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8/11/2019 48640 Lecture Note Week 2(2)
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8/11/2019 48640 Lecture Note Week 2(2)
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8/11/2019 48640 Lecture Note Week 2(2)
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APPLICATIONS
As the slider block A moves horizontally to the left with
vA, it causes the link CB to rotate counterclockwise. Thus
vBis directed tangent to its circular path.
Which link is undergoing general plane motion? Link AB
or link BC?
How can the angular velocity, , of link AB be found?
Planetary gear systems are used in
many automobile automatic
transmissions. By locking or releasing
different gears, this system can
operate the car at different speeds.
How can we relate the angular velocities of the
various gears in the system?
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Point A is called the base pointin this analysis. It generally
has a known motion. The x'- y frame translates with the
body, but does not rotate. The displacement of point B
ABAB ddd
/rrr
due to
rotation
aboutA
due to
translation
aboutA
due to translation
and rotation
x-y frame provides
significant convenience
to study the rotation of
point B about A.
Velocity Relationship between the velocities of pointsAand B
dt
d
dt
d
dt
dABAB /rrr
ABAB /vvv
Since the body AB is taken as rotating about A,
Here only has a kcomponent since the axis of rotation isperpendicularto the plane of translation.
or
ABABAB rdtrd
/// /
v
Sometimes it
may be
impossible to
adopt this
equationdirectly and
we have to go
back to
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When using the relative velocity equation, points A and B
should generally be points on the body with a known
motion. Often these points are pin connections in
linkages.
Point A on link AB must move along
a horizontal path, whereas point B
moves on a circular path.The directions of and areknown since they are always tangent
to their paths of motion.
Case A
vB= vA + rB/A
Furthermore, point B at the center of the wheel moves
along a horizontal path. Thus, has a known direction,e.g., parallel to the surface.
When a wheel rolls without slipping, point A is often
selected to be at the point of contact with the ground.
Since there is no slipping, point A has zero velocity.
ABAB /rvv
Case B
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3. If the solution yields a negative answer, the sense of
direction of the vector is oppositeto that assumed.
2. Express the vectors in Cartesian vector form(CVN) and
substitute them into vB = vA + rB/A. Evaluate thecross product and equate respective i and j
components to obtain twoscalar equations.
1. Establish the fixedx - ycoordinate directions and drawthe kinematic diagram of the body, showingthe vectors vA, vB, rB/A and . If the magnitudes areunknown, the sense of direction may be assumed.
Procedure for Analysis
VECTOR ANALYSIS
It is possible to determine the directions of some
parameters by observation even their magnitudes are
unknown.
It is OK to show the observed correct directions in the
kinematic diagram.
However, when we do vector analysis, we can always
assume the direction of these parameters that have
unknown parameters (no matter how we express them
in the kinematic diagram). The final answer from the
vector analysis can show the correct sense
automatically.
VECTOR ANALYSIS
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3. Write the scalar equations from the x and y
components of these graphical representations of the
vectors. Solve for the unknowns.
1. Establish the fixedx-ycoordinate directions and draw akinematic diagramfor the body. Then establishthe magnitude and direction of the relative velocity
vector /.
2. Write the equation vB = vA + vB/A. In the kinematic
diagram, represent the vectors graphically by showing
their magnitudes and directions underneath each
term.
Procedure for Analysis
SCALAR ANALYSIS
Kinematic Diagram(s)
Must Be Presented
in Solution
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READING QUIZ
1. When a relative-motion analysis involving two sets of coordinate
axes is used, thex- ycoordinate system will
A) be attached to the selected point for analysis.
B) rotate with the body.
C) not be allowed to translate with respect to the fixed frame.
D) None of the above.
2. In the relative velocity equation, vB/A is
A) the relative velocity of B with respect to A.
B) due to the rotational motion.
C) rB/A.
D) All of the above.
The link is guided by two blockAand B, which move in the
fixed slots. If the velocity of A is 2 m/s downward,
determine the velocity of B at the instant= 45.
Example I (Example 16.6)
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Solution
Vector Analysis
Kinematic DiagramSince points A and B are restricted to move and vA is
directed downward, vB must be directed horizontally to
the right.
This motion causes the link to
rotate CCW
Apply the velocity equation
to pointsAand Bin order to
solve for the two unknown
magnitudes vBand
ABAB/
vvv
SolutionVector Analysis
Velocity Equation
We have
Equating theiandjcomponents gives
ijji
jikji
rvv
45cos2.045sin2.02
)]45cos2.045sin2.0([2
/
B
B
ABAB
v
v
m/s2,rad/s1.1445sin2.020
45cos2.0
B
B
v
v
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The collar Cis moving downward with a velocity of 2 m/s.
Determine the angular velocity of CBat this instant.
Example II (Example 16.8)
Only know
Two unknown
and
so two equations expected
SolutionVector Analysis
Kinematic Diagram
CBandABrotate counterclockwise.
Velocity Equation
Link CB(general plane motion):
m/s2andrad/s10
2.020
2.0
2.02.02
)2.02.0(2
/
BCB
CB
CBB
CBCBB
CBB
CBCBCB
v
v
v
v
ijji
jikji
rvv
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Solution
Velocity Equation
Link AB(rotation about a fixed axis)
rad/s10
2.02
)2.0(2
AB
AB
AB
BABB
jki
rv
1. Which equation could be used to findthe velocity of the center of the gear, C,if the velocity vAis known?
A) vB= vA+ gearrB/A B) vA= vC+ gearrA/C
C) vB= vC+ gearrC/B D) vA= vC+ gearrC/A
vA
2. If the bars velocity at A is 3 m/s, whatbase point (first term on the RHS of the
velocity equation) would be best used to
simplify finding the bars angular velocity
when = 60?
A) A B) B
C) C D) No difference.
A
4 m
B
C
CONCEPT QUIZ
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Given:
Roller A is moving to the right at 3m/s.
Find:
The velocity of B at the instant
= 30.
1. Establish the fixed x-y directions and draw a kinematic
diagram of the bar and rollers.
2. Express each of the velocity vectors for A and B in
terms of their i,j, kcomponents and solve
Example III
ABAB /rvv
Plan:
Equating the iandjcomponents gives:
0 = 30.75
-vB=1.299
Express the velocity vectors in CVN
vB= vA + rB/A
-vBj= 3 i + [ k
(-1.5cos30i+1.5sin 30j)]
-vBj= 3 i1.299 j0.75 i
Solution:
Solving: = 4 rad/s or = 4 rad/s k
vB= 5.2 m/s or vB= -5.2 m/sj
y
Kinematic diagram:
y
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Given:
Crank rotates OA with an
angular velocity of 12 rad/s.
Find:
The velocity of piston B and the
angular velocity of rod AB.
Notice that point A moves on a circular path. The
directions of vA
is tangent to its path of motion.
Draw a kinematic diagram of rod AB and use
vB= vA+AB rB/A
Example IV
Plan:
By comparing the i, jcomponents:
i: 0 = -3.6 + 0.3 AB AB = 12 rad/s
j: vB = 0.5196AB vB = 6.24 m/s
Rod AB. Write the relative-velocity equation:
vB= vA+ AB rB/A
Since crank OA rotates with an
angular velocity of 12 rad/s, the
velocity at Awill be:
vA= -0.3(12) i= -3.6 im/s
Kinematic diagram of AB
vBj = -3.6 i+ AB k (0.6cos30 i 0.6sin30j)
vBj = -3.6 i+ 0.5196 ABj+ 0.3 AB i
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GROUP PROBLEM SOLVING
Given:
The shaper mechanism is designed togive a slow cutting stroke and a quick
return to a blade attached to the
slider at C. The link AB is rotating at
AB = 4 rad/s.
Notice that link AB rotates about a fixed point A. Thedirections of vB is tangent to its path of motion. Draw a
kinematic diagram of rod BC. Then, apply the relative
velocity equations to the rod and solve for unknowns.
Find:
The velocity of slider block C when = 60.
Plan:
Solution:
Draw a kinematic diagram of rod BC
Since link AB is rotating at AB = 4 rad/s,
the velocity at point B will be:
vB= 4 (300) = 1200 mm/s
At= 60, vB= -1200 cos 30i+1200 sin 30j
= (-1039i+600j) mm/s
Notice that the slider
block C has a
horizontal motion.xy
vB
vC
rC/BBC
45
30
B
C
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Apply the relative velocity equation in order to find the
velocity at C.
vC= vB+
BC rC/B
BC =4.8 rad/s
vC= 1639 mm/s = 1639 mm/sx
y
vB
vC
rC/BwBC
45
30
B
C
vCi= (- 1039i+600j)
+BCk (-125 cos 45i+125 sin 45j)
vCi= (1039125 BC sin 45)i+(600 125 BC cos 45)j
Equating the iandjcomponents yields:
vC= 1039125 BC sin 45
0 = 600 125 BC cos 45
1. If the disk is moving with a velocity at
point O of 15 m/s and = 2 rad/s,
determine the velocity at A.
A) 0 m/s B) 4 m/s
C) 15 m/s D) 11 m/s
2. If the velocity at A is zero, then determine the
angular velocity, .
A) 30 rad/s B) 0 rad/s
C) 7.5 rad/s D) 15 rad/s
2 mV=15 m/s
A
O
ATTENTION QUIZ