Computer Project 1. Nonlinear Springs
Goal:
Investigate the behavior of nonlinear springs.
Tools needed:
ode45
,
plot
Description:
For certain (nonlinear) springmass systems, the spring force is not given by
Hooke’s Law but instead satisﬁes
F
spring
=
ku
+
±u
3
,
where
k >
0 is the spring constant and
±
is small but may be positive or negative and
represents the “strength” of the spring (
±
= 0 gives Hooke’s Law). The spring is called a
hard spring
if
± >
0 and a
soft spring
if
± <
0.
Questions:
Suppose a nonlinear springmass system satisﬁes the initial value problem
(
u
00
+
u
+
±u
3
= 0
u
(0) = 0
,
u
0
(0) = 1
Use
ode45
and
plot
to answer the following:
1.
Let
±
= 0
.
0
,
0
.
2
,
0
.
4
,
0
.
6
,
0
.
8
,
1
.
0 and plot the solutions of the above initial value problem
for 0
≤
t
≤
20. Estimate the amplitude of the spring. Experiment with various
± >
0.
What appears to happen to the amplitude as
±
→ ∞
? Let
μ
+
denote the ﬁrst time the
mass reaches equilibrium after
t
= 0. Estimate
μ
+
when
±
= 0
.
0
,
0
.
2
,
0
.
4
,
0
.
6
,
0
.
8
,
1
.
0.
What appears to happen to
μ
+
as
±
→ ∞
?
2.
Let
±
=

0
.
1
,

0
.
2
,

0
.
3
,

0
.
4 and plot the solutions of the above initial value problem
for 0
≤
t
≤
20. Estimate the amplitude of the spring. Experiment with various
± <
0.
What appears to happen to the amplitude as
±
→ ∞
? Let
μ

denote the ﬁrst time the
mass reaches equilibrium after
t
= 0. Estimate
μ

when
±
=

0
.
1
,

0
.
2
,

0
.
3
,

0
.
4.
What appears to happen to
μ

as
±
→ ∞
?
3.
Given that a certain nonlinear hard spring satisﬁes the initial value problem
(
u
00
+
1
5
u
0
+
±
u
+
1
5
u
3
)
= cos
ωt
u
(0) = 0
,
u
0
(0) = 0
plot the solution
u
(
t
) over the interval 0
≤
t
≤
60 for
ω
= 0
.
5
,
0
.
7
,
1
.
0
,
1
.
3
,
2
.
0. Continue
to experiment with various values of
ω
, where 0
.
5
≤
ω
≤
2
.
0, to ﬁnd a value
ω
*
for
which

u
(
t
)

is largest over the interval 40
≤
t
≤
60.