MAT 274 Osillations
Worksheet
Consider the damped mass-spring system:
x
00
+ 2
x
0
+ 9
x
= 0 (1)
with initial onditions
x
(0) = 1,
x
0
(0) =
2. The best Maple syntax to solve
suh a dierential equation is
de
:=
diff
(
x
(
t
)
;
t
;
t
) +
2
diff
(
x
(
t
)
;
t
) +
9
x
(
t
) =
0
;
dsolve
(
f
de
;
x
(
0
) =
1
;
D
(
x
)(
0
) =
2
g
;
x
(
t
))
1. Determine the pseudo period of the osillations.
2. Find graphially:
(a) The time for the rst passage of the mass through the equilibrium
position. What is the veloity at that instane.
(b) The time for the rst maximal elongation. What is the veloity
at that instane ? What is the aeleration at that instane?
() The time for the rst minimal elongation. What is the veloity
at that instane ? What is the aeleration at that instane?
3. Determine all the above data exatly.
4. Get the amplitude phase form
re
t
os(
!t
Æ
) for the solution and
plot
re
t
together with the solution. Determine where the two urves
touh graphially and analytially.
5. Assume you have a measurement auray of 10
2
in x and determine
when the spring is at rest.
6. If you want to osillate the spring longer how would you have to hange
the damping onstant? Determine approximately a damping onstant
suh that the spring osillates twie as long as before.
7. Where are the maximum and minimum points of the osillation rela-
tive to the ontat points with the exponential envelope.
1