1

ME 302 DYNAMICS OF ME 302 DYNAMICS OF MACHINERYMACHINERY

Dynamic Force Analysis V

Dr. Sadettin KAPUCU

© 2007 Sadettin Kapucu

2Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

X

Z

Y

O

Inertial FrameX

Z

Y

O

X

Z

Y

O

X

Z

Y

O

X

Z

Y

O

X

Z

Y

O

axisYaboutrotationis oA 90axisXaboutrotationis o

B 90

axisYaboutrotationis oA 90 axisXaboutrotationis o

B 90BA

AB

3Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

Inertial FrameX

Z

Y

O

X

Z

Y

O

BA AB

Finite Rotation

4Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

X

Z

Y

O

ABBA

X

Z

Y

O

Xaboutthen

axisYabout

B

A

X

Z

Y

O

Yaboutthen

axisXabout

A

B

5Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

tt

0lim

Infinitesimal rotation can be treated as vector

tt

0lim

6Gaziantep University

Rotation of a Rigid Body with Rotation of a Rigid Body with One Point FixedOne Point Fixed

http://hyperphysics.phy-astr.gsu.edu/HBASE/top.html

7Gaziantep University

PreliminaryPreliminary Kinematics of a Rigid Body Kinematics of a Rigid Body

2

1 21

8Gaziantep University

PreliminaryPreliminary Kinematics of a Rigid Body Kinematics of a Rigid BodyZ

Y

Xx

y

z Body coordinate

2

1

x2xyzXYZ dt

d

dt

d

21

21221 x

xyzxyz dt

d

dt

d

9Gaziantep University

Rotation of a Rigid Body with Rotation of a Rigid Body with One Point FixedOne Point Fixed

10Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

X

Z

YP

x

z

y

O

O’

Inertial Frame

Body coordinate system it rotates at the same

angular velocity as the body

Arbitrary point in the body

Rigid body angular

velocity wrt inertial frame

11Gaziantep University

oR

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

x

Z

YP

x

z

y

O

O’

r

R

Position of P

kpjpipr zyx

�

The position of P wrt inertial coordinate frame

rRR o

The absolute velocity of P is

dt

rd

dt

Rd

dt

RdV o

12Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

oR

X

Z

YP

x

z

y

O

O’

r

R

dt

kdp

dt

jdp

dt

idpk

dt

dpj

dt

dpi

dt

dp

dt

rdzyx

zyx

�

rRR o

The absolute velocity of P is

dt

rd

dt

Rd

dt

RdV o

dt

kdpk

dt

dp

dt

jdpj

dt

dp

dt

idpi

dt

dp

dt

rdz

zy

yx

x

Becomes zero because body is rigid

}}rx

rxdt

rd

13Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

rRR o

The absolute velocity of P is

dt

rd

dt

Rd

dt

RdV o

oo V

dt

Rd

rxdt

rd

rxVV o

oR

X

Z

YP

x

z

y

O

O’

r

R

14Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

rRR o

The absolute velocity of P is

rxVV o

Acceleration of P wrt inertial coordinate system is

dt

rxd

dt

Vd

dt

Vda o )(

dt

rdxrx

dt

daa o

rxxrxaa o

oR

X

Z

YP

x

z

y

O

O’

r

R

15Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The 0.8 m arm OA for a remote-control mechanism is pivoted about the horizontal x-axis of the clevis, and the entire assembly rotates about the z-axis with a constant speed N=60rev/min. Simultaneously the arm is being raised at the constant rate . For the position where =60o determine (a) angular velocity of OA, (b) the angular acceleration of OA, (c) the velocity of point A, and (d) the acceleration of point A.

srad /4

sradNz /283.660/)60(260/2

sradkizx /283.64

xx k

283.62/13.254283.6 sradjixk

z

x

kjr

4.0693.0

smkji

kji

rxV

/77.260.135.4

4.0693.00

283.604

sradx /4

16Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The 0.8 m arm OA for a remote-control mechanism is pivoted about the horizontal x-axis of the clevis, and the entire assembly rotates about the z-axis with a constant speed N=60rev/min. Simultaneously the arm is being raised at the constant rate . For the position where =60o determine (a) angular velocity of OA, (b) the angular acceleration of OA, (c) the velocity of point A, and (d) the acceleration of point A.

srad /4

sradkizx /283.64

2/13.254283.6 sradjixk

z

x

kjr

4.0693.0

Vxrx

rxxrxa

)(

2/40.644.3811.20

77.260.135.4

283.604

4.0693.00

013.250

smkji

kjikji

a

17Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The electric motor with an attached disk is running at a constant low speed of 120 rev/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.

srad /4

)sin(cos kj

srado /460/)2(120

srad /260/)60(2

Kkoo

sradkj

kj

kj

kjk

oo

o

o

/)0.53(

)30sin24()30cos2(

)sin()cos(

)sin(cos

18Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.

srado /460/)2(120 srad /260/)60(2

sradkj /)0.53(

19Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.

srado /460/)2(120

srad /260/)60(2

x

sradiii

iio

o

o

/4.6830cos)4)(2()cos(

)cossin()cossincos( 2

kjxkj o

)sin()cos()sin(cos

20Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.

srado /460/)2(120 srad /260/)60(2

sradkj /)0.53(

21Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The electric motor with an attached disk is running at a constant low speed of 120 rey/mm in the direction shown. Its housing and mounting base are initially at rest. The entire assembly is next set in rotation about the vertical Z-axis at the constant rate N=60 rev/min with a fixed angle of 300. Determine (a) the angular velocity and angular acceleration of the disk, (b) the space and body cones, and (c) the velocity and acceleration of point A at the top of the disk for the instant shown.

srad /4

kjr

250.0125.0

smi

kji

rxV /1920.0

250.0125.00

530

Vxrxrxxrxa )(

2/83.116.26

)192.0()53()250.0125.0(4.68

smkj

ixkjkjxia

22Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

Br

X

Z

YA

x

z

y

O

B

BAr

Ar

coordinate system rotates with

this angular velocirty

Body coordinate frame rotates with this angular velocirty

BFLetting

rxrVV BA

Denote the angular velocity of the reference wrt the body frame, the angular

velocity of the body is related to that of the coordinate system

BF

The velocity of a point of the body may be represented by

rxVVrxrxVV relBB

FBA

23Gaziantep University

PreliminaryPreliminaryKinematics of a Rigid BodyKinematics of a Rigid Body

Br

X

Z

YA

x

z

y

O

B

BAr

Ar

coordinate system rotates with

this angular velocirty

Body coordinate frame rotates with this angular velocirty

BA

BA

BA

BABA rxxrxrxraa

2

Acceleration of a point of the body is obtained as:

The velocity of a point of the body may be represented by

BA

BF

BF

BA

BFrel rxxrxa

rel

BA

BF Vxrxx

22

BA

BA

BA

FB

BA

FB

FB

BA

FBBA

BA

BArelrelBA

rxxrxrxxrxxrxaa

rxxrxVxaaa

2

2

rxrVV BA

24Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The motor housing and its bracket rotate about the Z axis at the constant rate The motor shaft and disk have a constant angular velocity of spin with respect to the motor housing in the direction shown. If constant at 30o, determine the velocity and acceleration of point A at the top of the disk and angular acceleration of the disk.

srad /3

K

3

JrB

350.0

kjrB

A

120.0300.0

smiI

JxKrxV BB

/05.105.1

350.03

relBA VrxVV

smiiikjxKrxB

A /599.0)30sin36.0()30cos9.0()120.0300.0(3

smikjxjrxpVB

Arel /960.0)120.0300.0(8

smiiiiVA /689.0960.0599.005.1

sradp /8

25Gaziantep University

Kinematics of a Rigid BodyKinematics of a Rigid BodyExampleExample

The motor housing and its bracket rotate about the Z axis at the constant rate The motor shaft and disk have a constant angular velocity of spin with respect to the motor housing in the direction shown. If constant at 30o, determine the velocity and acceleration of point A at the top of the disk and angular acceleration of the disk.

srad /3

2/899.073.2)30sin30cos(15.3

15.3)350.03(3)(

smkjkj

JJxKxKrxxa BB

2/899.0557.1)599.0(3

)120.0300.0(33)(

smkjixK

kjxKxKrxxB

A

2/88.299.4)30sin30cos(76.576.5960.0)3(22 smkjkjJixKVx rel

sradp /8

BA

BArelrelBA rxxrxVxaaa

2

0

2/68.7))120.0300.0(8(8)( smkkjxjxjrxpxpaB

Arel

2/086.8703.0 smkjaA

222 /12.8086.8703.0 smaA 2/8.20)30cos24(0 sradii

)83(3 jKxKx