6-Exams Solution First Major Exam 041 Version 2

8/13/2019 6-Exams Solution First Major Exam 041 Version 2

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ME 413: System Dynamics & Control First Major Exam First Semester 2004-2005 (041)

PROBLEM#1: (30 marks)In the system shown below I Uo is the displacement of the dashpot and the input to thesystem. The displacement x of the mass is the output to the system.

rue0k b

1. Draw the Free Body Diagram FBD of the system.2. Derive a mathematical model differential equation for the system.3. Write the transfer function tf of the system.4. Represent the above tf in a block diagram and show the input and the output.5. For m= 1.0 kg I b = 2.0 N-s/m I and k = 5.0 N/m I write the explicit expression of

the response x t as a function of time for each of the following cases:

Uot Uo t

1

t t2 2

a b6. Use the Final Value Theorem FVT to find the steady state value Xss = lim x t .1 >00

10; F13]) QAu.s.b.o. (.J.J11

I(XL .A2 lltpply d(.uJI:Q.l1~. 2~' /4UJ

m ~ ~.. a ~mS~iL1... kJzI1Sl4h~Q : L~m ,.(>1 b -u)0..

L {. . I ..- K X_I~X - U0) =m)l6. Q,> 1.' vmX+ClXm+8.. ..X = - - - - - - _u-- - - (I) 482/13


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3D/ Lt:LLCL (L ~)bfmbQ 1:0... Sl~ ~mmm~ ~ ll) (Eot::z~rQ

bL~ bs + fL.). X(5J:...bS.. ~ (5)XeS) -, ) I .4n1S

J..IfI -1 --bs. . ' .. A1 $ ~.,.. h;$ .,. /(.

~

suhshi~.,..m...,m...::: :Jl.l=2..La ti#~)rnJ. 1 =5~(JlV/mxes) -Yo (5) S

2$---2$ r rUfJ(S):~ ..a.o ..i..bLpuLl~ ~n t ,. m~~w.l ~

' ,.In.. i:~, .,U::v;;L..mW~.~ Mo(6)= ..'~C6') ~. {~(t)}= Uf1j(S} = i... ...7h~cf:Uf:L.

X{.s.)::: Cfo(.s) .L- :::_. 5~ .$L+ 25-1' S- S 25+ IC~+I) 2-

-;;~ (S+I),._~ ~ --(S'H)~+2.l. (S+I}7L+ ,2.

..__.m.. . fJ..t1

x(5)=-

.c

f// X

,(/0 (S)M:> ~ Si;~,.(Lo(~J= .1,it) ...~. :Lf.1

2.5. ' . '.. '......$ 2. t,S-;1-' ( ( -t;-.X Ct): LX (l)I=.e.. .$111 ~ t---S ': ~ ?< (t,) -= ~/13 for 1,0~ ~~

t ~ jQ

A..

22....~4


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.~~~()B~~\..~.~~.(~9 .rr\arks) .Consider the following mechanical system consisting of a rigid bar with mass M, a damperwith a damper coefficient of b, and two springs with spring coefficients k] and k2. Weassume that the bar rotates around point P due to the applied force F. We also assumesmall angle of rotation. That is, () is small, as shown in the figure.

1. Obtain the equation of motion of the system.2. Find the transfer function of the system where the input is F t and the output is

B t .3. For M = .1 kg, I]=0.2 m, 12=0.4 m, k] = 10 N/m, k2 = 40 N/m, b = 5 N s/m.

Compute the natural frequency of the system.4. If we assume that the bar is massless (M = 0), write the equation of motion forthis case.

k,/2 xJlvI

b.~..

I] ;J \0 1 x. - I--2

- - .l.. =~h~J~:= Y~/7I ~I ..~ ...~c. I --I . c r ~ c--. ~ ~L.{_< .I ---

i'p ---, ., .~l,..::: ,::: ~..,g~. -- . .. -- -~ ';.L 1.)~~~...~'i~/Sf'~~~~~'..,.,..,~..,/&'R.'

~ d-::-- I -J:~-~ -- - ~~ll =.21.,~{~;i-:. i ~ ~ -= .'iJ:;.i_'~-;;-, 1,2;. ~ - . . ,--,- ::5/13 2.

b 1c:h Q'., . 'mw...w_w.. .~2.'mm.

:R~ - ~I-- 2-[ l e-~-~,20


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ME 413: System Dynamics ontrol irst Major Exam irst Semester 2004-2005 (041)

(f)I'.rl=~ 6.. }. n, J . Jl,,' ,.1.,.. -K{I.t:J,)R K(J 6) ~- 12~l.rJl- . /ttIQ 1.- :.. .Ili

21 , 12 2 1 2. .:J 2.. fh 02. / n2 I t t~ & +b...L8,...,+\K J( ,+ , I. ) ,,(;J=F~I _MDt Z- '.......r 2 12 21 r.J 2.( )

@

4- - --- - -- -TE : EJ S) l,

. .E S ) - ~ S 2.,. bl.~a.s ..,.(K,J: fl: )Noh'u -IAa.lweljno,.,e .1h( e.f~Gt.. Df~rtt. .1ydeL.il l ihc CO uhh '0 n of . 1'~1.~a 4,1U' 'pl ~

'1. . ... . .....

~M...f 2;Jl, ~ . :: J. ~ .'~(O.2 .,. Q,~) ~+ 1 if ( 0 .,.,- o. ') ~Il 2

= . ),3 ,.. ,.~,O .~ fJ.J,+_D./'2. 9,VI_O.OI.//2 Jf - 12 - 12 -..t.I.J 1)::

.

..

.

,.1.

..

.

'

.

'

. -Kit ~ +f(. t~.'f. . , *:q~VO ~o ... 22~- ... 3: o'oc/' p ,/5

@~ .. M :. 0 J ~=o;; ..~rb L~=~3.t2~ f_~.a6~~ = Ei~1

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PRO L M#~:. 30 marksFor the system shown, the pulley is solid and freeto rotate about its center. The rope around thepulley is inextensible and rolls without slipping. Themass of the bob mb is attached to the solid link Lthat is rigidly attached to the pulley. Assuming thatlink L has negligible mass and ) is small,determine;

1. Equation of motion of the system usingNewton s second law.2. Equation of motion using the Energy method.3. Compare the results and comment on them.4. Find the natural frequency of the system OJn

as a function of the system parameters.

;r..

~

n1d 1

x mb

EBD .~...S Jot.un _Ih,/2+ ullppl.yNt.IN ti,n S2 ~~.

lfLUJr u..w1 /.lAC m )h

.~ E ..u=m X

m9-Z = OJ 'Xorl' ...: m9 mX - - ..(IJ.

IJpplr. I~.hl'J~,rm...2 ~.~I am.fo ,. ihL ~ ~ ...bIrclL:Ic + /.) )t.{'\ .2 .T. :uuJ;8

K )

T u Yttu) I~tiuL.~.~

m -h L. + fl..).sin 6TRu_uK :x. ~ _m9 L.,.I?)SinlJb ..

= I if; ..0 ~~. (~~~.8/13


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13UI .IO=Jj sk +m {ol. :m J- .,. ..~No slifJfirJ,Co??dA'h.on .9 ..'X=...R.( JFe,. ..small. an,.9/fA .sin f:J ~&m

CI l4S:: . .ml

Tkr ~r rL ., EfI 2 Jcanhc w:tL He.n .. t. .t-o.... m. [r .., .. ~~. ~~~3)

S ubsb'h.d.t.. 7~om (j). A..iL.fom@a n cXmm=Ri:iY ~.~

(~;;~]i: '~4~;(~~~ ~lq) I..,. ,.. -

:301 [.It.. of rh(.En~JAlt...Tb ad :Lm+.. V = consJ.

lAJhue. T = ~ ~ tR ..+ ro 0

~r ,- I ..m~ - _. ,WJ..'t..2~:r( =i~ fJ1 ,.: , :- J:l l... 2.

/ LtIt)

v= ~ . + II + I~5pl'll1 .. m~o,m'lw~~~~io, :: (L +It ) (1- '-tJc fJ) ~ 39/13


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Vm~ m_mg x.(dfJ~l't(An.rJ)r 'mV ::Co ?tAt

T+ V :mm. m ~ 1 +. J.m~. .S ~+ J..mm{ 8m + .. .K'X2.+ (L+12) (1- Ub 9 m2. 2. f1I1 2 111 2. b- m9 X = . Go Jt o.JCtn d. frLllA. -tAL.eUl,..it.rA/i.' td-muJoso:IuJL'X: R f:J

sLm{7+ V) = 0oIt-;'..2~f~~~~m~

+(L+ ~ lS,tim 1 3mm-DI J R .=.O

l~~~~L_(f)-- - - ~E~ .:-(i:jJ ma:niil-tS;-an fk.J'a;7f'I ~ ; = ~. Xf~...c.;.g m ..m.mmm .. m...mmCoMt.rf.J7A.ia'~Th


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ME 413: System Dynamics Control First Major Exam First Semester 2004-2005 041

PRO L M4: (10 mar

1. The equivalent spring constant of two parallel springs with stiffnesses ~ and kzis/- ~+kz (b) 11 1-+-~ kz (c) 1 1-+-~ kz

3. The force associated with a damper element is given by:(a) f=bx (b) f =bv /e f=bx

4. A spring element with a spring constant k stores:(a) Kinetic energy ~. potential energy (c) dissipative energy

5. The period of oscillations T is related to the frequency of oscillations (1) by therelation:

(a) T=277:0 T=277:/0 (c) T = 0 /277:

6. A second order system is:a a system that has two degrees-of-freedom.b a system represented by two algebraic equations.c~ a system represented by a second-order differential equation.d a system that responds always with oscillatory motion.

11113

2. The natural frequencyof a systemwithmassm and stiffness k is given by:(a) k 4 (c) Hm


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ME 413: System Dynamics ontrol irst Major Exam irst Semester 2004-2005 041)7. Laplace Transform converts a given differential equation in time domain intoa)- .c)

a differential equation in s-domain.an algebraic equation in s-domain.a partial differential equation in t-domain.8. Find the Laplace transform of 2-31+2 sin 7t).

a);18

c)d)

2-6/ 3+14/ .1+49).2/ s-6/ 3+14/ .1+49).2/ s-6/ 3+14s/ .I+49).2/ s-3/ .1+14/ .1+49).

9. Find the Laplace transform of 8 e-4t+7 t. t-3), where t. t) denotes the unit stepfunction.

10. .

b)

c)

a)b)d)

8/ s-4)+7 e3s/s.8/ s+4)+ 7 e3s/ s.8/ s+4)+7 e-3s/ s.8/ s-4)+7 e-3s/ s.

The transfer function is defined as:the ratio of the Laplace Transform of the output to the Laplace transformof the input under the assumption that all initial conditions are zero.the ratio of the Laplace Transform of the input to the Laplace transform ofthe output under the assumption that all initial conditions are zero.

the product of the Laplace Transform of the input to the Laplace transformof the output under the assumption that all initial conditions are zero.

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6-Exams Solution First Major Exam 041 Version 2