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George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology ME3015 Test I Solutions Spring 2012
NAME__________ Key ____________
Total Score:_____________/
Please read this first:
- The test is closed-book and closed notes. A Laplace Transform table and some useful
formulas are given on the last page.
- There are ten (10) questions/problems on this test. Read all of the problems carefully and
start from the one that is easiest for you.
- Budget your time -- total exam time is 75 minutes.
- Make sure you have a clear presentation for the solutions. No answer without justification
is accepted.
- Sign the pledge of honor below -- any act of academic dishonesty may result in your
failing the course.
- Do not forget to write down your NAME.
I pledge on my honor that the work presented in this test is entirely my own; I have
neither given nor received any inappropriate aid in preparation of this test.
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M
1) In
an
a) y=
c) y =
2) Id
Input
x, Tw
Ts ,Ti,
3) W
f
f(t)=
F(s)=(
0
t
L
5) F
a) F
b) F
c) F
6) F
b ,
2
2
s
sY s
1 5 Test I Solu
ndicate wheth nd linear (L) =u 2 S_ ___
= τ τ
t u d 0
( ) _
dentify 3 inpu
Outpu
wi , Q (^) i , Q (^) f
Qw T (^) wo, Ω
T (^) s (can
What is the is
t
t t f t
(t)t-1(t-1)(t- (1-e -s )/s 2
The Laplac
τ
F
f dt
ind the invers
s
s F s ( 1
s
s F s
s
s F s ( 2
ind Y(s)=L (
, and c are co
as bY s
s sy y
utions
her the follow or nonlinear N__
D_ ___
uts and 3 outp
ut
f , Qe
Ωs n also be an outp
the Laplace t
? (3 pts.)
ce transform
2
−
s s
e
s
F s
s
se Laplace tra
s s
s 2
1 )( 2 )
s s +
s
2 2 ( 2 )
(y(t)) given th
nstant coeffic
s
c s a
y asY
N
wing input-out (N): (4 pts) b) y
L_ d) y
puts of the ho
ut )
transform of
of function f (
s
[
− 2 e f
t
L
ansform of ea
s 2
2 2
s
f s
2 2
hat y ( t ) satisfi
cients. (5 pts.)
2
s
s as c
s y b
NAME_____ _
tput relationsh
y + 3 y + 5 y =
y + sin y = u
ot water furna
( t ) is given by
f ( t ) ] = F ( s + 2
ach of the foll
t f ( t ) 2 e
− ⇒ =
2 2 s
f ( t )=( 1 − 2 t
es y + ay + b
)
2
s
Y s
s
bYs
______ Key __
hips are Dyna
= u D_
_D_____ __
ace system sho
y ( ) 2
−
s
e Fs
2
( 2
− +
s
e
s
owing functio
t t e
− 2 −
f ( t ) e
t ⇒ =
−
t t e
2 )
−
by = c , y ( 0 )=
2
2
s as b
s as c
amic (D) or S
L_
N__
own below: (
s
. Determin
)
f ( t )= 1 (
ons: (6 pts)
(cos 3 t +sin
= 1 , y ( 0 )= 0
tatic (S)
3 pts)
ne (3 pts)
( t − 1 )sin( t −
3 t )
, where a ,
M3015 Test I Solutions NAME_____ ______ Key ___________________
10) The differential equation of motion for the rack and roller system shown in the figure may be
expressed as (^) meff x (^) + kx = f. Assuming that each roller has radius r and moment of inertia J ,
and rolls without slipping about its fixed center, find m (^) eff in terms of m , J , and r. (6 pts)
The equations of motion of mass m and pinion J are given by
θ= − θ
J Fr kr
mx F F
p
p
Where F (^) p is the contact force between each pinion gear and mass m. Using that x=rθ
yields
x kx F r
J
m
meff
⎟ +^ =
2
x
m
f k
J
J
r
r
