The Past, Present and Future of Vibration Analysis
This paper was initially presented at a 2003 conference of the Canadian Machinery Vibration Association (CMVA) in Halifax, Nova Scotia, Canada. It has been republished here with permission from the author.
International Measurement Solutions, Baie d’Urfe, Quebec
Vibration analysis of rotating machinery has been around for several decades, and although its basic principles have not necessarily changed, the evolution of vibration measuring instrumentation has had a profound impact on the way it is performed today, as opposed to its early days. Vibration waveforms can generate huge amount of data and current technology is still not at the stage where transducer performance, memory size and processing speed are sufficient to always provide ideal conditions to collect and analyse data. In addition to instrumentation considerations, analysis results are often limited by our understanding of signal processing and/or machine dynamics. The future holds much in store for the evolution of this science, and this paper will attempt to describe, through a short historical account of the progress accomplished so far, the past, present and future of vibration analysis.
ANTIQUITY AND PRE-MODERN ERA
Although the word vibration already existed around sixth century B.C. and was mostly used in the context of music (evidence shows the existence of musical instruments in China over 13,000 years B.C.), one of the earliest known vibration research laboratories is probably the one founded by the Greek mathematician and philosopher Pythagoras of Samos (570-497 B.C.), who was inspired by the sound emitted by hammers in a metal worker’s shop (Figure 1), and unified in a rigorous manner the known music theory of the time to his theory of numbers1. He observed, for instance, that the pitch of the sound emitted by hammers hitting the metal did not depend on the strength used by the workers, but rather on the weight of the hammers themselves, and went on to establish that the hammer weight ratio required to generate a consonance one octave lower was 2 to 1, thus laying out one of the first axioms of vibration theory. Experimenting also with various types of weights, pipes, shells and strings, he also invented the MONOCHORD, the first known scientific instrument specifically designed to provide a standard for vibration measurements (figure 2). Other great thinkers of Ancient Greece such as Daedalus (many attribute to him the invention of the pendulum), Archimedes (famous for his work on static equilibrium), Platon (resonance and sympathetic vibration) and Aristoteles (one of the first treaty on mechanics – Mechanics and the laws of motion – Physics) contributed to the evolution of vibration theory as a formal field of science, but limitations in mathematical and physics theory of the time (the notion of inertia was not well understood then and there was no differential calculus) prevented a thorough understanding of the subject. This did not prevent its use with ancient machinery. The term “balancing”, which was also known at that time, was used in the sense of “tuning” for ensuring that the catapults were stretched to the optimal pre-stress level to achieve maximum strain energy storage. Vibration transducers also existed in other parts of the world. Herodotos describes in one of his short stories the use of a vibration transducer made of a shield coated with a thin sheet of bronze, to detect the digging of underground tunnels during the Persian siege of a North African town situated in today’s Libya. Figure 3 shows the principle of an instrument using falling balls and based on a three-meter-long pendulum invented by Chang Cheng around the second century A.D. to measure the direction and magnitude of earthquakes2. The story goes that while the inventor was trying to convince the authorities of the usefulness of his invention, it recorded an earthquake more than 1000 kilometers away of such small magnitude that nobody had noticed it. When the news came from the distant province, the public became convinced of the “magic” properties of his instrument (the natural frequency of the pendulum was around 0.3 Hz, a frequency too low to be detected by most people).
By the late 19th century, machines started to turn considerably faster, more particularly due to the development of locomotives and steam turbines. The great development of theoretical mechanics in the 18th and 19th centuries gave a very strong base for further development of vibration theory. Dr. G.P. de Laval, a Swedish engineer who invented in 1874 the milk separator, a machine which worked at speeds between 6000 and 10,000 RPM, went on to experiment with much higher speeds. His early units were geared step-up but he soon realized the need for a direct drive and the steam turbine was re-born (Archimedes and DaVinci were also working with steam driven rotating mechanisms). When de Laval introduced his steam turbine at the 1893 Chicago world exposition, it marked the beginning of high speed machinery. His turbines exceeded 40,000 RPM, way above their first critical speed and he had already worked out many practical aspects essential to achieve and maintain those speeds. The 20th century, which could be labeled the era of vibration applications, saw the advent of steam and gas turbines which operated above their 1st critical speed at much higher stresses and loads than previously, and vibration analysis and monitoring would take a new meaning, being essential for the safe operation of these turbomachines. Mechanical vibration became an engineering discipline around the publication in 1910 of W. Hort’s authoritative book titled “Technische Swingungslhere”5, which became a classic text book used by contemporary vibration engineers, followed by other influential written work by Timoshenko6 and den Hartog7. Rankine8 calculated critical speeds and anticipated whirl in 1869, while Föppl9 solved the whirling problem in 1928. Rotor dynamics evolved rapidly with important contributions from engineers such as Newkirk, Kimball, Stodola10 (gyroscopic effects and the influence of fluid bearings), Frahm and Holzer11 (shaft and beam mode frequencies and shapes), van den Dungen, Prohl and Thomson12 (transfer matrix method). Non-linear system theory was discussed by Duffing13 and van der Pol14, and a general theory of system instability was brought forward by Liapounov15 in 1907. Since the 1960s, more has been published on vibration analysis and its applications than from the beginning of humanity up to that period.
MODERN ERA: CONDITION MONITORING AND CONDITION-BASED MAINTENANCE ERA
One of the milestones of Vibration Measurement and Machine Condition Severity Assessment was the publication in 1939 of a paper by T.C. Rathbone titled “Vibration Tolerance” in Power Plant Engineering18. The paper included a severity chart (Figure 4) based on casing displacement measurements from 1 to 120 Hz (60 to 7200 cpm) in order to provide guidelines for judging the condition of a machine based on its vibration level. Rathbone’s empirical values, which he generated for the Fidelity and Casualty company of New York, roughly approximated constant velocity curves around the running speed of his machines and were based mostly on his experience with steam turbine generators, as he was at the time the Chief Engineer of the Turbine and Machinery Division. Severity charts are closely associated with the instrumentation available at the time of publication. Prior to the 1940s, the only practical vibration transducer available was the mechanical displacement transducer (see Figure 6).
After the 1950s, velocity transducers and accelerometers became available, and Rathborne’s chart evolved to constant velocity criteria which led in 1974 to ISO 2372 and 3945, and subsequently to ISO 10816, released in 1996 (Figure 5), and based on RMS velocity levels from 2 to 1000 Hz.
Evolution of Vibration Transducers Up to Now
Prior to the 1940s, only mechanical contact displacement transducers were available to the practioner. The hand vibrograph, which is shown in Figure 6, could not only measure displacement, but also plot the time waveform on pressure sensitive waxed paper, so a skilled vibration analyst could tell if the main vibration component was at running speed, and maybe a little more. In the 1940s, the velocity pick-up, based on a moving magnet inside a coil became available. It offered advantages over the contact displacement transducer in terms of ruggedness, extended dynamic and frequency range, and the fact that its self generating output could be measured electronically. Early piezoelectric accelerometers appeared around the same time17, but were initially used mostly for shock testing in aerospace applications, as they required more expensive electronics and were very sensitive to low frequency thermal transient and base bending. Eventually in the 1970s, better accelerometer designs were achieved (Figure 7), such as models working on the shear principle rather than compression, and with the reducing cost of electronics and the inclusion of built-in electronics, accelerometers became in the 1980s the transducer of choice for most seismic measurement applications, offering much larger dynamic and frequency range, durability and linearity than the velocity transducer.
The early manufacturers of piezoelectric accelerometers were Bruel & Kjaer (Denmark), Columbia Research Laboratories (Woodlyn, PA), Gulton Manufacturing (Metuchen, NJ), Endevco (Pasadena, CA) and Kistler Instruments (Buffalo, NY). Brüel & Kjaer released its model 4303 in 1945, which represented probably the first commercial piezoelectric accelerometer. They also released the first shear accelerometer in 1972 (model 8307), closely followed by the highly acclaimed DeltaShear design patented in 1974. The first company to claim fame for buit-in electronics was Kistler with its Piezotron, patented in 1968. Wilcoxon Research Inc., formed in 1960, is known for its early work on Impedance Head and now concentrates on building rugged industrial accelerometers. Emerging from Kistler in 1967, PCB Piezotronics registered the ICP trademark (Integrated Circuit Technology), their first ICP accelerometer being model 308A04, released in 1973. Accelerometers with built-in electronics are known under many different trade names depending on the manufacturer (e.g. DeltaTron, LIVM), but most are compatible and the generic name used today is IEPE (Integrated Electronics Piezo Electric sensor). Today newer manufacturers have joined the league, including Analog Devices (air bags and structural accelerometers), Dytran (an off-shoot of PCB known for their excellent impact hammers), EG&G IC sensors, Kulite Semiconductor (diffused silicon beam accelerometer), Entran Devices (Kulite spin-off), Vibra-metrics (sensor highway), Metrics PCM/Beta (Houston, TX), Oceana Sensors, Techkor (wireless transducers), and the list continues to expand even today.
Toward the late 1960s, as machines grew larger and faster, it was realized that casing measurements were insufficient when dealing with large high speed machines with stiff casings and light rotors operating on fluid film bearings, such as high-pressure compressors. Shaft riders (Figure 8) were already popular with steam turbine manufacturers, who often supplied them with their machine for continuous monitoring, but a new type of transducer was to emerge, the non-contact displacement probe. Two operating principles (capacitive and Eddy current) were contenders for this type of probe, but after many years of experimentation, the clear winner was the Eddy current probe (Figure 8). Early models were offered by Bently-Nevada and Helm Instruments, followed later by a number of other manufacturers (Indikon, CML, and a few others). This kind of probe gained wide acceptance for on-line protection of turbomachines, perhaps partly because of very good marketing and support from the leading manufacturers. Recently, VibroSystM released a non-contact capacitance probe which almost eliminates the main drawback of Eddy current probes, electrical and mechanical run-out, while surviving adverse conditions in the field. Today, it is generally accepted that both casing and shaft relative displacement measurements are necessary in order to adequately monitor high speed machinery such as gas turbines.
In the late 1980s, laser transducers based on the Doppler effect appeared for non-contact velocity measurements (Brüel & Kjaer, Polytec, Ometron) of translational and angular vibration. Holography and interferometry methods are often used in turbine disc vibration and are found mostly in the labs. For torsional measurements, torsiographs, strain gauges with telemetry and frequency demodulation of toothed wheels are commonly used. All these transducers remain up to now fairly expensive, and, together with some other specialized transducers such as sound and structural intensity probes, have not yet gained wide popularity in the field.
Evolution of Analyzers and Analysis Techniques Up to Now
Early vibration instruments were large, relatively expensive, and therefore were confined mostly to research laboratories (Figure 9). In 1948, Art Crawford designed and built a portable vibration analyzer with phase analysis capability which was very successful and led to the foundation of IRD (International Research and Development Corporation) in 1952, of which he remained the owner until 1958. Up to the mid 1970s, Fourier analysis was performed with either heterodyne filters (fixed linear bandwidth), swept filters (constant percentage bandwidth) or fixed filters (Figure 10). Portable field signature recorders were mostly based on fixed filters16 or swept filter analysis, and were painfully slow (Figure 11). It could take as much as half a day to obtain a few signatures on a piece of equipment. In the mid 1970s, the first real-time Fourier analyzers made their appearance, and were based either on the principle of time-compression or analog fixed filters (octave and fractional octaves). Hardwired FFT analyzers, which were based on an algorithm first published by Cooley & Tukey in 1965, appeared on the market in the early 1980s, and were often used with instrumentation tape recorders (Figure 13). Recursive digital filter analyzers also made it in the market place around the same time, but were mostly used in the field of acoustics.
In the early 1980s, several companies started to implement condition monitoring programs and sales of vibration instrumentation experienced new heights. The emergence of the PC (Personal Computer) as a serious tool in industry in the late 1980s revolutionized the way data was collected in the field, with the wide dissemination of the vibration data collector. The first models with overall and octave band levels were available in the early 80s (Figure 14) and by the late 80s, most models incorporated 400 lines FFT capability. Although they were relatively slow and the lack of resolution was really a limitation, they made the process of collecting vibration data so much more easier than manual methods (Figure 12) that, together with the report generating capability of the PC, they became extremely popular in condition-based maintenance programs. This popularity brought commercial opportunities which were responsible for the creation and expansion of new companies such as Beta Monitors & Controls, Palomar, Entek, CSI, SKF Condition Monitoring division, Vitec, DLI, Diagnostic Instruments, Stell Diagnostics, and a few others. Among some of the techniques that became
the backbone of fault detection for condition monitoring were CPB spectrum comparison (Brüel & Kjaer) and Spectral Bands Alarming (Entek and Technical Associates of Charlotte). On the other side ot the arena, Bently-Nevada was promoting the use orbits, Bode, Polar and waterfall plots for run-up and coast-down of turbomachinery. Another analysis method that became very popular was the identification of rolling-element bearing defect frequencies using narrowband analysis, following a paper by Taylor19. Later, enveloping techniques, which were marketed under various names (Brüel & Kjaer’s envelope analysis, Spike-Energy from IRD, Peak-View from CSI, ESP from Diagnostic Instruments, SEE from SKF, DREAM from Barkov’s Russian team) also created a small comotion in rolling-element bearing fault detection. Eventually, it was realized that more lines of resolutions were necessary to diagnose components such as gears or to separate vibration from magnetic origin on electric motors, so from the 1990s, speed and memory of these instruments were improved to reach 20kHz real-time speed and 6400 lines spectrum (Figure 15) at the onset of the 21st century.
Many other important breakthroughs also occurred in the 80s and 90s as a spin off of the digital era, namely the release of two channel FFT’s (introducing the concepts of correlation, coherence and frequency response functions), modal analysis techniques including ODS (Operational Deflection Shapes – Figure 16), cepstrum analysis (Brüel & Kjaer), digital tracking for run-up and coast-down analysis (Figure 17), phase demodulation techniques using the Hilbert transform and intensity analyzers (for mechanical power flow measurements). All these powerful tools were initially aimed at the specialist but eventually, with progress in computers and instrumentation, appealed to a wider crowd of vibration analysts.
Automated Diagnosis and Expert Systems
The next logical step after being able to acquire and process large amount of vibration data was to try to speed up the detection and diagnosis process by applying computerized diagnostic methods. One of the first machinery expert system using vibration analysis was Amethyst, introduced by IRD in the late 1980’s. Around the same time, Predict DLI Seattle also developed an expert system for use by the US Navy in their aircraft carrier condition monitoring program. In the early 90s, the Canadian Government awarded a contract to DMSI in Vancouver BC to develop an expert system for use by the Canadian coastguard (the system was eventually licensed through SKF CM). CSI also introduced their own expert system. All these systems were rule-based. Burrows Electronics was also awarded a contract from Iron Ore Canada Inc. in the 90s to develop, build and install an expert system on electric locomotives based on artificial neural network technology combined with Octave Band Velocity Profiles20 (Figure 18). Although some level of success was reported with all of these systems, more research is still going on today, especially in the area of neural networks, in order to improve their reliability. One of the advantages of artificial neural networks over rule-based systems is the ability to learn from previous data, and therefore adapt to new situations unforeseen by the designer.
Evolution of Vibration Standards Up to Now
As mentioned previously, Rathbone published the first severity chart in 1939. This chart was modified and extended by IRD in 1964, resulting in constant velocity criteria curve covering a frequency range from 100 RPM to 100,000 RPM. While the chart worked well with most equipment, it was very misleading for high-speed turbomachines with very stiff casings and shaft displacement levels were often used instead, measured first with forked shaft sticks (also known as shaft riders) and then later with non-contact displacement probes. Note that shaft riders measure the absolute movement of the shaft while non contact displacement probes only measure relative displacement of the journal relative to the bearing housing, and therefore do not give exactly the same results. There were problems associated with both types of transducers, but
in the end, Eddy current probes were deemed to be more reliable. By combining Eddy current probe data with accelerometer data, absolute shaft displacement data can be produced. In the 1960s, Blake, who worked at Monsanto Chemical Company and formed the Vibration Institute in 1972, revisited the Rathbone and IRD charts to include constant velocity curves between 20 and 1000 Hz, while reducing the allowable velocity levels for a given criteria both at the low and high frequency ends, in order to account for his intuition that force depends increasingly on displacement as frequency is reduced and on acceleration as frequency is increased. In addition to that, he incorporated the notion of service factors into his new chart, which was very well received by the community. In the 1970s, Dresser Clark introduced a chart for shaft displacement which also became widely used. At about the same time, the API (American Petroleum Institute) was working on a series of guidelines for turbomachine design, vibration acceptance and sensitivity to rotor unbalance which led to the release of API 670, 617 and 618, which are well recognized standards throughout industry. The first ISO (International Standard Organisation) standards on machine vibration, 2372 “Mechanical vibration – Measurement on Non-Rotating Parts and Evaluation” and 3945 published in 1974 under the auspices of TC 108 (Technical Committee 108 – Shock and Vibration) SC2 (Subcommittee 2), were based on RMS velocity levels measured on non rotating parts, and divided machines in four categories, from small to very large, identifying four severity zones, from A (less severe) to D (unacceptable). Many improvements were made over the years to these standards, resulting in ISO 10816 “Mechanical vibration – Evaluation of machine vibration by measurements on non-rotating parts” (1996). ISO 10816 currently includes 6 parts devoted to different kinds of machines (general guidelines, large turbo-generator sets exceeding 50 MW, industrial machines >15 kW turning between 120 and 15000 RPM, gas turbines except the ones from aeroderivative type, generators and hydraulic pumps, reciprocating machines > 100 kW.), and a seventh part which is currently at the draft stage, to address vibration of rotodynamic pumps. There is also an ISO standard based on shaft displacement measurements, ISO 7919 “Mechanical Vibration of Non-Reciprocating Machines – Measurement on Rotating Shafts and Evaluation”, that was instigated by Canada and the United States, and where both absolute and relative shaft displacement are considered for large steam turbine generator sets, gas turbines and hydraulic machines. Many other vibration standards exist within ISO, including ISO 2041 Use of Vocabulary for Vibration and Shock, ISO 1940 Balancing Quality of Rigid Rotors, and ISO 2954, 5348 and 5347 which relate to vibration instrumentation. In 1992, SC5 was formed in order to provide further standardization in condition monitoring of rotating machines, and many new standards have already started to emerge from that group, including the new ISO 18436 (part 1 and 2) “Training and Accreditation in the Field of Condition Monitoring and Diagnostics of Machines” which was a Canadian initiative largely influenced by the CMVA and the Vibration Institute. For more information on ISO standards related to vibration of machines, refer to their internet site, which keeps a very up-to-date list of current and upcoming standards as well as their current status. In 1994, John S. Mitchell spearheaded the development of MIMOSA (Machinery Information Management Open Systems Alliance), a group aimed at developing a means for the data exchange of vibration information between various interveners in machinery management and process control33.
WHAT ABOUT THE FUTURE?
Although we have witnessed unprecedented developments in vibration analysis technology during the last 20 years (Figure 19), everything seems to indicate that the next two decades will be even more exciting in terms of technological achievements. Memory size and prices as well as processing speeds are always improving, which is really a necessity in order to accommodate the more complex signal processing algorithms required to perform better fault detection and diagnosis. Many machinery vibration problems still remain unsolved to this day, and research continues in order to find better methods for solving these problems, given the very complex nature of vibration signals. Here are some of the developments the author foresees in the next few years.
Connectors and cabling systems are a major concern when installing permanent monitoring equipment. The wireless accelerometer promises to solve many problems in this area. It would also simplify measurements on moving parts, generating a whole new range of applications (rotating parts, linear motion moving parts such as crossheads of reciprocating machines). Wireless accelerometer systems are now available for measurements up to 10 kHz. They are however still expensive and industry is still awaiting a cost competitive system, which should not be too far down the line. One of the advantages of the accelerometer is that it is in general easy to electronically integrate the signal into velocity and displacement. Electronic differentiation is generally more difficult and not a viable alternative. A parameter such as jerk, which is the derivative of acceleration, has proven useful when analyzing small shocks from rolling-element bearings. To the author’s knowledge, there is currently no piezoelectric jerk transducer available on the market, but this could be remedied soon. Such a transducer, together with digital integration, should provide more sensitivity to high frequency phenomena such as friction, cavitation and micro-shocks. Other improvements for accelerometers should include better phase matching for tri-axial accelerometers, in order to provide more accurate measurements of motion in space, as well as higher temperature and self-generating power accelerometers with built-in electronics. We should also witness new transducers for measurements in the angular domain, cost reduction and performance improvements in laser based transducers, the emergence of new transducers for mechanical power flow measurements and improvements in non-contact displacement probe design to reduce problems associated with electrical or mechanical run-out, as well as high temperature limitations.
Vibration Analyzers and Analysis Techniques
Analog to digital conversion will eventually migrate to 24 bit technology in order to provide larger dynamic range, a feature quite desirable when integrating accelerometer signals. Other desirable advancements will be higher speed for acquiring multi-channel data in order to cover a wider frequency range in multi-channel systems, more FFT line for tackling signals with low frequency modulation of a high frequency carrier, rolls with nearly identical speed, and gearboxes with large number of teeth, and improved methods using frequency and amplitude demodulation techniques. The long awaited small battery powered instrument for collecting a large number of long time waveforms for post-processing in the PC will certainly completely change the way data collection is performed today. It will speed up data acquisition by a factor of at least 4 and post-processing will allow much more computations to take place in the PC, including the playback of signals, and the possibility to use new analysis methods, whenever they become available (Figure 20).
Accelerometer signals can cover a range over 5 decades in frequency and 7 decades in amplitude. This leads to a vast number of possibilities when it comes to signal processing, as shown in Figure 21, which illustrates only some of the most common signal processing techniques in use today21.
Although this already constitutes a vast array of signal processing techniques, many others, which are still mostly at the research or exploratory stage at present, should gain popularity in the future. A few of the most promising ones are briefly summarized below.
Since the early 90s, many new time-frequency analysis techniques have been proposed for vibration analysis applications, in particular the wavelet transform, the Wigner-Ville distribution and the time-frequency worm. These three techniques hold many promises to extract useful information from the signal, by providing simultaneously time and frequency information. Cyclic machines generate many events which are more readily identified by looking at the time domain. The following example, taken from a 6 cylinder in-line cylinder marine diesel engine22, illustrates the usefulness of wavelet time-frequency maps. Figure 22 shows an average of 10 cycles in the time-frequency domain using 1/3rd octave wavelets displayed over one cycle of the engine. In the top trace, the engine was running under normal condition and on the bottom trace, misfiring on cylinder #1 was created by blocking the fuel line. The fault can be clearly seen on Figure 23, which was derived by subtracting the two traces of Figure 22. Difference in various events can be more readily identified when both time and frequency information is present.
Figure 24 shows a wavelet analysis, as well as the time waveform and resulting frequency spectrum on a large gear mounted on the drum of a hoist lifting a bucket on a large mining shovel. Because the gear wheel does not even make a full turn during the lift of the bucket, there is not enough time signal to calculate a frequency spectrum where the gear faults will show. The wavelet analysis shows both domains, identifying clearly the faults. The wavelet transform was developed in the 1980s by the French geophysicist J.P. Morlet for analyzing reflections in seismic signals. More recently, another type of time-frequency analysis, the time-frequency worm23, was proposed by F. Léonard, from IREQ (Institut de Recherche de l’Hydro-Québec). This type of transform, although computationally much more intensive than the previous ones, is more suited to observe fast changes in resonance frequencies or pure frequencies than the Fourier or wavelet transform and should find many applications in vibration analysis of transient events. The dichotomy of the time-frequency structure for the three transforms applied to a varying frequency sinusoid is illustrated on Figure 25. The example on Figure 26 shows a comparison between the results obtained from a worm transform and the spectrogram, where the worm transform clearly outshines the spectrogram.
Spectral Correlation, Bispectrum and Trispectrum (Cyclo-Stationarity and Non-Linearity)
There is often a correspondence between many of the frequencies generated by non-linear phenomena inside a machine (this is certainly the case with bearing and gearing systems), although these frequencies might not be harmonically related. Spectral correlation, according to the definition given below,
is the frequency equivalent of the two-dimensional correlation function and allows to study the interaction between these frequencies. Many cyclic events such as gearmeshing or shocks in rolling-element bearings can be assessed with this technique (Figure 27) as each type of event can lead to different patterns in the SC diagram. For instance, Figures 28 and 29 show the difference that can occur between bearing and gear faults. The horizontal smear in the frequency axis for the bearing signal is due to the fact that bearing shocks are only quasi-periodic with a random component as opposed to the gearmeshing process, which is considered purely deterministic. This promising technique, which is still at the research stage, is currently under evaluation by the U.S. Navy for its helicopter gearbox condition monitoring program.
Spectral correlation methods are often compared to higher order spectral analysis such as the bispectrum24. The bispectrum (Figure 30) can be viewed as a decomposition of the third moment of a stationary random variable and can detect non symmetric non linearities while the trispectrum is a decomposition of the fourth order moment (kurtosis), and can be used to analyze symmetric non linearities. Since the trispectrum is a function of three frequency variables, it requires four dimensions to display it in its entirety, leading to more exotic graphs such as Figure 31 taken from reference 32. Both functions are part of the generalized concept of polyspectra25. Since in practical engineering situations, many non-linearities can be approximated by only considering the quadratic and cubic terms, both bispectrum and trispectrum are necessary in order to study symmetric and non symmetric non linearities.
State Space Methods
These methods use state variables and geometry as a mean to simplify complicated motions by extracting some of their dimensional characteristics. The Poincaré map for instance, tends to get fuzzy in the presence of chaotic motion, and has a fractal structure which depends on the amount of chaos in the system. The model shown in Figure 32 illustrates its use26.
Blind Source Separation
Usually, many sources contribute to a signal measured on a particular location on a machine. The blind source separation method, which is analogous to SDV (single value decomposition) methods, enables one to separate the individual contribution of each source in the overall signal.
Deconvolution, De-Reverberation and Noise Removal
A signal measured away from the source can be quite different from the original signal, because of modifications due to transmission path effects or reflections in the surrounding structure. The degradation of the signal due to the transmission path can be removed by deconvolution methods, such as short-pass liftering of the complex cepstrum followed by inverse Fourier transforms. Similarly, reflections can be removed using de-reverberation techniques. Lyon28 has published a paper describing his deconvolution and de reverberation method to recover the pressure inside a cylinder using accelerometer measurements performed on the outside of the engine by the use of cepstral windowing. Noise removal (removal of random or deterministic components in a signal) can be achieved by techniques such as SANC (self adaptive noise cancellation) described in a paper given by R.B. Randall29 at the 2001 CMVA annual seminar in Edmonton. This area of signal processing is still fairly new and more techniques are expected to be available in the future.
Amplitude and Phase Demodulation Techniques for Angular Domain and Gearbox
Phase demodulation by the use of the Hilbert transform provides easy access to the angular domain30. The instantaneous speed is a very powerful tool to investigate faults in drive mechanisms such as electric motors and reciprocating engines (Figure 35). Other applications include drive synchronization in multi-drive systems, torsional vibration analysis, and gearbox vibration evaluation (Figure 34).
Profiling is a visualization technique aimed at simplifying the understanding of a signal generated by a rotating component (Figure 36). Generally, the time-averaged signal is plotted using polar coordinates over one revolution of the component. The technique, which is currently used to display roll geometry in press monitoring of paper machines, can also be applied to the detection of faults in magnetic poles pattern, gear wheels, and rotor bars.
Mechanical Power Flow
Mechanical power flows in structures such as beams and piping very much like water in waterways. By using a transducer such as the one shown in figure 37, power flow can be estimated using the relation P = F.V, where F is the force and V the velocity, both vector quantities. The technique is still at the exploratory stage, but once refined, could provide a very powerful tool to locate vibration sources. Sound intensity probes can be substituted for non-contact measurements.
The purpose of standardization in vibration analysis and machine condition monitoring is to address several issues such as:
a) providing guidelines and description of methodologies on how to obtain values and descriptors which are relevant or work best for the intended purpose
b) suggesting measurement techniques to ensure consistency of the data so that data obtained from one source can be easily compared to the one from another source
c) providing a consensus on what constitutes an acceptable or unacceptable vibration values for a given application or condition assessment
d) warning of potential pitfalls or shortcomings of specific measurement techniques
e) providing general guidelines on what type of vibrations could be harmful to a specific machine
f) describing protocols or recommended practices in various activities related to VA
Under ISO, there is currently no severity chart for rolling-element bearing vibration, or vibration specifically related to gearing or coupling systems. This gap will certainly be corrected in the future, once the international community agrees on the proper measurement parameter(s) to use for these components. There is also an urgent need to provide guidelines on how to specify frequency bands for alarming purposes. Work is currently underway in order to address the specific requirements of many types of machines. It is probably too early to address topics such as mobility, torsional measurements, time-frequency and demodulation methods by an international standard, but they will certainly be brought back into the scene at some time in the future. With the proliferation of measurement data and techniques, MIMOSA, upcoming ISO standards on data formats and presentation, and the vast possibilities of the ever expanding internet should provide a solid basis for the exchange of information between field personnel, vibration specialists, and managers of machinery.
The author would like to thank Robert Randall, John Burrows, Art Crawford, Ron Eshleman, François Léonard, and John S. Mitchell for their precious assistance and discussions during the preparation of this paper.
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