Understanding the Role of Linearity in Vibration Analysis

Introductory overview illustrates how awareness of the behavior of linear and nonlinear systems provides fuller understanding of machine health when analyzing vibration data.

The analysis of a vibration spectrum of a machine in the context of linearity and nonlinearity provides an additional basis for understanding why spectra look as they do and how the appearance of a spectrum relates to machine health. Here is an overview of the concept, augmented with straightforward illustrations and examples.

Linear systems
If a linear system is thought of as a black box, it can be said that what comes out of the box is directly proportional to what goes in. This concept is called proportionality. In Fig 1. we can see that the output motion is directly related to the input force. If the input force increases, the resulting motion also increases proportionally (click for Figs. 1-8 ).

Another quality of linear systems is superposition as demonstrated in Fig. 2. Superposition means that if we have two or more input forces, the output motion will be proportional to the sum of the input forces. In other words, nothing new is created. If we add a whole bunch of forces at the input, the output motion will still be directly proportional to the sum of those forces.

Nonlinear systems
Consider a dense metal cube sitting on ice. If you push the cube, it will slide proportionally to how hard you push it. This is a linear response. Now consider that the cube is made out of gelatin. When you give the gelatin a push it may slide a bit, but it also will wiggle and wobble. This is an example of a nonlinear response. The gelatin does not move only in the direction of the push, it also wiggles around in different directions. Therefore, we can say that the output motion is not directly proportional to the input force and therefore the gelatin block is nonlinear (Fig. 3).

Nonlinear systems also do not follow the law of superposition. This means that the output response is not proportional to the sum of the input forces. In a nonlinear system, the inputs combine with each other and produce new things in the output that were not present in the input (Fig. 4).

When one plays a stereo at a relatively low volume, the music comes out clearly. If one raises the volume slightly, the music comes out of the speaker more loudly, but still sounds good. This is a linear response.

We reach a point, however, where if we make the stereo loud enough, the music becomes distorted, and we begin to hear new sounds that were not recorded on the CD. This is a nonlinear response. The key again to understanding when something is nonlinear is that the output contains things that were not present in the input.

Linearity and nonlinearity in vibration
Now that we have described the basic concepts of linearity and nonlinearity, it is time to discuss them in terms of vibration signals. Simple mass-spring systems as shown in figures 5 and 6 will be used for this discussion.

An ideal mass-spring system (Fig. 5) can be described by the equation

F = kX

where F is the input force, k is the spring stiffness, and X is the resulting displacement of the spring. This is a linear system. If we input a sinusoidal force, the resulting displacement is also sinusoidal and proportional to the input.

If the stiffness of the spring changes as it is stretched and compressed (Fig. 6), the system is nonlinear. When we input a sinusoidal force, the resulting displacement is not sinusoidal, and thus this is a nonlinear system in which we get out something that looks different from what we put in.

If we remember the basic rules of vibration and the Fast Fourier Transform, the displacement sine wave in Fig. 5 will produce a single peak in a vibration spectrum. The displacement wave in Fig. 6 will produce a peak in the spectrum with harmonics or multiples. This brings up another important point—the harmonics in this case are the result of nonlinearity.

Machinery vibration
When we look at the vibration spectra for a machine in the context of linear and nonlinear systems, we can make a very general statement that as machines deteriorate and develop faults they become less linear in their responses. We also can say that many machine faults create nonlinearity. Therefore, also in very general terms, we can expect the spectra from a healthy machine to be relatively simple compared with the spectra from a machine with faults. If we consider mechanical looseness as a common machine problem, we can demonstrate this.

When the machine is not experiencing looseness and is in good health, its spectra may look like that in Fig. 7, which shows the shaft rate peak (the big one on the left) and a couple of harmonics of the shaft speed. The same machine with a looseness problem (Fig. 8) might show considerably more shaft rate harmonics at higher amplitudes. This is very similar to the example of the two mass-spring systems in that when the mass-spring system was linear, only one peak was produced in the spectrum, i.e., the output looked like the input. When the mass-spring system was nonlinear, the output waveform was not sinusoidal and therefore produced harmonics in the spectrum.

If we take a step back, we can consider that the mechanical input forces in a simple rotating machine are coming from the rotating shaft. If the shaft is rotating perfectly (i.e., there is no looseness) and the response of the machine structure is perfectly linear, then we would expect to see only a single peak in our spectrum corresponding to the shaft rate. In other words, the output would look like the input. No machines are perfect, however, and shafts do not typically rotate perfectly around their centers; this is why we expect to see some harmonics in machine spectra (Fig. 7). However, as the machine becomes more nonlinear, due to a condition such as looseness, foundation cracks, or broken mounting bolts, more harmonics with higher amplitudes appear (Fig. 8).

Note that if one views a spectrum with a linear amplitude scale, one may not see the harmonic content of the spectrum if the harmonics are much smaller in amplitude than the shaft rate peak. If one views the data using a logarithmic amplitude scale, more harmonic content will be visible on the graph.

Sidebands
Sidebands in a spectrum are another result of nonlinearity. Sidebands are produced by amplitude modulation.

The top waveform in Fig. 9 is an example of a modulated waveform. What we have here is a wave that repeats itself with a frequency X; however, the amplitude of this wave goes up and down at the frequency Y of the wave on the bottom of the diagram. The bottom wave is simply included to demonstrate the frequency at which the amplitude of the top wave goes up and down.

If one wishes to visualize this in mechanical terms, consider a set of gears where one gear is not centered on its shaft. We will say that the noncentered gear has 32 teeth. In one revolution of the noncentered gear we will see 32 tooth impacts. This would relate to frequency X. Since this gear is not centered on its shaft, the amplitude of the tooth impacts will go up and down as the gear moves closer and farther away from the second gear. It will take one revolution of the noncentered gear for the level of the impacts to go from maximum to minimum and back to maximum again. So, the frequency with which the levels of the impacts change (or are modulated) is the rotation rate of the noncentered gear. This would relate to frequency Y in Fig. 9.

The spectrum of these gears (Fig. 10) shows a peak at frequency X with one peak on either side of it Y distance away. Stated another way, we will see a peak at frequency X, another at X+Y, and a third at X-Y. The peaks at X+Y and X-Y are called sidebands.

Why is this system nonlinear? Because X+Y and X-Y are not found anywhere in the input signal but they do appear in the output. The only thing in the input is X or the rate of the teeth impacting. These impacts go up and down in amplitude at a rate Y, but there is certainly no X+Y or X-Y in the input.

The off-centered gear also may cause frequency modulation because the effective radius of the off-center gear changes as it moves closer and farther from the other gear. As the effective radius changes, the rate of tooth contact speeds up and then slows down repetitively. Frequency modulation is similar to amplitude modulation in that it also results in sidebands. In amplitude modulation, the amplitude of the impacts goes up and down in level repeatedly. In frequency modulation, the rate of impacts gets faster and slower repetitively. In this example, both would result in the same pattern in the spectrum.

Nonsynchronous tones
Rolling element bearing wear, gear defects, and motor-bar defects will produce sidebands. Rolling element bearings also will create nonsynchronous tones. These are new peaks that are not exact multiples (harmonics) of the shaft rate.

Figure 11 shows a machine with a serious bearing problem. Compare this with Fig. 7 and note the peaks that are not related to the shaft speed (1x). The two peaks with circles on them are bearing tones and the peaks with the arrows are sidebands. In terms of linear systems, we can say that this spectrum represents a very nonlinear response and suggests the machine has faults (which it does).

To understand why rolling element bearings create nonsynchronous tones and sidebands, consider the case of a horizontal machine with an inner-race bearing fault. As the shaft and inner race spin, a certain number of balls will impact the fault on the inner race and will produce a peak in the spectrum equal to the number of impacts per revolution of the shaft. This peak is called a bearing tone. The number of impacts will almost never be an integral amount. In other words, there will be 3.1 or 4.7 impacts per revolution, but rarely exactly 3 or 5 impacts. Thus, the peaks will not be direct multiples of the shaft rate and are therefore termed nonsynchronous. The higher peak marked with a circle in Fig. 11 is an example of a bearing tone at 3.1x the shaft rate.

Considering this example further, we also can see that the weight of the shaft will cause the impacts against the fault to be greater in amplitude when the fault is below the shaft. As the fault on the inner race rotates to the top of the shaft, the impacts will be smaller because there is less weight (load) on the fault. In one revolution of the shaft the fault will travel around one time—into the load zone, out of the load zone, and back into the load zone. Therefore, the frequency of the change of amplitude in this case is equal to the shaft rate and this also will coincide with the spacing of the sidebands around the bearing tone (the peaks with the arrows in Fig. 11).

A similar phenomenon occurs if there is a fault on a ball or roller. We will see a bearing tone at a frequency equal to the number of impacts the fault on the ball makes with the races in one revolution of the shaft. This peak also will be nonsynchronous and is called a bearing tone. The fault on the ball or roller also travels in and out of the load zone; however, it travels at the cage rate, not the shaft rate. Therefore, the sideband spacing around the bearing tone will be equal to the cage rate, which is usually in the neighborhood of 0.3x the shaft rate.

Vibration trending recommended
The concept of linear and nonlinear behavior gives us another way to think about a vibration spectrum and how its appearance relates to machine faults. Healthy machines should respond more linearly than machines with faults; in other words, as machines develop faults they likely will respond less linearly. As they become less linear we begin to see more and larger harmonics and/or sidebands in the spectra.

Because we may not know all of the details about the design of a machine or how its spectra will appear when it is healthy, it is still best to trend information over time. Look for more and larger harmonics and new peaks that were not there before as an indication that the health of the machine is deteriorating. MT

Alan Friedman has worked in software development, expert system development, data analysis, training, and installation of predictive maintenance programs at DLI Engineering, 253 Winslow Way West, Bainbridge Island, WA 98110; (206) 842-7656 . The author wishes to thank Glenn White who contributed to this article.

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