Download Induced Current Measurement of Rod Vibrations

Induced Current
Measurement of Rod
Vibrations
Charles A. Sawicki,
North Dakota State University, Fargo, ND
T
he longitudinal1 normal modes of vibration
of rods are similar to the modes seen in pipes
open at both ends. A maximum of particle
displacement exists at both ends and an integral number (n) of half wavelengths fit into the rod length.
The frequencies fn of the normal modes is given by
Eq. (1), where L is the rod length and V is the wave
velocity:
nV
fn = ᎏᎏ.
2L
(1)
Many methods have been used to measure the velocity of these waves. The Kundt’s tube method commonly used in student labs will not be discussed
here. A simpler related method has been described
by Nicklin.2 Kluk3 measured velocities in a wide
range of materials using a frequency counter and
microphone to study sounds produced by impacts.
Several earlier methods4,5 used phonograph cartridges complete with needles to detect vibrations in
excited rods. A recent interesting experiment6 used
wave-induced changes in magnetization produced in
an iron rod by striking one end. The travel time,
measured as the impulsive wave reflects back and
forth, gave the wave velocity for the iron rod. In the
method described here, a small magnet is attached to
the rod with epoxy, and vibrations are detected using
the current induced in a few loops of wire. The
experiment is simple and yields very accurate velocity
values.
Figure 1 shows the apparatus used in these measurements. In part (a), a small disk-shaped NdFeB
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Fig. 1. (a) gives a schematic diagram of the end of a rod
with attached NdFeB magnet and detector coil. (b)
gives a schematic of the circuit used to amplify the coil
output. Although the figure shows supply voltages of
⫾15 V, a pair of 9-V batteries will also work. The 10 K⍀
offset adjustment potentiometer is used to zero the
output with pin 2 shorted to ground. Without this
potentiometer, the offset voltage for this circuit is typically 1 V.
DOI: 10.1119/1.1533965
THE PHYSICS TEACHER ◆ Vol. 41, January 2003
Fig. 2. The frequency content of the signal produced by
tapping the end (far from the detector coil) of a 0.95-cm
diameter aluminum rod 1.045 m long. Peaks are labeled
with the mode order n, and data for n = 6 through 9
have been multiplied by a factor of 10 so that these
mode peaks are clearly visible.
Fig. 3. Normal mode frequencies for the data of Fig. 2
are plotted (as diamond-shaped points) versus mode
order n. Errors are smaller than the points. The solid
straight line represents a best fit to the data.
magnet7 is shown attached to one end of a rod with
epoxy. The detector coil (5 to 10 turns) is positioned
very close to the magnet. Longitudinal motions produced by vibrations move the magnet relative to the
loosely attached coil, producing an induced current.
This signal was amplified using the LF411 JFET amplifier shown in Fig. 1(b). Amplifier output was
recorded using a LabPro interface and Logger Pro
software.8 Finite Fourier transforms (ffts) of the data
were calculated with Logger Pro to determine the frequencies of normal mode vibrations excited in the
rod.
Figure 2 shows the fft of data produced by tapping
an aluminum rod (0.95-cm diameter, 1.045 m long)
that I held with two fingers at the center. In this experiment I collected 10,000 data points at 50,000
points/s. Even n modes are seen to make a smaller
contribution than the next higher n odd modes. At
the rod center, odd n modes have nodes while even n
modes have maximum displacement. The pattern of
peaks seen in Fig. 2 results because holding the rod at
the center damps even n modes more quickly than
odd n modes. More technical details of the experiments are discussed in the Appendix.
Figure 3 plots normal mode frequency versus mode
order n for the data seen in Fig. 2. A best-fit straight
line is plotted with the data points. This best fit gives
V/(2L) = 2411 ⫾ 3 Hz and V = 5040 ⫾ 6 m/s. This
agrees well with the tabulated result9 5000 m/s for
very thin aluminum rods. At this level of difference,
wave velocity depends upon the details of composition and material processing. This method works well
for a wide range of materials. For example I obtained
V = 1798 ⫾ 4 m/s for a nylon rod 2.54 cm in diameter and 1.002 m long. This compares well with the
standard value9 1800 m/s. For materials with much
stronger damping, like wood, fewer and broader normal mode peaks are produced, and less accurate velocity values are obtained.
THE PHYSICS TEACHER ◆ Vol. 41, January 2003
References
1. Waves produced by striking the end of a thin rod are
not simple longitudinal waves (sound waves). Longitudinal waves only occur in large volumes of material. In
thin rods, the rod surface also moves in the radial direction as a wave passes. For example, in the case of aluminum the sound velocity is 6420 m/s in bulk material,
while the velocity of similar waves in thin rods is 5000
m/s.
2. R.C. Nicklin, “Measuring the velocity of sound in a
metal rod,” Am. J. Phys. 41, 734 (May 1973).
3. Michael T. Frank and Edward Kluk, “Velocity of sound
in solids,” Phys. Teach. 29, 246–251 (April 1991).
4. Oakes Ames, “A direct measurement of the speed of
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5.
6.
7.
sound in rods,” Am. J. Phys. 38, 1151–1152 (Sept.
1970).
Nobuo Naba, “Observation of longitudinal vibration of
metal rods,” Am. J. Phys. 40, 1339–1340 (Sept. 1972).
David Potter, “The speed of sound in an iron rod,”
Phys. Teach. 40, 56–57 (Jan. 2002).
For the 0.95-cm diameter aluminum rod, I used a 3/8in diameter 1/16-in thick NdFeB magnet. Magnets
were obtained from Force Field, 2606 W. Vine Drive,
Fort Collins, CO 80521; 877-944-6247. They typically have small quantities of magnets other than the 40 or
so for sale at their website at http://www.wondermagnet.com.
Appendix: Details and an Added
Experiment
I wound the detection coil on the rod using five
turns of enameled 22-gauge magnet wire. To prevent rapid damping of the excited vibrations, the
wire should not grip the rod tightly. With a properly adjusted coil on a metal rod, the vibration signal typically decreased to one-fourth of its initial
amplitude during the 0.2-s recording time. The
coil was connected to the LF411 amplifier with 2
m of twisted pair 26-gauge wire to reduce noise
pickup. As shown in Fig. 1(b), the amplifier gain
was 106 ⍀ /510 ⍀ = 1960 in these experiments.
This gain should be adjusted so that a light tap on
the rod produces a signal of more than 1 V at the
amplifier output.
I made a small metal hammer (total mass 0.037
kg) using 3/8-in diameter steel rod to tap the ends
of rods. Because the detector coil was relatively
loose on the rod, it is best to use a small hammer
that transfers a relatively small amount of momentum to the rod. Jerking of the rod can produce a
low- frequency offset contribution in the recorded
voltage signal. This can be removed with a highpass filter placed before the amplifier, but I found
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8.
9.
The LabPro interface with Logger Pro software from
Vernier Software was used to collect data in this experiment, 13979 SW Millikan Way, Beaverton, OR 970052886; 503-277-2299; http://www.vernier.com.
CRC Handbook of Chemistry and Physics (CRC Press
Boca Raton, FL, 1985), p. E-43.
Charles Sawicki has been a member of the physics faculty at North Dakota State University since 1979. He
received a B.S. from the California Institute of Technology
and a Ph.D. from Cornell. Physics Department, North
Dakota State University, Fargo, ND 58105-5566; charles.
[email protected].
the added complication unnecessary. A hard steel
hammer produces a short (time duration) impact
with a metal rod that excites more higher frequency modes than a softer wood hammer. The frequency spectrum of the driving force (impact) determines the relative intensities of the normal
modes initially excited. A short-duration impact
from a hard hammer includes high-frequency
components and excites high-frequency
vibrations.
For students interested in music, this apparatus
can be used for an added set of experiments. The
principle discussed above applies to percussion instruments such as the xylophone. The hardness of
the hammer used affects the frequency content of
the sounds produced by striking. Using this apparatus with metal, wood, and hard rubber hammers
striking an aluminum rod provides a nice experimental demonstration of this principle. With a
wood hammer, mainly the n = 1 mode is excited,
while with a rubber hammer it is very difficult to
excite even the n = 1 mode. For a material like nylon, which is easier to compress, the wood hammer is nearly as effective as the steel hammer in exciting higher frequency modes, and a hard rubber
hammer easily excites the n = 1 mode.
THE PHYSICS TEACHER ◆ Vol. 41, January 2003