This month, as an example of measuring everyday objects, we will introduce a case study of measuring the vibration of a pendulum.
While there are various ways to evaluate the magnitude of vibration, such as seismic intensity, vibration level, vibration acceleration, vibration velocity, and vibration displacement, vibration displacement (amplitude) is perhaps the most intuitive and easiest to understand. The vibration displacement (amplitude) of things like the pendulum of a pendulum clock, a hanging lamp, or a guitar string can be measured by eye or with a ruler to determine the approximate peak-to-peak amplitude in millimeters.
On the other hand, piezoelectric Accelerometer are commonly used as sensors for measuring vibrations. Since the signal obtained from the sensor is vibration acceleration, many people may not be able to visualize the vibration state just by looking at the acceleration signal. Furthermore, the method of double integration of the acceleration signal is susceptible to low-frequency noise, so it is necessary to determine whether the obtained waveform and measurement values are correct not only by looking at the displacement signal but also by looking at the acceleration and velocity signals and their spectra.
In this measurement column, we will present the results of measuring the vibration of a pendulum, along with the waveforms of vibration velocity and vibration displacement obtained using the integration function of a data analysis tool, as well as video footage from a video camera.
Example of a measurement system
- ONO SOKKI NP-3211 Accelerometer with Built-in Preamplifier
- ONO SOKKI DS-3000 Series Data Station
- ONO SOKKI Time Series Data Analysis Tool Oscope 2
(Configuration: OS-2720 FFT analysis pack, OS-0261 FIR filter option)
OS-0281 (Video Playback Option) - Digital video camera
Measurement target
The object of this measurement is a science toy called "Newton's Cradle," also known as a collision ball or balance ball. In the photo, we are measuring with three balls, but originally there are five metal balls lined up, and when you pull and release the ball on the far right, at the moment it collides with the stationary balls, only the ball on the far left will fly off in an arc to the left.
Acceleration, velocity, and displacement acting on the pendulum weight
Figure 1 shows the state of the pendulum and the acceleration and velocity acting on the weight.
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Figure 1: State of the pendulum and the acceleration and velocity acting on the weight.
The acceleration sensor attached to the right side of the weight detects acceleration in the left-right direction (the direction in which the pendulum swings). Acceleration to the right will be a positive value.
When the weight is at the left end (angle θ), the acceleration is at its maximum value, which is g・sin(θ) (where g is the acceleration due to gravity). At the left end, the weight's velocity is 0. When the weight is in the center, the acceleration is 0 and the weight's velocity is at its maximum. When it is at the right end, the acceleration is at its maximum value in the leftward direction (negative).
If the amplitude of the pendulum is small, the acceleration (a), velocity (v), and displacement (x) are all sinusoidal and can be expressed by equation 1, where f is the frequency. Also, the relationship between the amplitude of acceleration (A), the amplitude of velocity (V), and the amplitude of displacement (X) is given by equation 2.
.................................(1)
.................................(2)
Figure 2 shows the waveforms of acceleration, velocity, and displacement signals acting on an imaginary pendulum weight. The acceleration, velocity, and displacement signals are phase-shifted by 90 degrees.
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Figure 2: Acceleration, velocity, and displacement acting on an ideal pendulum weight.
Acceleration, velocity, and displacement signals when there is one weight.
Figure 3 shows the results of measuring the acceleration signal with only one weight in Newton's cradle, and calculating the velocity and displacement signals using the calculus function of Oscope 2. Since the acceleration signal also contains high-frequency vibrations, the acceleration signal with a 20 Hz low-pass filter applied is also shown.
Note that a BPF (Band-Pass Filter) of 0.2 Hz to 20 Hz was applied when calculating the velocity and displacement signals. Figure 4 shows the power spectra of each signal. The displacement signal is relatively close to a sine wave, but the acceleration signal and others are considerably distorted. The amplitude of the acceleration signal is 0.925 m/s² (pp), the amplitude of the velocity signal is 0.0931 m/s² (pp), and the amplitude of the displacement signal is 11.722 mm² (pp). The period for all is approximately 1.31 Hz, and 2πf = 8.2, but the measured ratio of acceleration to velocity is approximately 9.94, which deviates slightly from the theoretical value.
Possible causes for this include the accelerometer not being perfectly oriented sideways but being tilted, and the occurrence of vibrations other than pendulum oscillations, such as twisting of the weight. The power spectrum shows not only the first order (1.31 Hz), but also second order (2.63 Hz) and third order (3.94 Hz) components, indicating that it is not a simple oscillation.
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Figure 3. Acceleration, velocity, and displacement signals (for one weight)
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Figure 4 Power spectra of acceleration, velocity, and displacement signals (for one weight)
I used the FIR filter function of the Oscope2 (OS-0261 FIR filter option) to extract the frequency band of the signal I was interested in from this data.
Please refer to the following figure. There is a slight phase shift, but the result is as expected.
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Figure 5. Acceleration, velocity, and displacement signals (for one weight)
The FIR filter function cuts out frequencies below 0.5 Hz and above 2 Hz.
Furthermore, using the statistical processing function to examine the amplitude, we find an acceleration of 0.74 m/s². 2 Speed 0.089 m/s, change
The result was 11.1 mm, which was the expected value.
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Figure 6. Acceleration, velocity, and displacement signals (for one weight)
Amplitude values obtained using statistical processing functions
Acceleration, velocity, and displacement signals when there are three weights.
Figure 7 shows the acceleration, velocity, and displacement signals when three weights are used.
In the original "Newton's cradle," when a weight collides from the left, the weight on the far right bounces up, returns to its original position, and then comes to rest when it collides with another weight from the right. However, in this measurement, similar waveforms were observed regardless of whether the collision came from the left or the right. Possible reasons for this include an imbalance caused by the addition of an acceleration detector, or the fact that the amplitude of the repeated collision and rest motion was considerably smaller than the acceleration at the time of the collision and therefore could not be detected.
The experimental results show that an acceleration of approximately 400 m/s² is generated during the collision. This is 400 times the value when a single weight is swung. Generally, very large accelerations are generated during collisions. When weights are released a few centimeters apart and collided, an acceleration of 8400 m/s² is generated. This is a value greater than the maximum operating acceleration of the NP-3211 accelerometer (4900 m/s²).
As impact drop tests and similar experiments generate very high accelerations, it is necessary to use acceleration detectors specifically designed for such tests. Similarly, if the acceleration detector is mishandled and dropped onto a hard surface, a large acceleration will also be generated, so care must be taken when handling it.
Velocity and displacement signals are waveforms that have been integrated due to undershoot caused by AC coupling. They are far removed from the actual velocity waveform. This is important to note, as acceleration detectors often use AC coupling for input.
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Figure 7 Acceleration, velocity, and displacement signals (with 3 weights)
summary
In this experiment, we expected that the displacement signal obtained by double-integrating the acceleration signal would match the movement of the weight, but the actual measurements showed a significant discrepancy. The integration process emphasizes low-frequency noise, and the pendulum's period of 1.31 Hz is close to the lower limit of the accelerometer's frequency range (1 Hz to 10 kHz, ±5%), so we anticipated that observing the displacement waveform by double integration would be difficult, and indeed, the measurement proved challenging. We felt that measures such as precisely fixing the accelerometer, reducing background vibrations, and using an appropriate high-pass filter (to avoid low-frequency noise) would be necessary to obtain accurate measurements.
The method of measuring displacement by double-integrating the acceleration signal is susceptible to low-frequency noise, so it is necessary to judge the accuracy of the measurement results not only by examining the displacement signal but also by examining the acceleration signal, velocity signal, and their spectra. Furthermore, in order to perform accurate measurements, measures such as precisely fixing the acceleration detector, reducing background vibrations, and using an appropriate high-pass filter (to avoid low-frequency noise) are likely to be necessary.
Revised: July 3, 2012
(Excerpt from the email newsletter issued on June 21, 2012)
