Lecture #9 Control Engineering REVIEW SLIDES Reference: Textbook by Phillips and Habor
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VDB1023
Engineering Mechanics
Lecturer
Dr Zubair Imam Syed
Email: [email protected]
Ph: 05 368 7313
Room: 14.03.13
Week 14 Review Lecture
mailto:[email protected]
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Force
Resultant of Forces
Components of a force
Position vector
Force in Cartesian vector form
Can you find the magnitude and direction of the resultant of the force system given
below
Force and Moment
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30m
6
12
8
A man is pulling the cord with a 300 N force. Represent this force as a Cartesian
vector and determine its direction.
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Exercise If F3 = 15 kN, =35, and = 40, determine the magnitude and coordinate angle of the resultant force acting on the ball-and-socket joint.
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Moment of a Force Vector Formulation
zyx
zyxO
FFF
rrr
kji
FrM
Determine the moment produced by each of the forces
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Equilibrium Conditions and applications
Fx = 0
Fy = 0
Fz = 0
These equations are the three scalar equations of equilibrium.
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Reactions on a Rigid body
Two force member
Reactions at hinge and fixed supports
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Example
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Truss Analysis
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Problem
Find the member forces in members KJ, CD and CJ
for the forces shown in the figure below.
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Centre of Gravity, Centroid
A y dA
A dA
y =
~
A x dA
A dA
x =
~
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Centroid of Complex Shapes
i i
i
x Ax=
A
i i
i
y Ay=
A
x
y
i i
i
z Az=
A
Centroid Location of Complex Shapes
x
y
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Centroid for an area is determined by the first moment of an area about an axis
Second moment of an area is referred as the moment of inertia
Moments of Inertia
Consider area A lying in the x-y plane
Be definition, moments of inertia of the differential plane area dA about the x and y axes
For entire area, moments of
inertia are given by
Ay
Ax
yx
dAxI
dAyI
dAxdIdAydI
2
2
22
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y
y
x
C
x
50
35
10
50
35 20 20
10
10 10
Calculate the moment of inertia about the centroid
axis,
Problem
x
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oo
oo
o
ssavv
attvss
atvv
2
2
1
22
2
Rectilinear Kinematics
dt
dsv
dt
dva dvvdsa
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Curvilinear Motion
nt uua nt aa
2va
vdvdsava
n
tt
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Equations of Motion
maF
dt
dvmmaF tt
2vmmaF nn
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Solution
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Conservation of Linear Momentum
smv
v
vmmvmvm BABBAA
/5.0
)27000()75.0)(12000()5.1)(15000(
)()()(
2
2
211( )
Momentum and Impulse
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Plan:
1) Determine the speed of the crate just before the collision using projectile
motion or an energy method.
2) Analyze the collision as a central impact
problem.
Problem A 2 kg crate B is released from rest, falls a distance h = 0.5 m, and
strikes plate P (3 kg mass). The coefficient of restitution between B and
P is e = 0.6, and the spring stiffness is k = 30 N/m. The velocity of crate
B just after the collision.
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0 + 0 = 0.5(2)(v2)2 + (2)(9.81)(-0.5)
T1 + V1 = T2 + V2
0.5m(v1)2 + mgh1 = 0.5m(v2)
2 + mgh2
v2 = 3.132 m/s
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Apply conservation of momentum to the system in the
vertical direction:
Analyze the collision as a central impact problem.
+ mB(vB)1 + mP(vP)1 = mB(vB)2 + mP(vP)2
(2)(-3.132) + 0 = (2)(vB)2 + (3)(vP)2
(vB)2
(vP)2 (vP)1 = 0
(vB)1 = 3.132 m/s
B
P
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Using the coefficient of restitution:
+ e = [(vP)2 (vB)2]/[(vB)1 (vP)1]
0.6 = [(vP)2 (vB)2]/[-3.132 0]
-1.879 = (vP)2 (vB)2
Solving the two equations simultaneously yields
(vB)2 = -0.125 m/s and (vP)2 = -2.00 m/s
Both the block and plate will travel down after the collision.
(vB)2
(vP)2 (vP)1 = 0
(vB)1 = 3.132 m/s
B
P