Homework 6
—due 5:00 p.m. Friday, March 12
These problems are concerned with the motion of a mass
m
suspended by three springs of
stiffness
k
in a moving frame:
4
k
k
m
k
¯
y
(
t
)
¯
x
(
t
)
y
(
t
)
3
x
(
t
)
The suspended mass has displacements
x
(
t
)
and
y
(
t
)
relative to an inertially fixed reference
frame, and the outer supporting frame has displacements
¯
x
(
t
)
and
¯
y
(
t
)
relative to the inertially
fixed reference frame. One spring is horizontal, one is vertical, and the third is inclined so that
it has a slope of 4/3, as shown. Ignore gravity for this entire homework assignment.
1. Derive equations of motion that govern
x
(
t
)
and
y
(
t
)
.
2. Find natural frequencies and natural modes of free vibration for the mass
m
. Show each
mode visually and describe it in words.
3. Transform the equations of motion using the modes, so that they are decoupled.
4. Find the free response to these initial conditions:
(a)
x
(0) =
x
0
,
y
(0) = ˙
x
(0) = ˙
y
(0) = 0
. Plot this response in the
x

y
plane over the
first two periods of the lowerfrequency mode. (This will be a parametric plot, with
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 Spring '10
 Staff
 Mass, Special Relativity, Rectangular function, initial displacements

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