Over the past four sessions, we have discussed topics such as natural frequency and damping ratio through the lens of "Fundamentals of Vibration Measurement," including free vibration, forced vibration, vibration transmissibility, and vibration damping.
This time, we'll go back to the basics and talk about the amplitude, frequency, and phase of vibration waveforms, as well as how to represent vibrations (acceleration, velocity, displacement).
Vibration phenomena can be represented as waves, which are movements that repeat over time around a central point, like a pendulum or a swing. The most basic type of vibration waveform is the sinusoidal wave.
As shown in Figure 1, when a weight is attached to a spring and the weight is pulled and then released, it vibrates up and down repeatedly, and when the time change of the weight is plotted as a waveform, it becomes a sine wave.
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Figure 1: Oscillation waveform representing the time change of the weight
If we now consider the length of the pull (displacement) to be A and the repetition time to be T, then the time waveform is:
x(t)=Asin(2πft+φ)
This is how it works. f is called frequency (or vibration frequency), and f = 1/T, which is the speed at which the wave repeats per second. Its unit is Hz (Hertz), so for example, if the period T is 0.1 s, then the frequency is 10 Hz.
Next, the height of the peak in the vibration waveform (A in equation (1)) is called the amplitude (or peak value), and it represents the intensity of the vibration. This will be explained in more detail later.
Another parameter is the phase (φ in equation (1)), which is the relative position of the peak from a given reference time, and is usually expressed as an angle with one period being 360° (2π radians).
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Figure 2. Phase difference between two weights (the time for the left weight is shown by the red dotted line on the right).
Figure 2 shows the relative positions of the two weights (A and B);
Figure 2-(a) Weight B is 90° behind weight A in phase.
Figure 2-(b) Weight B is 180° behind (out of phase) weight A.
Figure 2-(c) Weight B is in phase with weight A.
That's how it is.
Phase plays an important role in balancing measurements of rotating bodies and in determining the vibration modes (shapes) of mechanical structures.
In summary, in a vibration waveform, frequency represents the rate of repetition, amplitude represents the intensity of the vibration, and phase represents the delay (time difference) of the peak from a certain reference point. These three parameters (frequency, amplitude, and phase) are called the three elements of vibration.
For periodic vibration waveforms like sinusoidal oscillations, the intensity of the vibration can be expressed almost entirely by the amplitude (peak value) as described above. However, how should the intensity of an irregular vibration waveform, such as the one shown in Figure 3, be expressed?
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Figure 3: Example of an irregular vibration waveform
In such vibrations, the amplitude cannot be clearly defined, and the peak value in the figure tends to be an overestimation; therefore, the effective value is defined by equation (2) below:
This is often used as a measure of vibration intensity.
The physical meaning of the RMS value is the square root of the mean square (power), which is the time average of the squared value (instantaneous energy) of the vibration waveform, and is a quantity corresponding to the power of the signal. Furthermore, the crest factor FC, defined by equation (3) below, is also frequently used as an evaluation quantity for impulsive vibrations.
Various amplitude values are shown using sinusoidal vibration waveforms for easier understanding.
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Figure 4 Various Amplitude Parameters of a Sine Wave
If we express the sine wave in Figure 4 using equation (1), then:
② Amplitude (single amplitude, peak value): A
② Total amplitude (peak-peak value): 2 A
③ Average value:
④ Effective value:
⑤ Waveform rate:
⑥ Wave height ratio:
As explained in the measurement column "Frequency Analysis from the Basics (22) - "Fundamentals of Vibration Measurement," vibration quantities can be expressed as displacement, velocity, and acceleration, and these are related by differential and integral calculus.
Furthermore, the physical units for typical translational vibrations are m for displacement, m/s for velocity, and m/ s² for acceleration.
Now, if we consider equation (1) as a vibration waveform of displacement and rewrite it with displacement amplitude X (m) and angular frequency ω (= 2πf), then:
x(t)=Xsin(ωt+φ)
It will be.
From this, the velocity v(t) and acceleration a(t) are:
In other words, if the displacement amplitude is X, the velocity amplitude can be calculated as ωX (m/s) and the acceleration amplitude as ω 2X (m/s²). Similarly, the integral can be calculated by dividing by ω, and frequency differential and integral calculations in actual FFT analyzers are performed by multiplication and division by ω (= 2πf).
These relationships can be summarized in Figure 5 below.
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Figure 5 Interrelationship between displacement, velocity, and acceleration (frequency calculus)
If displacement is kept constant regardless of frequency, the velocity and acceleration will exhibit frequency characteristics as shown in Figure 6, and each frequency band will have a corresponding range for three types of vibration quantities.
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Figure 6. Coverage range for three types of vibration intensity.
Displacement is the apparent amplitude of the vibration and is mainly used when evaluating the amount of vibration in the low frequency band. Typically, single-amplitude values (peak values) or total amplitude values (peak-peak values) are commonly used.
Velocity is used in the midband when energy is a concern, and in particular, the RMS value of velocity is used in vibration severity (ISO 2954).
Acceleration is used in cases where the magnitude of the force is important, such as impact force, or for diagnosing vibrations with high-frequency components, such as bearing damage vibrations, and the RMS value is usually used.
Finally, here's a summary.
- A system with a weight suspended from a spring will exhibit sinusoidal oscillations.
- The rate at which vibrations repeat is called frequency, the intensity of the vibration is called amplitude, and the relative position of the peak from a certain reference time is called phase. These are known as the three elements of vibration.
- In the case of irregular vibrations, the intensity of the vibration is evaluated by its RMS value, and the RMS value corresponds to the power of the vibration.
- Vibration can be expressed using displacement, velocity, and acceleration, and these are related by differential and integral calculus.
- For each of the displacement, velocity, and acceleration parameters, the displacement amplitude, velocity amplitude, and acceleration amplitude can be calculated.
【keyword】
Sine wave vibration, frequency, period, amplitude, peak value, phase, out-of-phase, in-phase, balancing, vibration modes, three elements of vibration, RMS value, instantaneous energy, mean square, power, crest factor, crest factor, single amplitude, total amplitude, peak-to-peak value, mean, form factor, displacement, velocity, acceleration, translational vibration, displacement amplitude, angular frequency, velocity amplitude, acceleration amplitude, vibration severity
【reference】
"Technologies and Laws for Pollution Prevention," edited by the Editorial Committee for Technologies and Laws for Pollution Prevention, Japan Industrial Pollution Prevention Association (1990).
(Excerpt from the email newsletter issued on May 26, 2016)
