Unformatted text preview: Chapter III: Moving Reference Frame Kinematics Homework III1 Chapter III Moving Reference Frame Kinematics
Homework III2 Chapter III: Moving Reference Frame Kinematics Homework Chapter III: Moving Reference Frame Kinematics Homework III3 Problem III.1
Given: Particle P moves within a straight slot cut into a rotating disk with the xposition of P
increasing at a constant rate of x. The disk is rotating with counterclockwise sense at a speed of
˙
Ω, with changing at a rate of ω . The xyz coordinate system is attached to the disk with its origin
˙
at the center O of the disk.
Frame Kinematics Homework Problems ME 274 Find: For this problem: • Determine the velocity and acceleration of particle P. Express your answers as vectors in
terms of their xy components. oves within a straight slot cut into a rotating disk with the xposition of P
! The disk is rotating with counter acceleration vectors
constant rate of x . Make a sketch of the velocity andclockwise sense at a found above.
•
!
ith ! changing at a rate of ! . The xyz coordinate system is attached to
ts origin at the center O of the disk.
y
ine the velocity and acceleration of
e P. Express your answers as vectors in
f their xy components. !
x sketch of the velocity and acceleration
found in a) above. P h x
O Use the analysis: x = 0.8 ft h = 0.6 ft , x = x f s ,
˙
!
ing parameters in yourfollowing parameters ,in your analysis: 15=t1/.5f t, h = 0.6f t, x = −10f t/sec, Ω = 6rad/sec
˙2 = −4rad/sec2 .
and Ω
!
and ! = "5 rad / s . III4 Chapter III: Moving Reference Frame Kinematics Homework Problem III.2
Given: A telescoping arm OP is pinned to a cart at end O. The cart moves along a horizontal
˙
surface with a constant speed of vO . The angle of the arm, θ, is increasing at a constant rate of θ
˙ . The xyz coordinate system is attached to the telescoping arm with
and is extending at a rate of L
its origin at end O of the arm. ing Reference Frame Kinematics Homework Problems ME 274 Find: For this problem: blem III2
• Determine the velocity and acceleration of particle P. Express your answers as vectors in
elescoping arm OP is pinned theircart at end O. The cart moves along a horizontal
terms of to a xy components.
face with a constant speed of vO . The angle of the arm, !, is increasing at a constant
• Make a sketch !
!
L The xyz coordinate system vectors found above.
of ! and is extending at a rate of of .the velocity and accelerationis attached to the
scoping arm with its origin at end O of the arm.
a) Determine the velocity and
acceleration of particle P.
Express your answers as vectors
in terms of their xy
components.
b) Make a sketch of the velocity
and acceleration vectors found
in a) above. P x L
y
O !
vO ˙
Use the following parameters in your analysis: θ = !90◦ , vO = 6m/sec, θ = 5rad/sec, L = 2m,
the following ˙parameters in your analysis: ! = 0 , vO = 10 m / s , ! = "3 rad / s ,
¨ = 3m/sec2 .
L = 0 and L
!!
!
4 m , L = !3 m / s and L = 0 . Chapter III: Moving Reference Frame Kinematics Homework III5 Problem III.3
Given: An Lshaped arm OBE is pinned to ground at O (section OB is perpendicular to section
ME 274
is known to be rotating in the counterclockwise sense with a constant
rotational speed of Ω. Rigid link AP is pinned to ground at A, and with P able to slide within a
slot that is cut into section BE of arm OBE, as shown. The xyz coordinate system is attached to
III3
arm OBE with its origin at pin O. For the position shown, section OB is aligned with the direction
ped arm OBE is pinned to P is at aat O (sectionyOB is perpendicular to section
of OA, and ground position of P within the slot.
rence Frame Kinematicsof OBE).Problems
BE Homework Arm OBE E). Arm OBE is known to be rotating in the counterclockwise sense with a
rotational speed of !.For this position: pinned to ground at A, and with P able to
Find: Rigid link AP is
in a slot that is cut into section BE of arm OBE, as shown. The xyz coordinate
• Determine the angular velocity of link AP. Write your answer as a vector.
attached to arm OBE with its origin at pin O. • Determine the angular acceleration of link AP. Write your answer as a vector. osition shown, section OB is aligned with the direction of OA, and P is at a
f yP within the slot. For this
E
etermine the angular
locity of link AP. Write
P
y
ur answer as a vector.
etermine the angular
celeration of link AP. Write
ur answer as a vector. L yP O B ! x A b
Use the following parameters in your analysis: Ω = 4rad/sec, yP = 5f t, L = 2f t and b = 3f t. ollowing parameters in your analysis: ! = 2 rad / sec , yP = 2 ft , L = 2.5 ft
ft . III6 Chapter III: Moving Reference Frame Kinematics Homework Problem III.4
Given: Arm OB is pinned to ground at end O. Rigid link AP is pinned to ground at A, with end
P able to slide within a straight slot that is cut into arm OB. Arm AP is rotating in the clockwise
Reference Frame Kinematics Homework Problems of Ω. The xyz coordinate system is attached to arm OB with its origin
ME 274
sense with a constant rate
at pin O. For the position shown, link OB is perpendicular to the direction of OA, with arm AP
at an angle of θ measured from OB. lem III4
B is pinned to ground at this position: link AP is pinned to ground at A, with end P
Find: For end O. Rigid
o slide within a straight slot that is cut into arm OB. Arm AP is rotating in the
wise sense with a constant rate of !. The xyz coordinate systemWrite your answer as a vector.
• Determine the angular velocity of arm OB. is attached to arm
ith its origin at pin O. • Determine the angular acceleration of arm OB. Write your answer as a vector. e position shown, • Determine the values ofto P and xP . of OA, with arm AP at
link OB is perpendicular x the direction
˙
¨
le of " measured from OB. For this position:
Determine the angular
velocity of arm OB.
Write your answer as a
vector. A ! Determine the angular
acceleration of arm OB.
Write your answer as a
vector.
Determine the values of
!
xP and !!P .
x L
y " P O B x xP
Use the following parameters in your analysis: Ω = 2rad/sec, L = 3m and θ = 36.87◦ . e following parameters in your analysis: ! = 8 rad / s , L = 2 m and ! = 53.13° . Chapter III: Moving Reference Frame Kinematics Homework III7 Problem III.5
Given: A disk and shaft OC are mounted in a clevis that rotates about a ﬁxed vertical axis at a
rate of Ω. The shaft and disk rotate with respect to the clevis with a rate of ω0 in the direction
shown below, with the angle θ held constant. The XYZ coordinate system is ﬁxed with the Zaxis
aligned with the ﬁxed rotation axis of the clevis. The xyz coordinateEsystem is attached to the disk
ng Reference Frame Kinematics Homework Problems
M 274
with the yaxis aligned with OC for all time. For the position shown below, the x and Xaxes are
aligned. blem III5
Find: For the in a clevis that
isk and shaft OC are mountedposition shown: rotates about a fixed vertical axis at a
of !. The shaft and disk rotate with respect to the clevis with a rate of "0 in the
• Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz
ction shown below, with the angle # held constant. The XYZ coordinate system is
components.
d with the Zaxis aligned with the fixed rotation axis of the clevis. The xyz coordinate
em is attached to theDetermine the yaxis aligned with OC the disk. Write your answer as a vector in terms of its
• disk with
angular acceleration of for all time. For the
ition shown below, the x and Xaxes are aligned.
xyz components.
the position shown: z a) Determine the angular
velocity of the disk. Write
your answer as a vector in
terms of its xyz
components. Z
y b) Determine the angular
acceleration of the disk.
Write your answer as a
vector in terms of its xyz
components. O C
x
Y X Use the following parameters in your analysis: θ = 36.87◦ , Ω = 4rad/sec = constant and ω0 = the following parameters in your analysis: ! = 53.13° , ! = 3 rad / s = constant and
6rad/sec = constant.
= 5 rad / s = constant . III8 Chapter III: Moving Reference Frame Kinematics Homework Problem III.6
Given: A disk and shaft OC are mounted in a clevis that is attached to a ﬁxed vertical axis
(Ω = 0). The shaft and disk rotate with respect to the clevis with a rate of ω0 in the direction
˙
shown below, with the angle θ increasing at a constant rate of θ. The XYZ coordinate system
is ﬁxed with the Zaxis aligned with the ﬁxed vertical direction.ME 274 xyz coordinate system is
The
ng Reference Frame Kinematics Homework Problems
attached to the disk with the yaxis aligned with OC for all time. For the position shown below,
the x and Xaxes are aligned. blem III6
Find: For the in a clevis that
isk and shaft OC are mountedposition shown: is attached to a fixed vertical axis
= 0 ). The shaft and disk rotate with respect to the clevis with a rate of !0 in the
• Determine the angular velocity of the disk. Write! your answer as a vector in terms of its xyz
ction shown below, with the angle " increasing at a constant rate of ! . The XYZ
components.
rdinate system is fixed with the Zaxis aligned with the fixed vertical direction. The
coordinate system is attached to the disk with the yaxis aligned with OC for all time. as a vector in terms of its
• Determine the angular acceleration of the disk. Write your answer
xyz components.
the position shown below, the x and Xaxes are aligned.
the position shown:
z a) Determine the angular
velocity of the disk. Write
your answer as a vector in
terms of its xyz
components. Z
y b) Determine the angular
acceleration of the disk.
Write your answer as a
vector in terms of its xyz
components. O C
x
Y X ˙
Use the following parameters in your analysis: θ = 0, θ = 1.5rad/sec, ω0 = 2rad/sec and ω0 =
˙ !
the following 5rad/sec2 . in your analysis: ! = 0 , ! = 8 rad / s and ! 0 = 6 rad / s
parameters
!
! 0 = 2 rad / s 2 . Chapter III: Moving Reference Frame Kinematics Homework III9 Problem III.7
Given: Arm OB rotates about a ﬁxed axis with a constant rate of Ω. A disk rotates about its
central axis BC with a constant rate of ω0 relative to arm OB. The XYZ coordinate system is ﬁxed
with Homework Problems
erence Frame Kinematics the Yaxis aligned with the ﬁxed rotation axis of OB. The 274 coordinate system is attached
ME xyz
to the disk with the zaxis aligned with BC for all time. For the position shown below, the xyz
axes are aligned with the XYZ axes. III7
Find: For with a constant rate
rotates about a fixed axis the position shown: of !. A disk rotates about its
xis BC with a constant rate of "0 relative to arm OB. The XYZ coordinate
• Determine the angular acceleration of the disk. Write your answer as a vector in terms of its
fixed with the Yaxis aligned with the fixed rotation axis of OB. The xyz
xyz components.
te system is attached to the disk with the zaxis aligned with BC for all time. For
ion shown below, • Determine are aligned with of point Aaxes. disk. Write your answer as a vector in terms of
the xyz axes the acceleration the XYZ on the
its xyz components. osition shown:
etermine the angular
elocity of the disk. Write
our answer as a vector in
rms of its xyz
omponents.
etermine the angular
cceleration of the disk.
rite your answer as a
ector in terms of its xyz
omponents. "0 d
R h !
Use the following parameters in your analysis: Ω = −2rad/sec, ω0 = 3rad/sec, d = 10inches,
h = 3inches and R = 5inches. ollowing parameters in your analysis: ! = "3 rad / s and ! 0 = 5 rad / s . III10 Chapter III: Moving Reference Frame Kinematics Homework Problem III.8
Given: Yoke A rotates about a ﬁxed axis with a constant rate of ω1 . A disk rotates about its
center C with a constant rate of p relative to yoke A. The XYZ coordinate system is ﬁxed with the
Xaxis aligned with the ﬁxed rotation axis of yoke A. The xyz coordinate system is attached to the
disk with the zaxis aligned with the rotation axis of the disk for all time. For the position shown
below, the xyz axes are aligned with the XYZ axes. erence Frame Kinematics Homework Problems ME 274 Find: For the position shown: III8
• Determine the angular velocity and angular acceleration of the disk. Write your answers as
rotates about a fixed vectors in a constant rate xyz!1. A disk rotates about its
axis with terms of their of components.
with a constant rate of p relative to yoke A. The XYZ coordinate system is fixed
• Determine the acceleration of A. The of the disk. Write your
Xaxis aligned with the fixed rotation axis of yokepoint B xyz coordinate system answer as a vector in terms of
its xyz components.
d to the disk with the zaxis aligned with the rotation axis of the disk for all
the position shown below, the xyz axes are aligned with the XYZ axes.
osition shown:
etermine the angular velocity
nd angular acceleration of the
isk. Write your answers as
ectors in terms of their xyz
omponents. d C etermine the acceleration of
oint B of the disk. Write your
nswer as a vector in terms of
s xyz components. R
B b
O Use the following parameters in your analysis: ω1 = 2rad/sec, p = 8rad/sec, d = 1.5m, b = 0.5m
and R = your analysis: ! = 3 rad / s , p = 6 rad / s , d = 0.5 m ,
ollowing parameters in 0.25m. and R = 1 m . 1 Chapter III: Moving Reference Frame Kinematics Homework Moving Reference Frame Kinematics Homework Problems ME 274 III11 Problem III.9 Problem Shaft AB rotates about a ﬁxed axis with a constant rotational speed of Ω. A tube is hinged
Given: III9
˙
Shaftshaftrotates about a fixed axis withtheconstant rotational speed of !.constantis
on AB AB with the angle θ between a tube and shaft increasing at a A tube rate of θ. Particle
P moves within the tube at a constant rate the tube and to the tube. The a constant
˙
hinged on shaft AB with the angle " between of xP relativeshaft increasing atXYZ coordinate system
is ﬁxed Particle Xaxis aligned the the ﬁxed rotation axis x the shaft AB. The xyz
!
rate of ! . with theP moves within withtube at a constant rate ofof! P relative to the tube. coordinate
system is attached to the tube with the xaxis aligned with the tube for all time. For the position
The XYZ coordinate system is fixed with the Xaxis aligned with the fixed rotation axis
shown below, the z and Zaxes are aligned.
of the shaft AB. The xyz coordinate system is attached to the tube with the xaxis aligned
with the tube for all time. For the position shown below, the z and Zaxes are aligned.
Find: For the position shown: For th• position shown:
e Determine the angular velocity and angular acceleration of the tube. Write your answers as
vectors in the a of their xyz components.
a) Determine termsngular velocity and angular acceleration of the tube. Write your
answers as vectors in terms of their xyz components.
• Determine the acceleration of particle P. Write your answer as a vector in terms of its xyz b) Determine the acceleration of particle P. Write your answer as a vector in terms
components.
of its xyz components.
x
Y
y xP " P
X O
A !
B U the following parameters in your analysis: ! = = ad / s , ! θ = , ◦! θ = r 4rad/sec
Use se the following parameters in your analysis: Ω 4 r3rad/sec, = 30°60!, =˙ "2 −ad / s , , xP = 2m
!
xPand3xP and 2m/sec. m / sec .
= ˙m = − xP = !4 III12 Chapter III: Moving Reference Frame Kinematics Homework Problem III.10
Given: The collar and clevis A are given a constant downward velocity of z < 0. This motion
˙
causes end B of bar AB to slide outward within a radial slot of the rotating disk. The disk has a
constant rotational speed of Ω about a ﬁxed axis passing through the center of the disk. The XYZ
eference Frame Kinematics Homework Problems
M rotation axis of the disk. The xyz
coordinate system is ﬁxed with the Zaxis aligned with the ﬁxed E 274
coordinate system is attached to bar AB. For the position shown below, the xyz axes are aligned
with the XYZ axes. m III10
Find: For the position shown:
!
llar and clevis A are given a constant downward velocity of z < 0 . This motion
• Determine the angular velocity of bar AB. Write your answer
end B of bar AB to slide outward within a radial slot of the rotating disk. The diskas a vector in terms of its xyz
components.
onstant rotational speed of ! about a fixed axis passing through the center of the
he XYZ coordinate system is fixed angular acceleration of bar AB. Write your answer as a vector in terms of its
with the Zaxis aligned with the fixed rotation
• Determine the
the disk. The xyz coordinate system is attached to bar AB. For the position shown
xyz components.
the xyz axes are aligned with the XYZ axes.
z, Z position shown:
Determine the angular velocity
of bar AB. Write your answer
as a vector in terms of its xyz
components.
Determine the angular
acceleration of bar AB. Write
your answer as a vector in
terms of its xyz components. L y, Y
x, X ! Use the in your parameters in your / s , L = Ω
˙
following parametersfollowinganalysis: ! = 3 rad analysis: 2.5=ft5,rad/sec, ft =ndf t, z = 1.5f t and z = −3m/sec.
z = 1.5 L a 2
ft / sec . Chapter III: Moving Reference Frame Kinematics Homework III13 Problem III.11
Given: Block A slides within a straight slot cut in a rotating disk with the position of A given by
x = 2sin (4π t), where x and t are given in inches and seconds, respectively. The angular orientation
of the disk is given by θ = 0.2sin (8π t), where and t are given in radians and seconds, respectively.
Find: Determine the acceleration of block A when:
• x = Problems ˙
e Frame Kinematics Homework 0 with x 11 (5/167) >0 • x = 2inches es within a straight slot cut in a rotating
position of A given by x = 2 sin 4! t ,
t are given in inches and seconds,
The angular orientation of the disk is
0.2 sin 8" t , where ! and t are given in
econds, respectively. Determine the
of block A when !
with x > 0 .
inches. ME 274 III14 Chapter III: Moving Reference Frame Kinematics Homework Problem III.12
Given: Link OA rotates with a constant angular speed of ωOA .
Find: Determine:
• the angular velocity of link BC. me Kinematics Homework Problems (5/183) ME 274 • the acceleration of pin A as seen by an observer on link BC. with a constant angular speed of ! OA .
ar velocity of link BC.
ration of pin A as seen by an observer on = 10rad/sec (CW) and θ = 30◦ . g parameters in your analysis: ! OA = 10 rad / sec ( CW ) and ! OA30° .
=
Use the following parameters in your analysis: ω Chapter III: Moving Reference Frame Kinematics Homework III15 Problem III.13
Given: A motor is attached to a platform that is rotating with a constant rate of N about a ﬁxed
vertical axis. The body of the motor pivots about a moving horizontal axis at a constant rate of γ
˙
with the shaft of the motor rotating at a constant rate of ω0 .
Find: Determine:
• the angular acceleration of the disk attached to the shaft of the motor. rame Kinematics Homework Problems (7/12) ME 274 • the velocity of point A on the disk when A is at the top of the disk. hed to a platform that is rotating with
f N about a fixed vertical axis. The
or pivots about a moving horizontal
nt rate of !! with the shaft of the motor
stant rate of ! 0 .
ne the angular acceleration of the disk
to the shaft of the motor.
ne the velocity of point A on the disk
is at the top of the disk. Use the following parameters in your analysis: N = 0, γ = 30◦ , γ = 3π and ω0 = 120rpm.
˙ ng parameters in your analysis: N = 0 , ! = 30° , !! = 3" rad / sec and
in . III16 Chapter III: Moving Reference Frame Kinematics Homework Problem III.14
Given: System shown.
rame Kinematics Homework Problems (7/17) ME 274 Find: Determine the angular velocity and angular acceleration of the jaws of arm O1 O2 . Assume
all rotation rates to the constant. ngular velocity and angular
the jaws of arm O1O2 . Assume all
the constant. ng parameters inse theanalysis: ! 3 = ! 4 = ! 5 =your!1 = 2 rad ω3 e= ,ω4 = ω5 = 0, ω1 = 2rad/sec, θ = 60◦ and
U your following parameters in 0 , analysis: / s c
˙
1.5 rad / sec .θ = 1.5rad/sec. Chapter III: Moving Reference Frame Kinematics Homework III17 Problem III.15
Given: System shown.
Frame Kinematics Homework Problems 5 (7/25) ME 274 Find: Determine the angular velocity and angular acceleration of shaft B. Assume all rotation
rates to the constant. angular velocity and angular
f shaft B. Assume all rotation rates
t. Use the following parameters in your analysis: Ω = 4rad/sec, ω0 = 3rad/sec, γ = 30◦ and
ing parameters˙ in your analysis:. ! = 4 rad / sec , ! 0 = 3 rad / sec ,
γ = (π /4) rad/sec ! = " / 4 rad / sec . III18 Chapter III: Moving Reference Frame Kinematics Homework Problem III.16
Given: System shown.
K inematics Homework Problems /51) ME 274 Find: Determine the angular acceleration of the rotor of the gyroscope. Assume all rotation rates
to the constant. ular acceleration of the rotor of the
Assume all rotation rates to the constant. Use the following parameters in your analysis: N = 20rev/sec, γ = 30◦ and γ = 4rad/sec.
˙ parameters in your analysis: N = 20 rev / sec , ! = 30° and Chapter III: Moving Reference Frame Kinematics Homework III19 Problem III.17
Moving Reference Frame Kinematics Homework Problems ME 274 ˙
Given: Person A is sitting on a merrygoround that is rotating at a constant rate θ. Person B is
running in a straight line at a constant speed vB along the sidewalk. Problem III17
!
Person A is sitting this problem: round that is rotating at a constant rate ! . Person B is
Find: For on a merrygo
running in a straight line at a constant speed vB along the sidewalk. ! • Determine the velocity and acceleration of person B as seen by the moving observer, person
A, at the instant shown. Use vB = 10m/sec θ = 3rad/sec, θ = 30◦ ,
(a) Determine the velocity and acceleration of person ,B˙ as seen by the movingr = 1.5m, d = 8m, and
h = 4m in A, at the instant
!
observer, person your analysis. shown. Use v = 10 m/s, ! = 2 rad/s, ! = 60°, r
B = 1 • ,Show5thatand h answerin your ﬁrst part is independent of the location of observer A. That is,
m d = m, your = 3 m in the analysis.
(b) Show showythatanswer in the first part is independent of velocity and acceleration of the runner B,
that our any observer on the disk sees the same the location of observer
A. That is, show of the observer’s location.disk sees the same velocity and
regardless that any observer on the
acceleration of the runner B, regardless of the observer's location. !
! ! III20 Chapter III: Moving Reference Frame Kinematics Homework Problem III.18Frame Kinematics Homework Problems
Moving Reference ME 274 Given: The mechanism shown below consists of a disk pinned at its center (point A), which rotates
!
with a constant18
Problem III angular velocity, ωAB , a slotted arm CD that is pinned at C, and a bar that can
oscillate within the sguides below consists of a disk pinned at its peg at (point A), whichthe slotted
The mechanism hown at E and F. As the disk rotates, the center B moves within
arm, causing itato rock back and forth. As ! , a slotted arm CD thata slow advance and a quick
rotates with constant angular velocity, the arm rocks, it provides is pinned at C, and a
AB
return to the reciprocating bar due to the change in distance between C and B. bar that can oscillate within the guides at E and F. As the disk rotates, the peg at B
moves within the slotted arm, causing it to rock back and forth. As the arm rocks, it
Find: For the position shown, determine:
provides a slow advance and a quick return to the reciprocating bar due to the change in
d• the angular velocity and For the position shown,the slotted armthe angular velocity
istance between C and B. angular acceleration of determine: (i) CD.
and angular acceleration of the slotted arm CD and (ii) the velocity and acceleration of
• bar. Use ! = 30°, !AB = 40 rpm, bar.
thethe velocity and acceleration of theR = 0.1 m, h = 0.2 m, and d = 0.24 m in your
analysis.
! ! Use the following parameters in your analysis: θ = 36.87◦ , ωAB = 30rpm, R = 0.1m, h = 0.3m
and d = 0.24m. Moving Reference Frame Kinematics Homework Problems ME 274 Chapter III: Moving Reference Frame Kinematics Homework
! III21 Problem III19
At the III.19
Probleminstant shown, the shaft rotates with a constant angular velocity !P . At the same
instant, the disk spins about its axle with a constant angular velocity !S . Determine the
Given: angular instant shown, the shaft the velocity of constantat this instant.
total At the velocity of the disk and rotates with a point C angular velocity ωP . At the same
instant, the disk spins about its axle with a constant angular velocity ωS . Use ! = 10 rad/s and ! = 6 rad/s in your analysis. P
S
Find: Determine the total angular velocity of the disk and the velocity of point C at this instant.
! ! Use the following parameters in your analysis: ωP = 15rad/sec and ωS = 5rad/sec. Moving Reference Frame Kinematics Homework Problems !
III22 ME 274 Chapter III: Moving Reference Frame Kinematics Homework Problem III20!
At the instant shown, the shaft rotates with an angular velocity !P and has an angular
Problem III.20
!
acceleration !P . At the same instant, the disk spins about its axle with an angular
!
velocity !S and has an angular acceleration !S . an angular velocity angular acceleration
Given: At the instant shown, the shaft rotates with Determine the total ωP . At the same instant,
theodisk spins and the accelerationan angular velocity ωS .
f the disk about its axle with of point C at this instant.
Find: Determine the! = 1angular ! = 3 rad/s2, and disk and the ivelocityanalysis. C at this
!P
!
Use !P = 8 rad/s, total 2 rad/s, acceleration of the !S = 2 rad/s2 n your of point
S
instant.
! ! Use the following parameters in your analysis: ωP = 10rad/sec, ωS = 8rad/sec, ωP = −2rad/sec2
˙
and ωS = 3rad/sec2 .
˙ Moving Reference Frame Kinematics Homework Problems ME 274 !
Chapter III: Moving Reference Frame Kinematics Homework
Problem III21! III23 At the instant shown, the helicopter is moving upwards with speed vH and acceleration
aH . At the same instant, the frame H, not the horizontal blade, rotates about a vertical Problem III.21 axis At a constant shown, the helicopter . moving upwards rotates with a and acceleration
Given:with the instant angular velocity of !H is If the tail blade Bwith speed vH constant
aH angular velocity !B / H ,the frame H, (not the H, determine the velocityabout a vertical axis with
. At the same instant, measured relative to horizontal blade) rotates and acceleration of
a constant angular velocity of ωH . The tail blade B rotates with a constant angular velocity ωB/H ,
point P, located on the end of the blade, at the instant the blade is in the vertical position.
measured relative to H.
Use v = 4 m/s, a = 2 m/s2, ! = 0.6 rad/s, and ! = 200 rad/s in your analysis. H
H
B
Find: Determine the velocity andHacceleration of point/ HP, located on the end of the blade, at the
instant the blade is in the vertical position. Use the following parameters in your analysis: vH = 6m/sec, aH = 3m/sec2 , ωH = 0.5rad/sec and
ωB/H = 250rad/sec. Moving Reference Frame Kinematics Homework Problems III24 ME 274 Chapter III: Moving Reference Frame Kinematics Homework Problem III22! A particle P (having a mass of m) slides along bar OA with a constant speed of vrel
Problem III.22
relative to OA. OA is pinned to a vertical shaft that is rotating about a vertical shaft with Given: Particle Pof !. In addition,OA with a constant speed thevvertical shaft with a is pinned
a constant rate slides along bar OA is rotating relative to of rel relative to OA. OA
to a vertical shaft that is rotating about a vertical shaft with a constant rate of Ω. In addition, OA
constant rate of " In vertical shaft with a you are rate of θ . In moving observer who
is rotating relative to .the solving this problem,constant asked to˙ use asolving this problem, you are
is to use a moving observer who is attached to bar OA, as shown below. origin at pin O.
askedattached to bar OA, as shown below. Let xyz be attached to OA with its Let xyz be attached
to OA with its origin = 3 rad/sec, " = 5 rad/sec, R = 2 meters, v = 10 m/sec and " = 0.
Use m = 20 kg, ! at pin O. ˙ ˙ rel i
Find:) For Determine the angular velocity and angular acceleration of bar OA in terms of
the position shown, determine:
their xyz components. • the angular velocity and angular acceleration of bar OA in terms of their xyz components. ii) Determine the acceleration of particle P in terms of its xyz components. • ithe acceleration theparticle P acting onof its xyzin terms of its xyz components.
ii) Determine of net force in terms particle components. observer A P
y x R
" vrel O
! ˙
Use the following parameters in your analysis: Ω = 3rad/sec, θ = 5rad/sec, R = 2m, vrel =
10m/sec and θ = 0. Chapter III: Moving Reference Frame Kinematics Homework
ference Frame Kinematics Homework Problems m III23 III25
ME 274 Problem III.23 ˙
˙
A particle P travels in a tube P travels 6 ft /tube with R = The tube.isThe tube is being raised at a constant
Given: Particle with R = in a sec = constant . constant being
˙rate of ! = 3 rad / sec . In addition, the a vertical shaft which is rotating about the ﬁxed Y
˙
raised at a crate of θ. In addition, the tube is attached to tube is attached to a
onstant
axis withis rotating aboutof ω .fixed observer is attached to the tube with the xyz axes also attached
a constant rate the An Y axis with a constant rate of
vertical shaft which
to . An observer is origin at point O.
! = 4 rad / secthe tube with its attached to the tube with the xyz axes also
attached to the tube with its origin at point O.
Find: For the position shown, determine: You are asked • the angular velocity vector of the observer.= 5 feet and ! =
to find the acceleration of particle P when R
36.87°. In doing so, answer the following questions.
a)
b)
c)
d)
e) Find the • ngular velocity vector of vector of the. observer.
a the angular acceleration the observer
Find the • ngular acceleration vectorseen byobserver.
a the velocity of point P as of the the observer.
What is the velocity of point P as seenas seen observer?
• the acceleration of point P by the by the observer.
What is the acceleration of point P as seen by the observer?
• the acceleration of point P using the above results.
Using the results from a)d) above, find the acceleration of point P.
Y x
P y R
! O X " ˙
˙
Use the following parameters in your analysis: R = 5f t, θ = 36.87◦ , R = 6f t/sec, θ = 3rad/sec
and ω = 4rad/sec. Moving Reference Frame Kinematics Homework Problems ME 274 III26 Chapter III: Moving Reference Frame Kinematics Homework Problem III24
Given: Arm AB is pinned to ground at point A. A block that is pinned to AB
Problem III.24
at B is constrained to slide within a slot on a disk. The disk is rotating
clockwise about a shaft at O a constant rate of 10 rad/sec. At the instant
Given:
shown, theArm ABhorizontal, arm ABat point A. A fblock5that iswith the AB at B is constrained
slot is is pinned to ground is at angle o ! = 3.13° pinned to
to slide
horizontalwithin a slot ft. a disk. The disk is rotating about a shaft at O with a constant rate of
and d = 0.5 on
ωdisk . At the instant shown, the slot is horizontal. Find: At Find: At this position, determine the angular velocity angular
this position, determine the angular velocity and and angular acceleration of arm AB. Write
acceleration of arm AB. Write your answers as vectors.
your answers as vectors.
" B O d
L ! A Use the following parameters in your analysis: ωdisk = 10rad/sec (CW), θ = 53.13◦ , L = 3f t and
d = 0.5f t. Moving Reference Frame Kinematics Homework Problems ME 274 Chapter III: Moving Reference Frame Kinematics Homework III27 Problem III25
Given: The mechanism shown below is driven by a motor attached at B. End A of
Problem III.25
link AB is free to slide within a straight slot within link OC. The angular
elocity of !AB i a constant 2 is driven by a motor attached
Given: vThe mechanism sshown below rad/s in the direction shown. at B. End A of link AB is
free to slide within a straight slot within link OC. The angular velocity of ωAB is in the direction
shown. For the position shown below, determine
Find:
Find: For a) tposition shown, determine:
the he angular velocity of OC. b) the velocity of A relative to an observer positioned on OC and rotating
with OC. • the angular velocity of OC. • the velocity of A relative to an observer positioned on OC and rotating with OC.
c) the angular acceleration of OC.
• the angular acceleration of OC. elative to an observer positioned on OC and
d) the acceleration of A r rotating with OC. • the acceleration of A relative to an observer positioned on OC and rotating with OC. Express your answers as vectors in terms of either their xyz or their XYZ components.
Y B !AB
30 ° y # L 0.17 m x
C " 60 °
O A
0.1 m X Use the following parameters in your analysis: ωAB = 2rad/sec, L = 0.17m, θ = 30◦ and φ = 60◦ . Moving Reference Frame Kinematics Homework Problems ME 274 Problem III26
Chapter III: Moving Reference Frame Kinematics Homework
Given: A mechanism is made up of links AB, BD and DE. This mechanism is pinned
to a shaft that is rotating at a constant rate of ! = 4 rad/sec about a fixed
Problem III.26
horizontal axis. All pins in the mechanism allow rotation about axes that are
perpendicular to the plane containing links AB, BD and DE. When " = 90°, it
Given: A mechanism is made up of links AB, BD and DE. This mechanism is pinned to a shaft
˙
is known that ! = 5 rad / Ω about a ﬁxed horizontal axis. All pins in the mechanism
that is rotating at a constant rate of sec = cons tan t . Shown in the figure below are the
X about that that are to ground and the xyz axes containing links to link
allow rotationYZ axesaxes are fixed perpendicular to the plane that are attached AB, BD and DE.
˙
A,
When θ = 90◦B.it is known that θ = constant. Shown in the ﬁgure below are the XYZ axes that
III28 are ﬁxed to ground and the xyz axes that are attached to link AB. For the position of " = 90°, determine the Find: Find: For the position of θ = 90◦ , determine: a) angular velocity of link AB.
• the angular )velocity of acceleration of link AB.
b angular AB.
c acceleration AB.
• the angular)acceleration of of pin B. • the acceleration in vector
Express ALL answersof pin B. form. In using the rotating reference frame kinematics
equations, please state to which body your observer is attached. x
B Y y d
D b
A " e
h E X !
˙
Use the following parameters in your analysis: Ω = 4rad/sec, θ = 5rad/sec, b = 2f t, d = 2.5f t,
e = 1f t and h = 0.5f t. A shaft rotates about a fixed vertical axis at a constant rate of ! = 5 rad/sec,
as shown below. A straight bar OA, having a length of L = 2 meters, is pinned
to point O on the shaft, with O being on the rotation axis of the shaft. At the
Chapter III: Moving Reference Frame Kinematics Homework
instant when " = 0, bar OA is being raised at a rate of 4 rad/sec from the
horizontal plane. In addition, the rate at which the bar is being raised is
Problem decreasing at a rate 3 rad / sec 2 .
III.27 Given: III29 A set of xyz coordinate axes is attached to bar OA with its origin at O. A Given: A shaft rotates about a ﬁxed vertical axis at a constant rate of Ω, as shown below. A
second having a length axes, is pinned to point O on At shaft, with O being
straight bar OA, set of coordinate of L, XYZ, are fixed to ground. the the instant when " on the
= , the yz and XYZ a instant when θ with each other.
˙
rotation axis0ofthe xshaft. At the xes are aligned = 0, bar OA is being raised at a rate of θ from the
¨
horizontal plane, with this rate changing at a rate of θ. A set of xyz coordinate axes is attached
Fto bar OAFwith its originwhen " = 0:
ind:
or the instant at O. A second set of coordinate axes, XYZ, are ﬁxed to ground. At the
instant when θ = 0, the xyz and XYZ axes are aligned with each other. a) find the angular velocity and angular acceleration of bar OA. Find: For the ) find the acceleration of point A on the bar.
b instant when θ = 0, determine:
• the angular be expressed in vector form. You may OA.
All answers shouldvelocity and angular acceleration of bar choose to write these vectors
in terms ofacceleration xyzpointYZ on the bar.
• the either their of or X A components. Y A y x L "
vertical shaft bar O ! bearing X ˙
¨
Use the following parameters in your analysis: Ω = 5rad/sec, θ = 4rad/sec, θ = −3rad/sec2 and
L = 2m. Problem III28
Given: A robotic manipulator is made up of two links OA and ADB as shown in the
figure below left. Link OA has a fixed length of L = 4 ft, and the length link
III30
ADB is increasing at a cChapter rIII: Moving Reference Frame Kinematics links
onstant ate of 6 ft/sec. The orientation angles of Homework
OA and ADB are increasing at constant rates of d!/dt = 2 rad/sec and d"/dt =
3 rad/sec, respectively.
Problem III.28
Given: A robotic manipulator is made up of two links OA and ADBaas shown0°, the ﬁgure below
Find:
For the position shown below right with b = 3 ft, ! = 0 nd " = 9 in
˙
left. Link OA etermine the acceleration ofthe lengthnlink ADB is changing at a constant rate of b.
d has a ﬁxed length of L, and point B o the manipulator.
Find: For In solving this problem, use an observer that is attached ,to AD and with acceleration of
HINT: the position shown below right with θ = 0 and φ = 90◦ determine the a set of
point B on coordinate axes also attached to AD.
the manipulator. x
x
B
B b y D " L
O ! D A
O 90° y
A ˙
˙
Use the following parameters in your analysis: b = 3f t, b = 6f t/sec = constant, θ = 2rad/sec =
˙
constant, φ = 3rad/sec = constant and L = 4f t. a) find the angular velocity vector of the boom OP.
b) find the angular acceleration vector of the boom OP. ( ) ( ) c) evaluate the Frame aO , v P / O
, a P/O
Chapter III: Moving Reference vectorsKinematics Homework rel and r P / O in the
rel III31 following equation for the acceleration of point P:
a P = aO + a P / O
+ ! " r P / O + 2# " v P / O
+ # " # " r P/O
Problem III.29
rel
rel ( ) ( ) ( ) Note that you are NOT asked to evaluate the right hand side of this Given: A crane is moving to the right with a constant speed of vC and is rotating about a vertical
equation for a P .
˙
˙
axis with a constant rate of Ω. The boom is being raised at a rate of θ with θ changing at a rate
¨.this problem, use an observer attached to the boom and with a moving set of xyz
of θ
For coordinate axes also attached to the boom, as shown in the figure below. XYZ are fixed
Find:
axes. Determine the acceleration of end P of the boom.
Y ! P x d
vC L
y
"
O X Use the following parameters in your analysis: vC = 0.5f t/sec, Ω = 1.5rad/sec, θ = 36.87◦ ,
˙
¨
θ = 0.6rad/sec, θ = 1.2rad/sec2 , L = 20f t and d = 4f t. ...
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 Fall '09
 Cartesian Coordinate System, Acceleration, Velocity, reference frame

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