Frequency-Domain Analysis of Vibrating Metal Beams: Attenuation of Artifacts

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Frequency-Domain Analysis of Vibrating Metal Beams: Attenuation of Artifacts

Frequency-Domain Analysis of Vibrating Metal Beams: Attenuation of Artifacts

BE 309

R3

Carl Anku

Kyung Chung

Seth Kramer

Sachin Puranik

December 16, 2004

Abstract:

The Hall Effect was utilized in order to determine the ideal configuration of magnets and Hall sensors that would produce the greatest linear relationship between the output voltage of the sensor and the displacement of the magnet. In addition to this linear relationship, it was also used to determine a set-up that would produce a frequency spectrum that omitted all artifacts. In determining the linearity relationship, the best configuration was found to occur when the magnet was attached to the bar in such a way that positive and negative displacement readings from the center of the magnet could be taken, which resulted in a linear range of 15 to 25 mm. However, the most dependable magnet – sensor configuration, and therefore the one most easily used in future studies, is one where the south pole of a magnet points directly at the branded face of the Hall Effect sensor, and vibration is parallel to the direction of the pole. For the determination of the best frequency-domain spectrum, this was found to occur when the magnet was facing the hall sensor placed at a position of 55 cm from the bottom of the bar, or any other position not corresponding to a node of one of the lower natural frequencies of the beam. In addition to these findings, it was also determined that magnet strength has minimal effect on the appearance of artifacts in the frequency spectrum.

Introduction:

In an earlier report, the vibrational modes of a metal beam were studied [1]. The procedure involved clamping a metal beam at one end and striking it to cause a vibration. The vibration was then measured by using a hall sensor to detect the flux of the magnetic field caused by a magnet placed on the end of the beam. Finally, the output of the hall sensor was analyzed in frequency domain, through the use of an FFT algorithm, in order to find the vibrational modes of the beam. However, the method for vibration detection used was not optimized. High amplitudes of vibration, a difficult to control parameter, caused unexpected non-linearities in the output of the hall sensor. These non-linearities caused artifacts in the frequency-domain that complicated the analyses and determination of mode frequencies.

The configuration of beam, magnet and hall sensor must be changed to produce a more dependable linear displacement relationship. This will create a more sinusoidal time-domain input, and therefore a clearer frequency-domain spectrum.

Objectives:

 Determine output voltage of the hall sensor with different magnets for a range of displacements.

 Find an experimental configuration of the magnet – hall sensor complex that yields a frequency-domain clear of harmonics and other artifacts

Background:

Hall Effect:

The sensors used in this study—Allegro Microsystsems’ 3515/6 Ratiometric, Linear Hall Effect Sensors—are, as the name suggests Hall Effect sensors. The Hall Effect is a special application of the Lorentz force:

(1)

which states that the force on a charge moving through an electric field is proportional to its charge, q, times the cross product of its velocity, v, and the strength of the magnetic field, B. The Hall Effect uses this principle in the context of a flat uniform semiconductor. The current flowing through the semiconductor can be expressed as the flow of charged particles, e-, of specific drift velocity, vd:

(2)

Combining equations (1) and (2) gives us the force on the electrons moving through a semi conductor in terms of the charge density, n, the cross-sectional area, A, and the magnetic field, B:

(3)

This force drives the flowing electrons to one side of the semiconducting plate, as seen in Figure 1, creating a potential difference between the two sides of the plate.

(4)

Where VH is the induced potential difference and w is the width of the semiconductor.

Figure 1 - The Hall Effect is measurable when a semiconductor of known thickness and charge density, with current I flowing through it, is exposed to a perpendicular magnetic field. (Figure from [2])

At equilibrium, the two forces FB and FE are balanced, this yields a direct relationship between the induced potential difference, an experimentally measurable variable, and the force of the magnetic field perpendicular to the plate in terms of the charge density of the semiconductor, a constant for each material; the thickness of the semiconducting plate; the charge of an electron; and the current through the plate:

(5)

APPARATUS:

  • Hall Effect Sensor
  • Impulse Piston (Hammer)
  • Bar support (attached to lab bench) with clamp and clamping plate
  • NdFeB Magnets
  • G-Clamps
  • Long Metal beam
  • Meter-long Rule
  • Biopac

METHODS:

Different configurations of a magnet – Hall Effect sensor complex were examined. This was done to determine which configuration produced the greatest linear relationship between the output voltage of the sensor and the displacement of the magnet causing the change in the sensor’s magnetic flux density. The output voltage was representative of the relative magnetic flux density produced by the interaction between the sensor and the attached magnet. The three configurations used are described below:

Configuration 1- South pole field towards branded face of sensor. Displacement parallel to pole direction.

In Error! Reference source not found., an NdFeB magnet was attached to the end of a meter-long metal beam with tape. The beam was clamped horizontally above a flat table surface with the south pole of the magnet directly facing the branded face of the sensor. The initial recorded distance (D) between the sensor and magnet was 0 mm. The beam was incrementally moved away from the magnet while taking successive readings of the distance and the output voltage produced. These values were recorded and the voltage was plotted against distance from the sensor.

Configuration 2 – Hall sensor perpendicular to pole direction. Displacement parallel to pole direction. Magnet and sensor are not in the same plane.

In Error! Reference source not found. the NdFeB magnet and beam were attached in such a way that positive and negative displacement readings from the center of the sensor could be taken.

The beam was positioned above the top of the sensor and moved horizontally, first with a positive displacement and then with a negative displacement, while readings of output voltage were taken. This allowed the graph produced to be centered on zero displacement. The effective air gap between the magnet and Hall Effect sensor was effectively 0 mm. The addition of the effective air gap above the sensor and the positive and negative displacement readings were the main differences between configurations 1 and 2.

Configuration 3 – Hall sensor parallel to pole direction. Displacement parallel to

In Configuration 3, the Hall sensor was flattened such that the horizontal beam and NdFeB magnet could slide over the sensor without contact, with the poles of the magnet moving perpendicularly to the poles of the sensor. The air gap between the sensor and the magnet was effectively 0 mm. Measurements of positive and negative displacement and output voltage were again taken by incrementally moving the clamped beam.

After determining the configuration that would produce the greatest linearity reading, the next objective was to determine how best to find the vibrational frequency readings of the beam using the Hall Sensor. This set-up would be determined by its ability to produce an FFT that clearly displayed the fundamental frequencies of the vibrating beam, while also omitting any type of harmonic frequency that comes as a consequence of the wave being introduced by the beam vibration as not being a perfect sine wave. By omitting these harmonics, it is much easier to find the fundamental frequencies of the beam.

The first set-up is similar to that of the set-up used in an earlier vibration experiment. A weak magnet is secured to the end of a metal beam with the south pole facing out (Setup 1Error! Reference source not found.). A Hall Sensor is then placed next to the beam so that it is in the configuration that was found to produce the most effective reading. The beam is then struck at a point along the length that does not correspond to a modal node and a reading is taken using Biopac. The sampling frequency of the system is set to 2000 samples ∙ sec-1 to reduce the chances of aliasing from higher frequency vibrations. This reading is then transformed using the FFT function of Biopac, which produces a frequency-domain analysis of the reading. In addition to Setup 1, where the magnet – sensor complex is placed at the end of the beam, the magnet is attached to the beam at positions 20 and 55 cm above the bottom with the Hall Sensor again set up facing the south pole of the magnet (Setup 2Error! Reference source not found.). This is repeated with both a weak stirring magnet, and a strong NdFeB magnet.

Setup 1 Setup 2

RESULTS:

Figure 2 - A plot of voltage vs displacement for Configuration 1

A plot of output voltage versus displacement for Configuration 1 resulted in Figure 2. The linear portion of the graph occurred between 0 and 10 mm representing a range of 10 mm over a voltage range of 0.73V.

Figure 3 - Displacement vs Voltage for Configuration 2

A similar plot for Configuration 2 gave the curve shown in Figure 3 with the linear portion occurring over a displacement range of 10mm and a voltage range of 1.07V.

Figure 4 - Displacement vs Voltage for Configuration 3

In Configuration 3, the plot of voltage versus displacement resulted in Figure 4 with the most linear portion occurring between 10 and 20mm representing a range of 10mm over a voltage range of 1.2V. This occurred with positive displacement away from the sensor as seen in the positive half of the graph.

Figure 5 - Frequency-domain of Setup 1 with Configuration 1

One normal mode frequency (5.86 Hz), with its six harmonics (11.72 Hz, 17.46 Hz, etc.), and the powerline frequency at 60.06 Hz is observed in this Fast Fourier Transform (Figure 5) of the signal from the hall sensor corresponding to the vibration of the beam with the magnet attached at the bottom. No other natural frequency can be distinguished within the frequency data.

Figure 6 - Frequency-domain of Setup 2 at 20 cm with Configuration 1

One normal mode frequency (5.86 Hz), one harmonic at 11.72 Hz, and the powerline frequency at 60.06 Hz is observed in this FFT (Figure 6.) of the signal corresponding to the vibration of the beam with the magnet attached 20 cm from the bottom of the beam. No other natural frequency is visible.

Figure 7 - Frequency-domain of Setup 2 at 55 cm with Configuration 1

The first three normal mode frequencies (5.86, 36.5, & 101.93 Hz) and the powerline frequency at 60.06 Hz are clearly observed in the Fast Fourier Transform (Figure 7.) of the voltage output from the vibration of the aluminum beam. No harmonics of the natural frequencies are visible in the FFT graph. The placement of the magnet was 55 cm from the bottom of the beam, a point along the length that does not correspond with the nodes of any of the first 3 vibrational modes.

Distance from bottom (cm) / Observed vibrational frequencies (Hz)
Weak / 0 / 5.86 / 11.72 / 36.5 / 60.06
magnets / 20 / 5.86 / 11.72 / 59.94
55 / 5.86 / 36.38 / 60.06 / 101.81
NdFeB / 0 / 5.86 / 11.72 / 17.46 / 30.52 / 36.25 / 42.11 / 47.97 / 60.06
magnets / 20 / 5.86 / 11.72 / 60.06
55 / 5.86 / 36.5 / 59.94 / 101.93
Powerline frequency
Natural frequencies
Harmonics

Table 1 - A chart of observed vibrational frequencies in the frequency-domain analysis of several magnet – Hall sensor placement points along the beam

Vibration frequencies of the aluminum beam, with an effective length of 80 cm, are displayed in Table 1. The first normal mode frequency and the powerline frequency are present in all trials. Harmonic frequencies were captured for trials with the magnet at 0 and 20 cm from the bottom of the beam.

Estimates of natural frequencies (Hz) at L = 80cm
w1 / w2 / w3 / w4 / w5
6.05 / 37.93 / 106.20 / 208.12 / 344.00

Table 2 - A table of natural frequencies for a beam of length 80 cm

The estimated normal mode frequencies for the aluminum beam with an effective length of 80 cm and a cross-sectional area of 2.55 cm2 (width = 5.1 cm, depth = 0.5 cm) are shown in Table 2. The experimental data exhibited an average error of 3.6% when compared to the estimates.

Discussion:

Linearity of Displacement:

The displacement of the magnet relative to the Hall sensor maintained a 10 mm range of linearity for all three configurations. Configuration 1 has a linear range from 0 to 10 mm, the range for configuration 2 is 15 to 25 mm, and configuration 3, 10 to 20 mm. The coefficient of linear regression, R2, was highest for configuration 3 at 0.99, while configurations 1 and 2 had coefficients of 0.98 and 0.84 respectively.

Configuration 1 was used to find the normal mode frequencies for the aluminum beam. This setup was favorable since the voltage signal monotonically decreased with distance from the hall sensor. Although configuration 2 had a greater region of linearity, the voltage output initially decreases and then increases beyond 1.5 cm from the sensor. This would not make it possible to obtain the normal frequencies of the beam. The FFT would include other frequency components that correspond to neither the normal frequencies nor its harmonics. These components would be a combination of the voltage output from two points, one from the negative slope and the other from the positive slope. The same problem would be encountered by configuration 3 since the signal increases up to 20 cm and decreases thereafter. Therefore, because configuration 1 exhibits a one-to-one relationship between the distance of the magnet from the sensor and the voltage output signal, it would be the best configuration. In other words, configuration 1 was chosen because it has the most dependable linear range – for the other configurations, positioning the Hall sensor too far or too close would cause non-linearities, however, with configuration one, there is only one unbounded end where non-linearities occur: the far one.

Determination of Normal Frequencies:

Using this information, the next objective was to determine the best setup to produce a FFT of which contains the fundamental frequencies of the metal beam, while excluding any artifacts. The first setup used (Setup 1) resulted in an FFT that contained many artifacts which were mainly the harmonics of the fundamental frequencies (figure 5). This result offers no improvement over the results obtained in the previous frequency analysis experiment. However, Setup 2, with the magnet – sensor complex placed 55 cm from the bottom of the beam, produced a much cleaner frequency-domain, mostly devoid of unwanted artifacts. This occurs because the amplitude of vibration of the magnet is well within the linear range of configuration 1. Also, the ratio between the amplitudes of the first mode to the other modes is not as great as in Setup 1, where the amplitude of the first mode was relatively large.

Recommendations for future beam-vibration studies:

When trying to find the normal mode frequencies, increased linearity of the output does not seem to help as data is transformed through FFT. Taking data in the range where the signal is monotonic in the direction both toward and away from the sensor seems to be the key in getting a sinusoidal output graph. Therefore, it would be advised to use Configuration 1, where the south pole magnetic field is pointing at the branded face of the Hall Effect probe. This would allow of the sensor to obtain the displacement of the beam within the linear range and exclude other combined frequency components.

The strength of the magnet does not appear to have a clear effect on the frequencies picked up by the hall sensor. One result of using the NdFeB magnet seems to be the accentuation of the harmonic frequencies as shown in the figure 1, where six harmonics of the first natural frequency are displayed. Also, using NdFeB magnets allowed for the use of lower gains as opposed to x20,000 or x50,000 for the weaker ones, its main effect being a increase in amplitude of the output voltage.

We recommend placing the magnet – sensor complex as shown in Setup 2, however, placement coinciding with nodes and antinodes of the natural frequencies would cause higher order modes to be attenuated. In the earlier study [1], the magnet was placed at the bottom of the beam. This position clearly allows the harmonics of the first and second normal mode frequencies to be picked up by the sensor. Placing the magnet higher up along the bar, but away from normal nodes, results in attenuation of the harmonic frequencies and accentuation of the modes. Therefore a position 55 cm above the bottom of the beam would be advised as opposed to 20 cm.

Reference:

[1] Kramer, S., Chung, K. J., Puranik, S., Anku, C. 2004. Analysis of Vibrational Frequencies of a Beam Using FFT. Unpublished.

[2] Nave C. R. 2003. Hyperphysics [online]. Available at: [http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/hall.html]