tenth edition - welcome to apl100 -...
TRANSCRIPT
VECTOR MECHANICS FOR ENGINEERS:
DYNAMICS
Tenth Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
Phillip J. Cornwell
Lecture Notes:
Brian P. Self California Polytechnic State University
CHAPTER
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15 Kinematics of
Rigid Bodies
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2 - 2
Quiz Date 17 August
Time: 6:15 p.m.
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Contents
15 - 3
Introduction
Translation
Rotation About a Fixed Axis: Velocity
Rotation About a Fixed Axis:
Acceleration
Rotation About a Fixed Axis:
Representative Slab
Equations Defining the Rotation of a
Rigid Body About a Fixed Axis
Sample Problem 5.1
General Plane Motion
Absolute and Relative Velocity in Plane
Motion
Sample Problem 15.2
Sample Problem 15.3
Instantaneous Center of Rotation in
Plane Motion
Sample Problem 15.4
Sample Problem 15.5
Absolute and Relative Acceleration in
Plane Motion
Analysis of Plane Motion in Terms of a
Parameter
Sample Problem 15.6
Sample Problem 15.7
Sample Problem 15.8
Rate of Change With Respect to a
Rotating Frame
Coriolis Acceleration
Sample Problem 15.9
Sample Problem 15.10
Motion About a Fixed Point
General Motion
Sample Problem 15.11
Three Dimensional Motion. Coriolis
Acceleration
Frame of Reference in General Motion
Sample Problem 15.15
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Applications
15 - 4
How can we determine the velocity of the tip of a turbine blade?
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Applications
2 - 5
Planetary gear systems are used to get high reduction ratios
with minimum weight and space. How can we design the
correct gear ratios?
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Applications
2 - 6
Biomedical engineers must determine the velocities and
accelerations of the leg in order to design prostheses.
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Vector Mechanics for Engineers: Dynamics
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n Introduction
15 - 7
• A rigid body is made of infinite points and
distance between any two points does not
change during motion.
• Kinematics of rigid bodies: relations
between time the positions, velocities, and
accelerations of the particles forming a rigid
body.
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Introduction
15 - 8
• Classification of rigid body
motions:
- general motion
- motion about a fixed point
- general plane motion
- rotation about a fixed axis
• curvilinear translation
• rectilinear translation
- translation:
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The rate of change of a scalar A is
independent of frame of reference.
If F is a fixed frame of reference and m
is a moving coordinate system
ȦǀF = Ȧǀm
Scalar
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1. Rate of change of vector in
translating frame
2. .Rate of change of vector in a
rotating frame
3. Relation between velocities and
accelerations of points on a rigid
body.
10
Topics
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Translating frames
•Let F & T be the two reference frames
such that frame T translates w.r.t F i.e
no rotations are involved.
•During translatory motion, the
coordinate axes in the two frames
retain same orientation w.r.t each
other.
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xyz translate relative to XYZ.
i,j,k as seen from XYZ remain constant in
magnitude and direction.
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Vector Mechanics for Engineers: Dynamics
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n Rate of change of vector in a translating frame
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Velocity of a point moving in a translating
frame
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Velocity of a point moving in a translating
frame
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Acceleration of a point in translating frame
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Rate of change of vector in rotating frame
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XYZ fixed frame F
X’Y’Z’ fixed frame F’ with Z’ oriented in
direction of w
xyz moving frame with angular velocity w. z
is also oriented in direction of w
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Vector Mechanics for Engineers: Dynamics
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Rate of change of vector in fixed frame
=
Rate of change of vector in moving
frame + w x A
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• For a vector fixed in m, i.e.,
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2nd derivative
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• Relation between velocity of two points
on a rigid body in fixed frame of
reference.
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Vector Mechanics for Engineers: Dynamics
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Vector Mechanics for Engineers: Dynamics
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n Relation between acceleration of two points on a rigid
body in fixed frame of reference.
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2 - 31
While Solving problems se will use vectors.
We will not use graphical method,
i.e. velocity or acceleration diagrams
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Concept Quiz
15 - 32
What is the direction of the velocity
of point A on the turbine blade?
A a) →
b) ←
c) ↑
d) ↓
Av rw
x
y
ˆ ˆAv k Liw
ˆAv L jw
w
L
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Concept Quiz
15 - 33
What is the direction of the normal
acceleration of point A on the turbine
blade?
A a) →
b) ←
c) ↑
d) ↓ 2
na rw
x
y
2 ˆ( )na Liw 2ˆ
na L iw
w
L
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Group Problem Solving
15 - 34
A series of small machine components being moved by a
conveyor belt pass over a 150 mm-radius idler pulley. At
the instant shown, the velocity of point A is 375 mm/s to the
left and its acceleration is 225 mm/s2 to the right. Determine
(a) the angular velocity and angular acceleration of the idler
pulley, (b) the total acceleration of the machine
component at B.
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Group Problem Solving
15 - 35
SOLUTION:
• Using the linear velocity and
accelerations, calculate the angular
velocity and acceleration.
• Using the angular velocity,
determine the normal acceleration.
• Determine the total acceleration
using the tangential and normal
acceleration components of B.
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Group Problem Solving
15 - 36
v= 375 mm/s at= 225 mm/s2
Find the angular velocity
of the idler pulley using
the linear velocity at B.
375 mm/s (150 mm)
v r
2.50 rad/sw
B
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Group Problem Solving
15 - 37
v= 375 mm/s at= 225 mm/s2
2225 mm/s (150 mm)
a r
21.500 rad/s
B Find the angular
acceleration of the idler
pulley.
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Group Problem Solving
15 - 38
v= 375 mm/s at= 225 mm/s2 B Find the normal
acceleration of point B.
2
2(150 mm)(2.5 rad/s)
na r
2937.5 mm/san
What is the direction of the normal
acceleration of point B?
Downwards, towards the center
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Group Problem Solving
2 - 39
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2 - 40
Video
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Knee joint
2 - 41
video
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Group Problem Solving
2 - 42
In the position shown, bar AB
has an angular velocity of 4 rad/s
clockwise. Determine the angular
velocity of bars BD and DE.
Which of the following is true?
a) The direction of vB is ↑
b) The direction of vD is →
c) Both a) and b) are correct
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Group Problem Solving
2 - 43
In the position shown, bar AB has an angular
velocity of 4 rad/s clockwise. Determine the angular
velocity of bars BD and DE.
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Group Problem Solving
2 - 44
How should you proceed?
Determine vB with respect to A, then work
your way along the linkage to point E.
Determine the angular velocity of bars
BD and DE.
wAB= 4 rad/s
(4 rad/s)AB kw
/ /–(175 mm) (–4 ) (–175 )
(700 mm/s)
r i v k i
v j
B A B AB B A
B
r
Does it make sense that vB is in the +j direction?
x
y
/AB A AB B v v rw
Write vB in terms of point A, calculate vB.
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wAB= 4 rad/s
(4 rad/s)AB kw
x
y
/AB A AB B v v rw
Write vB in terms of point
A, calculate vB.
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wAB= 4 rad/s
Does it make sense that vB is in the +j direction?
x
y
/ –(175 mm)r iB A
/ (–4 ) (–175 )
(700 mm/s)
v k i
v j
B AB B A
B
r
w
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wAB= 4 rad/s
x
y
Determine vD with respect to
E, then equate it to equation
above.
,Dv
kwDE DEw
/ (275 mm) (75 mm)r i jD E
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wAB= 4 rad/s
x
y
Determine vD with respect to
E, then equate it to equation
above.
,Dv
/
( ) ( 275 75 )
275 75
v r
k i j
v j i
wD DE D E
DE
D DE DE
w
w w
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2 - 49
Determine vD with respect to
B. wAB= 4 rad/s
x
y
,Dv
kwBD BDw
/ (200 mm)r jD B
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2 - 50
Determine vD with respect to
B. wAB= 4 rad/s
x
y
,Dv
/
700 ( ) ( 200 )
700 200
v v r
j k j
v j i
wD B BD D B
BD
D BD
w
w
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Equating components of the two expressions
for vD
,Dv
j: 700 275 2.5455 rad/sDE DEw w
3: 200 75
8i BD DE BD DEw w w w
0.955 rad/sBD w
2.55 rad/sDE w
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Group Problem Solving
2 - 52
Knowing that at the instant
shown bar AB has a constant
angular velocity of 4 rad/s
clockwise, determine the angular
acceleration of bars BD and DE.
Which of the following is true?
a) The direction of aD is
b) The angular acceleration of BD must also be constant
c) The direction of the linear acceleration of B is →
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Group Problem Solving
2 - 53
Knowing that at the instant
shown bar AB has a constant
angular velocity of 4 rad/s
clockwise, determine the
angular acceleration of bars
BD and DE.
SOLUTION:
• The angular velocities were determined
in a previous problem by simultaneously
solving the component equations for
BDBD vvv
• The angular accelerations are now
determined by simultaneously solving
the component equations for the relative
acceleration equation.
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2 - 54
From our previous problem, we used the relative
velocity equations to find that: wAB= 4 rad/s
0.955 rad/sBD w2.55 rad/sDE w
0AB
We can now apply the relative
acceleration equation with
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2 - 55
wAB= 4 rad/s
2/A /AB A AB B AB Bw a a r r
2 2 2/ (4) ( 0.175 ) 2.8 m/sa r i iB AB B Aw
Analyze Bar AB
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2 - 56
wAB= 4 rad/s
Analyze Bar BD
2/ /
22.8 ( 0.2 ) (0.95455) ( 0.2 )
D B BD D B BD D B
BD
w
a a r r
i k j j
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2 - 57
(2.8 0.2 ) 0.18223a i jD BD
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wAB= 4 rad/s
Analyze Bar DE
2/ /
2
( 0.275 0.075 )
(2.5455) ( 0.275 0.075 )
a r
k i j
i j
D DE D E DE D E
DE
rw
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( 0.075 1.7819)
(0.275 0.486)
a i
j
D DE
DE
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(2.8 0.2 ) 0.18223a i jD BD
( 0.075 1.7819)
(0.275 0.486)
a i
j
D DE
DE
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wAB= 4 rad/s Equate like components of aD
j: 0.18223 (0.275 0.486)DE
22.43 rad/sDE
i: 2.8 0.2 [ (0.075)( 2.43) 1.7819]BD
24.18 rad/sBD
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Planetary gears
2 - 62
Video
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General Plane Motion
15 - 63
• General plane motion is neither a translation nor a
rotation.
• General plane motion can be considered as the sum
of a translation and rotation.
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General Plane Motion
15 - 64
• General plane motion can be considered as the sum
of a translation and rotation.
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General Plane Motion
15 - 65
• General plane motion can be considered as the sum
of a translation and rotation.
• Displacement of particles A and B to A2 and
B2 can be divided into two parts:
- translation to A2 and
- rotation of about A2 to B2 1B
1B
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Instantaneous Center of Rotation in Plane Motion
15 - 66
• The same translational and
rotational velocities at A are
obtained by allowing the slab to
rotate with the same angular
velocity about the point C on a
perpendicular to the velocity at A.
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Instantaneous Center of Rotation in Plane Motion
15 - 67
• Plane motion of all particles in a
slab can always be replaced by the
translation of an arbitrary point A
and a rotation about A with an
angular velocity that is independent
of the choice of A.
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Instantaneous Center of Rotation in Plane Motion
15 - 68
• The velocity of all other particles
in the slab are the same as
originally defined since the angular
velocity and translational velocity
at A are equivalent.
• As far as the velocities are
concerned, the slab seems to rotate
about the instantaneous center of
rotation C.
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Instantaneous Center of Rotation in Plane Motion
15 - 69
• If the velocity at two
points A and B are
known, the
instantaneous center of
rotation lies at the
intersection of the
perpendiculars to the
velocity vectors through
A and B .
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Instantaneous Center of Rotation in Plane Motion
15 - 70
• If the velocity vectors at A
and B are perpendicular to
the line AB, the
instantaneous center of
rotation lies at the
intersection of the line AB
with the line joining the
extremities of the velocity
vectors at A and B.
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Instantaneous Center of Rotation in Plane Motion
15 - 71
• If the velocity vectors are parallel and
equal ,
• the instantaneous center of rotation is
at infinity and
• the angular velocity is zero.
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Instantaneous Center of Rotation in Plane Motion
15 - 72
• The instantaneous center of rotation lies at
the intersection of the perpendiculars to the
velocity vectors through A and B .
w
cosl
v
AC
v AA
w
tan
cossin
A
AB
v
l
vlBCv
• The velocities of all particles on the rod are as if
they were rotated about C.
• The particle at the center of rotation has zero
velocity.
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Instantaneous Center of Rotation in Plane Motion
15 - 73
• The particle coinciding with the center of rotation
changes with time and the acceleration of the particle
at the instantaneous center of rotation is not zero.
• The acceleration of the particles in the slab cannot be
determined as if the slab were simply rotating about
C.
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Instantaneous Center of Rotation in Plane Motion
15 - 74
At the instant shown, what is the
approximate direction of the velocity
of point G, the center of bar AB?
a)
b)
c)
d)
G
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Group Problem Solving
2 - 75
vD
What direction is the velocity of B?
vB
What direction is the velocity of D?
wAB= 4 rad/s
( ) (0.25 m)(4 rad/s) 1m/sB ABAB w vWhat is the velocity of B?
1 0.06 mtan 21.8
0.15 m
Find
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Group Problem Solving
2 - 76
vD
vB
B
D
Locate instantaneous center C at intersection of lines drawn
perpendicular to vB and vD.
C
100 mm
1m/s (0.25 m) BDw
4 rad/sBD w
( )B BDv BC w
Calculate wBD
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Group Problem Solving
2 - 77
vB
B
vD
D
C
100 mm
Find wDE
0.25 m( ) (4 rad/s)
cosD BDv DC w
1 m/s 0.15 m( ) ; ;
cos cosD DE DEv DE w w
6.67 rad/sDE w
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Absolute and Relative Velocity in Plane Motion
15 - 78
• Any plane motion can be replaced by a translation of an
arbitrary reference point A and a simultaneous rotation
about A.
ABAB vvv
ww rvrkv ABABAB
ABAB rkvv
w
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Sample Problem 15.2
15 - 79
The double gear rolls on the stationary lower rack: the
velocity of its center is 1.2 m/s.
Determine (a) the angular velocity of the gear, and (b) the
velocities of the upper rack R and point D of the gear.
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Sample Problem 15.2
15 - 80
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Sample Problem 15.2
15 - 81
The double gear rolls on the stationary lower
rack: the velocity of its center is 1.2 m/s.
Determine (a) the angular velocity of the gear,
and (b) the velocities of the upper rack R and
point D of the gear.
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Sample Problem 15.2
15 - 82
SOLUTION:
• The displacement of the gear center in one
revolution is equal to the outer circumference.
Relate the translational and angular
displacements. Differentiate to relate the
translational and angular velocities.
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.2
15 - 83
• The velocity for any point P on the gear
may be written as
Evaluate the velocities of points B and D.
APAAPAP rkvvvv
w
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Sample Problem 15.2
15 - 84
x
y
SOLUTION:
w < 0 (rotates clockwise).
m0.150
sm2.1
1
1
r
v
rv
A
A
w
w
kk
srad8 ww
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Sample Problem 15.2
15 - 85
• For any point P on the gear,
APAAPAP rkvvvv
w
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Sample Problem 15.2
15 - 86
Velocity of the upper rack is equal to
velocity of point B:
ii
jki
rkvvv ABABR
sm8.0sm2.1
m 10.0srad8sm2.1
w
ivR
sm2
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Sample Problem 15.2
15 - 87
Velocity of the point D:
iki
rkvv ADAD
m 150.0srad8sm2.1
w
sm697.1
sm2.1sm2.1
D
D
v
jiv
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Sample Problem 15.2
15 - 88
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Sample Problem 15.6
15 - 89
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Sample Problem 15.6
15 - 90
The center of the double gear has a velocity
and acceleration to the right of 1.2 m/s and 3
m/s2, respectively. The lower rack is
stationary.
Determine (a) the angular acceleration of the
gear, and (b) the acceleration of points B, C,
and D.
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Sample Problem 15.6
15 - 91
SOLUTION:
• The expression of the gear position as a
function of is differentiated twice to define
the relationship between the translational
and angular accelerations. • The acceleration of each point on the gear is
obtained by adding the acceleration of the gear
center and the relative accelerations with
respect to the center. The latter includes
normal and tangential acceleration
components.
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Sample Problem 15.6
15 - 92
11 rraA
m 150.0
sm3 2
1
r
aA
kk 2srad20
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Sample Problem 15.6
15 - 93
• The acceleration of each point is obtained by
adding the acceleration of the gear center and
the relative accelerations with respect to the
center.
The latter includes normal and tangential
acceleration components.
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Sample Problem 15.6
15 - 94
jii
jjki
rrka
aaaaaa
ABABA
nABtABAABAB
222
222
2
sm40.6sm2sm3
m100.0srad8m100.0srad20sm3
w
222 sm12.8sm40.6m5 BB ajisa
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Sample Problem 15.6
15 - 95
jii
jjki
rrkaaaa ACACAACAC
222
222
2
sm60.9sm3sm3
m150.0srad8m150.0srad20sm3
w
jac
2sm60.9
iji
iiki
rrkaaaa ADADAADAD
222
222
2
sm60.9sm3sm3
m150.0srad8m150.0srad20sm3
w
222 sm95.12sm3m6.12 DD ajisa
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Vector Mechanics for Engineers: Dynamics
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12 - 96
Figure shows a cyclist on a bicycle. Assume
that he is pedaling at 90 rpm and
chain runs between the 12 cm radius front
chain ring and the 3 cm radius rear cog. The
wheels have a radius of 25 cm. Determine how
fast the cyclist is travelling? Assume no
slipping between the wheel and the road.
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12 - 97
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Vector Mechanics for Engineers: Dynamics
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12 - 98
Since, O only moves horizontally, b1
component must be 0.
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12 - 99
To keep in shape during the winter, cyclists often put their bikes on
rollers which allows them to pedal as hard as they wish without
going anywhere. Assume that the cyclist is pedaling at 4 rad/s and
then begins to accelerate so that the angular acceleration of the main
crank assembly is equal to 1.2rad/s2. After accelerating at this rate
for 2 s. what is the acceleration of the pedal spindle P? The
distance from O, the center of
the bottom bracket, to the pedal spindle is 17 cm, and the bottom
bracket center O is fixed in space.
Find the acceleration of a bike's pedal spindle.
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Vector Mechanics for Engineers: Dynamics
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12 - 100
First we will need to determine the rotation rate
after 2 seconds by integrating with respect to
time.
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Example – General Plane Motion
2 - 101
The knee has linear velocity and acceleration from both
translation (the runner moving forward) as well as rotation
(the leg rotating about the hip).
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Sample Problem 15.4
15 - 102
The double gear rolls on the
stationary lower rack: the velocity
of its center is 1.2 m/s.
Determine (a) the angular velocity
of the gear, and (b) the velocities of
the upper rack R and point D of the
gear.
SOLUTION:
• The point C is in contact with the stationary
lower rack and, instantaneously, has zero
velocity. It must be the location of the
instantaneous center of rotation.
• Determine the angular velocity about C
based on the given velocity at A.
• Evaluate the velocities at B and D based on
their rotation about C.
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Sample Problem 15.4
15 - 103
SOLUTION:
• The point C is in contact with the stationary lower rack
and, instantaneously, has zero velocity. It must be the
location of the instantaneous center of rotation.
• Determine the angular velocity about C based on the
given velocity at A.
srad8m 0.15
sm2.1
A
AAA
r
vrv ww
• Evaluate the velocities at B and D based on their rotation
about C.
srad8m 25.0 wBBR rvv
ivR
sm2
srad8m 2121.0
m 2121.02m 15.0
wDD
D
rv
r
sm2.12.1
sm697.1
jiv
v
D
D
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Golf Robot
2 - 104
w,
If the arm is shortened to ¾ of its
original length, what happens to
the tangential acceleration of the
club head?
A golf robot is used to test new
equipment. If the angular
velocity of the arm is doubled,
what happens to the normal
acceleration of the club head?
If the arm is shortened to ¾ of its
original length, what happens to
the tangential acceleration of the
club head?
If the speed of the club head is
constant, does the club head have
any linear accelertion ?
Not ours – maybe Tom Mase has pic?
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Instantaneous Center of Zero Velocity
2 - 105
What happens to the location of the instantaneous center of
velocity if the crankshaft angular velocity increases from
2000 rpm in the previous problem to 3000 rpm?
What happens to the location of the instantaneous center of
velocity if the angle is 0?
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12 - 106
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12 - 107
A disc C is mounted on a shaft AB in Figure.
The shaft and disc rotate with a constant angular
speed ω2 of 10 rad/sec relative to the platform to
which bearings A and B are attached. Meanwhile,
the platform rotates at a constant angular speed ω1
of 5 rad/sec relative to the ground in a direction
parallel to the Z axis of the ground reference XYZ.
What is the angular
velocity vector ω for the disc C relative to XYZ?
What are (dω/dt)xyz and (d2ω /dt2)xyz ?
Consider the vector w2. Note that this
vector is constrained in direction to be
always collinear with the axis AB of the
bearings of the shaft. This clearly is a
physical requirement.
Also, since w2 is of constant value, we may
think of the vector w2 as fixed to the
platform along AB.
121 www
121 www
12112 www
Example 15.1:
3
1211
1211
2
121
sec/250)105(5
)(0
sec/50105
jradjkk
radijk
w
www
wwww
w
www
Consider a position vector ρ between two points on
the rotating disc. The length of ρ is 100 mm and at
the instant of interest is the vertical direction. What
are the first and second time derivatives of ρ at this
instant as seen from ground reference?
r is fixed in a frame rotating at w1 + w2|1
rwwr )(121
sec/1000100)105( mmikjk r
rwwrwwr )()(121121
rwwrwwr )()0(121121
rwwrwwr )()(121121
rwwrwwr )()0(121121
2
2
sec/1010
sec/1000)105(100)105(
mmkj
mmijkkjk
r
r
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12 - 114
A tank is maneuvering its gun into position.
At the instant of interest, the turrent A, is
rotating at an angular speed of 2 rad/sec
relative to the tank and is in position . Also
at this instant , the gun is rotating at an
angular speed of 1 rad/sec relative to the
turret and forms an angle =300 with the
horizontal plane . What are and of the
gun relative to the ground?
ww , w
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12 - 115
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12wConsider Fixed in turret.
121 www
121121 0 wwwww
0 0
2 1(1)cos 20 i (1)sin 20 j
0.940 i 0.342 j
w
2
2k (0.940 i 0.342 j)
1.880 j 0.684 irad / sec
w
)()(01211
1211
www
wwww
At instant of interest:
3sec/368.176.3
)]342.0940.0()2[(2
radji
jikk
w
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Concept Question
15 - 119
The figure depicts a model
of a coaster wheel. If both
w1 and w2 are constant, what
is true about the angular
acceleration of the wheel?
a)It is zero.
b)It is in the +x direction
c)It is in the +z direction
d)It is in the -x direction
e)It is in the -z direction
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Sample Problem 15.11
15 - 120
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Vector Mechanics for Engineers: Dynamics
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2 - 121
The crane rotates with a constant
angular velocity w1 = 0.30 rad/s
and the boom is being raised with a
constant angular velocity w2 = 0.50
rad/s. The length of the boom is l
= 12 m.
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15 - 122
Determine:
• angular velocity of the boom,
• angular acceleration of the boom,
• velocity of the boom tip, and
• acceleration of the boom tip.
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Vector Mechanics for Engineers: Dynamics
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15 - 123
SOLUTION:
With
• Angular velocity of the boom,
1 2w w w
ji
jir
kj
639.10
30sin30cos12
50.030.0 21
ww
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Vector Mechanics for Engineers: Dynamics
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15 - 124
•Angular acceleration of the boom,
21
22221
ww
wwwww
Oxyz
•Velocity of boom tip,
rv
w
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125
•Acceleration of boom tip,
vrrra
www
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Sample Problem 15.11
15 - 126 jir
kj
639.10
50.030.0 21
ww
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Sample Problem 15.11
15 - 127
SOLUTION:
•Angular velocity of the
boom,
21 www
kj
srad50.0srad30.0 w
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2 - 128
•Angular acceleration of the
boom,
kj
Oxyz
srad50.0srad30.021
22221
ww
wwwww
i 2srad15.0
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Sample Problem 15.11
15 - 129
•Velocity of boom tip,
0639.10
5.03.00
kji
rv
w
kjiv
sm12.3sm20.5sm54.3
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Vector Mechanics for Engineers: Dynamics
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15 - 130
• Acceleration of boom tip,
kjiik
kjikji
a
vrrra
90.050.160.294.090.0
12.320.53
50.030.00
0639.10
0015.0
www
kjia 222 sm80.1sm50.1sm54.3
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Absolute and Relative Acceleration in Plane Motion
2 - 131
As the bicycle accelerates, a point on the top of the wheel will
have acceleration due to the acceleration from the axle (the
overall linear acceleration of the bike), the tangential
acceleration of the wheel from the angular acceleration, and
the normal acceleration due to the angular velocity.
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Concept Question
2 - 132
A) Energy will not be conserved when I kick this ball
B) In general, the linear acceleration of my knee is equal to
the linear acceleration of my foot
C) Throughout the kick, my foot will only have tangential
acceleration.
D) In general, the angular velocity of the upper leg (thigh)
will be the same as the angular velocity of the lower leg
You have made it to the kickball
championship game. As you try to kick
home the winning run, your mind
naturally drifts towards dynamics.
Which of your following thoughts is
TRUE, and causes you to shank the ball
horribly straight to the pitcher?
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Applications
2 - 133
Rotating coordinate systems are often used to analyze mechanisms
(such as amusement park rides) as well as weather patterns.
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Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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• Vector moving in moving frame of
reference
139
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Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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Vector Mechanics for Engineers: Dynamics
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Coriolis Acceleration
15 - 147
• Consider a collar P which is
made to slide at constant relative
velocity u along rod OB. The
rod is rotating at a constant
angular velocity w. The point A
on the rod corresponds to the
instantaneous position of P.
cPAP aaaa
F
• Absolute acceleration of the collar is
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Coriolis Acceleration
15 - 148
cPAP aaaa
F
0 OxyP ra F
uava cPc w22 F
• The absolute acceleration consists of the
radial and tangential vectors shown
2wrarra AA
where
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Coriolis Acceleration
15 - 149
uvvtt
uvvt
A
A
,at
,at
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Coriolis Acceleration
15 - 150
uvvtt
uvvt
A
A
,at
,at
• Change in velocity over t is
represented by the sum of three
vectors TTTTRRv
2wrarra AA
recall,
• is due to change in
direction of the velocity of
point A on the rod,
AAtt
arrt
vt
TT
2
00limlim www
TT
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Coriolis Acceleration
15 - 151
uvvtt
uvvt
A
A
,at
,at
• result from
combined effects of relative
motion of P and rotation of the
rod
TTRR and
uuu
t
r
tu
t
TT
t
RR
tt
www
w
2
limlim00
uava cPc w22 F
recall,
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Vector Mechanics for Engineers: Dynamics
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Concept Question
2 - 152
v w
You are walking with a
constant velocity with
respect to the platform,
which rotates with a
constant angular
velocity w. At the
instant shown, in which
direction(s) will you
experience an
acceleration (choose all
that apply)?
x
y
OxyOxyP rrrra
2
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Concept Question
2 - 153
v w
a) +x
b) -x
c) +y
d) -y
e) Acceleration = 0
x
y
OxyOxyP rrrra
2
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15 - 154
•Determine the
acceleration of the
virtual point D’.
•Calculate the Coriolis
acceleration.
•Add the different components to get the
overall acceleration of point D.
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155
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Vector Mechanics for Engineers: Dynamics
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Group Problem Solving
15 - 156
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 being
moved to the left at a
constant speed u =
400 mm/s relative to
the sleeve. Determine
the acceleration of
Point D.
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Vector Mechanics for Engineers: Dynamics
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2 - 157
SOLUTION:
•The absolute
acceleration of point D
may be expressed as
'D D D BC ca a a a
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Vector Mechanics for Engineers: Dynamics
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2 - 158
Given: u= 400 mm/s, w = (3 rad/s) j.
Find: aD
Write overall expression for aD
2DOxy Oxy
a r r r r
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Vector Mechanics for Engineers: Dynamics
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159
Do any of the terms go to zero?
2
D
Oxy Oxy
a r r
r r
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Vector Mechanics for Engineers: Dynamics
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2 - 160
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
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Vector Mechanics for Engineers: Dynamics
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2 - 161
Determine the Coriolis
acceleration of point D
2DOxy Oxy
a 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
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2 - 162
/
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 kD
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Vector Mechanics for Engineers: Dynamics
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15 - 163
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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Vector Mechanics for Engineers: Dynamics
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15 - 164
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
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2 - 165
Geneva
mechanism
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.9
15 - 166
Disk D of the Geneva
mechanism rotates
with constant
counterclockwise
angular velocity wD =
10 rad/s.
At the instant when = 150o, determine (a)
the angular velocity of disk S, and (b) the
velocity of pin P relative to disk S.
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.9
15 - 167
SOLUTION:
• The absolute velocity of the point P
may be written as
sPPP vvv
• Magnitude and direction of velocity
of pin P are calculated from the
radius and angular velocity of disk D. Pv
• Direction of velocity of point P’ on
S coinciding with P is perpendicular to
radius OP.
Pv
• Direction of velocity of P with
respect to S is parallel to the slot. sPv
• Solve the vector triangle for the
angular velocity of S and relative
velocity of P.
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.9
15 - 168
SOLUTION:
• The absolute velocity of the
point P may be written as
sPPP vvv
• Magnitude and direction of
absolute velocity of pin P are
calculated from radius and
angular velocity of disk D.
smm500srad 10mm 50 DP Rv w
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.9
15 - 169
• Direction of velocity of P
with respect to S is parallel to
slot. From the law of cosines,
mm 1.37551.030cos2 2222 rRRllRr
4.42742.0
30sinsin
30sin
R
sin
r
6.17304.4290
The interior angle of the vector
triangle is
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.9
15 - 170
• Direction of velocity of point
P’ on S coinciding with P is
perpendicular to radius OP.
mm 1.37
smm2.151
smm2.1516.17sinsmm500sin
ss
PP
r
vv
ww
ks
srad08.4w
6.17cossm500cosPsP vv
jiv sP
4.42sin4.42cossm477
smm 500Pv
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.10
15 - 171
In the Geneva
mechanism, disk D
rotates with a constant
counter-clockwise
angular velocity of 10
rad/s. At the instant
when j = 150o,
determine angular
acceleration of disk S.
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.10
15 - 172
SOLUTION:
• The absolute acceleration of the pin P may
be expressed as
csPPP 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.
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.10
15 - 173
SOLUTION:
• Absolute acceleration of
the pin P may be expressed
as csPPP aaaa
• From Sample Problem
15.9.
jiv
k
sP
S
4.42sin4.42cossmm477
srad08.44.42 w
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Vector Mechanics for Engineers: Dynamics
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2 - 174
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.10
15 - 175
• Considering each term in the
acceleration equation,
jia
Ra
P
DP
30sin30cossmm5000
smm5000srad10mm500
2
222w
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.10
15 - 176
jia
jira
jira
aaa
StP
StP
SnP
tPnPP
4.42cos4.42sinmm1.37
4.42cos4.42sin
4.42sin4.42cos2
w
note: S may be positive or negative
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.10
15 - 177
sPv
• The direction of the Coriolis
acceleration is obtained by
rotating the direction of the
relative velocity
by 90o in the sense of wS.
ji
ji
jiva sPSc
4.42cos4.42sinsmm3890
4.42cos4.42sinsmm477srad08.42
4.42cos4.42sin2
2
w
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.10
15 - 178
• The relative acceleration
must be parallel to the
slot.
sPa
• Equating components of
the acceleration terms
perpendicular to the slot,
srad233
07.17cos500038901.37
S
S
kS
srad233
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.15
15 - 179
For the disk mounted
on the arm, the
indicated angular
rotation rates are
constant.
Determine:
• the velocity of the point P,
• the acceleration of P, and
• angular velocity and angular acceleration of the
disk.
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.15
15 - 180
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.15
15 - 181
SOLUTION:
• Define a fixed reference frame OXYZ at O and a moving reference frame Axyz or F attached to the
arm at A.
j
jRiLr
1w
k
jRr
D
AP
2ww
F
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.15
15 - 182
• With P’ of the moving reference frame coinciding
with P, the velocity of the point P is found from
iRjRkrv
kLjRiLjrv
vvv
APDP
P
PPP
22
11
www
ww
FF
F
kLiRvP
12 ww
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.15
15 - 183
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.15
15 - 184
• The acceleration of P is found from
cPPP aaaa
F
iLkLjraP
2111 www
jRiRk
ra APDDP
2
222 www
ww
FFF
kRiRj
va Pc
2121 22
2
wwww
F
kRjRiLaP
21
22
21 2 wwww
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Vector Mechanics for Engineers: Dynamics
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Sample Problem 15.15
15 - 185
• Angular velocity and acceleration of the disk,
FDww
kj
21 www
kjj
211 www
ww
F
i
21ww
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Vector Mechanics for Engineers: Dynamics
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15 - 186
Knowing that at the instant shown the length of
the boom is 6 m and is increasing at the constant
rate u= 0.5 m/s determine the acceleration of Point
B.
The crane shown
rotates at the constant
rate w1= 0.25 rad/s;
simultaneously, the
telescoping boom is
being lowered at the
constant rate w2= 0.40
rad/s.
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Vector Mechanics for Engineers: Dynamics
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Group Problem Solving
15 - 187
SOLUTION:
• Define a moving reference frame Axyz or F
attached to the arm at A.
• The acceleration of P is found from
'B B B ca a a a F
• The angular velocity and angular acceleration of
the disk are
Bw w
w w
F
F
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Vector Mechanics for Engineers: Dynamics
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Group Problem Solving
15 - 188
Given: w1= 0.25 rad/s,
w2= -0.40 rad/s. L= 6 m,
u= 0.5 m/s. Find: aB.
Equation of overall
acceleration of B
2DOxy Oxy
a r r r r
Do any of the terms go to zero?
2DOxy Oxy
a r r r r
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Vector Mechanics for Engineers: Dynamics
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Group Problem Solving
15 - 189
Let the unextending
portion of the boom AB
be a rotating frame of
reference.
What are and ?
2 1
(0.40 rad/s) (0.25 rad/s) .
w w
i j
i j
1 2
1 2
2(0.10 rad/s ) .
w w
w w
j i
k
k
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Vector Mechanics for Engineers: Dynamics
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15 - 190
Find
2DOxy Oxy
a r r r r
20 0 0.10 (0.3 m/s )
0 3 3 3
i j k
r iB
r
/
(6 m)(sin30 cos30 )
(3 m) (3 3 m)
r r
j k
j k
B A B
Determine the position vector rB/A
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Vector Mechanics for Engineers: Dynamics
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15 - 191
Find r
2 2 2
(0.40 0.25 ) (0.40 0.25 ) (3 3 3 )
(0.3 m/s ) (0.48 m/s ) (1.1561 m/s )
r i j i j j k
i j k
B
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Vector Mechanics for Engineers: Dynamics
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15 - 192
2DOxy Oxy
a r r r r
Determine the Coriolis
acceleration – first define
the relative velocity term
/ (sin30 cos30 )
(0.5 m/s)sin30 (0.5 m/s)cos30
v j k
j k
B F u
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Vector Mechanics for Engineers: Dynamics
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15 - 193
/
2 2 2
2 (2)(0.40 0.25 ) (0.5sin30 0.5cos30 )
(0.2165 m/s ) (0.3464 m/s ) (0.2 m/s )
Ω v i j j k
i j k
B F
Calculate the Coriolis
acceleration
Add the terms together
2 2 2(0.817 m/s ) (0.826 m/s ) (0.956 m/s )a i j kB