PHYS-1500 PHYSICAL MODELING
Class 21: Driven Mechanical Oscillator
FALL 2006
NAME ________________KEY_____________
In this exercise, you will examine a mechanical oscillator that is driven by a sinusoidal force.
The oscillator is just a mass, m, on a spring of spring constant k, which you have treated before.
However, for a more realistic model, and to keep the system from getting out of control, a
damping, or frictional, force has been added. The damping force is proportional to velocity, and
directed opposite to the velocity, with a constant of proportionality b. The driving force is written
as Fd = F sin 0t. Then, since the net force equals mass times acceleration, we have,
d2x
dx
d2x
k
b dx Fd
m 2   kx  b  Fd sin  0 t or
x

sin  0 t .
2  
dt
m
m dt m
dt
dt
This is the equation that is in the spreadsheet.
Consider the reaction of the system as 0 and b are changed. First set x0 = 0, v0 = 0, and F/m =
1.0 dyne/g. Then set b = 0.2 dynes/(cm/s) and find the maximum value of x (the amplitude) for
0 = 0.2, 1.0, 1.6, 2.0, 2.4, 3.0, and 4.0 rad/s. Then do the same for b = 0.5 and 1.0 dynes/(cm/s).
Enter the results in the table shown, and graph x(max) vs. 0 for all three values of b on the same
graph.
o
x (max)
0.5
0.25
0.37
0.61
0.97
0.7
0.37
0.19
0.2
0.26
0.41
0.84
2.24
1.01
0.45
0.19
b
0.2
1.0
1.6
2.0
2.4
3.0
4.0
1.0
0.25
0.32
0.42
0.5
0.44
0.27
0.17
Sketch the graph below.
2.5
x max (cm)
2
0.2
1.5
0.5
1
1
0.5
0
0
1
2
3
4
5
 0 (rad/s)
Turn the paper over. There is more on the back
Now set F/m = 0, and x0 = 2.0 cm. Set b = 0.2 and sketch the graph of x vs. t. Then do the same
for b = 0.5 and 1.0.
b = 0.2
2.5
2
1.5
x (cm)
1
0.5
0
-0.5
0
5
10
15
20
25
30
35
20
25
30
35
20
25
30
35
-1
-1.5
-2
t (s)
b = 0.5
2.5
2
1.5
x (cm)
1
0.5
0
-0.5
0
5
10
15
-1
-1.5
-2
t (s)
b = 1.0
2.5
2
x (cm)
1.5
1
0.5
0
0
5
10
15
-0.5
-1
t (s)