chapter 15 last 1

7/27/2019 Chapter 15 Last 1

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Vector Mechanics for Engineers: DynamicsT en t h

E d i t i on

Rate of Change With Respect to a Rotating Frame

15 - 2

• Frame OXYZ is fixed.

• Frame OXYZ rotates

about fixed axis OA

with angular velocity

• Vector functionvaries in direction

and magnitude.

t Q





E d i t i on


15 - 3

k Q jQiQQ z y xOxyz

• With respect to the fixed OXYZ frame,

k Q jQiQk Q jQiQQ z y x z y xOXYZ

• With respect to the rotatingOxyz frame, k Q jQiQQ z y x

•

rate of change with respect to rotating frame.

Oxyz z y x Qk Q jQiQ

T





Ed i t i on


15 - 4

• If were fixed withinOxyz then is

equivalent to velocity of a

point in a rigid bodyattached to Oxyz and

OXYZ Q

Qk Q jQiQ z y x

Q

• With respect to the fixed OXYZ frame,

QQQ Oxyz OXYZ

T E





Ed i t i on

Coriolis Acceleration

15 - 5

• Frame OXY is fixed and frameOxy rotates with angular

velocity .

• Position vector for the

particle P is the same in both

frames but the rate of change

depends on the choice of

frame.

P r

T E





Ed i t i on


15 - 6

• The absolute velocity of the particle P is

OxyOXY P r r r v

• Imagine a rigid slab attached tothe rotating frame Oxy or F for

short. Let P’ be a point on theslab which corresponds

instantaneously to position of

particle P . T E


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Ed i t i on


15 - 7

Oxy P r v

F velocity of P

' P v absolute velocity of point P’

on the slab

• Absolute velocity for the particle P may be written as

F P P P vvv

along its path on the slab

T E




Ed i t i on


15 - 8

F P P

Oxy P

vv

r r v

• Absolute acceleration for the particle P

is OxyOXY P r dt

d r r a

OxyOxy P r r r r a

2

OxyOxyOxy

OxyOXY

r r r dt

d r r r

but,

fT E




Ed i t i on


15 - 9

F P P

Oxy P

vv

r r v

Oxy P

P

r a

r r a

F

• Utilizing the conceptual point P’ on the slab,

• Absolute acceleration for the

particle P becomes

22

2

F

F

F

P Oxyc

c P P

Oxy P P P

vr a

aaa

r aaa

Coriolis

accelera

tion

V t M h i f E i D iT E




Ed i t i on


15 - 10

• Consider a collar P which ismade to slide at constant relative

velocity u along rod OB. The

rod is rotating at a constantangular velocity w . The point A

on the rod corresponds to the

instantaneous position of P .

c P A P aaaa

F

• Absolute acceleration of the collar is





Ed i t i on


15 - 11

c P A P aaaa

F

0 Oxy P r a F

uava c P c w 22 F

• The absolute acceleration consists of the

radial and tangential vectors shown

2w r ar r a A A

where





Ed i t i on


15 - 12

uvvt t

uvvt

A

A

,at

,at

• Change in velocity over t is

represented by the sum of three

vectors T T T T R Rv

2w r ar r a A A

recall,

• is due to change in

direction of the velocity of

point A on the rod,

A At t

ar r t

vt

T T

2

00limlim w ww

T T

V t M h i f E i D iT e

E



Vector Mechanics for Engineers: DynamicsTen t h

Ed i t i on


15 - 13

uvvt t

uvvt

A

A

,at

,at

• result fromcombined effects of relative

motion of P and rotation of the

rod

T T R R and

uuu

t

r

t u

t

T T

t

R R

t t

w w w

w

2

limlim00

uava c P c w 22

F

recall,


E




Ed i t i on

Concept Question

2 - 14

v w

You are walking with a

constant velocity with

respect to the platform,

which rotates with a

constant angularvelocity w. At the

instant shown, in which

direction(s) will youexperience an

acceleration (choose all

that apply)?

x

y

OxyOxy P r r r r a

2


E




Edi t i on

Concept Question

2 - 15

v wa) +x

b) -x

c) +y

d) -ye) Acceleration = 0

x

y

OxyOxy P r r r r a

2


E d




Edi t i on

Sample Problem 15.9

15 - 16

Disk D of the Genevamechanism rotates

with constant

counterclockwiseangular velocity w D =

10 rad/s.

At the instant when f = 150o, determine (a)

the angular velocity of disk S , and (b) the

velocity of pin P relative to disk S .


E d



Vector Mechanics for Engineers: Dynamicsen t h

Edi t i on

Sample Problem 15.9

15 - 17

SOLUTION:

• The absolute velocity of the point P

may be written as

s P P P vvv

• Magnitude and direction of velocity

of pin P are calculated from theradius and angular velocity of disk D. P v

• Direction of velocity of point P ’ on

S coinciding with P is perpendicular to

radius OP.

P v

• Direction of velocity of P with

respect to S is parallel to the slot. s P v

• Solve the vector triangle for the

angular velocity of S and relative

velocity of P.

Vector Mechanics for Engineers D namicsT e

E d




di t i on

Sample Problem 15.9

15 - 18

SOLUTION:

• The absolute velocity of the

point P may be written as

s P P P vvv

• Magnitude and direction of

absolute velocity of pin P arecalculated from radius and

angular velocity of disk D.

smm500srad10mm50 D P Rv w

Vector Mechanics for Engineers: DynamicsT eE

d




di t i on

Sample Problem 15.9

15 - 19

• Direction of velocity of P with respect to S is parallel to

slot. From the law of cosines,

mm1.37551.030cos2 2222 r R Rl l Rr

From the law of cosines,

4.42742.030sinsin30sin

R sin

r

6.17304.4290

The interior angle of the vector

triangle is


d




di t i on

Sample Problem 15.9

15 - 20

• Direction of velocity of point P ’ on S coinciding with P is

perpendicular to radius OP.

From the velocity triangle,

mm1.37

smm2.151

smm2.1516.17sinsmm500sin

s s

P P

r

vv

w w

k s

srad08.4w

6.17cossm500cos P s P vv

jiv s P

4.42sin4.42cossm477

smm500 P v


d




di t i on

Sample Problem 15.10

15 - 21

In the Genevamechanism, disk D

rotates with a constant

counter-clockwiseangular velocity of 10

rad/s. At the instant

when j = 150o

,determine angular

acceleration of disk S .


d



Vector Mechanics for Engineers: Dynamicsn t h

di t i on


15 - 22

SOLUTION:

• The absolute acceleration of the pin P may be expressed as

c s P P P 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 .


d




di t i on


15 - 23

SOLUTION:

• Absolute acceleration of

the pin P may be expressed

as c s P P P aaaa

• From Sample Problem

15.9.

jiv

k

s P

S

4.42sin4.42cossmm477

srad08.44.42 w

Vector Mechanics for Engineers: DynamicsT en

E d




di t i on


15 - 24

• Considering each term in theacceleration equation,

jia

Ra

P

D P

30sin30cossmm5000

smm5000srad10mm500

2

222w

jia

jir a

jir a

aaa

S t P

S t P

S n P

t P n P P

4.42cos4.42sinmm1.37

4.42cos4.42sin

4.42sin4.42cos2

w

note: S may be positive or negative


E d





i t i on


15 - 25

s P v

• The direction of the Coriolisacceleration is obtained by

rotating the direction of the

relative velocity by 90o in the sense of w S.

ji

ji

jiva s P S c

4.42cos4.42sinsmm3890

4.42cos4.42sinsmm477srad08.42

4.42cos4.42sin2

2

w


E d





it i on


15 - 26

• The relative accelerationmust be parallel to the

slot.

s P a

• Equating components of the acceleration terms

perpendicular to the slot,

srad233

07.17cos500038901.37

S

S

k S

srad233


E d i





it i on

Group Problem Solving

15 - 27

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 beingmoved to the left at a

constant speed u =

400 mm/s relative tothe sleeve. Determine

the acceleration of

Point D.


E d i





it i on


15 - 28

SOLUTION:

• The absolute

acceleration of point D may be

expressed as

' D D D BC ca a a a

• Determine theacceleration of the

virtual point D’.

• Calculate the Coriolis

acceleration.

• Add the differentcomponents to get the

overall acceleration of

point D.


E d i





t i on


2 - 29

2 DOxy Oxy

a r r r r

Given: u= 400 mm/s, w = (3

rad/s) j. Find: aD

Write overall

expression for aD

Do any of the terms go to zero?

2 DOxy Oxy

a r r r r


E d i





t i on


2 - 30

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


E d i t





ti on

2 - 31

Determine the

Coriolis acceleration

of point D

2

D Oxy Oxya 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


E d i t




Vector Mechanics for Engineers: Dynamics t h

ti on

2 - 32

/

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 k D

Vector Mechanics for Engineers: DynamicsT en t

E d i t




Vector Mechanics for Engineers: Dynamicsth

ti on

15 - 33

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

Vector Mechanics for Engineers: DynamicsT en t

E d i t



Vector Mechanics for Engineers: Dynamicsth

ti on

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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