engineering mechanics: statics chapter 2: force systems
TRANSCRIPT
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Engineering Mechanics: Engineering Mechanics: Statics Statics
Chapter 2: Force Systems
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ForceForce SystemsSystems
Part A: Two Dimensional Force Systems
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ForceForce 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
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Vector Components 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
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2D force systems•Most common 2D resolution of a force vector
•Express in terms of unit vectors ,
Rectangular ComponentsRectangular Components
F
x
y
i
xF
yF
j i j
ˆ ˆ
cos , sin x y x y
x y
F F F F i F j
F F F F
2 2x yF F F F
1tan y
x
F
F
Scalar components – can be positive and negative
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2D Force Systems2D Force Systems Rectangular components are convenient for finding
the sum or resultant of two (or more) forces which are concurrent
R
1 2 1 1 2 2
1 2 1 2
ˆ ˆ ˆ ˆ ( ) ( )
ˆ ˆ = ( ) ( )
x y x y
x x y y
R F F F i F j F i F j
F F i F F j
Actual problems do not come with reference axes. Choose the most convenient
one!
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Example 2.1Example 2.1
The link is subjected to two forces F1 and F2. Determine the magnitude and direction of the resultant force.
2 2236.8 582.8
629 N
RF N N
1 582.8tan236.8
67.9
NN
Solution
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Example 2/1 (p. 29) Example 2/1 (p. 29)
Determine the x and y scalar components of each of the three forces
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Unit vectors
• = Unit vector in direction of
cos direction cosinex
x
V
V
Rectangular componentsRectangular components
V
n
x
y
i
xV
yV
j
ˆ ˆˆ ˆ
ˆ ˆ cos cos
x y yx
x y
V i V j VVVn i j
V V V V
i j
n
V
x
y
2 2cos cos 1x y
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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.
Problem 2/4Problem 2/4
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MomentMoment 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 is known as the moment M of the force 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
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MomentMoment 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!
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MomentMoment A vector approach for moment
calculations is proper for 3D problems. Moment of F about point A maybe
represented by the cross-product
where r = a position vector from point A to any point on the line of action of F
M = r x F
M = Fr sin = Fd
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Example 2/5 (p. 40)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.
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Problem 2/43 Problem 2/43
(a) Calculate the moment of the 90-N force about point O for the condition = 15º. (b) Determine the value of for which the moment about O is (b.1) zero (b.2) a maximum
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CoupleCouple Moment produced by two equal, opposite,
and noncollinear forces = couple
Moment of a couple has the same value for all moment center
Vector approach
Couple M is a free vector
M = F(a+d) – Fa = Fd
M = rA x F + rB x (-F) = (rA - rB) x F = r x F
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CoupleCouple Equivalent couples
Change of values F and d Force in different directions but parallel plane Product Fd remains the same
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Force-Couple SystemsForce-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
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Problem 2/67Problem 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
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ResultantsResultants The simplest force combination which can
replace the original forces without changing the external effect on the rigid body
Resultant = a force-couple system
1 2 3
2 2
-1
, , ( ) ( )
= tan
x x y y x y
y
x
R F F F F
R F R F R F F
R
R
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ResultantsResultants 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 MO
Find the line of action of R by requiring R to have a moment of MO
( )
= O
O
R F
M M Fd
Rd M
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Problem 2/79Problem 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.
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ForceForce SystemsSystems
Part B: Three Dimensional Force Systems
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Rectangular components in 3D
•Express in terms of unit vectors , ,
• cosx, cosy , cosz are the direction cosines
• cosx = l, cosy = m, cos z= n
Three-Dimensional Force Three-Dimensional Force SystemSystem
ˆ ˆ ˆ x y zF F i F j Fk
2 2 2x y zF F F F
i j k
cos , cos , cosx x y y z zF F F F F F
ˆ ˆ ˆ ( )F F li mj nk
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Rectangular components in 3D
• If the coordinates of points A and B on the line of action are known,
• If two angles and which orient the line of action of the force are known,
Three-Dimensional Force Three-Dimensional Force SystemSystem
2 1 2 1 2 1
2 2 22 1 2 1 2 1
ˆ ˆ ˆ( ) ( ) ( )
( ) ( ) ( )F
x x i y y j z z kABF Fn F F
AB x x y y z z
cos , sin
cos cos , cos sinxy z
x y
F F F F
F F F F
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Problem 2/98Problem 2/98 The cable exerts a tension of 2 kN on the fixed bracket at
A. Write the vector expression for the tension T.
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Dot product
Orthogonal projection of Fcos of F in the direction of Q Orthogonal projection of Qcos of Q in the direction of F
We can express Fx = Fcosx of the force F as Fx =
If the projection of F in the n-direction is
Three-Dimensional Force Three-Dimensional Force SystemSystem
cosP Q PQ
F i
F n
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ExampleExample Find the projection of T along the line OA
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Moment of force F about the axis through point O is
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 Mo is normal to the plane in the direction
established by the right-hand rule
Evaluating the cross product
Moment and CoupleMoment and Couple
MO = r x F
ˆ ˆ ˆ
O x y z
x y z
i j k
M r r r
F F F
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Moment about an arbitrary axis
known as triple scalar product (see appendix C/7)
The triple scalar product may be represented by the determinant
where l, m, n are the direction cosines of the unit vector n
Moment and CoupleMoment and Couple
( )M r F n n
x y z
x y z
r r r
M M F F F
l m n
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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 Mz of T about the z-axis passing through the base O.
Sample Problem 2/10 Sample Problem 2/10
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A force system can be reduced to a resultant force and a resultant couple
ResultantsResultants
1 2 3
1 2 3 ( )
R F F F F
M M M M r F
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Any general force systems can be represented by a wrench
Wrench ResultantsWrench Resultants
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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
Problem 2/143Problem 2/143
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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
ResultantsResultants
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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
Problem 2/142Problem 2/142