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7/27/2019 Chapter 15 Last 1
http://slidepdf.com/reader/full/chapter-15-last-1 1/34
7/27/2019 Chapter 15 Last 1
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsT en t h
E d i t i on
Rate of Change With Respect to a Rotating Frame
15 - 2
• Frame OXYZ is fixed.
• Frame OXYZ rotates
about fixed axis OA
with angular velocity
• Vector functionvaries in direction
and magnitude.
t Q
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Vector Mechanics for Engineers: DynamicsT en t h
E d i t i on
Rate of Change With Respect to a Rotating Frame
15 - 3
k Q jQiQQ z y xOxyz
• With respect to the fixed OXYZ frame,
k Q jQiQk Q jQiQQ z y x z y xOXYZ
• With respect to the rotatingOxyz frame, k Q jQiQQ z y x
•
rate of change with respect to rotating frame.
Oxyz z y x Qk Q jQiQ
T
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Rate of Change With Respect to a Rotating Frame
15 - 4
• If were fixed withinOxyz then is
equivalent to velocity of a
point in a rigid bodyattached to Oxyz and
OXYZ Q
Qk Q jQiQ z y x
Q
• With respect to the fixed OXYZ frame,
QQQ Oxyz OXYZ
T E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 5
• Frame OXY is fixed and frameOxy rotates with angular
velocity .
• Position vector for the
particle P is the same in both
frames but the rate of change
depends on the choice of
frame.
P r
T E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 6
• The absolute velocity of the particle P is
OxyOXY P r r r v
• Imagine a rigid slab attached tothe rotating frame Oxy or F for
short. Let P’ be a point on theslab which corresponds
instantaneously to position of
particle P . T E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 7
Oxy P r v
F velocity of P
' P v absolute velocity of point P’
on the slab
• Absolute velocity for the particle P may be written as
F P P P vvv
along its path on the slab
T E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 8
F P P
Oxy P
vv
r r v
• Absolute acceleration for the particle P
is OxyOXY P r dt
d r r a
OxyOxy P r r r r a
2
OxyOxyOxy
OxyOXY
r r r dt
d r r r
but,
fT E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 9
F P P
Oxy P
vv
r r v
Oxy P
P
r a
r r a
F
• Utilizing the conceptual point P’ on the slab,
• Absolute acceleration for the
particle P becomes
22
2
F
F
F
P Oxyc
c P P
Oxy P P P
vr a
aaa
r aaa
Coriolis
accelera
tion
V t M h i f E i D iT E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 10
• Consider a collar P which ismade to slide at constant relative
velocity u along rod OB. The
rod is rotating at a constantangular velocity w . The point A
on the rod corresponds to the
instantaneous position of P .
c P A P aaaa
F
• Absolute acceleration of the collar is
V t M h i f E i D iT E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 11
c P A P aaaa
F
0 Oxy P r a F
uava c P c w 22 F
• The absolute acceleration consists of the
radial and tangential vectors shown
2w r ar r a A A
where
V t M h i f E i D iT E
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Vector Mechanics for Engineers: DynamicsT en t h
Ed i t i on
Coriolis Acceleration
15 - 12
uvvt t
uvvt
A
A
,at
,at
• Change in velocity over t is
represented by the sum of three
vectors T T T T R Rv
2w r ar r a A A
recall,
• is due to change in
direction of the velocity of
point A on the rod,
A At t
ar r t
vt
T T
2
00limlim w ww
T T
V t M h i f E i D iT e
E
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Vector Mechanics for Engineers: DynamicsTen t h
Ed i t i on
Coriolis Acceleration
15 - 13
uvvt t
uvvt
A
A
,at
,at
• result fromcombined effects of relative
motion of P and rotation of the
rod
T T R R and
uuu
t
r
t u
t
T T
t
R R
t t
w w w
w
2
limlim00
uava c P c w 22
F
recall,
V t M h i f E i D iT e
E
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Vector Mechanics for Engineers: DynamicsTen t h
Ed i t i on
Concept Question
2 - 14
v w
You are walking with a
constant velocity with
respect to the platform,
which rotates with a
constant angularvelocity w. At the
instant shown, in which
direction(s) will youexperience an
acceleration (choose all
that apply)?
x
y
OxyOxy P r r r r a
2
V t M h i f E i D iT e
E
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Vector Mechanics for Engineers: DynamicsTen t h
Edi t i on
Concept Question
2 - 15
v wa) +x
b) -x
c) +y
d) -ye) Acceleration = 0
x
y
OxyOxy P r r r r a
2
V t M h i f E i D iT e
E d
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Vector Mechanics for Engineers: DynamicsTen t h
Edi t i on
Sample Problem 15.9
15 - 16
Disk D of the Genevamechanism rotates
with constant
counterclockwiseangular velocity w D =
10 rad/s.
At the instant when f = 150o, determine (a)
the angular velocity of disk S , and (b) the
velocity of pin P relative to disk S .
V t M h i f E i D iT e
E d
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Vector Mechanics for Engineers: Dynamicsen t h
Edi t i on
Sample Problem 15.9
15 - 17
SOLUTION:
• The absolute velocity of the point P
may be written as
s P P P vvv
• Magnitude and direction of velocity
of pin P are calculated from theradius and angular velocity of disk D. P v
• Direction of velocity of point P ’ on
S coinciding with P is perpendicular to
radius OP.
P v
• Direction of velocity of P with
respect to S is parallel to the slot. s P v
• Solve the vector triangle for the
angular velocity of S and relative
velocity of P.
Vector Mechanics for Engineers D namicsT e
E d
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Vector Mechanics for Engineers: Dynamicsen t h
di t i on
Sample Problem 15.9
15 - 18
SOLUTION:
• The absolute velocity of the
point P may be written as
s P P P vvv
• Magnitude and direction of
absolute velocity of pin P arecalculated from radius and
angular velocity of disk D.
smm500srad10mm50 D P Rv w
Vector Mechanics for Engineers: DynamicsT eE
d
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Vector Mechanics for Engineers: Dynamicsen t h
di t i on
Sample Problem 15.9
15 - 19
• Direction of velocity of P with respect to S is parallel to
slot. From the law of cosines,
mm1.37551.030cos2 2222 r R Rl l Rr
From the law of cosines,
4.42742.030sinsin30sin
R sin
r
6.17304.4290
The interior angle of the vector
triangle is
Vector Mechanics for Engineers: DynamicsT eE
d
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Vector Mechanics for Engineers: Dynamicsen t h
di t i on
Sample Problem 15.9
15 - 20
• Direction of velocity of point P ’ on S coinciding with P is
perpendicular to radius OP.
From the velocity triangle,
mm1.37
smm2.151
smm2.1516.17sinsmm500sin
s s
P P
r
vv
w w
k s
srad08.4w
6.17cossm500cos P s P vv
jiv s P
4.42sin4.42cossm477
smm500 P v
Vector Mechanics for Engineers: DynamicsT eE
d
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Vector Mechanics for Engineers: Dynamicsen t h
di t i on
Sample Problem 15.10
15 - 21
In the Genevamechanism, disk D
rotates with a constant
counter-clockwiseangular velocity of 10
rad/s. At the instant
when j = 150o
,determine angular
acceleration of disk S .
Vector Mechanics for Engineers: DynamicsT eE
d
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Vector Mechanics for Engineers: Dynamicsn t h
di t i on
Sample Problem 15.10
15 - 22
SOLUTION:
• The absolute acceleration of the pin P may be expressed as
c s P P P aaaa
• The instantaneous angular velocity of Disk
S is determined as in Sample Problem 15.9.
• The only unknown involved in the
acceleration equation is the instantaneous
angular acceleration of Disk S .
• Resolve each acceleration term into the
component parallel to the slot. Solve for
the angular acceleration of Disk S .
Vector Mechanics for Engineers: DynamicsT eE
d
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Vector Mechanics for Engineers: Dynamicsn t h
di t i on
Sample Problem 15.10
15 - 23
SOLUTION:
• Absolute acceleration of
the pin P may be expressed
as c s P P P aaaa
• From Sample Problem
15.9.
jiv
k
s P
S
4.42sin4.42cossmm477
srad08.44.42 w
Vector Mechanics for Engineers: DynamicsT en
E d
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Vector Mechanics for Engineers: Dynamicsn t h
di t i on
Sample Problem 15.10
15 - 24
• Considering each term in theacceleration equation,
jia
Ra
P
D P
30sin30cossmm5000
smm5000srad10mm500
2
222w
jia
jir a
jir a
aaa
S t P
S t P
S n P
t P n P P
4.42cos4.42sinmm1.37
4.42cos4.42sin
4.42sin4.42cos2
w
note: S may be positive or negative
Vector Mechanics for Engineers: DynamicsT en
E d
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Vector Mechanics for Engineers: Dynamicsn t h
i t i on
Sample Problem 15.10
15 - 25
s P v
• The direction of the Coriolisacceleration is obtained by
rotating the direction of the
relative velocity by 90o in the sense of w S.
ji
ji
jiva s P S c
4.42cos4.42sinsmm3890
4.42cos4.42sinsmm477srad08.42
4.42cos4.42sin2
2
w
Vector Mechanics for Engineers: DynamicsT en
E d
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Vector Mechanics for Engineers: Dynamicsn t h
it i on
Sample Problem 15.10
15 - 26
• The relative accelerationmust be parallel to the
slot.
s P a
• Equating components of the acceleration terms
perpendicular to the slot,
srad233
07.17cos500038901.37
S
S
k S
srad233
Vector Mechanics for Engineers: DynamicsT en
E d i
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Vector Mechanics for Engineers: Dynamicsn t h
it i on
Group Problem Solving
15 - 27
The sleeve BC is welded to an arm that
rotates about stationary point A with a
constant angular velocity w = (3 rad/s) j.
In the position shown rod DF is beingmoved to the left at a
constant speed u =
400 mm/s relative tothe sleeve. Determine
the acceleration of
Point D.
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E d i
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Vector Mechanics for Engineers: Dynamicsn t h
it i on
Group Problem Solving
15 - 28
SOLUTION:
• The absolute
acceleration of point D may be
expressed as
' D D D BC ca a a a
• Determine theacceleration of the
virtual point D’.
• Calculate the Coriolis
acceleration.
• Add the differentcomponents to get the
overall acceleration of
point D.
Vector Mechanics for Engineers: DynamicsT en
E d i
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Vector Mechanics for Engineers: Dynamicsn t h
t i on
Group Problem Solving
2 - 29
2 DOxy Oxy
a r r r r
Given: u= 400 mm/s, w = (3
rad/s) j. Find: aD
Write overall
expression for aD
Do any of the terms go to zero?
2 DOxy Oxy
a r r r r
Vector Mechanics for Engineers: DynamicsT en
E d i
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Vector Mechanics for Engineers: Dynamicsn t h
t i on
Group Problem Solving
2 - 30
Determine the normal
acceleration term of the
virtual point D’
2
(3 rad/s) (3 rad/s) [ (100 mm) (300 mm) ]
(2700 mm/s )
a
j j j k
k
D
r
where r is from A to D
Vector Mechanics for Engineers: DynamicsT en
E d i t
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Vector Mechanics for Engineers: Dynamicsn t h
ti on
2 - 31
Determine the
Coriolis acceleration
of point D
2
D Oxy Oxya r r r r
/
2
2
2(3 rad/s) (400 mm/s)
(2400 mm/s )
a v
j k
i
C D F
w
Vector Mechanics for Engineers: DynamicsT en
E d i t
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Vector Mechanics for Engineers: Dynamics t h
ti on
2 - 32
/
2 2
(2700 mm/s ) 0 (2400 mm/s )
a a a a
k i
D D D F C
Add the different
components to obtain
the total acceleration
of point D
2 2(2400 mm/s ) (2700 mm/s )a i k D
Vector Mechanics for Engineers: DynamicsT en t
E d i t
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Vector Mechanics for Engineers: Dynamicsth
ti on
15 - 33
In the previous problem, u and w were both constant.
What would happen if u was increasing?
a)The x-component of aD would increase
b)The y-component of aD would increase
c)The z-component of aD would increase
d)The acceleration of aD
would stay the same
w
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E d i t
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Vector Mechanics for Engineers: Dynamicsth
ti on
What would happen if w was
increasing?
a)The x-component of aD would increase
b)The y-component of aD would increase
c)The z-component of aD would increase
d)The acceleration of aD would stay the same
w