Condition-Based Diagnostic Approach for Predicting the Maintenance Requirements of Machinery

doi:10.4236/eng.2009.13021

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Engineering, 2009, 1, 177-187

doi:10.4236/eng.2009.13021 Published Online November 2009 (http://www.scirp.org/journal/eng).

Condition-Based Diagnostic Approach for Predicting the

Maintenance Requirements of Machinery

C. I. UGECHI1, E. A. OGBONNAYA2, M. T. LILLY1, S. O. T. OGAJI3*, S. D. PROBERT3

1Department of Mechanical Engineering, Rivers State University of Science and Technology,

Port Harcourt, Nigeria

2Department of Marine Engineering, Rivers State University of Science and Technology,

Port Harcourt, Nigeria

3School of Engineering, Cranfield University, Bedfordshire, U K

E-mail: [email protected]

Received January 10, 2009; revised February 21, 2009; accepted February 23, 2009

Abstract

Wise maintenance-procedures are essential for achieving high industrial productivities and low energy ex-

penditure. A major part of the energy used in any production process is expended during the maintenance of

the employed equipment. To ensure plant reliability and equipment availability, a condition-based mainte-

nance policy has been developed in this investigation. In particular, this project explored the use of vibration

parameters in the diagnosis of equipment failure. A computer-based diagnostic tool employing an artificial

neural-network (ANN) was developed to analyse the ensuing machinery faults, their causes and conse-

quences. For various categories of this type of machinery, a vibration-severity chart (ISO 12372 / BS 4675:

1971) appropriately colour coded according to defined mechanical faults, was used in training of the ANN.

The model was validated using data obtained from a centrifugal pump on full load and fed into the program

written in Visual Basic. The results revealed that, for centrifugal pumps within 15 to 300kw power range,

vibration-velocity amplitude of between 0.9 and 2.7mm/s was within acceptable limits. When the values rose

to between 2.8 and 7.0mm/s, closer monitoring and improved understanding of the equipment condition was

needed. The evolved diagnostic and prognostic model is applicable for other rotary equipment that is used

within the same power limits.

Keywords: Condition Based, Diagnostic Model, Predictive Maintenance, Machinery,

Centrifugal Pumps

1. The Challenge

Maintenance, although requiring the expenditure of sig-

nificant amounts of energy, is usually required in order

to keep (or restore) facilities at an acceptable operational

standard [1]. For most plants, maintenance practice is

predominantly based on routine-scheduled prevention as

well as previously unanticipated reactions to overcome

faults. Predictive maintenance (PdM) procedures, such as

that devised in this project, are evolving and results in

less wasted effort. According to Ogbonnaya [2], Con-

treras et al. [3] and Salva et al. [4] condition monitoring

(CM) an aspect of PdM is defined as the use of appropri-

ate technologies to determine the operational state of the

considered machinery. For instance, it may involve vi-

bration measurements, infrared thermography, and/or oil

analyses etc.

For decades, conventional wisdom suggested that the

best way to optimise the performance of physical assets

was to overhaul or replace them at fixed interval (PM).

This was based on the premise that there is a direct rela-

tionship between the amount of time (or number of cy-

cles) equipment spends in service and the likelihood that

it will fail. Moubray [5], stated that this relationship be-

tween running time (age) and failure is true for some

failure modes, but that it is no longer very productive as

equipment are now much more complex than it was even

fifteen years ago. He pointed out that fixed interval

overhaul ignores the fact that overhauls are extraordinar-

ily invasive undertakings that massively upset stable

C. I. UGECHI ET AL.

178

systems. As such, they are likely to induce infant mortal-

ity, and so cause the very failure, which they seek to

prevent.

This has led to startling changes in the patterns of

equipment failure. Unless there is a dominant age-related

failure mode, fixed interval overhauls or replacements do

little or nothing to improve the reliability of rotary

equipment [5]. There is no gain in overhauling a machine

that has nothing wrong with it [6]. Moubray [5] con-

cluded that “in the absence of any evidence to the con-

trary, it is more realistic to develop maintenance strate-

gies which will assume that equipment failure can occur

at any time and not at fixed amount of time in service”.

2. Maintenance Management

Direct on-line real-time continual monitoring and analy-

sis of machinery behavior is the most reliable way to

achieve a high productivity [3]. If an abnormal situation

can be detected early, when defects are minor and have

not affected machine output, with the cause of the fault

diagnosed while the machine is still running, then the

downtime for associated repairs can be reduced and other

attendant advantages achieved.

Figure 1 shows the various maintenance methods/

techniques/strategies. Reactive maintenance is usually

only implemented following an unforeseen event leading

to a partial or total failure of the system. Preventive

maintenance (PM) is initiated according to a predeterm-

ined time-schedule in order to try to avoid the occurrence

of failure. Predictive maintenance (PdM) is laun- ched as

a result of behaviour of the equipment/ machinery before

total failure, whereas proactive maintenance may require

redesigning and/or modification of the adopted mainte-

nance-procedure where necessary.

Each of these techniques has merits and frailties, but

PdM is the most advantageous [7]; it combines the advan-

tages of preventive and proactive strategies. Its basic

Figure 1. Maintenance procedures.

concept is shown in Figure 2. Predictive maintenance is

summarized as involving actions taken to improve one or

more of the following machinery characteristics: avail-

ability, reliability, maintainability, safety, efficiency etc

as well as reduce energy waste and environmental pollu-

tion [4]. As a result, the implementation of PdM usually

enables one to have sufficient lead-time to plan, schedule

and make necessary repairs before the equipment would

otherwise fail. So major breakdowns and costly down-

time can then be avoided.

2.1. Condition Monitoring

This has long been practiced by maintenance personnel

who relied on their innate senses of hearing, touch and

sight, but the judgment and conclusions were often not

reliable. All physical structures and machinery, that are

associated with rotating components, give rise to vibra-

tion. The vibrations so generated by machinery have be-

come a well-utilized parameter for assessment in CM. It

is one of the most versatile techniques, which is capable

of detecting about 70% of common mechanical faults

associated with rotating machinery [6].

Machinery vibrations are complex, but can be meas-

ured, processed and their interpretation simplified in or-

der to facilitate the implementation of recommended

action [8]. According to Okah-Avae [9], rotating ma-

chinery produce vibration patterns, which repeat peri-

odically and so have been found to be amenable to

analysis.

Figure 2. Basic behaviour of a failing system (machinery) [5].

C. I. UGECHI ET AL. 179

2.2. Vibration Monitoring and Analysis

Even though the wise maintenance of industrial equip-

ment may require the monitoring of additional parame-

ters, such as temperature, pressure, flow, voltage, electric

current, horsepower, torque, etc, vibration data usually

contain more information about a machine’s health and

operating characteristics than any other parameter – see

Table 1. This informed the choice of vibration monitor-

ing and analysis over other condition monitoring tech-

niques in this research.

Measurements of vibration parameters are important

in many industrial applications. The parameters desired

may be displacement, velocity, or acceleration; in time or

frequency domain. These quantities are useful in pre-

dicting the fatigue failure of a particular component of

machine and play important role in analysis, which are

used to reduce equipment vibration [8]. According to

Ralph [10], when measurement of both amplitude and

frequency are available, diagnostic methods can be used

to determine the magnitude of a problem and its probable

cause.

Vibration severity is a function of displacement and

frequency of rotation of the component. Measurements

of vibration-velocity take into account both displacement

and frequency: “vibration-velocity amplitude is a direct

measure of vibration severity” [11]. Vibration-velocity

gives an indication of vibration severity over a wide

range of frequencies and hence is extensively applied in

condition monitoring [9].

Each mechanical defect generates vibration in its own

unique way [11]. This makes it possible to identify a

mechanical problem by measuring and noting its vibra-

tion signature. When vibration measurements and analy-

sis are performed systematically and intelligently, they

ill not only allow determination of machine health but w

also permit the prediction of the mechanical fault and

when such condition most likely will have reached un-

acceptable levels [12].

Vibrations occurring in the 600 to 60,000 cpm fre-

quency range are generally described and measured by

their vibration-velocity amplitudes [11]. In practice, the

following relationships apply:

Displacement of vibrating component

(x) = a / (2πƒ)2 (1)

Velocity of vibrating component

(v) = a /2πƒ (2)

Acceleration of vibrating component

(a) = 2πƒv (3)

3. Research Methodology

The identification of incipient faults in a machine, in

order to diagnose an impending problem and locate the

fault while the machine is still running, through an inter-

pretation of its unique vibration characteristic (i.e. sig-

nature) is the main aim of PdM [13]. A good vibration

survey program sets different limits for different ma-

chines, as well as different limits for different regions of

the frequency domain spectra for the same machine.

The delineation of severity limits for good and bad

bearing conditions are best determined by “comparison”

or “trending” methods [11]. In establishing a program for

checking the spike energy conditions of rolling element

bearings; a “comparison method is used. The spike en-

ergy levels of similar machines are measured and any

level which significantly departs from the average are

singled out for further analysis of potential bearing prob-

lems. This method has led to the establishment of criteria

levels which distinguished good and bad bearings.

Table 1. Parameters indicating the occurrence of faulty conditions in a rotating machine.

PARAMETER MEASURED

DETECTED CONDITION

TEMPERTURE

OF MACHINE

PRESSURE

PROCESS

FLUID

FLOW

FLUID

OIL ANALY-

SIS

SPIKE EN-

ERGY OF

BEARING

VIBRATION

MACHINE

OUT- OF - BALANCE X

MISALIGNMENT X X

BENT SHAFT X X

BALL-BEARING DAMAGE X X X X

JOURNAL-BEARING DAMAGE X X X X X

GEAR DAMAGE X X

MECHANICAL LOOSENESS X

MECHANICAL RUBBING X X

NOISE X

CRACKING X

C. I. UGECHI ET AL.

180

Various ranges of vibration velocity amplitude and

spike energy were represented with colour codes for

corresponding level of vibration severity: green for

good/normal condition, blue for acceptable condition,

yellow for fair condition/ improvement required, and red

for unacceptable condition. The use of a real-time recur-

rent simulation was therefore adopted in this investiga-

tion in order to develop an artificial neural-network

(ANN) for the analysis of the vibration data [4].

3.1. Artificial Neural-Networks (Anns)

Ogbonnaya [8] showed that ANN is a promising tool to

articulate and analyze the numerous data associated with

catastrophic failures in rotating machinery. According to

Agbese and Mohammed [14], since ANN, a branch of

Artificial Intelligence (AI), are modelled after the bio-

logical neurons of the human brain, they hold consider-

able promise as building blocks in actualising the ulti-

mate aim of AI systems. Out of the various architecture

with which ANN is conveyed; the back propagation al-

gorithm has proved most promising and accurate for

analyzing machine vibration data [8]. Also of important

is training the neuron of the network on the basis of pat-

tern recognition; especially when there are large amount

of data to handle.

Simulated neural networks are software models de-

signed through suitable interpretation of the structure and

basic function of the biological neuron of the human

brain. Therefore the more physiology of the brain is un-

derstood the better the ability to design ANNs that will

handle more complex problems. According to Carlton et

al [15]; Agbese and Mohammed [14], the artificial neu-

ron is called the processing elements or nodes, which are

capable of handling information in response to external

input. It has many input parts and combines the input

Legend: ∆P – change in active power of driver; ∆F – change in equip-

ment’s frequency; ∆V – change in equipment’s vibration-amplitude;

∆SE – change in spike energy of bearing

Figure 3. Triple hidden-layer network.

values it receives usually by summation. The combined

input is then modified by a transfer function, which can

be chosen to suit a particular application. This new value

becomes the output and can be connected to the inputs of

other processing elements through weighting functions,

which correspond to the synaptic strength of biological

neural connection [15,16].

As in the biological brain, the neural network learns

by altering the value of its weights. In a simulated neural

network, the weights are altered as to reduce the error

between the outputs the network produces in relation to a

particular input pattern and the actual required outputs

[15]. This is an iterative process, carried out as the pat-

terns to be learnt are presented; an algorithm calculates

the error and changes the value of the weights accord-

ingly.

Typically, an engineering application of ANN tech-

nology consists of a set of input nodes that forms the

input layer and one or more hidden layers. This type of

ANN is called a multilayer perceptron, and usually a

popular back-propagation algorithm is used to train the

network [17].

The triple-hidden layer ANN shown in Figure 3 was

designed with 4 nodes in the input layer. Hidden layer 1

is to be used for processing of the measured values; the

summation is then passed to hidden layer 2 for exact

fault-classification, while hidden layer 3 is designed to

issue task specifications for achieving possible solutions.

The output layer is therefore able to determine and dis-

play the nature of the exact fault and provide a solution

for the fault to be overcome, thereby optimizing the use

of energy and human resources.

A vibration-severity chart for various classes of ma-

chinery, as illustrated in Table 2, was used in the training

of the network. Its inputs were vibration-velocity ampli-

tude, motor power, equipment frequency and spike en-

ergy of the equipment. A computer program in Visual

Basic (VB) was developed from the flowchart shown in

Appendix 1. Further details of it are available from the

authors. The faults considered included misalignment,

imbalance, bent shaft, mechanical looseness, and poor-

bearing condition.

The diagnostic model is programmed according to

various colour codes for corresponding pump conditions,

diagnosed faults and appropriate task instructions on how

to avert catastrophic failure of the vibrating equipment (in

the considered case, a pump). The software flagged up

defined information once the vibration values were within

a specified range. The solutions obtained from the diag-

nostic model were used to determine how unwanted vi-

bration problems could be eliminated or reduced to al-

lowable limits.

When analysing vibration severity of a machine to

pinpoint particular problem, it is essential to know the

C. I. UGECHI ET AL 181

Table 2. Ranges of vibration severity for various classes of machinery (iso 12372 or bs 4675: 1971).

Range of Vibration SeverityMaximum Values

Class of Vibration of Machine

Range

Classification

Effective

Velocity:

RMS

(mm/s)

Vibration

Velocity

(mm/s)

Vibration

Displacement

(μm) Class I

Class II

Class III

Class IV

0.28 0.28 0.4 1.25

0.45 0.45 0.63 2

0.71 0.71 1.0 3.15

Good

1.12 1.12 1.6 5

Good

1.8 1.8 2.5 8

Acceptable /

Allowable

Good

2.8 2.8 4.0 12.5

Acceptable /

Allowable

Good

4.5 4.5 6.3 20

Improvement

Required

Acceptable /

Allowable

7.1 7.1 10 31.5

Improvement

Required

Acceptable /

Allowable

11.2 11.2 16 50

Improvement

Required

18.0 18 25 80

Improvement

Required

28.0 28 40 125

45.0 45 63 200

71.0

Not

Acceptable Not

Acceptable Not Acceptable Not

Acceptable

Legend:

Class I: Small machines; electric motors up to 15kW power.

Class II: Medium-size machines; electric motors of 15 to 300kW power.

Class III: Large prime-movers or machines on rigid foundations; electric motors of above 300kW power.

Class IV: Large prime-movers and other machines, Turbo Machines.

Good: Colour coded green.

Acceptable/Allowable: Colour coded blue.

Improvement Required: Colour coded yellow.

Not Acceptable: Colour coded red.

vibration frequency. Knowing the frequency helps in

identifying the exact nature of the problem and the loca-

tion of the faulty machine-component. Although all of the

frequencies in a complex vibration signal can be of con-

cern for analyzing machinery problems, the fundamental

and dominant frequencies are of special importance. The

fundamental frequency is equal to the speed of rotation of

the rotating element – first harmonic (1* RPM). The

dominant frequency is the frequency at which the largest

vibration amplitude occurs. The fundamental and the

dominant frequencies are not always the same. Where the

dominant frequency differs from 1* RPM (fundamental

frequency), the dominant frequency is usually more in

dicative of the trouble.

Therefore, during the analysis of the vibration data, in-

terest was devoted primarily to measuring the dominant

vibration amplitudes and determining the frequencies at

which they occurred. This helped in the identification of

the problem and isolation of the faulty machine compo-

nent. High vibration amplitudes occurring at integral mul-

tiples of the machine’s fundamental frequency (e.g. 2*

RPM, 3* RPM, 4* RPM, etc.) are associated to different

failure modes.

3.2. Instrumentation

In undertaking this investigation, the following instru-

ments were used: Vibration Data Collector (Model: IRD

880); Vibration Pick-Up Pen / Ear Piece; Laser Align-

ment Tools; Balancing Machine; Strobe Light; and a

Computer System.

The vibration analyser performs the function of meters

C. I. UGECHI ET AL.

182

Legend: = Measurement Locations/Points, i.e. A, B, C and D

 = Plain bearing

= Anti-friction bearing

Figure 4. The tested pump assembly and location measur-

ing points.

and monitors, and is capable of carrying out more com-

plex operations. Vibration meters, monitors and analys-

ers, uses vibration transducers. This is often referred to

as vibration sensors or pick-up. The heart of the meas-

urement system is the transducer; it is a sensing device

which converts one form of energy to another. The vi-

bration transducer converts mechanical vibration energy

into electrical signal. The sensitivity of a velocity trans-

ducer is constant over a wide range of operating frequen-

cies [11], but there are few limitations, which are above

the scope of this work. The data collector is used in ac-

quiring vibration-velocity, spike energy at variable fre-

quencies. The rated power and frequency of the driver

are used in the analysis as power at full load and funda-

mental frequencies. The data were collected manually

and fed into the computer model for analysis.

4. Results and Discussions

Interpretation of the field-vibration data and the subse-

quent diagnosis of the failure mode, constituted the most

difficult tasks in running the vibration-based program.

Much depended on the experience and skill of the analyst.

In undertaking maintenance, the need to avoid costly

mistakes, minimize energy expenditure and achieve the

benefits of PdM, led to the model developed for this in-

vestigation.

Vibration-velocity data, presented as root means

square (RMS) values were collected, with the pump at

full load - see Figure 4 and Table 3. The numerical val-

ues in Table 3 and trends on the associated graphs in

Figure 5 displayed high axial and radial vibrations at

locations D7 and C5 respectively, suffered by the pump

bearings. Bearings A and B (see Figure 4) for the electric

motor also experienced significant vibrations; although

of lower amplitudes. Significant vibrations of the motor

bearings could be transmitted through the shaft from

bearings C and D. Points A to D shown on Table 3 are

the location points where vibration values were taken,

while positions 1 to 8 represent the sequence in which

data were collected on the same equipment at different

frequencies.

Results of the analysis of data presented in Table 3

using the software model were displayed on the com-

puter screen as in Figure 6. This indicated significant

vibration amplitudes (depicted by the red and yellow

colours). The program then proceeded to the second

phase of the analysis in order to reveal the fault classify-

cations and task instructions, as shown in Figure 7. The

analysis indicated the presence of high axial and radial

vibrations at 1RPM, 2RPM, and 3RPM, which suggests

misalignment, while the high spike energy at B was in-

dicative of a defective bearing.

The misalignment originating at the driven end of the

pump assembly was seen as the source of the failure be-

cause the vibration amplitude was largest there. The mis-

aligned shaft and bearings at C and D led to the damage

of the bearing at B

Table 3. Vibration-analysis data sheet for unit 1800-01A pump.

PUMP MAKE: GIABBIONETA

POWER: 36.5kW

RPM: 2950 / 2945

DATE: 09/10/06

ANALYZER MODEL: IRD 880

MS/LMS/SFrequencyVelocity (VH)Velocity (VV)Velocity (VA)Spike EnergyMultiple of Fundametal

cpm

mm /sec

mm/sec mm/sec g-SEFrequency (cpm/2950)

(RMS) (RMS) (RMS)

C5 3,0128.21.11.00.09671 * RPM

D7 3,0664.71.69.80.0761 * RPM

B3 3,8343.71.11.30.5571 * rpm

A1 5,9467.50.91.20.0942 * rpm

D8 8,0073.02.70.90.11173 * rpm

C6 12,7301.93.61.10.13134 * rpm

A2 13,6861.91.11.30.0975 * rpm

B4 33,6762.01.01.61.3811 * rpm

C. I. UGECHI ET AL 183

0.0

2.0

4.0

6.0

8.0

10.0

12.0

05,00010,000 15,000 20,000 25,000 30,000 35,000 40,000

Frequency (cycles per minute)

Vibration Velocity (mm /sec)

V HV VV A

Figure 5. Vibration velocities for the 1800-01A pump.

Figure 6. Computer screen presentation for1st phase of the analysis.

Figure 7. Computer screen presentation for 2nd phase of the analysis.

C. I. UGECHI ET AL.

184

Table 4. Vibration-analysis data sheet for unit 1800-01A pump (AFE).

PUMP MAKE: GIAB BION ETA

POWER: 36.5kW

RPM: 2950 / 2945

DATE: 17/10/06

ANALYZER MODEL: IRD 880

Frequen c yVelocity (VH)Velocity (VV)Velocity (VA)Spike En ergyMultiple of Fundametal

MS/LMS/ S cpm

mm /sec

mm/sec mm/sec g-SEFrequency (cpm/2950)

(RMS) (RMS) (RMS)

A1 1,5002.5 2.6 1.30.070.5*RPM

B2 2,9452.42.11.30.1171 * RPM

B3 6,0202.62 .31.50.112 * RPM

C4 9,0002.32 .31.20.153 * RPM

D5 12,2002.32.61.10.044 * RPM

D6 15,1702.42.61.10.435 * RPM

0.0

0.5

1.0

1.5

2.0

2.5

3.0

02,0004,0006,0008,00010,00012,000 14,000 16,000

Frequency (cycles per minute)

Vibration V elocity (mm/sec)

V HV VV A

Figure 8. Vibration velocities for the 1800-01A pump (AFE).

Figure 9. Computer screen presentation for the analysis after fault elimination (AFE).

C. I. UGECHI ET AL.

185

Appendix 1. predictive maintenance program flow-chart.

The task instructions were executed and the data col-

lected after the fault elimination shown in Table 4. The

associated graph (i.e. Figure 8) showed almost smooth

trends with a maximum of 2.6mm/sec radial vibration

Did data exceed

limit?,

e.g ≤ 2.8mm/s

Input measured vibration

data

START

Define all variables used

Analyze status of equipment

Did data exceed

limit?,

e.g ≤ 2.8mm/s

Set values to ascertain vibra-

tion-fault class & task specifica-

tion

Generate report for the status of

the equipment

Proffer solutions & task in-

structions

Yes

STOP

C. I. UGECHI ET AL.

186

velocity amplitude and 1.5mm/sec amplitude in the axial

direction, which suggested an acceptable working condi-

tion, had been achieved.

Results of the analysis of these data are shown in Fig-

ure 9 and confirmed that the condition of the pump was

within acceptable range. This is evident in the displayed

green and blue colours. Therefore the program did not

proceed to a second phase of the analysis. Also compar-

ing the data in Table 4 with specified maximum vibra-

tion-level of 3.0mm/sec for the pump, as recommended

by the manufacturer, showed that the vibration values

were within the acceptable range.

5. Conclusions

A diagnostic condition-based model that can be used for

the PdM of rotary equipment has evolved from this study.

The complexities involved in the analysis of vibration

data have been simplified for the vibration analyst and

PDM personnel. The high level of human error associ-

ated with the analysis of vibration data could also be

reduced through this procedure. Faults of the rotating

machine, identified through this analysis of its vibration

characteristics, can be displayed numerately and graphi-

cally.

The results obtained from the model, which was de-

veloped using an ANN, revealed that the approach is

well suited to the diagnosis of vibration-based faults in

centrifugal pumps. Though the model was validated us-

ing vibration data obtained from a centrifugal pump, it

can be used to analyze vibration faults in other categories

of rotating equipment. The model can also therefore be

used for continuous real-time on-line condition monitor-

ing.

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Appendix

Abbreviations and Nomenclature

AFE After fault elimination

AI Artificial Intelligence

ANN Artificial neural-network

a vibration acceleration

BS British Standard

CM Condition monitoring

cpm Cycles per minute

ƒ vibration frequency

G Large machines having electric motors of above

300kW power

g-SE Unit of spike energy

ISO International Standards Organization

K Small machines having electric motors of up to

15kW power

M Medium machines having electric motors of be-

tween 15 and 300kW power

MS/L Measurement location

MS/S Measurement sequence

N Number of hidden layers

PdM Predictive maintenance

PM Preventive maintenance

RMS Root mean square

RPM Revolutions per minute

T Turbo machines

VA Vibration velocity in axial direction (mm/sec)

VB Visual Basic

VH Vibration velocity in horizontal direction (mm/sec)

VV Vibration velocity in vertical direction (mm/sec)

v vibration velocity (mm/sec)

x vibration displacement (mm)

Z Number of output layers

ΔF Change in vibration frequency

ΔP Change in active power

ΔSE Change in equipment spike energy

ΔV Change in vibration velocity amplitude

µm Micrometre

Glossary:

Dominant frequency: Frequency at which the largest vibra-

tion-amplitude occurs.

Field vibration-data: Measured vibration data collected from

running machines.

Fundamental frequency: Basic repetition of the rotating equip-

ment; i. e. the first harmonic.