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horizontal plane. The mass M 2tn is considered to be a concentrated mass. Find the equatio, Cheat Sheet of Engineering

Read each problem carefully. There are three pages and three problems, each worth 10 points. Include units in your answers, as appropriate. Total points: 30. 1. (10 pts) The uniform bar of mass m and total length L = 4a, sketched below, rotates by an angle 0 (assumed small) about a frictionless pin at point 0, and lies in the horizontal plane. The mass M 2tn is considered to be a concentrated mass. Find the equation of motion in terms of 0 and the parameters a, k, c, and in. (Note that for a uniform slender bar of length L, the mass moment of inertia about its center of mass 0 is

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Exam
1
ME461,
June
4,
2014
Name
Ozze
LAJ-0r
t~i
a
z\
~ct
i2~1
i
~
(21)
I
2
Read
each
problem
carefully.
There
are
three
pages
and
three
problems,
each
worth
10
points.
Total
points:
30.
1.
(10
pts)
The
T-bar below
is
a
single
rigid
body
formed
by
two
rigidly
welded
uniform
bars,
with
each
bar
of
mass
in
and
one
of
length
L,
the
other
of
length
2L.
The
center
of
gravity
G
of
the
T-bar
is
thus
located
a
distance
3L14
from
the
pin
at
0.
It
is
attached
to
a
pin
and
hangs
as
shown,
with
a
spring
attached,
depicted
in
the
undeformed
state.
(Note
gravity
acts
downward,
and
must
be
accounted
for.)
Find
(a)
the
differential
equation
of
motion
for
small
angles
0,
and
(b)
the
undamped natural
frequency
of
the
bar.
Note
that
the
mass
moment
of
inertia
of
each
independent
bar
of
length
L
about
its
center
of
gravity
is
‘G
=
j~~mL2.
a
k
L
L
fri0
I~9
-
z~j
\4)
4—I
\~~r
2
k(:fiL~9
4
—.
12.
4
IL
2.
c
4..
v~
L
~-2frn)3
(z
w~L~
ö
~
(s~
(ox)
(~)
3.
2
2k-
L
4-
YML.
t~.
j
3
/
~
9
I
10
7~L
(ii~
pf3

Partial preview of the text

Download horizontal plane. The mass M 2tn is considered to be a concentrated mass. Find the equatio and more Cheat Sheet Engineering in PDF only on Docsity!

Exam 1 ME461, June 4, 2014 Name Ozze LAJ-0r

t~i

a

z\ ~ct

i2~

i

I 2

Read each problem carefully. There are three pages and three problems, each worth 10 points.

Total points: 30.

  1. (10 pts) The T-bar below is a single rigid body formed by two rigidly welded uniform bars, with each bar of mass in and one of length L, the other of length 2L. The center of gravity G of the T-bar is thus located a distance 3L14 from the pin at 0. It is attached to a pin and hangs as shown, with a spring attached, depicted in the undeformed state. (Note gravity acts downward, and must be accounted for.) Find (a) the differential equation of motion for small angles 0, and (b) the undamped natural frequency of the bar. Note that the mass moment of inertia of each

independent bar of length L about its center of gravity is ‘G = j~~mL2.

a

k

L L

fri0 I~9^ -^ z~j^ \4)

4—I

~~r 2

k(:fiL~

IL

c

4.. v~ L

~-2frn)

(z

w~L~ ö ~ (s~

(ox)

(~)

2k- L 4- YML. t~. j

/ ~ 9 I —

10 7~L

(ii~

  1. (10 pts) The thin, half roller of mass in = 20 kg and radius R sketched below has the equation of

motion given. There is damping from the air and from the contact materials, which is represented by the

effective linear viscous damping term in the differential equation. In a free-vibration test, the oscillation

decayed such that over twenty cycles, the amplitude reduced by a factor of 3.51. That is X0JX20 = 3.51. The period of each oscillation was ta = 2~rf3 seconds. Find the value of the damping coefficient, c, as

defined in the equation of motion. (You can use small ~ approximations in the calculations, but ~ is

not zero.) (2~— 4) inR2O + cR2O + ~t~,ngR6 =

C

t

CL~

2(2~ ~4~)n~ çW

F’ rr

49 N N

hi

2 it

a

D

—- —

Z~Wq

S~\

‘ci

40w

fl C — ‘sJn

(2rr~4)wi

(‘in,

(2 ff-4) fr0L

rc

IT’

a.E.^ ¥^ EH^ %

n To