The MassSpring System
Math 308
This Maple session uses the massspring system to demonstrate the phase plane, direction fields,
solution curves (``trajectories''), the extended phase space, and more.
The differential equation is
=
+
+
m
°
±
²
²
³
´
µ
µ
∂
∂
2
t
2
( )
y
t
b
°
±
²
²
³
´
µ
µ
∂
∂
t
( )
y
t
k
( )
y
t
0
(I've used b instead of the usual
γ
for the coefficient of friction.)
We convert this to a system by defining
=
( )
v
t
∂
∂
t
( )
y
t
Then
=
∂
∂
t
( )
v
t


k
( )
y
t
m
b
°
±
²
²
³
´
µ
µ
∂
∂
t
( )
y
t
m
or
=
∂
∂
t
( )
v
t


k
( )
y
t
m
b
( )
v
t
m
This equation along with the equation that defines v(t) make up a system of two equations.
>
with(DEtools):
We first define the two equations that make up the system:
>
eq1 := diff(y(t),t) = v(t);
:=
eq1
=
∂
∂
t
( )
y
t
( )
v
t
>
eq2 := diff(v(t),t) = (k/m)*y(t)(b/m)*v(t);
:=
eq2
=
∂
∂
t
( )
v
t


k
( )
y
t
m
b
( )
v
t
m
Now define a variable that holds the system:
>
sys := [eq1,eq2];
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:=
sys
¶
·
¸
¸
¹
º
»
»
,
=
∂
∂
t
( )
y
t
( )
v
t
=
∂
∂
t
( )
v
t


k
( )
y
t
m
b
( )
v
t
m
We substitute in some numerical values for the parameters, so we can plot some solutions.
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 Spring '08
 McAllister
 Math, 2 m

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