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Unformatted text preview: ME 375 — Spring 2011
\,‘ Homework No. 2
Due: Friday, January 28 Pmbleru No. I (30%) Gears 1 and 2 (having masses of m and 2m. respectively) mesh at their common contact
point C (no slip). Flexible shafts having torsional stiffnesm of 2K and K connect gears 1
and 2, respectively, to ground. The mass W torque 1‘ acts on gear2asshownbelow.Let 9 representtbemtationofgear2.Treatgears 1 and2as
homogeneous rigid disks with radii of r and 21', respectively. Derive the equation of motion (EOM) for this singledegreeoffreedom system in terms
of the output rotation 0 and input T. Based on your EOM, what is the effective mass
moment of inertia and the effective rotational stiﬁ'ness for this system? You need to provide complete free body diagrams in the derivation of your EOM. gear 2
gear 2
t. f 2»:
0
C
m_ .\
1 gear 1 gear ENDVIEW .\ ‘/ S. r) 3 P a
, u L
I
‘4
I
n
a .. =
A
m A4 . O
. 9—
r ’h
t. . H
. A
A
A ._
I
g .
A A Problem No. 2
PM A)  15% Consider System I shown below made up of six springs, two masslecs connectors A and
B, and a block of mass 111. You are asked to represent this system by an equivalent
System II made up of a single spring of stiffness keg and the same block of mass m. Determine the equivalent spring stiffness has in System 11. SYSTEM! SYSTEM II __I ' I _ '
“I*I_'
"I—I : L
’ '4 ﬂ 4'. 'A'
C
 A. l A. : ‘
_ , ' , 5 d—
_, vi K
! IA ‘4 '14 .d. 0A X5 X5 —_
_—_'
——
:L  A
3% “M
=7.  = Problem No. 2
Part B)  15%
Particle P (of mass m) slides within a smooth circular trough of radius R. a) Derive the differential equation of motion for the system in terms of the output
variable 9. b) Develop the linearized form of the EOM found in a) above. Assume that 9 << 1
for all motion of the particle. c) Based on your linearized EOM from b) above, what is the natural frequency of
oscillations describing the small amplitude motion of P? You need to provide complete free body diagrams in the derivation of your EOM. D
0 " ' AA
’ I 
 IA
.A ‘r A
v. 0" . .
AA.‘ A.
no
A =
"4 
I
..
,‘
_,
j I
a '3 .n.‘
‘ I
.
A ~ .0
._».— A:
“
‘3 A . 4‘ it: 4’ 4"; r'. .‘fn
: "
9‘ ‘3 ‘\" IA ' 9 = .
¢ A’. .‘IA 0.4: 501..
A‘
—7‘A ~si~9
v
“A
U A.# a e
0 44 are: 3
.0
. L : a a ., =. ‘
. ‘I‘ A IA A nu 0 AA 0'
O A
\ u .
.A0 A * ‘A A + .. A '5
1.“? I A I + ‘1‘ :
mm— 0‘
—l_ '
' — I .\ DAIA Pmblem No. 3 (40%)
Consider the two—degreeoffreedom system shown below.
a) Derive the differential equations of motion (EOMs) for the system in terms of the
coordinates xI and x2 .
b) Suppose that the BOMs in a) above are written in the following vector form: [Ml{f}+[C1{i}+[Kl{x}={f} Whataretheelements ofthe matrices [M]. [C] and [K] andthevector {f} if m? c) Using your BOMs of a) above, develop the inputoutput differential equation for
the system corresponding to an output of x2 and an input f (t) .
(1) Put your EOMs in a) above in terms of their statevector form using the following
1:.
state vector: {z} = x'
x2 32
You need to provide complete free body diagrams in the derivation of your EOMs.
x, x2
l—+ ,.. #9 ... ‘3 k f (t) ' k
\\ I ” 2k
c _
_ smooth ‘ \\\\\\\\\\\\\ \\\\\\\ ‘§\\\\”\\ \\ \ \ '////,, ‘ '0’» 4‘ a g
’7 u . 4'
' L.. L
o __
n . "
.— O.
3 AA A
1‘. ‘1
O.
i M A .A * Q
+ 
O
4 . .—.
D
II,
  ~ A
 3 O
“ 1:. =7 :
A
‘ x o ' In I
z .'.o '
IA " A , + ‘A A .\ O... 9 z 2 _‘ _ *
m M H
6) . = " ‘ .. .
‘ M M M
 A f‘ on Inn—Ini
EﬂEmIMI‘ll ...
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 Fall '10
 Meckle

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