Pqe Vibration 09 Solution 1

8/10/2019 Pqe Vibration 09 Solution 1

http://slidepdf.com/reader/full/pqe-vibration-09-solution-1 1/15

COLLEGE OF ENGINEERING

MECHANICAL ENGINEERING

Ph.D. Preliminary Qualifying Examination

Signature PageVibration Examination (Modify)

January 26 2!!" (Monday)##Modify"$!! am % &2$!! noon

Room 2145 Engineering Building

'or identifiation uro*e* lea*e fill out the follo+ing information in in,. -e *ureto rint and *ign your name. hi* o/er age i* for attendane uro*e* only and+ill be *earated from the re*t of the exam before the exam i* graded. 0rite your*tudent number on all exam age*. Do 1 +rite your name on any of the otherexam age* be*ide* the o/er age.

Name (print in INK)

Signature (in INK)

Student Number (in INK)

Do all your +or, on ro/ided *heet* of aer. 3f you need extra *heet* lea*ere4ue*t them from rotor. 0hen you are fini*hed +ith the te*t return the examlu* any additional *heet* to the rotor.



COLLEGE OF ENGINEERING

MECHANICAL ENGINEERING

Ph.D. Preliminary Qualifying Examination

5o/er Page

"$!! am % &2$!! noonRoom 2145 Engineering Building

E1E789 31S7:531S$

hi* examination ontain* fi/e roblem*. ;ou are re4uired to *elet and *ol/e four of the fi/e roblem*. 5learly indiate the roblem* you +i*h to be graded. 3f youattemt *ol/ing all of them +ithout indiating +hih four of your hoie the four

roblem* +ith the +or*t grade* +ill be on*idered. 1ote that Problem number 5 is mandatory.

Do all your +or, on the ro/ided *heet* of aer. 3f you need extra *heet* lea*ere4ue*t them from the rotor. 0hen you are fini*hed +ith the te*t return theexam lu* any additional *heet* to the rotor.



Mehanial Engineering Ph.D. Preliminary Qualifying ExaminationVibration % January 26 2!!"

You are required to wor !our o! t"e !i#e problem$% one o! w"i&" i* Problem 1o. <' learlindi&ate w"i&" problem$ ou are &"oo$ing' S"ow all wor on t"e e*am $"eet$ pro#ided and write

our $tudent per$onal identi!i&ation (+I,) number on ea&" $"eet' ,o not write our name on an$"eet'

Your +I, number-............................

Que*tion =&

/ uni!orm bar o! lengt" L and weig"t W i$ $u$pended $mmetri&all b twoun$tre&"able $tring$ a$ $"own in t"e !igure' I! t"e bar i$ gi#en $mall initial rotation aboutt"e #erti&al a*i$%

a' ,raw t"e !ree bod diagram o! t"e bar during it$ !ree o$&illation' b' 0rite down t"e equation o! motion !or $mall angular o$&illation about a*i$ '&' ,etermine t"e period o! t"e !ree o$&illation o! t"e bar'

O

O L

a

h



You are required to wor !our o! t"e !i#e problem$' learl indi&ate w"i&" problem$ ou are&"oo$ing' S"ow all wor on t"e e*am $"eet$ pro#ided and write our $tudent per$onal

identi!i&ation (+I,) number on ea&" $"eet' ,o not write our name on an $"eet'

Your +I, number-............................

Que*tion =23"e $$tem $"own in t"e !igure i$ in it$ $tati& equilibrium po$ition (SE+)' It &on$i$t$ o! auni!orm rod o! ma$$ m and lengt" L and i$ $upported b $pring o! $ti!!ne$$ k andda$"pot o! &oe!!i&ient c '

a' ,raw t"e !ree bod diagram o! ea&" $$tem a$ it o$&illate$ about t"e SE+' b' ,eri#e t"e equation o! motion o! ea&" $$tem u$ing Newton$ $e&ond law'

&' ,etermine t"e undamped natural !requen&'d' ,etermine t"e damping ratio% t"e &riti&al damping &oe!!i&ient% and t"e dampednatural !requen&'



You are required to wor !our o! t"e !i#e problem$' learl indi&ate w"i&" problem$ ou are&"oo$ing' S"ow all wor on t"e e*am $"eet$ pro#ided and write our $tudent per$onalidenti!i&ation (+I,) number on ea&" $"eet' ,o not write our name on an $"eet'

Your +I, number-............................

Que*tion =>3"e $$tem $"own &on$i$t$ o! a &linder o! ma$$ m wit" a pi$ton% w"i&" impart$re$i$tan&e proportional to t"e #elo&it o! a linear #i$&ou$ damping c % t"e &linder i$re$trained b a $pring o! $ti!!ne$$ k

(a) draw t"e !ree bod diagram o! t"e &linder%

(b) write down t"e equation o! motion u$ing Newton$ $e&ond law% and

(&) determine t"e re$pon$e amplitude and p"a$e angle u$ing 5omlex 8lgebra'



You are required to wor !our o! t"e !i#e problem$' learl indi&ate w"i&" problem$ ou are&"oo$ing' S"ow all wor on t"e e*am $"eet$ pro#ided and write our $tudent per$onalidenti!i&ation (+I,) number on ea&" $"eet' ,o not write our name on an $"eet'

Your +I, number-............................

Que*tion =?

3"e $$tem $"own below &on$i$t$ o! two rotor$ &oupled b a di$&ontinuou$ $"a!t o!

modulu$ o! rigidit i$ 211'5 16 7 7G lb in rad = × -

• ,raw t"e !reebod diagram o! ea&" rotor%

• ,eri#e t"e equation$ o! motion%

• ,etermine t"e natural !requen&ie$ o! !ree tor$ional o$&illation$ and pro#ide t"e

p"$i&al meaning o! ea&" #alue%

• ,raw t"e normal mode $"ape and e#aluate t"e #alue o! t"e twi$t at t"e 8un&tion o!

t"e two $"a!t$% i'e' 17nθ θ or 27nθ θ



M81D87; P7-9EM (E9ERYNE IS RE:;IRE, 3 S<9E 3=IS +RB<E>) Your +I, number-............................

Que*tion =<

on$ider a rigid bod o! ma$$ m and ma$$ moment o! inertia J c wit" re$pe&t to &enter o! gra#itC g ' Suppo$e t"at t"e bod i$ $upported b two $pring$ o! $ti!!ne$$ k t"at are atta&"ed at di$tan&e$2l and l wit" re$pe&t to t"e &enter o! gra#it C g a$ $"own in ?igure 5a' <et m @ 16 g% J c @ 5 gm2

k @ 166 N7m% and l @ 1 m'

+art I-

(a) ,eri#e t"e equation$ o! motion !or t"i$ bod u$ing &oordinate$ x and θ '

(b) ,etermine t"e natural !requen&ie$ o! t"e $$tem'

(&) ,raw t"e natural mode $"ape$ o! t"e $$tem'

(5a) (5b)

Ne*t% &on$ider t"e $ame rigid bod a$ $"own in ?igure 5b'

+art II-

(d) ,eri#e t"e equation$ o! motion !or t"i$ bod u$ing &oordinate$ x1 and x2'

(e) ,etermine t"e natural !requen&ie$ o! t"e $$tem'

(!) ,raw t"e natural mode $"ape$ o! t"e $$tem'

+art III-(g) State i! t"ere are di!!eren&e$ in t"e equation$ o! motion% natural !requen&ie$ and mode

$"ape$ obtained in ea&" &a$e and e*plain w"'

(") 0"at are t"e &oupling$ in equation$ o! motion% re$pe&ti#el% in t"e$e two &a$e$A



+roblem 1

/ uni!orm bar o! lengt" L and weig"t W i$ $u$pended$mmetri&all b two $tring$ a$ $"own in t"e !igure' I! t"e bar i$ gi#en $mall initial rotation about t"e #erti&al a*i$%

d' ,raw t"e ?B, o! t"e bar during it$ !ree o$&illation'e' 0rite down t"e equation o! motion !or $mall angular

o$&illation about a*i$ '!' ,etermine t"e period o! t"e !ree o$&illation o! t"e bar'

?igure 5'Solution-

'-D

?rom t"e $tati& equilibrium po$ition we write

23 mg=

(1);nder !ree #ibration o! t"e bar and in an arbitrar po$ition θ % t"e bar will be rai$ed up

$lig"tl% and will be di$pla&ed b a di$tan&e ( 7 2)a θ !rom t"e it$ $u$pended $tring' 3"e

$tring al$o be tilted b an angle ϕ !rom t"e #erti&al $u&" t"at% ( 7 2) "a θ = ϕ ' 3"i$

geometri& relation gi#e$ ( 7 2")aϕ = θ '

Now writing t"e equation o! motion b taing moment$ about a*i$ % gi#e$

6I 2(3$in )( 7 2) 3 3

2"

aa a a

θ = − ϕ ≈ − ϕ = − θ ÷

&&

(2);$ing equation (1)% equation (2) tae$ t"e !orm

2

6 n

6

mg mgI 6 6 6

4" 4I "

2 2a aθ + θ = → θ + θ = → θ + ω θ =&& && &&

()

O

O L

a

h



w"eren 22

6

mg mg g

4I " "<<4 m "

12

2 2 2a a aω = = =

÷

(4)

+roblem 2

'-D

(b) ?rom t"e !reebod diagram% we write t"e equation o! motion !rom t"e $tati&equilibrium po$ition u$ing Newton$ $e&ond law o! moment$ about t"e "inge a*i$

( ) ( )2

2 2

66

mL I cL L ka a cL kaθ θ θ θ θ θ = − − → + + =&& & && & (1)

(&) 3"e undamped natural !requen& i$ obtained b di#iding bot" $ide$ o! t"e equation o!

motion (1) b t"e &oe!!i&ient o! θ &% i'e'%

2 2

2 2 6

n

c ka ka a k

m mL mL L m

θ θ θ ω + + = → = =&& & (2)

(d) 3"e damping ratio i$ obtained b writing t"e equation o! motion in t"e !orm22 6

n nθ ζω θ ω θ + + =&& & ()

3"u$ we &an write km

2

2 2 2

n

n

c c c cL

m m a k kmam

L m

ζω ζ ω

= → = = = (4)

3"e &riti&al damping &oe!!i&ient i$ obtained !rom (4) a$

2

2 cr

cr

c cL kmac

c Lkmaζ = = → =

3"e damped natural !requen&% nd ω % i$ written in term$ o! t"e undamped natural

!requen&% nω % a$

2 2 2

2

2

C1 1 1

122 nd n

a k cL a k c L

L m L m kmakmaω ω ζ

= − = − = − ÷



+roblem D

;$ing Newton$ $e&ond law and wit" t"e

"elp o! t"e ?B,% t"e equation o! motion i$

( )mx kx c y x= − + −&& & & (1)

Rearranging

mx cx kx cy+ + =&& & (1a)

But ( ) $in Im i t y t Y t Y e Ω= Ω = (2)

/l$o ( ) &o$ Im i t y t Y t i Y e Ω= Ω Ω = Ω& ()

Note t"at one &an write( ) ( ) ( ) ( )

7 2 72 be&au$e &o$ 7 2 $in 7 2

i ii e e i i

π π π π = = + =

3"u$ one &an write equation (1a) in t"e !orm

2Im Imi t

i t mx cx kx icY e cY eπ Ω + ÷Ω + + = Ω = Ω&& &

(4)

3"e re$pon$e mu$t o$&illate at t"e $ame !requen& o! t"e e*&itation in t"e $tead $tate atamplitude and p"a$e angle to be determined% t"u$ one &an write t"e re$pon$e in t"e !orm

( ) w"erei t i i t i x t X Ime X Ime X X Imeϕ ϕ Ω − Ω −= = = (5)

0e need t"e !ir$t and $e&ond time deri#ati#e$ o! ( ) x t % i'e'

2( ) % ( )i t i t x t i X Ime x t X ImeΩ Ω= Ω = −Ω& && ()

Sub$tituting e*pre$$ion$ (5) and () into equation (4)% gi#e$

2 22 2i t

i t i t i t

n n n X Ime i X Ime X Ime Y Ime

π

ζω ω ζω

Ω + ÷Ω Ω Ω + Ω + = Ω ()

an&eling out i t Ime Ω !rom bot" $ide$ o! equation () and rearranging% gi#e$

2 2 2 22 2 2 % w"erei i

n n n n X i X X Y Ime Y Y Y Ime

π π

ζω ω ζω ζω −Ω + Ω + = Ω = Ω = (F)

Rearranging

( )2 2

2

2

n

n n

X

Y i

ζω

ω ζω

Ω=

− Ω + Ω (C)

>ultipling and di#iding b t"e &on8ugate o! t"e denominator% gi#e$

( )( )( )

2 2

2 2 2 2

22

2 2

n nn

n n n n

i X

Y i i

ω ζω ζω

ω ζω ω ζω

− Ω − ΩΩ= ×

− Ω + Ω − Ω − Ω



( ) ( )

( )2 2

2 22 2

22

2

n

n n

n n

iζω

ω ζω ω ζω

Ω = − Ω − Ω − Ω + Ω

(16)

,i#iding t"e numerator and denominator b2

nω % and $etting 7

nr ω = Ω % equation (16)

tae$ t"e !orm

( ) ( )( )2

2 22

21 2

1 2

X r r i r

Y r r

ζ ζ

ζ

= − −

− + (11)

>ultipling and di#iding ea&" e*pre$$ion b ( ) ( )2 22

1 2r r ζ − + gi#e$

( ) ( )

( ) ( )

( )

( ) ( ) ( ) ( )

2 22 2

2 2 2 22 2 22 2

2 1 2 1 2

1 21 2 1 2

r r r r X r i

Y r r r r r r

ζ ζ ζ

ζ ζ ζ

− + − = −

− + − + − +

(12)

0it" t"e "elp o! t"e $"own triangle equation (12)ma be written in t"e !orm

( ) ( )

[ ]2 22

2&o$ $in

1 2

X r i

Y r r

ζ ψ ψ

ζ

= −

− + (1)

E*pre$$ing X and Y in term$ o! t"eir original

,e!inition$ (5) and (F)% gi#e$

( ) ( )[ ]7 2 2 22

2

&o$ $in1 2

i

i

X Ime r

iY Imer r

ϕ

π

ζ

ψ ψ ζ

−

= −− +

( ) ( )[ ]

( ) ( )

72

2 22 22 2

2 2&o$ $in

1 2 1 2

i i

i X Ime r r i IM e

Y r r r r

ϕ π ψ ζ ζ

ψ ψ

ζ ζ

− −−= − =

− + − +% t"u$

( ) ( )

1

22 22

2 2% and 7 2 or 7 2 tan 72

11 2

r r X i i i

r r r

ζ ζ ϕ π ψ ϕ ψ π ϕ π

ζ

−= − − = − = − → = −−− +



+roblem 4

In$ert a #irtual di$ at t"e $"a!t di$&ontinuit o! moment o! inertia 6n

J = % t"e

&orre$ponding twi$t angle i$ nθ ' 3"e equation$ o! motion o! t"e t"ree degree$ o! !reedom

are

1 1 1 1( )n J K θ θ θ = −&& % 1 1 2 2( ) ( ) 6n n n n J K K θ θ θ θ θ = − − + − =&& % 2 2 2 2( )n J K θ θ θ = − −&&

!rom t"e $e&ond equation we "a#e- 1

1 2 1 22 2 2

( ) 6'15 16 6'6C41 16 ( 6'5CC) 6'65C 16n n K K K K θ θ θ

θ θ θ + = + → × = × − + ×

3"u$2

6'621

1

nθ

θ = −



+roblem 5Solution to +roblem 5-+art I-3"e equation$ o! motion &an be deri#ed a$ !ollow$-

∑ = xm F → ( ) ( )θ θ l xk l xk xm +−−−= 2

θ C C J M ∑ = → ( ) ( )θ θ θ l xkl l xkl J C +−−= 22

w"i&" lead$ to t"e !ollowing matri* equation$

=

−

−+

6

6

5

2

6

62

θ θ

x

kl kl

kl k x

J

m

C

(1)

/$$ume t X x ω $in= and t ω θ $inΘ= % and $ub$titute t"e a$$umed !orm $olution$ intoEq' (1)'

=

Θ

−−

−−

6

6

5

2

22

2 X

J kl kl

kl mk

C ω

ω % (2)

3"e natural !requen&ie$ &an be determined b $etting t"e determinant o! Eq' (2) to Gero

65

2det

22

2

=

−−−−

ω

ω

C J kl kl

kl mk % w"i&" ield$

( )( ) 652 22222 =−−− l k J kl mk

C ω ω ()

Equation () i$ t"e &"ara&teri$ti& equation o! t"e $$tem' Sub$titute J C @ 6'5ml 2 into ()

61F12 2242 =+− k mk m ω ω % (4)

3"ere!ore% t"e natural !requen&ie$ are

( ) 2'42B51 =−= mk ω (rad7$e&) and ( ) 1'162B5

2 =+= mk ω (rad7$e&)' (5)

Sub$tituting t"e natural !requen&ie$ into t"e "omogeneou$ part o! Eq' (2) gi#e$ t"enatural mode$ o! t"e $$tem-

( )

( ) 1'42B4

1

2 2

1

1

1

=+−

=−

=Θ ω mk

k

l

X and

( )

( ) 1'62B4

1

2 2

2

2

2

−=−−

=−

=Θ ω mk

k

l

X ' ()

1$t mode $"ape 2nd mode $"ape



+art II-

3"e equation$ o! motion &an be deri#ed a$ !ollow$-

∑ = xm F → 2121

B

2

Bkxkx

x xm −−=

+

θ C C

J M ∑ = → 2112 2

BBklxklx

l

x

l

x J C

−=

−

w"i&" lead$ to t"e !ollowing matri* equation$% a!ter $ub$tituting J C @ 6'5ml 2

=

−

+

− 6

6

12

BB2

2

1

2

1

x

x

k k

k k

x

x

mm

mm

()

/$$ume t X x ω $in11

= and t X x ω $in22 = % and $ub$titute t"e a$$umed !orm $olution$

into Eq' (1)'

=

−+−

−−

6

6

12

2BB

2

1

22

22

X

X

mk mk

mk mk

ω ω

ω ω

% (F)

3"e natural !requen&ie$ &an be determined b $etting t"e determinant o! Eq' (F) to Gero

612

2BBdet

22

22

=

−+−

−−ω ω

ω ω

mk mk

mk mk % w"i&" ield$

( ( ( ( 6122BB 2222 =−−+−− ω ω ω ω mk mk mk mk (C)

Equation (C) &an be $impli!ied to

61F12 2242 =+− k mk m ω ω % (16)

3"ere!ore t"e natural !requen&ie$ remain t"e $ame a$ be!ore' =owe#er% t"e natural mode$o! t"e $$tem-

( )

( )

(( )

4'62BB

2B2B

B

2B

2

1

2

1

1

2

1

1 =−−

−−=

−

−=

ω

ω

mk

mk

X

X and

( )

( )

(( )

4'22BB

2B2B

B

2B

2

2

2

2

2

2

2

1 −=+−

+−=

−

−=

ω

ω

mk

mk

X

X '

(11)

1$t mode $"ape 2nd mode $"ape



+art III- Natural !requen&ie$ remain t"e $ame% but equation$ o! motion and mode $"ape$are di!!erent' 3"e !ir$t &a$e i$ $tati& &oupling and t"e $e&ond i$ bot" dnami& and $tati&&oupling'

Documents

Pqe Vibration 09 Solution 1