kinematics of rigid bodies relative velocity …kisi.deu.edu.tr/emine.cinar/dynamics/fall...
Post on 12-Oct-2020
12 Views
Preview:
TRANSCRIPT
KINEMATICS OF RIGID BODIES
RELATIVE VELOCITY RELATIVE
ACCELERATION PROBLEMS
1. The wheel of radius R rolls without slipping and the
center O has an acceleration a0. A point P on the wheel
is a distance r from O. For given values of a0, R and r,
determine the angle q and the velocity v0 of the wheel
for which P has no acceleration in this position.
For given values of a0, R and r, determine q and the velocity v0 of
the wheel for which P has no acceleration in this position.
Rr
Rr
R
r
a
a
aaa
OtP
PP
1
0
/
0/0
sin
sin
0
q
q
0a
OtPa /OnPa /
q ,w
0a
OtPa /
OnPa /
r
aRv
r
aRv
avR
rR
vra
R
vwwRvrwa OnPOnP
q
cos,
cos
cos,,,
00
0
22
0
0
2
02
2
0/
00
2
/
x
y
2. The velocity of roller A is vA = 0.5 m/s to the right as shown,
and this velocity is momentarily decreasing at a rate of 2 m/s2.
Determine the corresponding value of the angular acceleration
of bar AB as well as the tangential acceleration of roller B along
the circular guide. The value of R is 0.6 m.
vA = 0.5 m/s, decreasing at a rate of 2 m/s2. Determine the
corresponding value of the angular acceleration AB and tangential
acceleration of roller B along the circular guide, R = 0.6 m.
iwjw
BB
ABABAABAB
BBBAA
jikwijosviv
jirrwvvvv
josvivvsmiasmiv
1446.0191.1
///
2
1446.0191.15.01515sin
1446.0191.1,
1515sin,)/(2,)/(5.0
c
c
x
y
w
R15sinR
15sin2
RR
Bv
)2.1(2 mR)1446.0(
241.0
m
R
)191.1(985.1 mRmR 6.0
vA = 0.5 m/s, decreasing at a rate of 2 m/s2. Determine the
corresponding value of the angular acceleration AB and tangential
acceleration of roller B along the circular guide, R = 0.6 m.
sradwwwvi
wwvwosvj
w
B
BB
/078.1,5.0464.0,1446.05.015sin
233.115cos
191.1,191.115
319.0
c
x y
w
R
Bv
tBa
nBa
)/329.1( smvB
222
/94.26.0
329.1sm
R
va B
Bn
ABAB rww
AnB
r
AtBABAABAB aaasmiaaaa
//
///
2
/ ,)/(2,
jaiaji
jaiajiaaa
BtBt
BtBtBtBnB
15cos15sin76.084.2
15cos15sin15sin94.215cos94.2
vA = 0.5 m/s, decreasing at a rate of 2 m/s2. Determine the
corresponding value of the angular acceleration AB and tangential
acceleration of roller B along the circular guide, R = 0.6 m.
x y
w
R
Bv
tBa
nBa
ji
ij
BtBt jikkijaiaji
168.0384.1
156.0284.1
1446.0191.1078.1078.1215cos15sin76.084.2
2
2
/89.8
/987.7
7307.1823.13191.1168.05397.0895.1276.0
191.1168.015cos76.0
5587.035.13,1446.0456.315sin
1446.0384.1215sin84.2
sma
srad
aj
aa
ai
Bt
Bt
BtBt
Bt
ij
jik
1446.0191.1
1446.0191.1
3. The hydraulic cylinder imparts motion to point B which causes link OA to rotate. For the instant shown where OA is vertical and AB is horizontal, the velocity vB of pin B is 4 m/s and is increasing at the rate of 20 m/s2. For this position determine the angular acceleration of OA.
vB = 4 m/s, aB = 20 m/s2. Determine the angular acceleration of OA.
4. At the instant represented the velocity of point A of the 1.2 m bar is 3 m/s to the right and is constant for an interval including the position shown, determine the tangential acceleration of point B along its path and the angular acceleration of the bar.
smvsradvi
vvj
ijijviv
jikrv
jvivjvivv
ivvvv
BB
BB
BB
ij
ABAB
BBBBB
AABAB
/38.4/24.3,25.035.0
35.1,17.1866.0
25.017.13866.05.0
25.017.1
866.05.060sin60cos
3,
35.1
25.017.1
//
/
vA = 3 m/s (cst), aB and .
30°
0.5cos60=0.25
0.25
0.50
x
y
Bv
ABv /
Bv
ABv /
Av
60°
0.25 m
1.2 m
1.17 m
30°
0.5cos60=0.25
0.25
0.50
x
y
tBa
tABa /
60°
nBa
nABa /
222//
62.228.12
//
222
/60.12)24.3(2.1
25.017.124.324.3
185.1923.3330sin37.3830cos37.38
/37.385.0
38.4
30sin30cos60sin60cos
smraor
jikkra
jijia
smr
va
jaiaajaiaa
ABAB
ji
ABAB
B
BB
BBBBBB
n
n
n
n
nnnttt
tn
tn
ABABAB
BBB
A
ABAB
aaa
aaa
a
aaa
///
/
0
vA = 3 m/s (cst), aB and .
30°
0.5cos60=0.25
0.25
0.50
x
y
tBa
tABa /
60°
nBa
nABa /
22
095.58
25.017.1
//
/78.23,/24.36
17.162.2433.029.36185.19,17.162.260sin185.19
5.09.41,25.028.1260cos23.33
25.017.162.228.1260sin60cos185.1923.33
25.017.1
smasrad
aj
aai
ijjijaiaji
jikra
t
t
tt
tt
t
B
B
BB
BB
ij
ABAB
vA = 3 m/s (cst), aB and .
5. The elements of a simplified clam-shell bucket for a dredge are shown. With the block at O considered fixed and with the constant velocity v of the control cable at C equal to 0.5 m/s, determine the angular acceleration of the right hand-bucket jaw when q = 45° as the bucket jaws are closing.
block at O considered fixed, vC =0.5 m/s (cst), determine the angular acceleration of the right hand-bucket jaw when q = 45° as the bucket jaws are closing
OB
OB
= 45°
67.5°
CB CB
= 22.5°
34.50
sin
500
5.67sin
600
x
y
x
y
sine theorem
CBOB BB
CBCBB
ji
CBCBCBCB
CCBCB
OBOBB
OBOBOBOBB
OOBOB
vv
ijjv
jikrv
smjvvvv
ijv
jikrvv
vvvv
19.046.05.0
5.22sin5.05.22cos5.0
/5.0,
38.046.0
38.046.0
0,
19.046.0
//
/
//
/
O
B
600 mm
C
67.5°
460 mm
380 mm
block at O considered fixed, vC =0.5 m/s (cst), determine the angular acceleration of the right hand-bucket jaw when q = 45° as the bucket jaws are closing
OB
OB
= 45°
67.5°
O
B
50.34° 600 mm
461.91 mm
38
2.9
4 m
m
CB CB
= 22.5°
sradsrad
j
i
ijjij
vv
OBCB
CBCBOB
CBOBCBOB
CBCBOBOB
BB
CB
CBOB
/362.0,/725.0
5.069.0,46.05.046.0
5.0,19.038.0
19.046.05.038.046.0
5.0
x
y
x
y
ijjia
jikra
smjia
jikkra
aaa
aaaa
OBOBB
ij
OBOBOBOB
OB
ij
OBOBOBOB
OBOBOB
OOBOB
OBOB
t
n
n
tn
38.046.00500.00604.0
38.046.0
/0500.00604.0
38.046.0362.0362.0
0,
38.046.0
//
2/
138.0167.0
//
///
/
block at O considered fixed, vC =0.5 m/s (cst), determine the angular acceleration of the right hand-bucket jaw when q = 45° as the bucket jaws are closing
OB
OB
= 45°
67.5°
CB CB
= 22.5°
22
21.1
/
19.046.0
//
2
/
069.0167.0
//
///
/
/121.0,/1.0
075.075.0,46.0025.046.00500.0
21.1,46.00604.038.00604.0
19.046.0025.00604.0
19.046.0
/025.00604.0
19.046.0725.0725.0
0,
sradsrad
j
i
aa
ijjiaa
jikra
smjia
jikkra
aaa
aaaa
OBCB
CBCBOB
CBOBCBOB
BB
CBCBCBB
ij
CBCBCBCB
CB
ij
CBCBCBCB
CBCBCB
CCBCB
CB
CBOB
CBCB
t
n
n
tn
O
B
50.34° 600 mm
461.91 mm
38
2.9
4 m
m
x
y
x
y
6. In the mechanism shown, the flexible band F is attached at E to the rotating sector and leads over the guide pulley. F is given a constant velocity of 4 m/s as shown. For the instant when BD is perpendicular to OA, determine the angular acceleration of BD.
vE = 4 m/s (cst), BD perpendicular to OA, determine the angular acceleration of BD.
7. At a given instant, the gear has the angular motion shown. Determine the accelerations of points A and B on the link and the link’s angular acceleration at this instant.
x
y AB
AB
ABv /
ABr /
C x
y
scmv
ivi
j
ijiiv
ivv
ijiv
jikirivvv
ijkrvvv
B
ABB
ABAB
ABABB
BB
ABABB
ji
ABABABABAB
CACACA
/6
93.66
0,40
93.646
93.646
60sin860cos866
66
93.64
//
//
0
Determine aA, aB and AB
x
y AB
AB
tABa /
ABr /
C x
y
Determine aA, aB and AB
2
22
/
/73,7212,3612108
)36)(1()12)(1()36(3
scmajiajija
jijjrirjraaa
AAA
CACA
22
2
93.64
//
0
//
////
/74.112,/74.112
)18(93.612
)(/18,4720
93.647212
93.64
scmiascma
ai
cwsradjj
ijjiiaa
jikra
ra
aaaaaaa
BB
B
ABAB
ABABBB
ij
ABABABBA
ABBABABBA
ABABABAABAB
ABAB
t
n
tn
8. The disk with a radius r = 220 mm rolls on
the smooth horizontal surface without slipping
with an angular velocity of d = 3 rad/s (ccw).
End A of rod AB (length lAB = 500 mm) is fixed
on the disk. End D of rod BD (length lBD = 350
mm) is fixed on the collar which can slide
freely on the shaft. At the instant shown, the
velocity of collar D is constant and directed
downwards with a magnitude of vD = 8 m/s.
Also at this instant, the acceleration of the
center O of the disk has a magnitude of aO =
1.76 m/s2 directed to the left. Determine the
angular velocities of rods AB and BD (AB, BD)
and the angular accelerations of rods AB and
BD (AB, BD) at this instant. Take r0 = 180 mm,
q = 36.87°, = 60°, g = 45°.
r = 220 mm, d = 3 rad/s (ccw), lAB = 500 mm, lBD = 350 mm, vD = 8 m/s
(constant), aO = 1.76 m/s2. Determine the angular velocities of rods AB and
BD (AB, BD) and the angular accelerations of rods AB and BD (AB, BD)
at this instant. Take r0 = 180 mm, q = 36.87°, = 60°, g = 45°.
9. In the mechanism shown, collar C
follows a curvilinear path defined
by [m], where q is in radians
and b = 0.544. At the instant shown,
the radius of curvature of the path
followed by C is r = 0.8 m and the
velocity of C is vC = 2 m/s, which is
increasing at a rate of 3 m/s2.
Angles = 12° and = 27°.
Determine the angular accelerations
of bars AB and BC for the instant
represented.
q
2
bR
In the mechanism shown, collar C follows a curvilinear path defined by
[m], where q is in radians and b = 0.544. At the instant shown, the
radius of curvature of the path followed by C is r = 0.8 m and the velocity
of C is vC = 2 m/s, which is increasing at a rate of 3 m/s2. Angles = 12°
and = 27°. Determine the angular accelerations of bars AB and BC for the
instant represented.
q
2
bR
PROBLEMS
O q 45°
B
A
1.5 cm
2.5 cm
1 cm
vo
D y
x x=1.25 cm
1.5 cm
7 cm
10.The stepped disk that acts as a single unit rolls on the horizontal surface without
slipping. The center O of the stepped disk has a velocity of vO= 6 cm/s directed to the
right. For the instant represented, the acceleration of point B on the outer rim of the
disk has an acceleration given as (cm/s2). Point A on the disk is
connected to member AD by a pin. The roller at the end D of member AD can slide
freely along the parabolic slot. For the instant depicted, determine the angular velocity
and angular acceleration of member AD and the magnitude of the absolute acceleration
of point D.
jiaB
39.1275.57
4
2xy q37°
PROBLEMS
O q 45°
B
A
1.5 cm
2.5 cm
1 cm
vo
D y
x x=1.25 cm
1.5 cm
7 cm
vO= 6 cm/s (→), (cm/s2)
AD = ?, AD = ?, aD = ?
jiaB
39.1275.57
4
2xy q37°
top related