KINEMATICS OF RIGID BODIES RELATIVE VELOCITY …kisi.deu.edu.tr/emine.cinar/DYNAMICS/FALL 2018/G18...RELATIVE VELOCITY RELATIVE ACCELERATION PROBLEMS . 1. The wheel of radius R rolls

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

qq

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




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




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


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

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aj

aai

ijjijaiaji

jikra

t

t

tt

tt

t

B

B

BB

BB

ij

ABAB


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


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

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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

//

///

/


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°

Documents

KINEMATICS OF RIGID BODIES RELATIVE VELOCITY …kisi.deu.edu.tr/emine.cinar/DYNAMICS/FALL 2018/G18...RELATIVE VELOCITY RELATIVE ACCELERATION PROBLEMS . 1. The wheel of radius R rolls