Sinusoidal and Random Vibration Testing Primer
Posted on April 9, 2013 by DES
The most common types of vibration testing services conducted by vibration test labs are Sinusoidal and
Random. This primer is an explanation of the typical requirements found in vibration test specifications and
the parameters used to control the vibration tests. Both types of vibration tests are used to evaluate products
for ruggedness, durability and to expose vibration defects.
Sinusoidal or Sine Vibration Testing
Sinusoidal or Sine Vibration has the shape of a sine wave as seen in Figure 1. The parameters used to define
sinusoidal vibration testing are amplitude (usually acceleration or displacement), frequency, sweep rate and
number of sweeps.
Figure 1. Sinusoidal Vibration Wave Form
A typical sinusoidal vibration test profile is shown in Figure 2. The amplitude is defined over a range of
frequencies. The amplitude can be constant or variable. During a sine vibration test, the vibration wave
forms are swept through a range of frequencies, however they are of discrete amplitude, frequency and
phase at any instant in time. An important note is that the displacement increases as the frequency decreases
for a given acceleration. At low frequencies, the displacement could exceed the limits of the test
equipment. That is why some specifications use displacement for amplitude in the low frequency range.
Figure 2. Typical Sinusoidal Vibration Test Profile from MIL-STD-810G
Sine vibration is not usually found in the real world unless your product is attached to equipment such as a
motor or reciprocating compressor running at a fixed frequency. Why is it done? It is good to find
resonances (amplitude magnification in the device under test), it is a simple motion and it also produces a
constant acceleration vs. frequency. Also, it is probably carried over from old test methods prior to digital
computer controllers. However sine vibration does not correlate to a field life unless the product is exposed
only to fixed frequencies over its life.
Some parameters and definitions of a sine vibration test are:
Amplitude: The amplitude for a sine vibration test is usually specified as displacement or acceleration. In
DES’s experience, velocity is rarely used in a specification. As seen in Figure 1, amplitude can be
expressed as peak or peak-to-peak. When displacement is used to define amplitude, it is defined in either
peak units of inSA (mmSA) or peak-to-peak units of inDA (mmDA). SA stands for Single Amplitude
(peak) and DA stands for Double Amplitude (peak-to-peak). When acceleration is used to define amplitude,
its units are usually G’s or millimeter per second squared (mm/sec^2) or meter per second squared
Frequency: Frequency is defined as cycles per second. Its’ units are Hertz (Hz). Frequency is equal to the
reciprocal of the period.
G: One G is equal to the acceleration produced by earth’s gravity and is equal to 386.1 inches/sec^2 or 9.8
Octave: The interval between one frequency and another differing by 2:1.
Period: The time it takes to complete 1 cycle. Its’ units are seconds. Period is not typically used in the
definition of a sine test. It is listed here because of its relation to frequency. Period is the reciprocal of the
Resonance: A frequency at which an amplitude magnification occurs in the device under test when
compared to the vibration table amplitude. Usually a resonance is defined as a 2:1 or greater magnification.
Sweep and Sweep Cycles: A sweep is defined as a traverse from one frequency to another. A sweep cycle
varies from one frequency to another and then back to the staring frequency. For instance in Figure 2, a
sweep could be a traverse from either 5 to 500 Hz or from 500 to 5 Hz. A sweep cycle would traverse from
5 Hz to 500 Hz, then traverse back to 5 Hz. Some specifications require sweeps while others require sweep
cycles which causes confusion.
Sweep Rate: The rate at which the frequency range is traversed. The units for sweep rate are usually
Octave/minute or Hz/minute. Octave per minute is a logarithmic sweep rate while Hz/minute is a linear
Random Vibration Testing
Random Vibration is a varying waveform. It’s intensity is defined using a Power Spectral Density (PSD)
spectrum. Whereas sinusoidal vibration occurs at distinct frequencies, random vibration contains all
frequencies simultaneously. Also phase changes occur over time with random vibration. Sine and random
vibration testing cannot be equated.
Real world vibrations are usually of the random type. Vibrations from automobiles, aircraft, rockets are all
random. A random vibration test can be correlated to a service life if the field vibrations are known. Since
random vibration contains all frequencies simultaneously, all product resonances will be excited
simultaneously which could be worse than exciting them individually as in sine testing.
A typical random vibration test PSD is shown in Figure 3. The PSD is defined over a range of frequencies.
The square root of the area under the PSD curves yields the Grms. Specifying Grms only is not sufficient
because a wide variety of spectra can result in the same Grms.
Figure 3. Typical Random Vibration Profile from MIL-STD-810G
Some parameters and definitions of a random vibration test are:
Averages: Since random vibration constantly changes over time (that is why it is called random), the
controller takes samples or snap shots of the vibration data over time. Successive samples are averaged.
The averaging occurs in each band of resolution.
Average Weighting Factor: An exponential weighting factor that defines how fast the controller reacts to
changes. The controller reacts faster for small weighting factors vs. slower for larger weighting factors.
Statistical Degrees Of Freedom (SDOF): The number of independent values (measurements) used to
obtain a PSD estimate at a particular frequency. A higher SDOF means that more measurements are taken.
For a measurement channel, SDOF = 2*K,
For control channels, SDOF = 2*K*(2*N – 1) * n.
K = Number of averages per control loop
N = Averaging weighting factor
n = Number of control channels
Grms: Grms is used to define the overall energy or acceleration level of random vibration. Grms (root-
mean-square) is calculated by taking the square root of the area under the PSD curve.
Kurtosis: Fourth moment of the Probability Density Function (PDF). It measures the high G content of the
signal. The kurtosis of a Gaussian PDF is 3.
Number of Lines: The frequency range of the test divided by the band resolution equals the Number of
Lines. The vibration controller or spectrum analyzer will perform its calculations for each narrow band.
Open Loop/Closed Loop: Closed loop means the controller will continuously adjust the drive signal to
account for changes in the response of the device under test that are fed back from the control
accelerometers. Open loop means the drive signal will be fixed or the controller will stop adjusting the drive
signal regardless of changes in the response of the device under test.
PDF: A statistical Probability Density Function. A histogram showing the probability of occurrence and
the distribution of data.
Power Spectral Density (PSD) or Acceleration Spectral Density (ASD): Defines the intensity of the
random vibration signal vs. frequency. Its units are usually G^2/Hz or (m/s^2)^2/Hz.
Sigma (σ) & Sigma Clipping: Sigma is the standard deviation of a statistical PDF. A Gaussian PDF
distribution is assumed for random vibration which takes the shape of a bell shaped curve. Since the
amplitude or intensity of the random vibration will change over time, the time spent at different amplitude
excursions is measured using a PDF. Figure 4 shows a Gaussian PDF. The vertical axis would be 1/G, the
horizontal axis would be sigma and µ is the mean which is equal to zero for a shaker control. For a
Gaussian distribution, 68.2% of the peak G excursions occur between ± 1 sigma, 95.4% between ± 2 sigma,
99.7% between ± 3 sigma.
Figure 4. Gaussian PDF
Vibration controllers allow you to clip peak amplitude excursions using Sigma Clipping. Many
specifications allow the clipping to be set at ± 3 sigma.
Frequency, G, octave, period and resonance are the same as defined under sinusoidal testing.