3 D Kinetics Forces and Moments Sample Problems With Solutions(2)

8/10/2019 3 D Kinetics Forces and Moments Sample Problems With Solutions(2)

1/28

A uniform 1.6 kg rodAB is welded at its midpoint G to avertical shaft GD. Knowing that the shaft rotates with an

angular velocity of constant magnitude = 1200 rpm,determine the angular momentum HGof the rod about G.

Single rotation about a fixed point


2/28

x

y

( )

[ ]

22

cos sin , sin cos

21200 12!.66

60

sin cos "2.#$ 11$.1

1.6

1 1.6 0.610, 0."#61, 0, 0

12 12

0 0 0

0 0."#61 00 0 0

x x y y z z x y y z z x

x y z

i i j j i j

rad s rad s

j i j i j

m

I I ml I I I I

= = +

= =

= = + = +

=

= = = = = = = =

=

I

Find I and in inclined rotating frame

0%ew unit vectors 20

cos sin , sin cos

cos sin , sin cos

i i j j i j

i i j j i j

= = = +

= = +

Locate principal axes of the rod

Orient rotating coordinates along

principal axes

Find the transformations from

otating to ground frame and vice

versa


3/28

x

y

( ) [ ] { }

0 0 0 "2.#$

0 0."#61 0 11$.1

0 0 0 00."#61 11$.1 !$.!#

x y z x y zx y z

j j

= =

= =

GH I

0%ew unit vectors 20

cos sin , sin cos

cos sin , sin cos

i i j j i j

i i j j i j

= = = +

= = +

Find H in inclined rotating frame.

j

Transform to inertial frame

( ) ( )

cos sin , sin cos

!$.!# !$.!# sin cos 20.0" !!.06xyz

i i j j i j

j i j i j

= = += = + = +

GH


4/28

thin homogeneous dis! of mass m and radius r is mounted

on the hori"ontal axleAB. The plane of the dis! forms an

angle #$$% &ith the vertical. 'no&ing that the axle rotates

&ith an angular velocity( determine the angle formed by theaxle and the angular momentum of the dis! about G.

)etermine the rate of change HG of the angular momentum HGof the dis! for an arbitrary value of *( !no&ing that the dis! has

an angular velocity + , +i and an angular acceleration - , -i.

homogeneous ./ !g dis! is mounted on the hori"ontal shaft

AB. The plane of the dis! forms a #$0 angle &ith the yz plane

as sho&n. 'no&ing that the shaft rotates &ith a constantangular velocity + of magnitude 1$ rad2s( determine the

dynamic reactions at pointsA and B.

'no&ing that the assembly is initially at rest 3+ , $4 &hen a

couple of moment M$, 3#5.5 6.m4i is applied to the shaft(

determine 3a4 the resulting angular acceleration of the

assembly( 3b4 the dynamic reactions at pointsA and Bmmediately after the couple has been applied.

Single rotation about a fixed axis


5/28

y

x

2 2 2

, ,

%ew unit vectors

cos sin , sin coscos sin , sin cos

cos sin

1 1 1, , , 0

2 " "x x y y z z xy yz zx

i i j j i ji i j j i j

i i j

I mr I mr I mr I

= + = + = = +

= =

= = = =

thin homogeneous dis! of mass m and radius r is mountedon the hori"ontal axleAB. The plane of the dis! forms an

angle #$$% &ith the vertical. 'no&ing that the axle rotates

&ith an angular velocity( determine the angle formed by theaxle and the angular momentum of the dis! about G.

xes chosen

for constant I.

7alculations

are easier

[ ]{ }

2

2 2 2

2

10 0

2 cos1 1 1

0 0 sin cos sin" 2 "

01

0 0"

G

mr

i

H mr j mr i mr j

mr

= = =

I

( ) ( )2 2

2 2 2 2 2

1 1cos cos sin sin sin cos

2 "

1 1 1cos sin sin cos2 " "

GH mr i j mr i j

mr i mr i mr j

= + +

= + +

Find I and in inclinedrotating frame.

Find H8in rotating frame

Transform to

inertial frame


6/28

y

xxes chosen

for constant I.

7alculations

are easier

2 2 2 21 1 1 1 1cos sin cos sin sin cos2 " 2 " "

GH mr i j mr i i j = = + +



( ) ( ) ( )G G G xyzxyz x y zH H H = + & &

Sitting in xy" frame &e &ill not see Ichange but &e &ill

certainly see the disc spinning faster at

and angular momentum change as

( ) [ ]{ } [ ] { } [ ] { } [ ] { } { }

( )

2

2 2

2

0

10 0

2 cos1 1 1

0 0 sin cos sin" 2 "

01

0 0"

G x y zx y z x y zx y zx y z

G x y z

d d d

H dt dt dt

mr

i

H mr j mr i j

mr

= = + = +

= =

I I I I &

&

cos sini i j = =


7/28

y

xxes chosen

for constant I.

7alculations

are easier



( ) ( ) ( )G G G xyzxyz x y zH H H = + & &

( )2

2 2 2 2 2

1 1

cos sin2 "

1 1 1cos sin sin cos

2 " "

G x y zH mr i j

mr i mr i mr j

=

= + +

&

Transform firstcomponent to inertial

frame coordinates

9otating frame rotates at

( ) 2 2 2 2 21 1 1 1

cos sin sin cos sin cos2 " " "

G xyzH i mr i i j mr k

= + + =

This is not true for all probi lems =

Second component

( ) ( ) ( )

( )2 2 2 21 1 1 1cos sin sin cos sin cos

2 " " "

G G G xyzxyz x y z

G xyz

H H H

H mr i i j k

= +

= + + +

& &

&

6et rate of change of H8as seen from ground


8/28

y

xxes chosen

for constant I.

7alculations

are easier

( )

( )

( )

2

2

2 2

2 2

1 1cos sin

2 "

1 1cos sin

2 "

1 1 1 1cos sin cos sin2 " 2 "

1 1 1 1cos sin cos

2 " 2

G

G

H mr i j

d dH mr i j

dt dt

d dmr i j mr i jdt dt

d dimr i j mr

dt dt

=

=

= + = +

( ) ( )

( ) ( )

2 2

2

2

2 2

sin"

1 1 1 1cos sin cos sin

2 " 2 "


2 "


2 "

1 1cos

2

dj

dt

mr i j mr i i i j

mr i j i j

mr i i j i i j

mr i

= + = + +

+ + +

= + 2 2 2

2 2 2 2

1 1 1sin sin cos cos sin sin cos

" " 2 "

1 1 1 1cos sin sin cos sin cos

2 " " "

i j mr k k

mr i i j k

+ + = + + +

lternate method

2 2 2 21 1 1 1cos sin sin cos sin cos2 " " "

GH mr i i j k = + + +

&


9/28

y

x

homogeneous ./ !g dis! is mounted on the hori"ontal shaft

AB. The plane of the dis! forms a #$0 angle &ith the yz planeas sho&n. 'no&ing that the shaft rotates &ith a constant

angular velocity + of magnitude 1$ rad2s( determine the

dynamic reactions at pointsA and B.

( ) ( )0, 0X! X!

= =G G

v a'inematics of center of mass

8 in the inertial frame

6e&ton:s la&

( )

( )0 0

, 0

y z y z

y y z z

dm

dt

dA j A k B j B k m"j

dt

A B m" A B

=

+ + + = =

+ = + =

ext GF v

Az

mg9evolute ;oints

apply only x and

y reaction forces

and no moments

Ay

Bz

By


10/28

y

x

2 21sin cos

"GH mr k =&

( ) ( )0, 0X! X!

= =G G

v a9ate of change of H8


11/28

y

x

Az

mg9evolute ;oints

apply only x and

y reaction forces

and no moments

Ay

Bz

By

2 2

2 2

2 2

2 2

2 2

, 0

1sin cos , 0

"

1 1sin cos

2 $

1 1sin cos

2 $

0

0

&tatic reactions

1

2

1

2

'ence dynamic reactions

1sin cos

$

1

sin$

y y z z

y y z z

y

y

z

z

y

y

y

y

A B m" A B

mrB A A B

l

mrA m"

l

mrB m"

l

A

B

A m"

B m"

mrA

l

mr

B l

+ = + =

= =

=

= +

==

=

=

=

= cos


12/28

y

x

( ) ( )0, 0X! X!

= =G G

v a'inematics of center of mass

8 in the inertial frame

6e&ton:s la&

( )

( )0 0

, 0

y z y z

y y z z

dm

dt

dA j A k B j B k m"j

dt

A B m" A B

=

+ + + = =

+ = + =

ext GF v

Az

mg9evolute ;oints

apply only x and

y reaction forces

and no moments

Ay

Bz

By

'no&ing that the assembly is initially at rest 3+ , $4 &hen a

couple of moment M$, 3#5.5 6.m4i is applied to the shaft(

determine 3a4 the resulting angular acceleration of the

assembly( 3b4 the dynamic reactions at pointsA and B

immediately after the couple has been applied.


13/28

y

x

2 2 2 21 1 1 1cos sin sin cos sin cos2 " " "

GH mr i i j k = + + +

&

( ) ( )0, 0X! X!

= =G G

v a

9ate of change of H8


14/28

y

x

Az

mg9evolute ;oints

apply only x and

y reaction forces

and no moments

Ay

Bz

By

( ) ( )

2 2 2 2 0 00 2 2 2 2 2

2

2 2

2 2 2 2

2

, 0

1 1 " "cos sin

2 " 2cos sin 1 cos

1sin cos

"

1sin cos

"

1 1 1 1sin cos , sin cos

2 $ 2 $

1sin cos ,

$

y y z z

z z

y y

y y

z

A B m" A B

# ## mr mr

mr mr

A B mr

B A mr

mr mr A m" B m"

l l

A mr

+ = + =

= + = =+ +

=

=

= = +

=

( ) ( )

2

2 2 2 2

2 0 0

2 2

1sin cos

$&tatic reactions

1 1,

2 2

'ence dynamic reactions

1 1sin cos , sin cos

$ $

1 sin cos sin cossin cos ,

$ 2 1 cos 2 1 cos

z

y y

y y

z z

B mr

A m" B m"

mr mr A B

l l

# #A mr B

=

= =

= =

= = = + +


15/28

5=.5 !g advertising panel of length #a , #.# m and &idth #b

, 1.= m is !ept rotating at a constant rate 1about itshori"ontal axis by a small electric motor attached atA to frame

ACB. This frame itself is !ept rotating at a constant rate #

about a vertical axis by a second motor attached at C to thecolumn CD. 'no&ing that the panel and the frame

complete a full revolution in / s and 1# s( respectively(

express( as a function of the angle ( the dynamic reactionexerted on column CD by its support at D.

T&o rotations along intersecting axes

&how that (a) the dynamic reaction atD is independent

of the length 2a of the panel, (b) the ratio#1#2 of the

magnitudes of the couples e*erted by the motors atA and

$, respectively, is independent of the dimensions and

mass of the panel and is e+ual to # 1at any giveninstant.


16/28


( )

( )

1 2 1 2 2 2 1

2 2 2 2

, ,

%ew unit vectors

cos sin , sin cos

cos sin , sin cos

sin cos sin cos

1 1 1, , , 0

x x y y z z xy yz zx

i i j j i j

i i j j i j

k j k i j i j k

I m a b I ma I mb I

= + = + = = +

= + = + + = + +

= + = = =

( ) [ ] { }

( )

( )

2 2

2

2

2

12

2 2 2 2

2 2 1

10 0

sin1

0 0 cos

10 0

1 1 1

sin cos

G x y z x y z x y z

m a b

i

H ma j

k

mb

m a b i ma j mb k

+

= =

= + + +

I

( ) ( )

( ) ( ) ( )

( ) ( )

2 2 2 2

2 2 1

2 2 2 2

2 2 1

2 2 2 2 2 2 2 2 2

2 2 2 2 1

2

2

1 1 1sin cos

1 1 1sin cos sin cos sin cos

1 1 1 1 1sin cos cos sin sin cos

1sin cos

G x y zH m a b i ma j mb k

m a b i j ma i j mb k

m a b i ma i m a b j ma j mb k

mb

= + + +

= + + + + +

= + + + + +

= 2 2 2 22 2 11 1 1

sin

i mb j ma j mb k + + +

Find I and in inclined rotating frame.

Find H8in rotating frame


x

y


17/28


( )

( )

1 2 1 2 2 2 1

2 2 2 2

, ,

%ew unit vectors

cos sin , sin cos

cos sin , sin cos

sin cos sin cos

1 1 1, , , 0


i i j j i j

i i j j i j

k j k i j i j k

I m a b I ma I mb I

= + = + = = +

= + = + + = + +

= + = = =

( ) [ ] { } [ ] { } [ ] { }

( ) ( )2 2 2 2

2 2

2 2

2 2

12 2

1 10 0 0 0

sin sin1 1

0 0 cos 0 0 c

1 10 0 0 0

G x y z x y z x y z x y z x y z x y zx y z

d d dH

dt dt dt

m a b m a b

i id d

ma j madt dt

k

mb mb

= = +

+ +

= +

I I I &

( )

( ) ( )

1

2 2

1 2

2

1 2 1 2 1

2

2 2 2

1 2 1 2

os

10 0

cos1

0 0 0 sin , , 0, 0

01

0 0

1 1cos sin

G

x y z

j

k

m a b

i

ma j

mb

H m a b i ma j

+

= + = = =

= +

& & &Q

&

( ) ( )

( ) ( ) ( )

2 2 2

1 2 1 2

2 2 2

1 2 1 2

2 2 2 2

1 2 1 2 1 2

1 1cos sin


1 1 1cos cos sin

G x y zH m a b i ma j

m a b i j ma i j

ma i mb i mb j

= +

= + + +

= + +

&

Find I and in inclined rotating frame.

Find H8> in rotating frame


x

y


18/28


( ) ( ) ( )

( )

( ) ( )

2 2 2 2

1 2 1 2 1 2

2 2 2 2 2

2 1 2 2 2 1

2 2 2 2

1 2 1 2 1 2

2 2 2

2 1 2

2

1 2

1cos cos sin

1

sin cos sin

1 1 1cos cos sin

1 1sin cos

1sin

G G G xyzxyz x y zH H H

m a i b i b j

j k m b i b j a j b k

ma i mb i mb j

mb k mb i

mb

= +

= + +

+ + + + +

= + +

+

+

& &

2 2 2

1 2 1 2

2 2 2 2 2

1 2 1 2 2

1 1cos sin

2 2 1

cos cos sin sin cos

j mb i ma i

mb i mb j mb k

= +

x

y ( )2 1j k = +


19/28

homogeneous dis! of mass m = / !g rotates at the constantate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded to ahaft DCE rotating at the constant rate #, ? rad2s. )eterminehe angular momentum HAof the dis! about its centerA.

)etermine the angular momentum HDof the dis! about point D.

etermine the rate of change of HAof the disk

homogeneous dis! of mass m , !g rotates at the constant

ate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded tohaft DCE rotating at the constant rate #, , ? rad2s.

)etermine the dynamic reactions at D and E.

It is assumed that at the instant sho&n shaft DCE has an

angular velocity #, ? rad2s i and an angular acceleration#, / rad2s i. 9ecalling that the dis! rotates &ith a constantangular velocity 1, 1/ rad2sj( determine 3a4 the couplethat must be applied to shaft DCE to produce the given

angular acceleration( 3b4 the corresponding dynamicreactions at D and E.

T&o rotations along non intersecting axes


20/28

"

y

x

homogeneous dis! of mass m = / !g rotates at the constantate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded to ahaft DCE rotating at the constant rate #, ? rad2s. )eterminehe angular momentum HAof the dis! about its centerA.

2 2 2

, ,

2 1 2 1

, ,

1 1 1, ,

is the net angular ve

, 0

loci

" 2 "


i i j j k k

I mr I mr I mr I

i j i j

= = =

= = = =

= + = +

[ ]{ }

2

2

2 2 2

1 2 1

2

10 0

"1 1 1

0 02 " 2

01

0 0"

G

mr

i

H mr j mr i mr j

mr

= = = +

I

2 2

2 1

1 1

" 2AH mr i mr j = +

Find I and in rotating

frame. In this case aninclined frame is not

re@uired as principal axes

are already along xy"

Find Hin rotating frame

Transform to

inertial frame

is the cm of the dis!


21/28

"

y

x

2 2 2

, ,

2 1 2 1

, ,

1 1 1, ,

is the net angular ve

, 0

loci

" 2 "


i i j j k k

I mr I mr I mr I

i j i j

= = =

= = = =

= + = +

[ ]{ }

( ) ( ) ( )

2

2

2 2 2 2 2

1 2 1 2 1

2

2 2

2 1 2

2 2 2

2 1 2

10 0

"

1 1 1 1 10 0

2 " 2 " 20

10 0

"

1 1

" 2

1 1

" 2

A

D A A D A D$ BA $B BA $B

D BA $

mr

i

H mr j mr i mr j mr i mr j

mr

H H m mr i mr j l i l j l k m l k l j

H mr i mr j m l i l

= = = + = +

= + = + + + +

= + + +

I

r v

( )2B D$ BA D$ $Bi l l j l l k +

Find I and in rotatingframe. In this case an

inclined frame is not

re@uired as principal axes

are already along xy"

)etermine the angular momentum HDof the dis! about point D.


( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( ) ( )

( )

2 2

2

2 2 2

2 2 2

2

2 2

2 2

2 2 2 2

2

2 2'ere 0,

A A D D$ BA $Bxyz

A BA $Bxyz

A A D A DX!

D$ BA $B BA $B

BA $B BA $B

A $B BA BA $BX!

A BAX!

i i l i l j l k

l k l j

i i i

i l i l j l k i l k l j

l k l j l j l k

l l j l l k

l j

= = +

= +

= +

= + + +

= + + +

= + +

= = +

v r

v

a r r

a

a2

2 $Bl k

'inematics of center of

mass in the inertial

frame


22/28

"

y

x

( ) 2 2 2 22 1 2 11 1 1 1

" 2 " 2

A x y zH mr i mr j mr i mr j = + = + ( ) ( ) ( )A A A xyzxyz x y zH H H = +

& &

( ) [ ]{ } [ ] { } [ ] { } [ ] { } { }

( )

2 1 2 1 2 1

2

2

2 2

1 2 1

2

0

10 0

"

1 1 10 0

2 " 20

10 0

"

A x y zx y z x y zx y zx y z

x y z

A x y z

d d d

H dt dt dt

i j i j i j

mr

i

H mr j mr i j

mr

= = + = + = + = + = +

= = + =

I I I I

&

& 22 1

1 1

" 2mr i j

+

etermine the rate of change of HAof the disk

2i=

( )

( )

2 2 2

2 1 2 2 1

2

2 1 1 2

1 2

2

1 2

1 1 1 1

" 2 " 2

1 1 1

" 2 2

/n this problem 0

1

2

A xyz

A

xyz

H mr i j i mr i mr j

mr i j k

H mr k

= + + +

= + +

= =

=

&

&



23/28

"

y

x


homogeneous dis! of mass m , !g rotates at the constant

ate 1 , 1/ rad2s &ith respect to armABC( &hich is &elded toshaft DCE rotating at the constant rate #, , ? rad2s.)etermine the dynamic reactions at D and E.

Dz

Dy

Ez

Ey

( ) 2 1 21

2A xyz

H mr k =&

( ) ( )

( ) ( ) ( )

( )

2

2 2

2 2 2 2

2 2

2 2 2'ere 0,

A BA $Bxyz

A $B BA BA $BX!

A BA $BX!

l k l j

l l j l l k

l j l k

= +

= + +

= = +

v

a

a

'inematics of center of mass

in the inertial frame

9ate of change of H

6e&ton:s la&( )

2 2

2 2

2 2

2 2,

A A

y z y z BA $B

y y BA z z $B

d

m mdt

D j D k % j % k m l j m l k

D % m l D % m l

= = + + + = +

+ = + =

extF v a


24/28

"

y

x


Dz

Dy

Ez

Ey

( ) 2 1 21

2A xyz

H mr k =&9ate of change of H


25/28

"

y

x


Dz

Dy

Ez

Ey

22 2 2

1 2 2

2 2

2 2

2 2 2 2

2 2 1 2 2

22 2 2

2 2

1,

2

,

1

2

D$ $By D$ BA z

%D %D

y y BA z z $B

y BA y BA D$ BA

%D

D$ $Bz $B z $B

%D

m m l l % r l l %

l l

D % m l D % m l

mD m l % m l r l l

l

m l lD m l % m l

l

= =

+ = + =

= =

= =


26/28

"

y

x


Dz

Dy

Ez

Ey

( ) 2 2 1 1 21 1 1

" 2 2A xyz

H mr i j k = + +

&

( ) ( )

( ) ( ) ( )

2

2 2

2 2 2 2

A BA $Bxyz

A $B BA BA $Bxyz

l k l j

l l j l l k

= += + +

v

a

'inematics of center of mass in the inertial frame

9ate of change of H

6e&ton:s la&( )

( ) ( )

( ) ( )

2 2

2 2 2 2

2 2

2 2 2 2,

A A

y z y z $B BA BA $B

y y $B BA z z BA $B

dm m

dt

D j D k % j % k m l l j m l l k

D % m l l D % m l l

= =

+ + + = + +

+ = + = +

extF v a

It is assumed that at the instant sho&n shaft DCE has an

angular velocity #, ? rad2s i and an angular acceleration#, / rad2s i. 9ecalling that the dis! rotates &ith a constantangular velocity 1, 1/ rad2sj( determine 3a4 the couple

that must be applied to shaft DCE to produce the givenangular acceleration( 3b4 the corresponding dynamic

reactions at D and E.


27/28

"

y

x


Dz

Dy

Ez

Ey

9ate of change of H


28/28

"

y

x


Dz

Dy

Ez

Ey

( )

( ) ( )

( )

02 2 2

2 2 20 02 1 2 12 2 2 2

2 201 2 22 2

2

2

"

"

" 1 " 0

2" "

1 "

2 "

,

$A

z BA D$ $B D$ BA D$ $B D$

%D %D$A $A

y D$ $B D$ BA

%D $A

y y BA z

#

m r l

m # m # % l l l l r l l l l

l lm r l m r l

m #% r l l l l

l m r l

D % m l D

=+

= + = + =

+ + = +

+ + =

Q

2

2

2 2 2 2

2 2 1 2 2

22 2 2

2 2

1

2

z $B

y BA y BA D$ BA

%D

D$ $Bz $B z $B

%D

% m l

mD m l % m l r l l

l

m l lD m l % m l

l

+ =

= =

= =

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3 D Kinetics Forces and Moments Sample Problems With Solutions(2)