equal moments
about a given point
16ME101 - Division III Kaustubh Dasgupta
Moment of a Force
• Varignon’s Theorem
– Moment of a force about a point is equal to the
sum of the moments of the components of
the force about the same point
17ME101 - Division III Kaustubh Dasgupta
Moment of a System of Concurrent Forces
• Varignon’s Theorem
– Moment of the resultant of a system of
concurrent forces about a point is equal to the
sum of the moments of the of the individual
forces about the same point
18ME101 - Division III Kaustubh Dasgupta
kyFxFjxFzFizFyF
FFF
zyx
kji
kMjMiMM
xyzxyz
zyx
zyxO
The moment of F about O,
kFjFiFF
kzjyixrFrM
zyx
O
,
Rectangular Components of Moment
19ME101 - Division III Kaustubh Dasgupta
The moment of F about B,
FrM BAB
/
zyx
BABABAB
FFF
zzyyxx
kji
M
kFjFiFF
kzzjyyixx
rrr
zyx
BABABA
BABA
/
Rectangular Components of Moment
20ME101 - Division III Kaustubh Dasgupta
• Moment MO of a force F applied at
the point A about a point O,
FrMO
• Scalar moment MOL about an axis
OL is the projection of the
moment vector MO onto the axis,
FrMM OOL
Moment of a Force About a Given Axis
21ME101 - Division III Kaustubh Dasgupta
Application of Scalar Product
Moment of a Force About a Given Axis
• Significance of MOL
– Resolve into components
22ME101 - Division III Kaustubh Dasgupta
Moment of a Force About a Given Axis
• Significance of MOL
– MOL is a measure of the tendency of the F to
impart a rigid body rotation about the axis OL
• Moments of F about the coordinate axes
xyz
zxy
yzx
yFxFM
xFzFM
zFyFM
23ME101 - Division III Kaustubh Dasgupta
Moment: Example
24ME101 - Division III Kaustubh Dasgupta
Moment: Example
25ME101 - Division III Kaustubh Dasgupta
Moment produced by two equal, opposite and
noncollinear forces is called a couple.
Magnitude of the combined moment of the two
forces about O:
M = F(a+d) – Fa = Fd
Vector Algebra Method:
Moment of the couple
about point O:
FdrFM
Fr
Frr
FrFrM
BA
BA
sin
Moment of a Couple
The moment vector of the couple is
independent of the choice of the origin of the
coordinate axes, i.e., it is a free vector that can
be applied at any point with the same effect.
26ME101 - Division III Kaustubh Dasgupta
Two couples will have equal moments if
•2211 dFdF
• the two couples lie in parallel planes, and
• the two couples have the same sense or
the tendency to cause rotation in the same
direction.
Examples:
Moment of a Couple
27ME101 - Division III Kaustubh Dasgupta
• Consider two intersecting planes
P1 and P2 with each containing a
couple
222
111
planein
planein
PFrM
PFrM
• Resultants of the vectors also
form a couple
21 FFrRrM
• By Varignon’s theorem
21
21
MM
FrFrM
• Sum of two couples is also a couple that is
equal to the vector sum of the two couples
Addition of Couples
28ME101 - Division III Kaustubh Dasgupta
• A couple can be represented by a vector with magnitude
and direction equal to the moment of the couple.
• Couple vectors obey the law of addition of vectors.
• Couple vectors are free vectors, i.e., the point of application
is not significant.
• Couple vectors may be resolved into component vectors.
Representation of Couples by Vectors
29ME101 - Division III Kaustubh Dasgupta
Couple: Example
Moment reqd to turn the shaft connected at center of
the wheel = 12 Nm
• First case: Couple Moment
produced by 40 N forces = 12 Nm
• Second case: Couple Moment
produced by 30 N forces = 12 Nm
If only One hand is used F = 60N
Same couple moment will be produced
even if the shaft is not connected at the
center of the wheel
Couple Moment is a Free VectorL04
Equivalent Systems (Force-Couple Systems)
At support O:
WR = W1 + W2
(MR)O = W1d1 + W2d2
31ME101 - Division III Kaustubh Dasgupta
Equivalent Systems: Resultants
FR = F1 + F2 + F3
How to find d?
Moment of the Resultant force about the grip must be equal
to the moment of the forces about the grip
FRd = F1d1 + F2d2 + F3d3
Equilibrium Conditions
32ME101 - Division III Kaustubh Dasgupta