Transcript Engineering Mechanics: Statics
Engineering Mechanics: Statics
Chapter 2: Force Systems
Force Systems Part A: Two Dimensional Force Systems
Force
An action of one body on another Vector quantity External and Internal forces Mechanics of Rigid bodies: Principle of Transmissibility • • Specify magnitude , direction , line of action No need to specify point of application Concurrent forces • Lines of action intersect at a point
Vector Components
A vector can be resolved into several
vector components
Vector sum of the components must equal the original vector Do not confused vector components with perpendicular projections
Rectangular Components
F y j y
2D force systems
• Most common 2D resolution of a force vector • Express in terms of unit vectors ,
F F
F x F x
F y
F
x
y
ˆ
F y
F
i x F x
Scalar components – can be positive and negative
F x
2
F y
2 tan 1
F y F x
2D Force Systems
Rectangular components are convenient for finding are concurrent
R
R
F
1
F
2 = (
F
1
x
x
( 1
x F
1
y
1
y
ˆ
y
) ˆ
x
ˆ 2
y
ˆ ) Actual problems do not come with reference axes. Choose the most convenient one!
Solution
Example 2.1
The link is subjected to two forces F 1 and F 2 . Determine the magnitude and direction of the resultant force.
F R
236.8
N
629 N tan 1 582.8
236.8
N N
67.9
582.8
N
2
Example 2/1 (p. 29)
Determine the x and y scalar components of each of the three forces
Rectangular components
V y j y
y
x
V
Unit vectors
•
n
= Unit vector in direction of
V n i x n
V V
x
ˆ
V y
V x V i
ˆ
V y V j
ˆ
x i
ˆ cos
y
ˆ
j V x V x V
cos
x
direction cosine cos
x
2 cos
y
2 1
Problem 2/4
The line of action of the 34-kN force runs through the points A and B as shown in the figure. (a) Determine the x and y scalar component of F.
(b) Write F in vector form.
Moment
In addition to tendency to move a body in the direction of its application, a force tends to rotate a body about an axis.
The axis is any line which neither intersects nor is parallel to the line of action This
rotational tendency moment
M of the force is known as the Proportional to force F and the perpendicular distance from the axis to the line of action of the force d The magnitude of M is
M = Fd
Moment
The moment is a vector M perpendicular to the plane of the body. Sense of M is determined by the right hand rule Direction of the thumb = arrowhead Fingers curled in the direction of the rotational tendency In a given plane (2D),we may speak of moment about a point which means moment with respect to an axis normal to the plane and passing through the point.
+, - signs are used for moment directions – must be consistent throughout the problem!
Moment
A vector approach for moment calculations is proper for 3D problems.
Moment of F about point A maybe represented by the cross-product M = r x F where r = a position vector from point A to
any
point on the line of action of F M = Fr sin
a
= Fd
Example 2/5 (p. 40)
Calculate the magnitude of the moment about the base point O of the 600-N force by using both scalar and vector approaches.
Problem 2/43
(a) Calculate the moment of the 90-N force about point O for the condition = 15º. (b) Determine the value of maximum for which the moment about O is (b.1) zero (b.2) a
Couple
Moment produced by two equal, opposite, and noncollinear forces = couple M = F(a+d) – Fa = Fd Moment of a couple has the same value for all moment center Vector approach
M = r
A
x F + r
B
x (-F) = (r
A
- r
B
) x F = r x F Couple M is a free vector
Couple
Equivalent couples Change of values F and d Force in different directions but parallel plane Product Fd remains the same
Force-Couple Systems
Replacement of a force by a force and a couple Force F is replaced by a parallel force F and a counterclockwise couple Fd Example Replace the force by an equivalent system at point O Also, reverse the problem by the replacement of a force and a couple by a single force
Problem 2/67
The wrench is subjected to the 200-N force and the force P as shown. If the equivalent of the two forces is a for R at O and a couple expressed as the vector M = 20 kN.m, determine the vector expressions for P and R
Resultants
The simplest force combination which can replace the original forces without changing the external effect on the rigid body Resultant = a force-couple system 2
F
3
R x
F x
,
R y
F y
,
R
-1
R y R x
F F x
) 2
F y
) 2
Resultants
Choose a reference point (point O) and move all forces to that point Add all forces at O to form the resultant force R and add all moment to form the resultant couple M
O
Find the line of action of have a moment of
M O R
by requiring R to
R
F M O O
Problem 2/79
Replace the three forces acting on the bent pipe by a single equivalent force R. Specify the distance x from point O to the point on the x-axis through which the line of action of R passes.
Force Systems Part B: Three Dimensional Force Systems
Three-Dimensional Force System
Rectangular components in 3D • Express in terms of unit vectors , ,
F
F i
ˆ
F j
ˆ
z F x
F
F y
F
F z
F
cos
z F
F x
2
F y
2
F z
2 • cos x , cos y , cos z are the direction cosines • cos x = l, cos y = m, cos z = n
F
F li
mj
ˆ
nk
Three-Dimensional Force System
Rectangular components in 3D • If the coordinates of points A and B on the line of action are known,
F
Fn F
F AB AB
F
(
x
2 (
x
2
x i
ˆ
x
1 ) 2
y
2 (
y
2
y
1 ) 2 (
z
2 (
z
2
z
1 ) 2 • If two angles and f which orient the line of action of the force are known,
F xy F x
F F
f
F z
f
F y
F
sin f
F
cos sin
Problem 2/98
The cable exerts a tension of 2 kN on the fixed bracket at A. Write the vector expression for the tension T.
Three-Dimensional Force System
Dot product cos a Orthogonal projection of Fcos a Orthogonal projection of Qcos a of F in the direction of Q of Q in the direction of F We can express F
x
= Fcos
x
of the force F as F
x
= If the projection of F in the n-direction is
Example
Find the projection of T along the line OA
Moment and Couple
Moment of force F about the axis through point O is
M O
= r x F r runs from O to any point on the line of action of F Point O and force F establish a plane A The vector M
o
is normal to the plane in the direction established by the right-hand rule Evaluating the cross product
M
O
i
ˆ
r
x
F
x
j
ˆ
r
y
F
y
r
z
F
z
Moment and Couple
Moment about an arbitrary axis
M
) known as
triple scalar product
(see appendix C/7) The triple scalar product may be represented by the determinant
r
x
r
y
r
z
M
M
F l
x
F
y
F m n
z
where l, m, n are the direction cosines of the unit vector n
Sample Problem 2/10
A tension T of magniture 10 kN is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment M
z
of T about the z-axis passing through the base O.
Resultants
A force system can be reduced to a resultant force and a resultant couple 2
F
3 2
M
3
F
)
Wrench Resultants
Any general force systems can be represented by a wrench
Problem 2/143
Replace the two forces and single couple by an equivalent force-couple system at point A Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts
Resultants
Special cases • Concurrent forces – no moments about point of concurrency • Coplanar forces – 2D • • • Parallel forces (not in the same plane) – magnitude of resultant = algebraic sum of the forces Wrench resultant – resultant couple M is parallel to the resultant force R Example of positive wrench = screw driver
Problem 2/142
Replace the resultant of the force system acting on the pipe assembly by a single force R at A and a couple M Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts