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ME 563-Fall 2020

Test 1

Name______________________________________ Pledge________________________________

I have neither given nor received aid on this examination.

Instructions: • This is a closed-book, closed-notes exam.

• You are NOT allowed to use a programmable calculator during the exam.

• Please read the question on this exam carefully and answer only the questions I ask. Don’t waste your time doing extra work, and don’t skip any of the smaller questions within a problem. I can’t grade what I can’t see and I can’t give partial credit for something in your head.

KEY

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ME 563 - Fall 2020 Name_______________ Test Problem 1 -30points A bar is attached to a spring at pt O. The spring is constrained to deform purely in verical (x) direction. The bar has mass m and mass moment of inerita about its center of gravtiy of IG =1/12mL2. The coordinate x denotes the absolute position of the roller and θ the angular position of the bar.

a) Determine the expression for potential energy U in terms of the generalized coordinates x

and θ and determine the equilibrium positions of the system. b) Write down an expression for the kinetic energy T in terms of the generalized coordinates x

and θ and their time derivatives. From this expression, identify the elements mij , where:

c) Determine the mass matrix [M] and the stiffness matrix [K] corresponding small oscillations about the equilibrium state.

Datum

a He42km2 night 42050d KXmg.LUaomgLqsin0da1ge 0 dYuhg o

ka mga mga2 mgygs.in0 8 e NTIn 0 I 2 3

axe 0 andOle NT in Q l Z 3

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ME 563 - Fall 2020 Name_______________ Test Problem 1 Additional Page

bro Xi F Lycos it 425in

86 To Too x 142cos T t 42sin J

to fix sin 79 84105097 42mF HEINIEYg mini ni in Ea 42m98

My M

Mz Mei m sin andM22 m224

70c em m IIsince 435 L Iy b

Ke O14 321,1 k 142 311Ku Iffy mgtycoon mg42CIT

If ae.sn To mail

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ME 563 - Fall 2020 Name_______________ Test Problem 1 Additional Page

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ME 563 - Fall 2020 Name_______________ Test Problem 2 -20 points A single degree of freedom systems has system parameters of m =100 kg, k=25√2 N/m, and c =50 kg/s, The system has initial condition x0= 2cm, and v0=4cm/s.

a) Determine the natural frequency of the system. b) Determine the damping ratio of the system. c) Determine the damped natural frequency of the

system. d) Indicate the correct plot of the response of system.

On this label the initial conditions and the period of oscillations of the system, and the length of the period.

qfT 2

xa Yea Yet4k ke k 142 2512Nlm 17,67µm

Wn Fm F oo 0,4204rads

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ME 563 - Fall 2020 Name_______________ Test Problem 2 Additional Page

b 23am 4M

3 42man mrEaym rmkewC

42 Song 0,5946285rad

underdamped

c Was_Wnffge 0,42450.59762 0,3380nadyg

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ME 563 - Fall 2020 Name_______________ Test Problem 3-20 points A 2-DOF system has the follow equations of motion.

Determine the natural frequencies and mode shapes of the system.

Assume I Icke I we

m CITI 8 f W Cm 1 KT IeYW 0

Natural Frequenciesarm K K

a wannaE

Z

C w mek 2 Ck 2 w4mEw42mk IE IE O

204 2mkW2 W want 2m14 O

202 0 W2m2 zonk O

W 0 and we reran RigidBodymodeW O

Modal ShapesCarm 1k I KI 2 0 Iz W3ntk

ToN'MK 11

I 12 1a o L W Ekrem TIEE

l

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ME 563 - Fall 2020 Name_______________ Test Problem 3 Additional Page

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ME 563 - Fall 2020 Name_______________ Test Problem 4-20 points Consider the system below, whose motion is described by the absolute coordinates shown. Use the method of influence coefficients to develop the flexibility matrix [A] =[K]-1.

Numbering to Letters 30

A I4B 32 a

c 73 i

p 24

Apply a unit load a A

Kah NoforceherecMiffy Efx Kai 0 an_ya

k 009,2 9,3 9,4 0

Byreciprocity 92 921 0 9,3 8 0 ay _9,4 0

Apply unit load B

924to2k9g f EFxi Kau O aq Yzk

Me a Mf 923 0 932 925 0

921 0922

924 922 942 424

Apply unit load Cqz O as 0 952 0

4933cWf f 943 9,3 923 0

933 to Efx Kazz tf O

933 1,4234944

f ked L t KkpmKew Year E k tfkdl

Kew 312K keq 2 K

ISfx 213K f O 944 324

In summary94 14 9,2 0 9,3 0 914 0921 0 922 1

2c 923 0 924 42k931 0 932 0 933 14 934 0

941 0 942 42K 943 0 944 34

a Kc O O OO 42k O 42k

O O k O0 42k O 312K