Vibration Units & Relationship

THE RELATIONSHIP BETWEEN TIME AND FREQUENCY
Time
When we say that AC line frequency is 60 cycles per second, this means if a one second time period was observed, 60 cycles . However, it is not always practical to observe one second of time and count the number of cycles.

We can measure the time period for one cycle and calculate the frequency. We can also calculate the time period for one cycle if the frequency is known. Time and frequency are the reciprocal of each other.

If 60 cycles occur in one second and the time period for one cycle is 0.0167 seconds, the calculation can be verified by: F x T =1 or 60 x 0.0167 = 1.

Frequency
Frequency is the number of cycles that occur in one time period, usually one second. Until a few years ago, frequency was identified as cycles per second (CPS). CPS was changed to Hertz, honoring the man who developed the frequency theory. Today Hertz (cycles per second) is the standard measurement of frequency. Machine speed is measured in revolutions per minute (RPM), but the frequencies generated by those machines are measured in Hertz.

Example : What is the frequency of a time period of 50 milliseconds?
Answer: T = 50 ms x 0.001 =0.05 sec, F =1/t=1/0.05 = 20 Hz
In the above formula, when determining frequency in cycles per second, time must be in
seconds.
Example : What is the speed of a machine that generates a fundamental frequency of
29.6 Hz?
Answer: 29.6 Hz x 60 Sec/min = 1776 RPM
Example : What is the fundamental frequency a machine will generate if the machine
speed is 1180 RPM?
Answer: 1180 C.PM /60 Sec /min= 19.7 Hz

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AMPLITUDE MEASUREMENT
The four different ways to express the vibration amplitude level are: peak-to-peak, zeroto-peak, RMS, and average.

Peak-to-peak is the distance from the top of the positive peak to the bottom of the negative peak. This type of measurement is most often used when referring to displacement amplitude. Zero-to-peakor peak is the measurement from the zero line to the top of the positive peak or the bottom of the negative peak. The zero-to-peak value of the vibration level . This type of measurement is used to describe the vibration level from a velocity transducer /accelerometer.

The Root Mean Square (RMS)is the true measurement of the power under the curve. In the RMS value is the cosine of 45 degrees times peak (0.707 x peak only applies to pure sine waves). The true RMS value is calculated by the square root of the sum of the squares of a given number of points under the curve.

When calculating true RMS, the crest factor and duty cycle must be considered for signals that contain pulses. The crest factor (CF)is the ratio of the peak value to the RMS value with the DC component removed .

Analog meters measure average amplitude. Various constants are then used to calculate peak, peak-to-peak, or RMS. Most measurements that are not true RMS measurements are either overstated or understated.

When describing the vibration level of a machine, the RMS value should be used if possible. However, some cases require peak-to-peak measurements, for example, when measuring mils of displacement. Other cases require zero-to-peak displacement .

Average = 0.637 x Peak
Average = 0.90 x R M S
Peak to Peak = 2 x Peak
Peak = 1.414 x R M S
Peak = 1.57 x Average
RMS = 0.707 x Peak
R M S = 1.11 x Average

RELATIONSHIP BETWEEN VELOCITY, DISPLACEMENT, AND ACCELERATIO

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Velocity is the measurement of how fast an object is moving from zero-to-peak and is normally measured in tenths of one inch per second (IPS). The effective frequency range of most velocity transducers is from about 10 to 2,000 Hz. Velocity is the most accurate measurement because it is not frequency related. For example, 0.15 IPS is the same at 10 Hz as it is at 2,000 Hz.

Displacement is the measurement of how far an object is moving from peak-to-peak and is normally measured in thousandths of one inch (mils). Displacement is frequency related. Therefore, any measurement of displacement must be at a specified frequency. The effective frequency range of non contacting displacement transducers is from about 0 to 600 Hz. For contacting displacement transducers, the effective frequency range is about 0 to 200 Hz

Acceleration measures the rate of change of velocity from zero-to-peak and is normally measured in units of gravitational force (g’s). This means that high frequencies generate high g levels, and acceleration is frequency related. The effective frequency range for low frequency accelerometers is from about .2 to 500 Hz. The effective range of high frequency accelerometers is from about 5 to 20,000 Hz.

Please note the displacement curve is downward and outward sloping. This indicates low frequencies generate high levels of displacement and high frequencies generate low levels of displacement. Therefore, the displacement transducer most effectivelymeasures lower frequencies.

The frequency response of the velocity transducer is relatively flat from about 10Hz to about 2,000 Hz. It is the most accurate transducer to use in this frequency range.

The acceleration curve is an outward and upward sloping curve, which means that high frequencies generate high levels of acceleration. The accelerometer must be used for frequencies above 2,000 Hz and may not be as effective for frequencies below 100Hz.

The displacement of a body undergoing simple harmonic motion is a sine wave as we have seen. It also turns out (and is easily proved mathematically), that the velocity of the motion is sinusoidal. When the displacement is at a maximum, the velocity will be zero because that is the position at which its direction of motion reverses. When the displacement is zero (the equilibrium point), the velocity will be at a maximum. This means that the phase of velocity waveform will be displaced to the left by 90 degrees compared to the displacement waveform. In other words, the velocity is said to lead the displacement by a 90-degree phase angle.

Remembering that acceleration is the rate of change of velocity, it can be shown that the acceleration waveform of an object undergoing simple harmonic motion is also sinusoidal, and also that when the velocity is at a maximum, the acceleration is zero. In other words, the velocity is not changing at this instant. Then, when the velocity is zero, the acceleration is at a maximum — the velocity is changing the fastest at this instant. The sine curve of acceleration versus time is thus seen to be 90 degrees phase shifted to the left of the velocity curve, and therefore acceleration leads velocity by 90 degrees.

UNITS OF MEASUREMENT
The following symbols are used in the vibration field:
D = Displacement in inches peak-to-peak
V = Velocity in inches per second zero-to-peak
g = Acceleration in g’s zero-to-peak
F = Frequency in Hertz

RELATIONSHIPS
The relationships between these measurements and the conversion from one engineering
unit to another can be accomplished with the use of the following equations:

Example : The vibration level on a variable speed motor operating at 400 RPM is 0.12
IPS (measured with a velocity transducer).
The displacement in peak-to-peak can be calculated by using equation
First, frequency in Hertz must be determined.
F= 400 R.P.M/60 = 6.67 Hz
D= 0.3183(V/F) = 0.3183(0.12/6.67) = 5.7 mils

Example: If the vibration level for the motor in above Exampe is determined with an accelerometer to be 0.013 g’s, then the displacement can be calculated by using equation 6 .
D = 19.57 (g/f*f) = 19.57 (0.13/6.67*6.67 ) = 5.7 mils