Frequency Ai ilysis of RandomStationary Vibrations

ARTURO CHIESA

Abstract-The most powerful method of representing randomstationary vibrations is based upon the power density spectrum or themean square density spectrum. Among the many procedures devisedto obtain this spectrum is an analog method described here which isparticularly suitable for processes lasting some minutes or tens ofminutes with frequencies ranging from less than one cycle per secondto some thousands of cycles per second. First, the principle on whichthe procedure is based is described; then the block diagram is dis-cussed with some information about the band filter, the squarer, andthe integrating unit. The control logic unit is then described in de-tail; this unit permits complete automation of the chain. Lastly, someinformation about the performance of a chain practically built up isgiven and the degree of repeatability is discussed.

SOME DEFINITIONS ON THE MEAN SQUAREDENSITY SPECTRA

A RANDOM PROCESS can be ,entirely irregular,that is, it can be entirely unforeseen in its evolu-tion, but very often the process, observed over a

sufficiently long interval, becomes stationary, howevercomplex or random the causes.

It is known that a harmonic Fourier analysis is some-times used to describe random stationary vibrations byconsidering the process repetitive. This, however, apartfrom some theoretical inconsistencies, actually may beimpossible if the fundamental period is long and if har-monics of very high orders are involved.

In these cases it is more advisable, from both thetheoretical and the experimental points of view, to repre-sent the random stationary vibration through either itspower density or its square value density spectrum.Obviously the density is referred to the unit frequencyinterval (for example, one cycle per second).The mathematical properties of the mean square

density spectra have been studied by Crandall [1 ] andby Rice [2 ]. A fundamental property is that its integral,calculated along the entire frequency range, gives theoverall mean square value of the function

00

V2 = f S(f)df

where S(f) is the mean square spectral density.Once the mean square density spectrum is known,

some useful information about the detail of the processcan then be obtained. For example, the number of timesno that x(t) passes through zero per unit of time, on theaverage, is

Manuscript received November 30, 1964.The author is with the Rubber Labs., Pirelli, S.p.A., Milan, Italy.

4 f2S(f)df

nO=S(f)df

and the number of times n. that x(t) passes through a

particular value of x with positive slope per unit of time,on the average, is

no~ 2-2

n, =- e-x2v2

PROCEDURES TO OBTAIN THE MEAN SQUAREDENSITY SPECTRA

To study a random stationary process, one of themain properties to be investigated first is the minimumtime in which the process becomes stationary. In prac-

tice, processes exist which can be considered as sta-

tionary after fractions of a second, or at the other ex-

treme, only when observed for hours, days, or even

years. Another important feature is the extension of thefrequency range. Theoretically, this should be extendedfrom zero to infinity; however, in practice, it is alwaysformed from a more or less limited range.

It is evident that the procedures of frequency analysischange radically according to the particular values ofthese two fundamental characteristics. In fact, many

procedures have been devised to obtain the mean square

density spectra. Some are based on digital techniques[3], [4], and require that the signal first be convertedinto digital form. Others deal directly with the signal in

an analog form.A procedure basically of the analog type is described

here. This procedure has been developed for the studyof random oscillating processes, which have a frequencyspectrum ranging from a fraction of one cycle per secondto some hundreds or thousands of cycles per second and

for which the "samples" can be considered as stationaryafter some hundreds of seconds. A great many oscillatingprocesses can be gathered into this category, and above

all, mechanical and acoustical vibrations. In fact the

procedure was used to study the vibrations of movingcars [51], [6].

A PARTICULAR ANALOG PROCEDUREAND ITS REQUIREMENTS

In principle, the procedure comprises the followingsteps: the signal to be studied first of all is filtered

38

Chiesa: Random Stationary Vibrations

_ I

digital Iiconverter

main signal-.- -control and check signals

--- control impulses

~~~

princterlogic unit coverter

check

oscillosco

tape

to the digiatacom puter

Fig. 1. Block diagram of the analysis chain.

through a variable band-pass filter, then squared to ob-tain the instantaneous square value, and finally aver-

aged to obtain the square value of each band for theentire period of observation.

It must be pointed out that the last step is performedby means of an integration, which can last for many

minutes. Therefore, the procedure can, as required, alsobe applied to processes, which may be considered as

stationary only after considerable intervals of time. Atthis point the procedure is radically different fromothers, which evaluate the content of energy in every

band by means of almost instantaneous readings.It is advisable to consider in some detail each of the

three main steps of the procedure.The band-pass filter should have an equivalent band-

width1 as narrow as possible, to improve the accuracy

of the detail of the spectrum.However, the narrower the band, the longer the time

and the greater the complexity of operation, both be-cause the number of bands to be analyzed becomesgreater and because, at the shifting of every band, thetransient becomes greater. Furthermore, it is not con-

venient to keep the bandwidth constant, but on groundsof the nature of the processes to be studied and the val-ues of the relative errors, it is advisable to use a filterwith an equivalent bandwidth proportional to the fre-quency.

Insofar as the intermediate step is concerned, i.e., thesquarer, it should be noted that, due to the presence ofeven high frequencies, servomechanism multipliers can-

1 Equivalent bandwidth of a filter means the width which should bepresented by a correspondent ideal filter with rectangular character-istics, allowing the same portion of mean square value of white noiseto pass through.

not be used; generally, only electronic multipliers aresuitable.The third step requires an integrator for processes

lasting several minutes or tens of minutes. This problemhas already been discussed [7]; integrations of thesedurations can easily be accomplished by means of elec-tronic circuits, such as the chopper stabilized amplifiersof the modern analog computors.What has been outlined in the foregoing is referred to

in the determination of the average mean square valuein every band. Considering the fact that for a completeanalysis this operation must be repeated many times,it is evident that the whole procedure would cause theoperator a considerable amount of work. Therefore, ithas seemed necessary to devise another chain, a logicchain, for the automatic operation of the various partsof the analyzer.

DESCRIPTION OF THE ANALYSIS CHAIN ANDPARALLEL LOGIC CONTROL CHAIN

The process is recorded on magnetic tape. The tape isthen formed into a loop (Fig. 1) in order to make theprocess repetitive. The junction of the ends of the tapeprovides, through a lamp and a photocell, an electricalimpulse every revolution.

After amplification, the signal is fed to a band filterfor a period of time defined by a system of timers; at theend of every cycle, an electrical impulse advances thefrequency control of the filter one bandwidth and theprocess is then repeated until the whole selected fre-quency range is covered.The signal from the band filter is squared and in-

tegrated for a period given by the timer system. Forsimplicity, this period is held constant for every band.

39

Ik

40 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION

The result of the integration is then converted intodigital form, printed on an electric typewriter, andpunched on a paper tape for further processing with adigital computer.

This process is a division of the mean square valueobtained in every band by the equivalent bandwidth.The final results are the densities of the mean squarevalues throughout the whole frequency range.The period of the integration is determined by a

timer. Another timer is placed before this in order toallow the initial transient superimposed on the signal bythe switching of the band to become sufficiently dampedout.The integration unit requires that the value V of the

voltage attained at the end of the integration be withina given range Va < V< Vb. If V is smaller than Va, theerror, due to the noise and the drift of the integrator,becomes unacceptable. If, on the other hand, V isgreater than Vb, the response of the amplifier ceases tobe linear. Owing to the fact that the intensity of the sig-nals in every band can vary greatly, according to theoutline of the spectrum, it is necessary to adjust theinput gain beforehand in order to keep the output levelsof the integrator within the required limits.With this in mind, a special device was provided at

the input of the band filter which permits variation ofthe position of the input gain control step-by-step, bothclockwise and counterclockwise. The control of theproper gain position by the logic unit is performed in thefollowing way. The logic unit reads the final value of theintegrator; if this is comprised within the allowed inter-val, the logic unit operates 1) the printing of the result,2) the advancement of the filter to the next band, and3) the reset of the integrator. If, on the contrary, thefinal value of the integrator is outside the proper limits,then at the end of the cycle there is no band advance-ment, no print of the result, and the logic unit operatesone step of variation of the gain in the proper direction,that is, increases or decreases, depending on whether Vis smaller than Va or greater than Vb.The tape loop magazine consists of a frame with 21

free rotating pulleys, on which it is possible to wind upto 30 meters (100 feet) of tape of '-inch width. No shiftof the tape was observed despite the considerable speed,60 inches per second. Owing to the very large number ofrevolutions of each loop of tape (for instance, one thou-sand, if four tracks have been recorded), particular caremust be taken to avoid wear of the tape producing anunacceptable decrease of the SNR. To this end, it wasfound very useful to spray molybdenum sulfide on thetape every now and then. There are also tapes on themarket with surfaces already lubricated.The band filter, the electronic multiplier, the analog

integrator, and the electric typewriter are based on cir-cuits which are widely known and easily found on themarket. Listed are some particular specifications.

Filter: Auto-Analyzer Muirhead-Pametrada, modelD-940A, frequency range 10-19,000 c/s, used withbandwidth in position, five per cent. The actual "equiva-lent bandwidth" of every band was determined; thevalues of 1/Q =Af/f were comprised between 1/14 and1/15 for the entire frequency range.

Multiplier: Philbrick Universal Multiplier-Divider,model SV-5-M, linear within 3 dB up to 7.5 kc/s andwith very high stability (drift less than 1 millivolt dc in8 hours).

Integrator: a chopper-stabilized transistorized opera-tional amplifier, formed by the two Philbrick Units, P2and P5, with high stability (drift less than 0.05 volt perhour in the whole integration circuit). The linear rangeof the output voltage is 1+10 volts.

Electric Typewriter: Olivetti, model T2-CN with tapepunching device T2-PF.The design of the control logic unit is completely new

and it is described in some detail in the next section.A calibration oscillator and an oscilloscope are useful

auxiliary parts to check the regular operation of thevarious parts of the chain.

LOGIC CONTROL UNIT2The logic digital control unit is in the reset state until

a trigger signal arrives from the first timer. This signal("compute" signal) sets, in the operation state, a start-stop circuit, which opens the input gate of a binarycounter, and thus receives 50 c/s trigger pulses and de-livers a conversion signal to the analog-to-digital con-verter every 160 milliseconds. Therefore, the analog-to-digital converter makes a measurement of the analoginput coming from the integrator at the rate of 160milliseconds. This operating mode continues until an-other signal ("hold") arrives from the timer unlessstopped in advance. Three cases may develop.

1) The converter output is always lower than 1000,and when the hold signal arrives, it is not lowerthan 100. The hold signal then stops the conver-sion signal to the analog-to-digital converter andmakes the electric typewriter print the convertedvalue (three decimal digits) as well as the numberthat identifies the filter gain. Moreover, the unitgives an advance command to the frequency con-trol of the filter, and a reset command (about 0.2seconds) to the timer.

2) The converter output is lower than 100 when thehold signal arrives, that is, at the end of the integra-tion. Then no printing occurs; the unit gives anadvance command to the gain control of the filterby means of a stepping bidirectional actuator anda reset command to the timer. The measure is re-peated with the gain increased.

2 The design and the construction of this control unit are due toC.E.A., Perego, Milan, Italy, to whom the detailed descriptionreported here is also due.

March

Chiesa: Random Stationary Vibrations

3) The converter output exceeds 999 before the holdsignal arrives. As soon as this happens, the unitdelivers a backward command to the gain controlof the filter and a reset command to the timer. Themeasure is repeated with the gain decreased, theconverter full scale having been exceeded in theprevious measure.

In all the cases the digital control unit goes to thereset state and is ready to receive another compute sig-nal. From the functional point of view, the digital con-trol unit can be divided into the following buildingblocks:

1) A start-stop circuit that handles the compute andthe hold signals coming from the timer.

2) A fast, eight states, three flip-flop binary counterwhich, under the control of the start-stop circuit,receives 50 c/s trigger pulses derived from the linefrequency, and generates the 20-millisecond pulsesused to trigger the analog-to-digital converter andto control the printing on the typewriter.

3) A slow four flip-flop counter, which is activatedwhenever a printing is required, and receives apulse every cycle of the fast counter, that is, every160 milliseconds. It controls the sequence of thetypewriter operation: the printing of the threedigits of the converted values (000 to 999); theprinting of the two digits of the number that iden-tifies the filter gain (01 to 12), carriage return, linefeed, spacing, figure shift.

4) A decade counter, to count the number printed ona line of the paper. Whenever ten value-gain pairshave been printed, this decade counter indicatesthat a line is complete and delivers to the slowcounter a signal that causes the carriage returnand line feed before the next printing.

5) Logic circuits, to code the data to be delivered tothe typewriter in the international five-bit code.To do this, a part of these circuits receives at itsinput the digital output of the converter (12 bits,corresponding to three binary coded decimal dig-its) and the gain number (12 bits, only one ofwhich is 1 in correspondence to each gain), and,under the control of the slow counter, it gives, atits output, the same data in decoded form, a deci-mal digit after the other, when they must beprinted. These data, and the carriage return, linefeed, spacing, and figure shift commands, enter adiode matrix which codes them in the internationalfive-bit typewriter code. The five-bit output of thediode matrix is converted in serial form under thecontrol of the fast counter, and in such a way the20-millisecond pulses that drive the typewriter aregenerated.

6) Logic circuits, to generate the commands for the

put and compare it with the low and high limits(100 and 1000, respectively), and receive alsothe hold signal; their outputs are the advance andbackward commands to the frequency and gaincontrols of the filter and the reset command to thetimer, as previously described.

All the circuits are built with germanium diodes andtransistors and are assembled on plug-in printed cir-cuit cards.

PERFORMANCE OF A PARTICULAR CHAIN

Here some information is given on a chain whichhas been working for two years.

As stated, the chain is used to analyze mechanicaland acoustical vibrations present in a running car [5],[6]. The vibrations are recorded on a FM multitrackmagnetic tape recorder with a sufficiently wide fre-quency range. For the mechanical vibrations, the anal-ysis is usually extended from 0.5 to 500 c/s, for thesound from 20 to 1000 c/s in case of noise inside thecar, or from 20 to 10,000 c/s for noise outside.

Integration periods of about two minutes are in thesecases sufficient to insure that the process may be con-

sidered as stationary provided that the road pavementis uneven in a sufficiently uniform way along the wholetest run.

To speed up the successive operations of analysis, thetape is played back in laboratory with increased speed.In effect a speed of 7' inches per second or 33 inches per

second of recording is used, whereas the speed of play-back is usually 60 inches per second, hence the two min-utes of recording are reduced in the laboratory to 15or 7.5 seconds, respectively. The stabilization period ofthe first timer is held constant at 3 seconds.To give some general ideas about the time required

for the analysis, it can be noted that in a spectrum rang-

ing between 0.5 and 500 c/s ten octaves are comprised(in practice a more limited range is often considered,according to the nature of the process under study).

Using a filter with bandwidth 1/15 of octave, thereare 150 bands, and the time of analysis should be 2700or 1575 seconds, according to the recording speed. Inpractice, also taking into account the time required forthe repetition of the loop in the automatic gain adjust-ment, the overall duration of a complete analysis, thatis, from 0.5 to 500 c/s, is somewhat less than an hour or

more than half an hour, according to the recordingspeed. During the whole analysis operation no manualswitching is required.To shorten the time, it is possible to use more than

one chain actuated by the same timers, which simul-taneously operate the various signals coming from a

multitrack tape. In effect, a second chain, completelyequal to the first one, has been successively constructedin the described device. An overall view is presented in

filter and the timer. They sense the converter out-

1965 41

Fig. 2.

42 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS AND CONTROL INSTRUMENTATION

I ig. 2. Overall view of two chains practically built up to analyzesimultaneously two tracks of the tape. From left to right: thetape loop magazine, the tape transport mechanism and below thelogic units, the analog-digital converters, the timers, and in thetwo racks at right, the filters, the multipliers, the amplifiers, andthe integrators. The two typewriters are not shown.

:12

00

00.

X:2

Y A

To illustrate the degree of repeatability of the opera-tion, a two-minute record of the vertical acceleration ofthe vibrations present in a car has been analyzed sixtimes. For every frequency band there are six values ofthe density of mean square value from which the con-fidence interval 2a has been deduced.' The degree ofrepeatability can be considered good (Fig. 3), takinginto account the influence of the wear of tape for suc-cessive repetitions.

ACKNOWLEDGMENTThe author wishes to express his appreciation to F.

Celeri and L. Carenzi for several helpful discussions,for the construction of some parts of the described ap-paratus, and for the overall assembly. He also wishesto thank R. Martini of C.E.A., Perego, Milan, Italy,for the paragraph containing the description of thelogic unit.

REFERENCES1] Crandall, S. H., Random Vibration, New York: John Wiley, 1958.[2] Rice, S. O., Mathematical analysis of random noise, Bell System

Tech. J, vol 23, 1944, pp 282-332; vol 24, 1945, pp 46-156.[3] Rona, T. P., Random Vibration, New York: John Wiley, 1958,

ch VII, pp 145-185.[4] Blackman, R. B., and J. W. Tukey, The Measurement of Power

Spectra, New York: Dover Publications, 1958.[5] Chiesa, A., Experimental studies on noise inside cars, J. Sound

and Vib., London, vol 1, 1964, pp 211-225.[6] Chiesa, A., Une methode d'analyse des vibrations aleatories,

Rev. Franc. Mecan., nos. 7 and 8, 1963, pp 19-25.[7] Chiesa, A., Amplitude distribution analyzer by means of elec-

tronic integrators, Acustica, vol 11, no. 5, 1961, pp 335-341.

s The confidence interval represents the interval in which thereis the probabilty that 95 per cent of the determinations are included.

Fig. 3. Repeatability test of the same tape analyzed six times. Thecontinuous line is a mean square density spectrum obtained byaveraging band by band the values of the six repetitions; the twodotted lines represent the confidence interval 2of.