PASCO ME-6828 Dynamics Cart Magnetic Damping - Manual (Page 8)

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D y n a m i c s C a r t M a g n e t i c D a m p i n g E x p e r i m e n t 2 : C r i ti c a l D a m p i n g
8
b, the damping constant ( ) determined by the height of the magnets
, the natural frequency of the system in the absence of damping
, a quantity with units of frequency
If m and k are held constant, then the damping behavior of the system (whether it is
under-, over-, or critically damped) depends on the value of b.
Under damping
For the system to be under-damped, we must have a relatively low value of
γ. Let
γ = γ
u
such that . Then
Where .
Over damping
For over damping, the value of
γ must be higher. Let γ = γ
o
such that
. Then
Where .
Critical damping
Critical damping occurs at a specific value of
γ. Let γ = γ
c
such that
ω
0
2 − γ
c
2 /4 = 0.
Then
Modeling the cart’s motion
Graph the three equations above and compare them to your actual data. Either mea-
sure or estimate your experimental values of
x
0
, m, and k (or ω
0
) to put into the equa-
tions. Estimate values of b for under, over, and critical damping.
F
damping
bv – =
ω
0
k m ⁄ =
γ b m ⁄
=
ω
0
2 γ
u
2
4
⁄
– 0
>
x t ( ) x
0
e
γ
u
t – 2
⁄
ωt ( ) cos
γ
u
2
ω
------- ωt ( ) sin + =
ω ω
0
2 γ
u
2
4
⁄
– =
ω
0
2 γ
o
2
4
⁄
– 0
<
x t ( ) x
0
1
2
---
γ
o
4
β
------ –
 
  e γ
o
2 ⁄ β + ( )t –
x
0
1
2
---
γ
o
4
β
------ +
 
  e γ
o
2 ⁄ β – ( )t –
+ =
β γ
o
2
4
⁄ ω
0
2
– =
x t ( ) x
0
1
γ
c
t
2
------- +
 
  e
γ
c
t – 2 ⁄
=