User's Manual

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Dynamics Cart Magnetic Damping Experiment 2: Critical Damping
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
km⁄=
γ bm⁄=
ω
0
2
γ
u
2
4⁄–0>
xt() x
0
e
γ
u
t–2⁄
ωt()cos
γ
u
2ω
-------
ωt()sin+=
ωω
0
2
γ
u
2
4⁄–=
ω
0
2
γ
o
2
4⁄–0<
xt() x
0
1
2
---
γ
o
4β
------–
e
γ
o
2⁄β+()t–
x
0
1
2
---
γ
o
4β
------+
e
γ
o
2⁄β–()t–
+=
βγ
o
2
4⁄ω
0
2
–=
xt() x
0
1
γ
c
t
2
-------+
e
γ
c
t–2⁄
=