For a more quantitative analysis of the damping, we begin .... The plot and fit were done with Graphical Analysis. Vernier Software, http://www.vernier.com. 6.
Measuring the Damping Constant for Underdamped Harmonic Motion Michael C. LoPresto
and
Paul R. Holody,
Henry Ford Community College, Dearborn, MI
T
d 2x
dx = – kx – bᎏᎏ. mᎏᎏ dt2 dt 22
0.95
Position (m)
0.90 0.85 0.80 0.75 0
5
10
Time (s)
15
20
25
Fig. 1. Plot of position vs time of an underdamped oscillating mass-spring system with a fit of the data (blue line) to the form (dark line) of Eq. (2).1 A*exp(-E*t)* cos (B*t + C) + D, where A = 0.103 m, B = 4.99 s-1, C = -1.474, D = 0.0826 m, and E = 0.069 s-1.
0.95
0.90
Position (m)
he now widespread use of microcomputers in introductory physics laboratories allows for an addendum to the traditional mass-springsystem experiment that usually culminates with the determination of the spring constant k. A sonicmotion detector can be used to create plots of position, velocity, and acceleration as functions of time and even position versus velocity. Students can then be asked to compare these plots and observe the relationships between the variables. This, however, is only the beginning. The use of features of the data-collection software allows for further and even more informative analysis. Figure 1 is a plot of position as a function of time for a damped mass-spring system.1 A thin wooden square was taped to the bottom of the mass so the detector could more easily “see” the oscillating mass. This provided increased but insufficient damping. The damping was increased further by attaching an 8.5-x11-in piece of paper that was pierced at its center by the hook that attaches the mass to the spring. A simple exercise is to ask students to read about damping in their textbooks and determine from a plot similar to Fig. 1 what kind of damping is occurring. Another interesting exercise is to inspect a plot of position versus velocity for damped motion, such as that shown in Fig. 2, and attempt to explain the reason for its shape. For a more quantitative analysis of the damping, we begin with the equation of motion of a mass m attached to a spring with constant k, subject to a damping force:
0.85
0.80
0.75
-0.4
-0.2
-0.0
0.2
0.4
Velocity (m/s)
Fig. 2. Plot of position vs velocity for an underdamped oscillating mass-spring system.1
(1) DOI: 10.1119/1.1533960
THE PHYSICS TEACHER ◆ Vol. 41, January 2003
x1 = ebt / 2 m , x2
0.110 0.100
where T is the period. Taking the natural log of both sides and rearranging gives
0.0900
x (m)
0.0800
2m x1 b = ᎏᎏ ln ᎏᎏ . T x2
0.0700 0.0600 0.0500 0.0400 0.0300 0.0200 0.0
4.0
8.0
12.00
Manual Curve Fit: f(x)=A*exp(b*x) A = 0.103
(5)
16.0
20.0
24.0
28.0
t (s)
B = -0.0694
Mean Square Error: 9.57e - 06
Fig. 3. Plot and an exponential fit of the decay envelope of the oscillating system.4,5
For underdamped motion, the solution is of the form x = Ae –bt/2m cos t ,
冢 冣
(2)
(3)
with b being a “damping constant”2 of the motion. The cosine term in Eq. (2) represents the oscillatory motion and the exponential is the decay of the amplitude. A fit of the plot to Eq. (2) is also shown in Fig. 1. A value of b = 0.046 kg/s for an effective mass3 of 0.33 kg was determined by setting the argument of the exponential (which was 0.069 s-1) equal to b/2m. The positions and times of the peaks4 in Fig. 1 can be used to verify the value for b. For motion obeying Eq. (2), the ratio of two successive peaks is given by
x1 Ae − bt / 2 m cos t = x2 Ae − b (t +T ) / 2 m cos (t + T ) .
Applying Eq. (6) to each pair of peaks in the time interval shown in Fig. 1 gave an average value of b = 0.043 kg/s, similar to that obtained using the fit in Fig. 1. Although individual oscillations gave a range of values, most were within the standard deviation of = 0.021 kg/s. The same value for b also resulted when Eq. (6) was applied over the entire time interval. A fit of the peak locations to an exponential is shown in Fig. 3.5 The fit of the decay in Fig. 3 gave the same value for the coefficient of the argument of the exponential, 0.069 s-1, and thus the damping constant, b = 0.046 kg/s, as the fit in Fig. 1. For a decay constant this small, the first term domik nates in Eq. (3) and so 2 is very nearly equal to ᎏᎏ. m
where k b 2 = ᎏ ᎏ – ᎏ ᎏ 2, m 2m
(6)
(4)
The value of = 4.99 s-1 from the fit in Fig. 1 and the effective mass can then be used to determine the spring constant, in this case k = 8.3 N/m, which compares favorably (
