New Methods of Studying Noise and Vibration and Cybernetic Diagnosis of Machines and Mechanisms

8/13/2019 New Methods of Studying Noise and Vibration and Cybernetic Diagnosis of Machines and Mechanisms

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N A S A E C H N I C A LT R A N S L A T I O N

NEW METHODS OFS T U D Y I N G NOISE A N D V I B R A T I O NAND CYBERNETIC DIAGNOSISOF MACHINES AND MECHANISMSEdited by K . M . RagidskisAbstrczcts of paper s from the All-Union Symposizrmhe l d June 29 to JuZy 1, 1970 at KaanasPolytechnical Inst i tate, Kaunas, 1970

A T I O N A L A E R ON A U TI CS A N D S PA CE A D M I N I S T R A T I O N W A S H I N G T O N , D. C M A Y 1972


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TECH LIBRARY KAFB, NM

NEW METHODS OF STUDYING NOISE AND VIBRATION ANDCYBERNETIC DIAGNOSIS OF MACHINES AND MECHANISMS

Edi ted by K. M . Ragul'skis

Translat ion of Novyye Metody Issledovaniy Shumov i Vibratsiy iKibem eticheskay a Diagnostika Mashin i Mekhanizmov.Abstracts of pa pe rs f r om the All-Union Symposiumheld June 29 to July 1, 1970 at KaunasPolytechnical Institute, Kaunas, 1970

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION.. . .~For sale by the Nation al Technical Infor matio n Service, Springfield, Virgin ia 22151

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TABLE OF CONTENTSPageOptimal De term ina tion of Changes i n Pr op er ti es of Random Phenomena .....1References .................................................~...........2The Problem o f O ptim al S yn th es is of Dynamic Models By t h e Method ofLP Search .............................................................. 3References. ............................................................ 5Est5mation of Dominant Parameters of Dynamic Systems . . . . . . . . . . . . . . . . . . .6Adaptive Model For Evaluation of Weight Function and Level of Noiseof Linear Dynamic System ............................................... 8Study of S to ch as ti c Approximation Algorithm By th e "Monte C arlo"Method, Used To Es ti ma te The Weight Fu nc ti on ........................... 9Est imat ion of Parameters of Nonlinear Hammerstein Operator . . . . . . . . . . . .11Est imat ion of Parameters and Character ist ics of Nofl l inear Dynamic

Systems . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . 13I d e n t i f i c a t i o n A s A Task o f S e q ue n ti a l S t a t i s t i c a l A n al ys is . . . . . . . . . . .15Dif feren t ia t ion of Random Forced Osci l la t ions and Sel f -Osci l la t ionsPer tu rb ed By Random Act io ns ........................................... 16Determination of Par t ial Damping Coeff icients of Linear Systems FromMean Periods of Enveloped of Random Oscillations ...................... 1 7In t eg ra l Eq u a t io n s A s A Method of Determining The Spectrum of NormalModes of Background Vibrations ........................................ 18Appl ica t ion of Corre la t ion Analys is For Diagnosis o f the Condi t ionO f Mechanisms ......................................................... 19Method and Device For Quasi-Parallel Analysis of Sh.ort-TermA c o u s t i c a l S i g n a ~ s................................................... 2 1Use of d Correlat ion Based On Or th o go na l F i l t e r s I n Aco u s t i ca lMachine Diagnosis ..................................................... 22Determination of Equivalent Random Perturbation During Modelingof Complex Dynamic Systems ............................................ 23I d e n t i f i c a t i o n of Model Of Human Op era tor With Random V ibra t io nActions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Multichannel Diagnosis of Machines By Digital Computer ................27Cybernet ic Method of Studying Vibra t io n Acoust ica l C ha ra c t er i s t i csof Complex Structures ................................................. 29References ............................................................ 30Automation of Process ing of Exper imenta l Data and DevelopmentRecommendations For Se le ct io n of Optimal Vibra tion-A bsorbi ng C oat ing. .31References ............................................................ 33Random Osc i l la t i on s and I de n t i f ic a t io n of Parameters of P r e c i s i o nBall Bearings ......................................................... 34Use of Dig i ta l Computer For Determinat ion of Ef fec t o f Vibra t ion- -Damping Coat ing By Di sp la ce mh t of Resonant Frequenc ies of S tr uc tu re .3 6Fac to r s De te rmining Vib ra t io n Aco u s t i ca l Ch a r ac t e r i s t i c s o f Ba l lBearings ..............................................................37Study of Noise Formation In Throttling Devices For Measurement ofN o i s e C h a r a c t e r i s t i c s o f Fans ......................................... 38

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Pagee thod o f S tudy ing Relaxa t ion Osc i l l a t ions o f A Ball I n t h eS ep a r a t o r o f A Ball Bearing Unit ....................................... 40The S t a t i s t i c a l S t r en g t h Rese rv e of Discrete Syste,,1s . . . . . . . . . . . . . . . . . . .42Influence of Methods of Fas ten ing o f Accelerom eter On I t sFrequency Charac te r i s t i cs . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . .44S t u d y o f C o l l i s i o n s of Machine Elem ents On t h e Basis of Phenomenological Models of I n e l a s t i c M edia .................................... 45Methods of St udyin g Vi br at io ns of High-speed Diesels . . . . . . . . . . . . . . . . . . .46Di f f e r en t i a l Equa t ion For Hor izon ta l Loop o f S igna l Ca r r ie r . . . . . . . . . . . .48Shudy o f Flexural Oscillations of Magnetic Drums ....................... 49Vibra t ions o f A St or ag e Drum of A BESM-6 Computer ...................... 50Study o f Vibration and Dynamic Balancing o f Mechanisms. Including

S e v e r a l P a r a l l e l . S h a f t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Analog Computer Studies of Osci l la t ions of Sys tems Having Hysteres isC h a r a c t e r i s t i c s . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . 52Electromagnetic Impact Machines ........................................ 54Estimation and Normalization of Opera ting Modes of Vib rat ionTechnological Machines . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 56Anal ysis of t h e Dynamics of Multi-Mass Elas t i c Systems By A FrequencyMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Study of Repeated Impact Influence of P r es s u r e R o l l e r On MagneticTape . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . 58Diagnosis of S t a t e s o f P l an e t a ry Gear Reduction For Cer ta in Parameters .59Po ss ib i l i t y o f Tes t ing th e Techn ica l Condi t ion o f In te rn a l CombustionEngi nes On t h e Basis of Noise And Vibration Parameters . . . . . . . . . . . . . . . . .60P o s s i b i l i t y o f E st i m at i n g Wear By Vi br at io n- Ac ou sti ca l Methods . . . . . . . . .61Diagnos is of Rear Axle Reduction Gear o f ZIL-130 Motor Ve h icle ByAcoustical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Use of ' S t a t i s t i c a l Methods In th e I nv es t iga t io n o f V ibra t ions Andt h e Dynamics o f Mechanisms and Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Optimizat ion of th e Cross Sect io n of t he Frames of A Mult i s tageVibra t ing S tand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A Se lf Tuning Cont rol System For th e Motion of A Vibra t ing S tand . . . . . . .66Compensation of t h e In flu enc e of A n Electrodynamic Vib rat i ng StandOn Ch ar ac te r i s t i cs o f Mechan ical S t ruc tu r es In Vibra t ion Tes t s . . . . . . . . .68Low Frequency Vibrating Stand For Physical Modeling of the Movementsof Complex Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Opt imal iz ing Inpu t Con t ro l l e r o f O sc i l l a t i ng Ampl itude For Fa t igueTes t ing o f P ar t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Anal ysis of Mechanical System With Matched V ib ra to r .................... 72Muftichannel Apparatus For Diagnosis o f Vibrating Mode of Main ShipEngines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Problems of t h e Dynamics And S t a b i l i t y o f A C y li n d ri c a l So l id On AnA er o s t a t i c S up po rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74References . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 74

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PageStudy of Vibrat ions of A Magnetic D r u m With Pneumatic Drive AndPneumatic Suspension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Vibration Damping of Bearing Units .................................... 76I n v e s t i g a t i o n o f Vibrations With Random Amplitudes and Frequencies ....78Some Sp ec if ic s of t h e I nt er ac ti on o f Dynamic Systems With Di str ibu tedAnd Lumped Parameters ................................................. 79A Set o f Apparatus For Measurement of O sc il la ti on s of A Moving Tape ...81Study of Vibrat ion D i s k s With Magnetic Coatings ....................... 82Some Problems of Pr ac t i ca l Ana lysi s o f V ibra tion-Acous t ica l P roper t i esof Centrifugal Pumps .................................................. 84Determination of Primary Sources of Noise In Machines ByCorrelation Mathods ................................................... 85Study of O pt ic al Dynamic Models of Fl ex ib le Rotor Systems By t h e LP-Search Methods ....................................................... 86Study of Methods of Separating Useful Signals From General Noise ofDup lic ati ng Machines I n Solv ing The Problem of Automatic Recognitio nof Origin o f Noises I n Co ncret e Mechanisms Using D ig it a l Computers . . . .87St a t i s t i c a l S tud ies o f Random Vibra t ions o f A Railroad Car Body . . . . . . .89Method of Measurements During I n ve st ig at io n of Motorcycle Noise . . . . . . .91Determina tion and Calcu la t ion o f the C har ac t e r i s t i cs o f NoiseCreated by Turboprop Passenger Ai rc ra ft In t he Area o f A i r p o r t s . . . . . . .92Study of A i r Noise of Pneumatic Loom By S t a t i s t i c a l Methods ...........93A Water Spray A s A Source of Random Force For Acoustical Measurements.94References ............................................................ 95Noise A t Acoust ical Receiver Resul t ing From Spa t i a l ly Noncorrela tedSources Di s t r i bu te d Over A Sur face .................................... 96Study of t h e Noise o f A Diese l On A T r ac t o r ........................... 98Automatic Method of Studying Sound In su la ti on of Cy lin dri cal TubesWith Various Sound Radiation Conditions .............................. 100Sign i f ica nce o f In fo rmat ion On Cer ta in t y o f Unstable Noise In theProcess of I t s A ct io n On t h e H u m a n Organism .......................... 103The P roblem of th e S t a t i c Ca lcu la t ion of V i b r a t i o n I n s u l a t i o nSystems With S ix And Twelve Degrees o f Freedom ....................... 105Reduction of Noise And Vibration o f Pneumatic Loom UsingVibra t ion- Insu la t ing Suppor t s ........................................ 106Genera l iza t ion o f t he Hypothes is o f Y e . S . Soroki n To In clu deNonlinear E l a s t i c Damping Elements ................................... 107Os ci l l a t io ns o f E la s t ic a l l y Supended Body With Center of Gravi tyMismatched To Center of E l a s t i c i t y of Support ........................ 109One Method of Damping Parame tric O s c i l l a t i o n s of A Rod ConsideringA Damping Suspension.................................................110Study of Noise And Vibration Upon Impact .............................112Theory of Multichannel Compensation System For Oscillations InSt r uc t u re (F ie ld ) o f Arb i t r a ry Form .................................. 113Ce rt ai n Types o f Mu lticha nnel Systems For Compensation ofS t r u c t u r a l O s c i l l a t io n s .............................................. 115Measurement of t h e Atte nua tion Fac tor In Osc i l l a t i ng Sys tems .........117

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PageRef erences.......................................................... 118St udy of Dynam c Pul se Type Loads Tr ansm t t ed To Foundat i on ByNew Types of Shut t l el ess Looms ...................................... 119Dampi ng of Osci l l at i ons of Rods By El ect r omechani cal Feedback. . . . . . .120St udy of Noi se I n Vi br at i on At I ndust r i al Ent er pr i ses of t heLi t huani an SSR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Devel opment of Mul t i channel Vi br at i on Measur i ng I nst r ument..........124The Probl em of Per f or m ng St udi es of Har moni c Loads and LoadsWhi ch At t enuat e W t h Ti me........................................... 125New Devel opment s of Vi brat i on- Measur i ng Devi ces W t h ExpandedFr equency Range..................................................... 127St udy And El i m nat i on of I nf l uence o f Pr essur e Osci l l at i ons I nPi pe Syst ems On Operat i ng Pr ocess And Vi br at i on of El ement s ofPi st on Compr essor St at i on........................................... 129Anal ysi s of Hydr odynam c Noi se Ar i si ng As Li qui d Fl ows Over RoughSur f aces............................................................ 130Method And I nst al l at i on For St udy of Pr ocesses of Vi br at i on Movement sof Par t s And Accompanyi ng Phenomenas ................................ 131Experi ment al Determ nat i on of Changes o f Dynam c Char act er i st i csof Hi gh- speed Rotor s 0pera. ti ng I n Bal l - Beari ng Mount s W t h Passageof Ti me............................................................. 132Ref erences .......................................................... 132Cal cul at i on of Vi br at i ons of Di st r i but ed El ast i c Syst ems By Fi ni t eEl ement s Me~hod.................................................... ~ 3 3The Probl em of Decr easi ng Vi br at i ons of El ect r i c Machi nes ...........135Osci l l at i ons of A Rot or Resul t i ng Fr om I naccur acy of Manuf act ur eof Bal l Bear i ngs .................................................... 137St udy of t he I nf l uence of Resonant Twi st i ng Osci l l at i ons I n A Mot orVehi cl e Tr ansm ssi on On Vi br at i on And Noi se I n t he Cab..............138Appl i cat i on of Di mensi onal Anal ysi s To St udy of Vi br at i on Act i vi t yof Pi st on Engi nes ................................................... 139St udy of Vi bt at i ons I n t he Di r ecti on of I ncreasi ng And Decreasi ngTi me................................................................ 142Cont act l ess Met hod of Measur i ng Vel oci t y of I mpact i ng El ement s......144St udy of Mechani cal Osci l l at i ng Syst ems Usi ng Nat ur al Vi br ator s . . . . .146The Exci t at i on of Per i odi c And Random Tor si onal Osci l l at i ons I nRotat i ng Syst ems .................................................... 148I dent i f i cat i on of Charact er i s t i cs of Dynam c System of Cyl i ndr i cal Gr i nder By Bl ack Box Met hod Dur i ng Gr i ndi ngs...................... 149Spectr al - Cor r el at i on Anal ysi s of Vi br at i ons of Avi at i on Engi ne Under'Test St and Condi t i ons............................................... 150Regr essi on Anal ysi s of Noi se of An Ai rcr af t Engi ne .................. 152Some Pr obl ems I n t he Appl i cat i on of t he Two- Di mensi onal Pr obabi l i t yDi st r i but i on Funct i on For Anal ysi s of Acoust i cal Noi se and 6i gnal s..153One Cont act l ess Met hod of St udyi ng t he Nat ur al Osci l l at i ons ofEl ast i c St r uctuses.................................................. 155Desi gn of Vi br at i on- Pr ot ect i ve Syst ems W t h Random Vi br at i onActi on.............................................................. 157

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Pageert ai n Pr obl ems of Machi ne Acoust i cs.............................. 158Ref erences......................................................... 160Det erm nat i on of t he Par ameter s of Mechani cal Osci l l at i ng Syst emsOn t he Basi s of t he Ampl i t ude- Frequency Char act er i st i cs As A Meansof Vi br at i on Di agnosi s of Machi nes. . . . . ............................ 161Probl ems of I dent i f i cat i on of Par ameters of Dynam c Syst ems BasedOn Tape Dr i ves..................................................... 162New Met hods of Studyi ng the Dynam cs of I mpact Pr ocesses. . . . . . . . . . .164New Methods of St udyi ng And I ncreasi ng t he Dynam c Accur acy ofPr eci si on J I G Bor i ng Machi nes ...................................... 166

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NASA TT F - 6 9 4NEW METHODS OF STUDYING NOISE A N D VIBRATION ANDCYBERNETIC DIAGNOSIS OF MACHINES AND MECHANISMS

ABSTRACT. Brief summaries are presen ted of research workp er fo rm ed i n t h e a r ea o f t h e s t u d y of n o i s e and v i b r a t i o nand cy be rn et ic diag no sis of machines and mechanisms. Ther e p o r t s were or ig in al ly pr esen ted a t an All-Union Symposiumhe ld June 29, 1970 through Ju ly 1, 1970.

- . . _ _ .PTIMAL DETERMINATION OF. CHANGES I N~-ROPERTIES OF RANDOM PHENOMENAL. A. Telksnys (Vilnyus)

The problem i s st ud ie d of det erminin g th e most proba ble moments i n t ime /?*f o r a change in t he s t a t i s t i c a l p ro pe r t i es o f r andom phenomena on the ba s i s o fs i n g l e o r re pe at ed r e a l i z a t i o n s , f i xe d i n o b se rv at io n i n t e r v a l s o f f i n i t eleng th .

This problem a r i s e s i n many a r e a s , i n p a r t i c u l a r i n t h e i n v e s t i g a t i o n o fn o i s e an d v i b r a t i o n , a s we l l a s i n t h e d i ag n os i s of machines and mechanisms.

Statement of problem.Let

. P J r ) , t~ bo,u)X ( t )= L2) (t ) f E u , u2),be a random phenomenon with mathematical expectation

m 1 ) = m ( l ) ( t ) ,I E(Uo, u){ t n ( ( t ) , E u , u2),and wi th cor re la t ion func t ion

k 4, T) = { k1 (%, T); 8,T E ( U o . U )k8(8,7); 8 , T E ( U , up).

X(l) ( t ) , X(2) ( t ) a re normal. Eioment i n t i m e u, when random phenomenonX( t) , t E (u,u,) changes i t s s t a t i s t i c a l p r o pe r t ie s i s unknown o r random. Thea p r i o r i d i s t r i b u t i o n a ( u) a t i n s t a n t u e i t h e r i s known, o r it i s assumed thati t i s c o ns t an t o ve r t h e i n t e r v a l i n q u es t io n , i . e . , a ( u ) = cons t .

- . .- . ...- . .. .. . ..-- . . ..-.~~ ~~-~ . .- . . . - ... . . - - . . . - . ~ ~*Numbers i n t he marg in in d i ca te pag ina t ion i n t he fo re ign t e x t .

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I t i s r e q ui r e d, u s i ng one o r more r e a l i z a t i o n s x ( t ) , E ( u0 ,u 2) of randomphenomenon X(t) , Q ( u0 u2) , t o f in d th e most p ro b ab le e s t ima te u* o f p a r ame ter / Au. That i s , we must de te rm in e t h e most pro ba bl e moment i n ti me u* when randomphenomenon X(l ) ( t ) w i l l be co nv ert ed t o random phenomenon X ( 2 ) t).

The problem i s solved by determining the maximum of t h e a p o s t e r i o r id i s t r i b u t i o n d e n s i t y f u n c t i o n o f t h e d e s i r e d pa r am e te r u. I t i s found on theb a s i s of th e maximum o r minimum of t h e f un ct io n, r e l a t e d i n some manner t o t hea p o s t e r i o r i d i s t r i b u t i o n d e n s it y f u n ct i on 1 1 , 4 ] .

C a l c u l a t i o n s f o r d e t e r m i na t i o n of u* can be per formed i n p ra c t ic e bycomputer .

Examples of s ol ut io n of t he problem by computer a re pre sen ted .

REFERENCES1. Te l ' k sn y s , L. A . , V . Yu. Chernya uskas, "D ete rmi nat ion of Most Pro bab leMoment i n Time f o r Change i n t h e N ature of a Random Pr oc es s. Reporta t F i r s t All-Union Symposium on S t a t i s t i c a l Problems i n EngineeringCyb erne tic s," Moscow, 14-18 February 1967, on f i l e at All-Union Ins t i t u t e o f Sc ie n t i f i c and Tech n ica l I nfo rma tio n , 8 1 2-69 .2 . Telksnys , L. A . , V . Ce rn iau sk as , "Dete rmina tion of t h e Moments of Time ofChange i n S t a t i s t i c a l P r o p e r t i e s of Random Processes," IFAC Symposium,1970, I de n i t i f i ca t i on and Process Parameter Est ima t ion .3 . Telksnys, L. A . , "Dete rmina tion of Optimal Bayes Lear ning Algorithm i nDet erm ina tio n of Moments i n Time f o r Change of P ro p er t ie s of Random

S i g n a l s , ' 1 Avtomatika i TeZemekhanika, No. 6, 1969.4 . Telksnys , L . A . , "Dete rminat ion of Most Pr obab le Moments i n Time f o rChange i n Pro pe rt ie s of Random Si gna ls , IT Avtomatika i VychisZiteI nayaTekhnika, No. 1 , 1970.

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THE -PROBLEM OF OPTIMAL SYNTHESIS OF DYNAMIC MODELS B Y THE METHOD OF LP SEARCH. . - . . . .- - . . . . . - . - . - . ..- .M . D . Genkin, V . K . Grinkevich, I . M . , Sobol ' , R . B . Statnikov (Moscow)

The pr es en t work s tu di es a new uni ve rs al method f o r search f o r optimalparameters based on th e use of Haar f u n c t i o n s (2,4) .

This method can be called LP s ea r ch , s i n c e i n p l ace o f t h e random p o i n t si n a mult id imens ional cube, we u s e t h e p o i n t i n an LP T sequence ( 2 ) . Thesep o i n t s a r e n o t random and are di st r i b ut e d much more evenly t han random po in ts .However, ca lc ul at io n of t he se poi nt s by computer i s q u i t e simple.

For fu r t he r improvement of th e parameters of th e models, w e used o rd inarylo ca l methods (1 ) . Inc ide nta lly , i n many problems, no improvement was r eq u i r ed ,s i nc e a su ff ic ie nt ly good model was achieved i n the f i r s t s t a g e .

We note two more s i tu a t io ns , i n which th e use of random, and pa r t ic ul ar lyLP s ea r ch , i s q u i t e ex p ed i en t :

a) I f i t i s necessary t o a na ly ze t h e p o s s i b i l i t y of opt imiza t ion of a modelon t h e b a s i s o f s e v e r a l d i f f e r e n t c r i t e r i a o r t o s t ud y t h e i n f l u e n ce o f manyparameters , t h i s can be performed us ing t he same t e s t p o i n t s ;

b) I f cond i t io na l cond i t ion s sep ara te a nonconvex s e t i n th e space o f models ,convergence of l oc al methods may occur only i n a c e rt ai n a re a around th e optimum; t h i s ar ea can be reached by random o r LP search .

W s h a l l r e pr e s en t t h e i n e r t i a l and r i g i d i t y pa ra me te rs , as well a s t h edamping coeff ic ients , by c x l , . . . , c x r . P e rm i ss i bl e v a r i a t i o n s w i t h i n t h e l i m i t so f t h e d e f i n ed k i nem at i c p l an i n d i ca t e t h a t

Then the dynamic model RE ( wi th f i x ed k i n em at i c s t r u c t u r e E) i s determined byt h e p o i n t ( a l , . ..a ) i n an r -d i mens io na l p a r a l l e l e p i p e d ( 1 ) .

. r . - _ _ - . . ~-l T h i s f or mu la f o r RE c or re sp on ds t o t h e f o l l ow i ng d e f i n i t i o n : models r e t e r t otwo systems of o b j e c t s A and B y between which a homomorphic mapping of system Ai n a c e r t a i n s y st em A and a homomorphic mapping of B i n a ce r t a i n s ys tem B 'can b e e s t ab l i s h ed s uch t h a t A and B' are isomorphic . I t i s assumed that there la ti on sh ip of isomorphism (homomorphism, iso fun ct ion al ism ) ob ta ins between t h emodel and p r o t o t y p e ( I ) .

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' In cons t ru c t ive r e a l iz a t io n of th e model, th e fo l lowing p roblem ar ises :f rom the s e t of a l l RE, s e l e c t an optim al model which maximizes a c e r t a i n c r i t e r i a l func t i ona l , dependen t on the so lu t i on o f t h e sys tem, descr i b ing dynamicmodel R E I n t h e f i n a l a n a l y s i s , e ach su ch f u n c t i o n a l depend s on t h e p ar a meters a1,. . . ,CY Therefo re , w e write i t i n the fo rm of @

Y '(a) where the

v e c t o r CY = ( C X ~ , . . . , C Y ~ ) ,h i le y can be c a l l e d t h e q u a l i t y i n de x.Assuming t h a t i n a c e r t a i n c l o s ed area r b e lo n gi ng t o p a r a l l e l e p i p e d ( 1)

+CYhen parameter - w i l l b e r ef e r re d t o as the op t imal parameter under cond i t ion+r , whil e t h e co rrespo nding model RE( ) i s r e f er r e d t o as the optimal modelunder condi t ion r . F or b r e v i t y , t h e v a l u e s o f 4, ( a ) where &r w i l l b e w r i t t e nY - . -i n t h e form 4, ( a T I .Y -

U si ng t h e p l an p re s en t ed i n [ 2 ] (p. 2 1 9 ) , t h e p o i n t s Q, = (9, ,l , . . .qtYN)J ,were ca lcu la ted , forming th e LP sequence i n t he N-dimensional cube. For each /6p o i n t HQ,, a t e s t p o i n t -2) = (a12 ) ,.--,aNI ) ) was d et er mi ne d i n t h e p a r a l l e l e p i p e d [ l ] :

Then t he sys tem of equa t ions descr i b ing th e dynamic ob jec t be ing s t ud ie d wi thparameter O() was s ol v e d, and t h e q u a n t i t i e s o f i n t e r e s t were c a l c u l a t e d ,By looking through th e sequence of po in ts ( 2 = 0 , 1 , 2 , ...,k - l ) , t h e p"bes t" p o in t s were s e l e c te d , from which "improvementll of t h e para mete rs wasperformed by re gu la r methods. However, t he val ue of t h e f u n c t i o n a l s s t u d i e dwas improved only s l i g h t l y by r eg ul ar methods i n compar ison with t he b es t oft h e p p o i n t s .

T hu s, t h e s e s t u d i e s con fi rm ed t h e e f f ec t i v en es s o f g l o b a l LP s ea rc h i ncomparison wi th random se ar ch (t h e number of s t a t i s t i c a l t e s t s i s s i g n i f i c a n t l yr ed uced and t h e p r o b a b i l i t y o f f i n d i n g t h e g l o b a l ext reme f o r f u n c t i o n a l s wi thmult ip le extremes i s in cr ea se d) , and i n many cas es improvements are made i ncomparison with any regular search methods.

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REFERENCES1. Chzhu, S. Ya. and V. Prager, Latest Achievements in Optimal Planning ofStructures, Mekhanika, No. 6, (118), 1969.

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- - - -STIMATION OF DOMINANT P A F T E F S OF DYNAMIC SYSTEMS___ _ _ .I . I . Bal t runas (Vi l 'nyus)

State ment of Problem. The term "dominant parameters of a dynamic system"re fe r s t o : t h e t r a n s f e r c o e f f i ci e n t , t i m e constan t and pure de lay .

The prob lem co ns is ts of cons t ruc t ing a se l f - tu n in g mo d e l , i t s i n v e s t i g a t i on and es t i mat ion of th e dominant parameters o f t he dynamic sys tem.

Symbols. x( t) i s t h e i np ut s i g n a l ( i t i s assu med th a t x ( t ) i s a randomp r o ce s s , s t a b l e i n t h e b ro ad s e n se ) ; z ( t ) = y ( t ) + n ( t ) i s t h e o bse rv ed s i g n a la t t h e o ut p ut o f t h e o b j e c t ; y ( t ) i s nois e (assumed t o be a s t a b le Gauss ianp r o ce s s s i m i l a r t o " w hi te n o i s e" ) .

P ( t ) = A ( t ) + ( x ( t ) , g ( t ) ) i s t h e o u tp ut s ig n a l o f t h e mod el , A( t ) and B( t ) /zare i n t h e g en er al case t h e v ec to r s o f t h e tu ned p a ramete r s o f t h e model a t,. nmoment i n ti me t; A( t ) i s r e l a t e d t o y ) t ) l i n e a r l y and t o i ( t ) - -nonl inear ly ;,. ,.$ ( x ( t ) B ( t ) ) i s a known vector function of i t s argument x, B.y A(t ) , y g ( t ) are d e f i n e d f u n c t i o n s o f t ime ( i n p a r t i c u l a r , c o n s ta n t s );

y i s t h e in t e rv a l o f g l id in g summation, T i s t h e l e n g th o f r e a l i z a t i o n s o f pr oc e ss e s x ( t ) , z ( t ) .

Th e se l f - tu n in g a lg o r i th m i s ach iev ed u s in g th e fo l lo win g eq u a t io n s :

The q u a l i t y c r i t e r i o n Q, o f t h e a d a p ti v e p r o ce s s c o n s i s t s o f t h e f o l lo win g fu n c t io n a l s:

Q,(Ar, gr, t )= [z ( t )-A ( t ) p (x t ) , i ( t ) ) r,

6

-------.----..-.IIII~111111..11.1.111111111111.11111.1.1.l1ll111 111 1111111111111 111111I 1111111111111111 1111111111 111111


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One ch a r ac t e r i s t i c f e a t u r e o f pr ob lems o f t h i s c l a s s i s t h a t t h e v e c to rof parameters of ( f or example, th e t ime cons tant of t h e i n e r t i a l l e ng th ,delay) i s i n cl u de d n o n l i n e a r l y i n t h e e q ua t io n of t h e c i r c u i t .

The adaptive system i s s tud i ed by the method o f s t a t i s t i c a l modelingby d i g i t a l computer, i . e . , a disc re te ana log o f the con t inuous sys tem i ss t u d i ed . I n v es t i g a t i o n s a r e p er fo rm ed f o r :

1. Determinab i l i ty o f the dominating parameters of the dynamic system as af u nc t io n o f t he s t a t i s t i c a l c h a r a c t e r i s t i c s o f t he i n pu t s i g n a l x ( t ) .

2 . The accuracy of t he es t imat ion of th ese parameters as a func t ion of th e / Et y p e of ad ap t a t i o n a l g o ri t h m and t h e i n t en s i t y of n o is e n ( t ) . F o r t h i s purpose , the fo l lowing a lgor i thms are compared: t he gra di ent algori thm, algori thmwith gl i d i ng summation, s t oc ha s t ic approximat ion algor i t hm and the l ea s tsquares a lgor i thm.

3 . Cer ta in r ecommendations a re p resen ted on u t i l i z a t i o n o f t hese a lgor i thms .

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ADAPTIVE MODEL FOR EVA&UATION OF WEIGHT FUNCTION AND LEVEL OF NOISE OF LINEARDYNAMIC SYS-TKMA. A. Nyamura and K . I . Rashchyus (Kaunas)

A s tu d y i s p r e s e n t e d o f t h e p r o ce s s o f a d a p t a t i o n i n an adaptive model,d es ign ed fo r e s t ima t io n o f th e we ig h t f u n c t io n and n o i se l ev e l of a l i n e a rdynamic system. S in ce th e we igh t f u n c t io n o f t h e o b j ec t i s n o t a c c e s si b l e t od i r e c t o b se rv a t io n, ca l cu l a t i o n of th e we ig ht co e f f i c i e n t s o f th e model i sp er fo rmed on th e b a s i s o f r e s u l t s Of o b s e r v a t i o n o f p e r t u r b a t i o n of x ( t ) an d th eo u tp u t co o rd in a te s o f th e o b jec t z ( t ) by g r ad ien t methods and s to ch a s t i capproximation. Formulas a r e p resen ted f o r de terminat io n of t he speed of t head ap ta t io n p ro cess , i . e . , t h e p rocess of au to ma t ic ad ju stmen t of weight coe f f i c i e n t s o f th e model i n th e ca se o f a q u as i - s t ab le o b jec t and when n o i sen ( t ) a t i t s outpu t i s wh i te n o i se . I t i s d emo n s t r a t ed th a t t h e ad ap ta t io ntime i s l e s s wh en th e ad ap t iv e sy s t em o p e ra te s u s in g th e g r ad ien t a lg o r i th m.The durat ion of ad ap ta t io n o f a d i sc r e t e ad ap t iv e sys tem o p e ra t in g u s in g th ea lg o r i th m o f s t eep es t d escen t i s s t u d i e d .

A n experimental s tu d y i s presen ted of a d i sc r e t e ad ap t iv e sys t em b y th emethod o f s t a t i s t i c a l mod el in g u s in g th e UMI d i g i t a l co n t r o l machine . I n v es t i g a t i o n was pe rf or me d w i t h r e a l i z a t i o n o f t h r e e c h a r a c t e r i s t i c t y p e s o f i n p u t p e r t u r b a t i o n of x wit h th re e le ve ls of no is e n The dependence OS t h e r a t e o fi it u n in g o f th e we ig ht c o e f f i c i en t s on th e form of co r r e l a t io n fu n c t io n o f th einput s i g n a l a nd on t h e n o i s e l e v e l a t t h e o u t m t o f t h e o b jec t i s demonstrated.

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STUDY OF STOCHASTIC APPROXIMATION ALGORITHM B Y THE "MONTE CARLO"MEMOD, USED TO ESTIMATE THE WEIGHT FUNCTIONN . A. Arbachyauskene (Kaunas)

The es t im at i on of t h e weight fu nct ion of a l i n e a r dynamic system by th e 19method of s t oc ha s t ic approxiniat ion i s analyzed f o r th e case when t he numbero f r e s u l t s o f o b s e rv a t io n s i s f i n i t e . Estimates of t h e ve c to r o f weigh t co e f f i c i e n t s B are ca l cu l a t ed u s in g t h e f o r mu la :

&+, (I+ 1 ) = + 1 ( i)+y(i) [ z ( i + 1) -FT, , i) Bl+li)] F/+, i), (1)n T.where BZ,l i s an estimate o f t h e v e c to r we ig ht c o e f f i c i e n t B (bo,bl , . ..,bZ] ,

f f ( i ) i s t h e i n pu t s ignal; FZ+l( i ) = ( f ( i ) , f ( i - 1) ...f ( i - Z))T i s t h e v e c t o ro f t h e i np ut s i g n a l ; z z ( i ) i s th e ou tpu t s ig na l observed wi th no ise , i . e . ,z ( i ) = x ( i ) + n ( i ) ; n ( i ) i s t h e n o i s e a t t h e o u tp u t of t h e o b j e c t ; y ( i ) i sa s e r i e s o f p o s i t i v e num bers, s a t i s f y i n g t h e co n d i t i o n c y ( i ) -+ a ;m m Yz(i) < 03*i = O i =Oi i s the i t e ra t i on number, Z i s t h e number of components o f v ec to r B and F .

A s u f f i c i e n t l y p r e c i s e e s t i ma t e of v e c t o r 6 z+1 by th e method of s toc has t i c approximat ion , par t i cu la r l y wi th a high no ise l e ve l , can be produced on lya f t e r a tremendous number o f i t e ra t ions (10 -10 ) . The determinat ion o f t h i s5g r ea t number o f p o i n t s o f r e a l i z a t i o n s of i n p u t and o u tp u t s i g n a l s r eq u i r e s ag r e a t d e a l o f t im e . T h e re f or e , t h e p o s s i b i l i t y i s s t u d i ed o f u s i n g a r epea tedl i s t i n g a lgori thm, th e th eo re t i ca l bas i s o f which was p rov ided by B. M .Litvakov .

The accuracy o f r e s t o r a t io n o f a parameter i n a l im i ted number o f i t e ra t i o n s depends on t h e s t a t i s t i c a l c h a r a c t e r i s t i c s o f t h e i n p ut s i g n a l f ( i ) ,n o i s e l ev e l n ( i ) and t h e s e l e c t i o n o f a t u n i n g s t ep l en g t h y ( i ) . Only p a r a meter y ( i ) can be changed f re e l y .

Inves t i ga t ion by th e Monte Car lo method a l lowed ce r t a i n sp ec i f ic s o f th ea l go ri th m w i th r e p e at e d l i s t i n g t o b e e s t a b l i s h e d . The in f luence o f the

/Eol lowing fac to r s on the accuracy o f es t im at ion was de termined : a ) Inpu t s i gn a lco r r e l a t i o n f u n c t i o n ; b ) n o i s e a t t h e o u t p ut o f t h e o b j e c t ; c ) number o f re9


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I , ,, ... . . .. , ... ~ ~.

results of observations. Practical recommendations are given for selection ofthe tuning step length T(i) as a function of these factors.

10

____...-_..,.,... ... ,. ... .,._,. ,.,.,,., , .. I I I , I I , 1 1 1 1 I I, I I I I I 111 I I


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ESTIMATION O F PARAMETERS OF NONLINEAR HAMMERSTEIN OPERATOR.- .. .G . A . Rubikas (Kaunas)

A n o n l in e a r , u n i v a r i a t e , i n e r t i a l o b j e c t o f known s t r u c t u r e i s i d e n t i f i e dby a Hammerstein model of the following form:

m al,= Y k +uk = v j C i ( P ( X k - j ) + k=l, 2, .. (1)J -0 1-0

h e r e Ik i s the measured output of o b j ec t yk t o g e t h e r wi t h ad d i t i v e n o i s e 5a c t i n g on t h e o u tp u t ; { + i ( x ) ) i s a sys tem of l i ne ar ly independen t func t ionso f i n p u t s i g n a l x, v j '*-j = O , m , i = O,n a re unknown coe ff ic ie nt s , es t i mat i ng the correspondingp a r am e t e r s o f t h e o b j ec t s .

Var ious mathematical methods a l low us t o cr ea te var iou s a lgor i thms f o rde te rmina t ion of v j ' ci In o rder t o improve th e i t e r a t i o n p rocedure , we re p lac e the p a i red p roduc ts w .c by a s i n g l e c o e f f i c i e n t h j i . Ca lcu la t ion canJ ibe performed usin g th e fo llowin g two alg ori thms : 1) an algorithm produced bythe le as t squares method ( the M NK algorithm) and 2) an al gorith m produced byth e s t oc ha s t ic approximat ion method ( th e SA algor i thm) .

The MNK a lgor i thm:hv+l=r,-:IL+l

Ahere hv+l i s t h e v e c t o r h i n t h e (v + 1 )- t h i t e r a t i o n ,-1r w + l i s a mat r ix cons ider ing the inpu t s ig na l x in the (v + 1 ) - t h i t e r a t i o n

I ryl;:l = (1 + t ) r;l- (v + r;l pv+l)-lv r;r Q ~ + ~where

@v = pv (Pi.when

cp:'[cp (xv-o). . . ., 9. (Xv-0). cPoj+)r . *cpa(Xv-l), .. ., 9 0 (x v -m)9 * * * cP, (x . -31 ;

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h e r e Zu+l i s t h e o u t p u t o f t h e o b j ec t m easu red t o g e t h e r w i t h n o i s e uU + l 'L0 = 2 l 0 - l = aI, where I i s a u n i t m a t ri x ;0 3 0

a i s a c er ta in number, 1 2 u 2 1000.The SA a lgor i thm-

h e r e h U + l , q = 0, m , E , = O,n i s t h e En -th c o e f f i c i e n t i n t h e ( v + 1 ) t h e i t e r a t i o n5n{ + i ; ( x ) 1 i s a s ys tem o f l i n ea r l y i n depend en t f u n c t i o n s o f t h e i n p u t s i g n a l= >

u i s t h e j t e r a t i o n n umber, 0 < a < 1 ( a = 0.55) .The a lgor i thms sugges ted a l low th e c oe f f i c i en t s o f expres s ion (1) t o b e

d et er mi ned w i th a s i g n i f i c a n t l e v e l o f n o i s e a t t h e o u t p u t . I t should be notedt h a t t h e i t e r a t i o n sp ee d and q u a l i t y o f t h e M N K a lgor i thm i s b e t t e r t h an o f t h eS A algor i thm.

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ESTIMATION OF PARAMETERS AND CHARACTERISTICS OF NONLINEAR DYNAMIC SYSTEMS___ .- . .A. K . Nenorta (Kaunas)

Certain methods are s t u d i e d f o r d et er mi ni ng t h e c h a r a c t e r i s t i c s and p a r a meters of nonl inea r dynamic sys tems on th e bas is of o b s e rv a t i o ns of t h e i r i n p ut q u a n t i t i e s x ( t ) and o ut pu t q u a n t i t i e s y ( t ) .

A non lin ea r dynamic system can be desc ribe d approximately by a segment of / zt h e f u n c t i o n a l v o l t e r r a se r ie s

P 0)= wo( t ) + J Wl U x ( t- Xl) dh, +0

m m+J J wgLAJ x ( t -A,) x ( t- h8)~ h ,,0 0

where w0( t ) i s a f un c t i on , d ep en de nt on t h e i n i t i a l s t a t e o f the sys tem;w ( A1 1) i s a w ei gh t f u n c t i o n , d e f i n i n g t h e l i n ea r p o r t i o n of t h e o u tp u tq u a n t i t y ;w2(A1,A2) i s a weight fu nct ion , de fin in g th e second order component of

t h e o u t p ut q u a n t i t y ;y ( t ) i s t h e outp ut va lue of a model of t h e dynamic system .

I n o rd e r t o es t imate th e non l ine ar dynamic sys tem opera to r be ing s tu d ie d ,we assume w 0 ( t ) = 0. Est imates w i l l be sough t i n t he fo l lowing form:

where + j(A1) , (A2) i s a sys tem of known continuous funct i ons , sa t i s f yi ng t hecond i t ion $ . ( A1 1) = +P A 2) , i f j = p and A 1 = A 2 , th e method of s el ec t i on ofwhich i s d is cu ss ed i n t h i s r e p o r t ;

aj p ( j = 1 , 2 , ...,k ; p = 0 , l ,. .. , k ) ar e unknown coe f f ic ie n t s , e s t imate s ofwhich must be found. I t i s assumed t h a t th e weight fu nc tio ns wl(A1) andw2(A1,A2) can be p r e c i s e l y approximated by fu nc ti on s ;(Al) and w 2 ( A 2) res pec t ive ly . Cons ider ing (2 ) and 3), expres s ion (1) can be wr i t t en as fo l lows :

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where we r ep r e s en t

If i n t h e p ro ce ss o f p r o du c t i o n o f d a t a , a l l v a l u e s o f t h e i n p u t an d o u t p u t /Eq u a n t i t i e s x ( t ) and y ( t ) are p r e c i s e l y de f in e d, c a l c u l a t i o n of estimates ofc o e f f i c i e n t s a r e q u i r e s as many v a l u e s o f q u an t i t i e s x ( t ) and y ( t ) a s arej pr e q u ir e d t o d et er mi ne e s t i m a te s o f c o e f f i c i e n t s aj p ' But i n p r a c t i c e w e canm eas ure p r ec i s e l y o n l y i np u t q u a n t i t y x ( t ) . The measurement e r r o r n( t ) i no ut pu t q u a n t i t y y ( t ) w i l l b e co n si d e red ad d i t i v e n o i s e , n o r ma ll y d i s t r i b u t e d .

AThen estimate a can be sought by solving the redundant sys tem of equat ionsj p

by th e method o f l e a s t squares o r th e method of s tochas t ic approx imat ion ,where z ( i ) = y ( i ) * n ( i ) i s t h e r e s u l t o f o b s e rv a ti o n of v al ue y ( t ) i n t h ep r es ence o f n o i s e n ( t ) a t d i s c r e t e moments i n time t s = l , Z , . . .S

The accuracy of the estimates produced was determined by ex per imentals t u d ie s of th ese methods using t h e BESM di g i t a l computer by th e method ofs t a t i s t i c a l t e s t i n g .

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IDENTIFICATION .AS -A. -TASK OF _ _ ~. . . SEQUENTIAL STATISTICAL ANALYSIS-A . A . Nyamura (Kaunas)

For t h e pr ob lem o f i d en t i f i c a t i o n , it i s necessary f i r s t of a l l t od i f f e r e n t i a t e c l a s s e s wi th t h e f ol lo wi ng c h a r a c t e r i s t i c s : n o n l i n ea r or l i n e a r ,s t a b l e o r u n st a bl e , i n e r t i a l o r n o n i n e r ti a l . A f t e r a s u f f i c i e n t l y n arro w c l a s sof an oper ator has been def ined, th e op erat or i t s e l f becomes known with anaccuracy t o a c e r t a in vec t o r parameter c , i . e . , we have the fo l lowing functional dependence:

P ( r ) = A r y ( t ) ] = ~ . [ 1 1 ( S ) ,O < . s < r ; c ( s ) , p < s < t ; ?I,where A [ * ] i s th e mathematical op er ato r of th e system, unknown i n advance,requ i r ing de te rmina t ion ; u( t ) and y ( t ) a r e the inpu t and ou tpu t genera l izedvec tor coordina tes of th e sys tem, which may be d et er mi ni s t ic o r random funct ionsof scalar argument T. The l a s t t i n t h e f o rm ul a i n d i c a t e s t h a t t h e f u n c t i o n a l /14ependence @ [ ' I may change as t im e p a s s e s . V a r i ab l e u ( t > and y ( t ) a r e observedunder the inf luence of random noise . I t t h en rema in s t o estimate the unknownvec t o r parameter c .

C r i te r i a of o pt i ma l i t y of p lan s of observa t ion and some problems ofrecogn i t i on o f th e c las s o f opera to r A ar e s tu die d, and a complete ana lys isof methods of es t i mat i on of ve cto r parameter c i s performed.

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DIFFERENTIATION O F RANDOM FORCED OSCILLATIONS AND SELF-OSCILLATIONS PERTURBEDBY RANDOM ACTIONSA . A . Gorbunov and M. F . Dimentberg (Moscow)

The problem con s i s t s i n determining whether a measured narrow band randomprocess a t t h e o u t p ut o f a n o s c i l l a t i n g s ys te m c o n s i s t s o f f o r ce d o s c i l l a t i on s of t he sys tem under the inf lue nce of ex te rn al wide band random per tur bat io ns o r whether t he sys tem i s i n t h e s t a t e o f s e l f - o s c i l l a t i o n s , modeled byrandom noise as a r e su l t o f random per tu rb a t io ns a t t h e i n p u t o f t h e s y s t em .The so l u t ion t o t h i s p roblem i n many cases can i nd ic a t e what source o fe x c i t a t i o n i s making t h e b a s i c co n t r i b u t i o n t o t h e v i b r a t i o n o f t h e m ech an icalsystem.

Two recog n i t ion c r i t e r i a a re sugges ted fo r an ob j ec t which can be r epre sented by a second o rd er equ iva le n t qua s i - l in ea r sys tem, based on ana l ys i s o ft h e p r o b ab i l i t y d en s i t i e s o f t h e en ve l op e o f t h e o u t p u t p r o ce ss o r t h e p r o ce s si t s e l f . The f i r s t cr i t e r ion fo l lows f rom so lu t ion o f the Fokker -P lanckeq u a t i on , co rr e sp o n di n g t o an abbrev ia ted equa t ion fo r the ampl i tude . Thes ec on d c r i t e r i o n i s produced on th e b a si s of th e known in t eg ra l dependencebetween t h e p r o b a b i l i t y d en s i t i e s o f a narrow band random process and i t senvelope. The accuracy o f t he second c r i t e r io n has been tes te d by ana logcomputer modeling and was found t o be good even i n the case o f r a t he r s hor ti n t e r v a l s o f o b s e r v a ti o n o f t h e o u t p u t p r o ce s s .

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DETERMINATION OF PARTIAL DAMPING COEFFICIENTS OE LINEAR SYSTEMS FROM MEANPERIODS OF ENVELOPES OF RANDOM OSCILLATIONSM . F . Dimentberg and A. R. Abul'khanov (Moscow)

A formula i s d e r i v ed f o r c a l c u l a t i o n o f p a r t i a l damping co e f f i c i e n t s of gl i n e a r systems on the b as i s of v al ue s of t he r esona nt freque ncy and mean numbero f i n t e r s ec t i o n s o f t h e env e lo p e o f a narrow band random o sc i l la t i on with i t smathematica l exp ec ta t io n . The formula i s concluded i n the co ns idera t ion o f th ei n f l ue n c e o f t h e f i l t e r - a n a l y z e r u s ed t o s e p a r a t e t h e o s c i l l a t i o n s i n t h edegree o f f r eedom be ing s tud i ed . Resu l t s a re p resen ted from experimenta lchecks of the accuracy of th e method us ing e l ec t r i c a l models.

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INTEGRAL EQUATIONS AS A METHOD OF DETERMINING THE SPECTRUM OF NORMAL MODESOF BACKGROUND VIBRATIONSP . F . Ovchinnikov (Odessa)

Methods a r e s t u d i e d f o r d e te r m in i ng t h e s p ect ru m of n a t u r a l o s c i l l a t i o n sof systems, t h e o s c i l l a t i o n s o f which a r e d e s c r i b ed by l i n ea r , l oaded i n t eg r a leq u a t i on s w i t h an e s s e n t i a l l y p o s i t i v e k e r n e l and a non-monoton ic d i s t r ibu t ionf u n c t i o n . I n c o n t r a st t o a spectrum with a m o n o t o n i c d i s t r i b u t i o n f u n c t i o n ,t h e n a t u r a l s pec tr um of t h e s e eq u a t io n s f a l l s on both s ides of the number axisI t i s demonstrated t h a t n ei th er methods of composicion of f requency equat ionsand t h e i r conversion t o s ec u la r form, nor methods o f expansion o f the sec u la requat ion t o a polynomial can be extended t o equat io ns with non-monotonic d i s t r i b u t i o n f u n c t i o n s .

The po ss ib i l i t y o f us ing the method o f succe s s ive approx imat ion l o r ca l cu la t io n o f t he na tu r a l number wi th the lowes t abso l u te va lue i s demonstrated.

Exten sion o f t h e method o f wakes allows any na tu ra l number t o be def inedas t o a b s o lu t e v a l ue . Af te r demons t ra t ing t ha t th e p roblem of d e te rmina t ionof natural numbers can be r educed to a c e r ta in var ia t i on a l p rob lem, th e pos s i b i l i t y o f u s in g t h e method o f R i t z i s e s t a b l i s h e d .

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APPLICATION OF CORRELATION ANALYSIS FOR DIAGNOSIS OF THE CONDITION OF MECHAN SMSI . V . Alekseyev and E . P . Pyshkina (Moscow)

Knowledge o f such os c i l l a t in g ch ar ac te r i s t i c s o f pa r t s o f mechanisms as /Eth e damping fa c t o r , na tu ra l f r equency and spe c t ra l no i se power de ns i ty i so f i n t e r e s t f o r p ur po se s o f a c o u s t i c a l d ia g n o s is .

I n o r d e r t o d et e rm i ne t h e s e ch a r ac t e r i s t i c s and e s t i m a t e t h e i n f l u en ce o fdes ign parameters o f p ar t s on the p roces s o f no i se fo rmat ion , the Depar tmentof Engines of t h e Moscow Highway I n s t i t u t e has produced a moto rle ss modelingin s t a l l a t i on , a b lock d iagram of which i s shown on Figure 1. T h i s i n s t a l l a t i o n , us ing a co r r e l a t o r p roduced i n Department E - 9 of t h e Bauman MoscowHigher Technical School, has been used to d e te rm i ne th e ch a r ac t e r i s t i c s ofindividual par ts ment ioned above.

T h e p a r t s t e s t e d were modelsr of cy l i n d e r h eads o f i n t e r n a lcombustion engin es. Four ve rsi on so f head s were s t u d i ed , d i f f e r i n gfrom each o th er i n th ickness o fend wal l and co ns t ruct ion mater i a l . The cy l in der heads have th e

Figure 1. f o l l o w i n g r ad i a t i n g wall t h i ck nesses : 3 k 0.05 nun; 4 . 5 k 0 . 05mm; 5 . 2 5 k 0.05 mm; and 6 k 0.05 mm, and were made of type 45 and 18Kh s t e e l s .The s e l ec t i o n o f a d e si g n f o r a t h i n wall head was based on the followingcons idera t ions :

a) The end wal l o f t h e head i s made so t h a t i t s r i g i d i t y i s much less thant h e r i g i d i t y o f t h e r em ai ni ng el em en ts o f t h e d e s ig n o f t h e i n s t a l l a t i o n ,a l lowing the as sumption th a t the p r imary share o f a co us t ica l energy rad ia te d byt h e cover i s d e te rm i ned b y o s c i l l a t i o n s of i t s t h i n w a l l por t ion ;

/Eb) The design of t h e h ea d a ll ow s a n a l y t i c c a l c u l a t i o n of t h e n a t u r a l f r e

quency of i t s r a d i a t in g su r face , making it p o s si b l e t o e va lu at e t h e r e l i a b i l i t yo f t h e ex p er i m en t a l d a t a .

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Exper imental de termina t ion of t he damping f ac to r and na tu ra l f requencyof heads was performed on the ba s i s o f t he parameter s o f th e co r r e l a t io n funct i o n of t he random pr oce ss, measured usin g a v i b r a t i o n s e n s o r i n s t a l l e d ono ne s i d e of t h e r ad i a t i n g s u r f ace o f t h e h ead . The v i b r a t o r was i n s t a l l eds ym metr ic a ll y t o t h e v i b r a t i o n s en s o r on t h e o t h e r s i d e o f t h e h ead , a l lo w in gt h e form of e xc i t a t io n dur ing th e experiment t o approx imate th e form of ac tua le x c i t i n g f o r c es a c t i n g on t h e ra d i a t in g e lement and thus avo id ing undes i rab lea s y m m et r i c a l o s c i l l a t i o n s an d t h e i r co r r e s p o n d i n g n a t u r a l f r eq u en c i e s .

Voltage from a n o i s e g en e r a t o r was s en t t o t h e v i b r a t o r . The v o l t ag efrom the v ib r a t i on sen sor was se n t t o a c o r re la to r , which was used t o de t e r mine t h e p a ram et e rs o f t h e co r r e l a t i o n f u n c t i o n . The o s c i l l a t i n g ch ar ac t e r i s t i c s o f t h e h ead a r e un amb ig uo usl y r e l a t e d t o t h e p a ram e te r s o f t h eco r r e l a t i o n f u n c t i on . The an a l y s i s o f t h e o r e t i c a l and ex pe r im ent a l d a t a f o rdeterminat ion of the se cha ra ct e r i s t ic s showed good correspondence. The re s u l t s p roduced demonst rate t ha t t he use o f a c o r r e l a t i o n a n a ly z er s i g n i f i c a n t l yfa c i l i t a te s th e measurements per fo rmed.

The us e o f c o r r e l a t i o n an a l y s i s h a rd l y exc l ud es t h e u s e o f c l a s s i c a lmethods of a co us ti ca l measurements. The expediency of t he use of any givenmethod i s d et er mi ne d by t h e p r a c t i c a l s i t u a t i o n : a v a i l a b i l i t y o f a p pa r at u s,i t s complex i ty , cos t , na tu re o f t h e p ro c es s b e in g s t u d i e d , e t c .

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.. . .. ........ .. ... . .. . , ...

METHOD A N D DEVICE FOR QUASI-PARALLEL ANALYSIS OF SHORT-TERM ACOUSTICAL SIGNALSI . V . Alekseyev and V . N . Lukanin (Moscow)

We have developed a measurement channel f o r spe c tr a l analy s is of noi seand v i b ra t io ns o f eng ines dur ing opera t ion (F igure 1 ) . I t c o n s i s t s o f a s e tof 24 one- th ird octave f i l t e r s , t h e i n p u t s o f which r e c e i ve t h e s i g n a l t o b ean al y zed fro m t h e v i b r a t i o n s en s o r i n s t a l l ed o n t h e en g i n e an d f i r s t subj e c t e d t o a m pl i fi c a ti o n. A swi tch ing dev ice , co n t r o l l ed by pu lses marking th e /=t o p d ead cen t e r p o i n t , co nn ect s t h e o u t p u t s o f e ach o f t h e f i l t e r s i n t u rn t oth e recordi ng device , an 8-beam cathode ray osc i l lo sco pe with a photographicattachment. The swi tch ing devi ce used i s a s t ep p i n g s w i t ch , t h e co n t a c t s o fwhich a r e c losed i n pa i r s . Th is connec tion o f con tac t s a l lows the s i gn a l frome ach f i l t e r t o b e f e d t o t h e o s c i ll o sc o pe d ur in g two f u l l c ra nk s h a f t r o t a t i o n s , which co r re s po n ds t o t h e l en g t h o f t h e o p e r a t i n g cy c l e i n a f o u r s t r o k eeng ine . Thus, the s i gn a l be ing s tud ied i s r ecorded a f t e r hav ing been passedt hr ou gh t h e o n e- t h i rd o c t av e f i l t e r s .

The c o i l o f t he s tepp i ng swi tch i s connected t o the anode c i r cu i t o f athyratron, which i s opened by a pul se . Recordings of noi se , v ib ra t i on s , aswel l as t h e p r o ce s se s d e f i n i n g t h e o p e r a t io n o f t h e e n gi ne , t h e i n d i c a t o r d i a gram and t r av e l o f th e sp rayer need le , pe rformed us ing th i s channe l , demons t r a t e t h e p o s s i b i l i t y o f i t s u t i l i z a t i o n f o r d i ag n o st i c p u rp os es . The pulseso f v i b r a t i o n and no i s e r e s u l t i n g from t h e i m pa ct i n t e r a c t i o n s i n t h e s t r u c t u r e so f t h e en g i ne a r e s ep a r a t ed and can be an al y zed t o f i n d r e l a t i o n s h i p s w i t hparameter s char ac te r i z ing t he t echn ica l cond i t ion of the eng ine .

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USE OF A CORRELATOR BASED O N ORTHOGONAL FILTERS I N ACOUSTICAL MACHINE DIAGNOSISYu. I . Bobrovnitskiy, M . D . Genkin and K . D . Morozov (Moscow)

I t i s s u gg es ted t h a t t h e i n t e r n a l p a r ame t e rs o f m achi ne s ( d i ag n os i s p a r a meters) be measured usin g an or thogonal f i l t e r c o r r e l a t o r . I t i s demonstratedt h a t t h e p a r am e te r s o f the f i l t e r a t t h e i n p u t o f o ne o f t h e ch ann e l s can b es e l e c t e d s o t h a t a s i g n a l i s produced a t th e ou tpu t of t h e co r r e l a t o r whichi s prop or t iona l t o th e measured machine diagnos is parameter . The con s tru ct io nof such a f i l t e r r eq u i r e s t h a t c e r t a i n co r r e l a t i o n m easu rements b e p e rf or medi n advance wi th s eve ra l known va lues o f th e d iagno s i s parameter, i . e . , t h a t a" lea rn ing p rocess i s needed.

A model d i ag no s t ic experiment i s performed. The model of the diagnostics i gn a l was a mixture o f two random, s t r on g ly co r r e l a t ed s i gn a l s , th e ampli tudeof one of which was propor t iona l t o the measured d iagno s i s parameter , Afterl e a r n i n g , t h e co r r e l a t o r m easur ed t h i s p a r am e te r w i th an e r r o r n ev e r ex ceedi n g10%.

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DETERMINATION OF EQUIVALENT RANDOM PERTURBATION DURING MODELING OF COMPLEXDYNAMIC SYSTEMSL. A . Manashkin and N. G . Baranov (Dnepropetrovsk)

Optimizat ion of th e parameters of damping devic es requ ire s t ha t a model of /19the equipment be perturbed so th at the damping devices oper ate under nea rnormal con diti ons . This i s done by feeding a voltage from a n o i s e g en e r a t o rwi th a r a th er b road spec t rum t o a s y n t h e s i z e r . The signal formed by the synt h e s i z e r i s f ed t o an e l e c t r i c a l model. The ou tpu t qu an t i t i es o f the model a reformed s o th a t they can be compared wi th the r e su l t s o f exper imenta l s tu d ie sfrom a ( s tandard) a c tu a l dev ice under acu ta l cond i t io ns . The re su l t s o fi n v e s t i g a t i o n o f o s c i l l a t i o n s o f t h e l i n k o f i n t e r e s t i n t h e a c t u al d e vi cea r e r e pr e se n t ed a s t h e s p e c t r a l c h a r a c t e r i s t i c s o f t h e s e o s c i l l a t i o n s . Thee l ec t r i c a l v o l t ag e p ro du ced i n t h e m odel and co rr e sp o nd i ng t o t h e o s c i l l a t i o n so f t h e u n i t b e in g s t u d i e d i s s en t t o an an a l y ze r . T h e r e s u l t s o f an a l y s i sar e compared with th e f i xe d spectrum produced prel im ina r i l y dur ing exper imentals t u d i e s o f t h e o s c i l l a t i o n s o f t h e a c t u a l e q ui pm en t. By c o n t r o l l i n g t h e sy nt h e s i z e r , approximate correspondence of th e s pe ct ra produced exper ime ntal lyand by modelling can be achiev ed. The corresponding pe rt ur ba ti on w i l l beeq u i v a len t t o a c t u a l p e r t u r b a t i o n from t h e s t an d p o i n t o f t h e o p e r a t i n g co n di t i o n s o f t h e u n i t b e in g s t u d i e d .

A block diagram of the system as a whole, a block diagram of t he s ynthes i z e r and o f the ana lyzer a re developed.

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IDENTIF IC ATION OF MODEL OF HUMAN OPERATOR WITH R A N D O M VIBRATION ACTIONS. . . .__ . . . . . . . . . . . . . . . . . - . . . . . . . - . . . . - . .L. V. Goykhaman, B . A. Potemkin and K . V . Frolov (Moscow)

A dynamic human operator system influenced by wide band random (whiten o i s e ) p e r t u r b a t i o n i n t h e f r eq uency r an g e up t o 200 Hz i s s t u d i e d .

The problem i s so lved o f de te rmin ing th e parameter s of t h e dynamic humanope ra tor system and co nst ru cti on of a model f o r each po s i t i on . The body of the /20o p e r a t o r i s looked upon as a l i n e a r dynamic model co ns is t i ng of se r ie s -connected abso l u te ly r i g i d bod ies wi th masses mi connected by e l a s t i c anddamping elements k i and C i .

*. +----- - __ -~I 4 t t J,r ad / s ecFigure 1.

A method i s p r es e nt e d f o r s e l e c t i n g t h e l en g t h o f t h e r eco r d i n g o f t h ei n p u t pr o ce s s t o a s s u r e i t s s t a b i l i t y . D i s tr i b ut i o n r u l e s o f t h e o u t p u t p ro ce s s a r e f ou nd f o r e ach p o s i t i o n , and t h e co r r e l a t i o n f u n c ti o n s and s p ec t r a ld en s i t y f u n c t i o n s a r e c a l c u l a t ed f o r t h e i n p u t and o ut p ut p r o ce s s e s .

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Figure 2 .The number o f equ at io ns of motion of th e model i s determine d by th e form

o f t h e f re qu en cy c h a r a c t e r i s t i c f o r e ac h p o s i t i o n . The equat ions s t udi ed were /=u se d t o f i n d a n a l y t i c e x p r es s i on s f o r t h e a m pl it ud e- f re qu en cy r es p on s es :

IIc BIW'G ( O ) = i m (1)c A i m '

i 0The area l i m i t a t i o n s are f i x e d i n t h e form Emi = M , where M i s t h e mass

of the human operator .By se le c t in g N values of th e exper imenta l f requency response a t

f r equenc ies w and se t t i n g them equa l t o express ion (1) f o r ea ch o f t h e f r e jquenc ies , a syst em o f non l inea r a lge bra i c equa t ions i s p ro du ce d r e l a t i v e t oth e model para mete rs t o be de te rmined .

In th i s case , t he nodal f r equenc ies co r respond t o the ex tr eme po i n t s o ft h e e x p e ri m en t al f re qu en cy c h a r a c t e r i s t i c . Calc ula t i on of th e model para meters mi ci and k .1 was done on th e BESM-3M computer .

T h e p o s s i b i l i t y i s demonst ra ted of us ing an a lgor i thm al lowing calcu lat i o n o f parameters by t h i s method wi th a f i x e d a cc ur ac y, I n t h i s c a s e , t h e / z

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number of nodal f requencies N may exceed the number of unknown parameters.A s a r e s u l t o f th e work, dynamic models a r e produced f o r each o f t h e p o s i

t i on s wi th numer ica l va lues of th e parameters .

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MULTICHANNEL DIAGNOSIS OF MACHINES B Y DIGITAL COMPUTER. . . - * . . . . . . . - . . . . _ . . . _ __ _ _ _B. D. Tar takovsk iy , A. I . Vyalyshev and B. A . Kanayev (Moscow)

Exis t ing methods o f d iagnosis of machines and mechanisms based on measurement o f t h e i r v i b r a t i o n - a c o u s t i c a l c h a r a c t e r i s t i c s h ave t h e s i g n i f i c a n t s h o r t coming t h a t t h e c h a r a c t e r i s t i c s of t h e a c o u s t i c a l e le m en ts o f s t r u c t u r e s o fmachines and t h e base s on which they r e s t , a s w e l l a s s e c t o r s o f t h e s o un dconductors a long which v ib ra t i on s and ac ou st ic a l waves p ropagate f o r recep t ionb y th e t e s t i n s t ru m e n ts , i n f l u e n c e t h e r e co r de d r e s u l t s . T h is i n f l u e n c e ,b e in g e s se n t i a l l y dep en dent on th e p o i n t o f p l acement o f t h e t e s t v i b r a t i o nand sound sen sor s, i n most cases cannot be th eo re t i ca l l y ca lc u la t ed . However,exper imenta l checks o f th is in f luence have shown tha t i t i s p r a c t i c a l l y i m p o s s i b l e t o e l i m i n a t e it b y p r o p er s e l e c t i o n o f t h e po i n t .

One method of s uppr ess ing t h i s i nf lu en ce , which must be analyzed f o r mostc a s e s o f r e a l i z a t i o n o f d i a g n o s is s ys te ms as a random ca tegory , co ns is ts i nusin g mult ichanne l ac ou st ic al dia gnos is systems (MSD) . An exp erim enta l st udywas performed f o r se l ec t i on of MSD parameters , cons is t i ng i n th e de terminat ionof exped ient f requency bands and dis tan ce s between sens ors .

The m ac hin e s e l e c t e d f o r s t ud y was a d e f e c t i v e e l e c t r i c m ot or , i n s t a l l e don a long , r ig i d f rame. The sen s o r s u sed were p i ez o e le c t r i c v ib ro me te r s ,p laced on the frame a t var i ous d is tan ce s f rom t h e motor . The measurementsp er fo rm ed i n t h e s t a b l e mode u t i l i z e d f i l t e r s o f v a r io u s ty p es wi th v a r io u swid ths o f f requency band . The re s u l t s were p rocessed by d i g i t a l computer us inga s p e c i a l a l go r it h m b as ed on s t a t i s t i c a l m et ho ds .

V al ue s c h a r a c t e r i z i n g t h e a ve ra ge c ha nge i n v i b r a t i o n s r e s u l t i n g from t h eappearance of a d e f e c t were fo un d. The minimum band wi dt hs of f i l t e r s andminimum d is t ance s be tween v i br a t io n rece p tors p rov id ing t he oppor tun i ty t o / zan a ly ze th e s ig n a l v a lu es p ro du ced a t t h e c e r t a i n p o i n t s on ce r t a in f r eq u en cybands a s noncohe rent were dete rmined .

Th is all owe d th e maximum number o f inde pen den t va lu es of v i b r a t i o n -a c o u s t i c a l s i g n a l s o c c u r ri n g i n t h e e xp er im e nt t o b e d e te rm i ne d , and on t h i sb a s i s a ll ow ed t h e co nf id en ce i n t e r v a l s f o r s t a t i s t i c a l e s t im a t i on of t h e re l i a b i l i t y o f t h e mean d i f f e r e n c e be tw ee n s i g n a l s r e l a t e d t o a d e f ec t t o b e

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fo un d, i . e . , a ll ow ed e s t im a t i on o f th e r e l i a b i l i t y o f t h e d i a g n o s t i c e f f e c t .In t he experiment performed, th e random change i n t he mean sig na l l ev el res u l t i n g f rom no i s e a t i n d i v i d u a l p o i n t s was o nl y s l i g h t l y l e s s t h an t h e d i f f e r en ce b etween l ev e l s r e su l t in g from th e d e f ec t . However, t h e u se o f t h i smethod of da ta p roc ess i ng f o r d a t a p roduced us ing mul t ichannel measurementsa l lo wed th e ex i s t en ce o f a d e f e c t i n t h e e l e c t r i c m ot or t o b e e s t a b l i s h e d w it ha p r o b a b i l i t y o f no l e s s t h a n 0 . 8 .

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- -

CYBERNETIC METHOD OF STUDYING VIBRATION ACOUSTICAL CHARACTERISTICS OF COMPLEX. . . .. - . . ~~STRUCTURESB . D . Tartakovskiy and G . S . Lyubashevskiy (Moscow)

The ess enc e o f an al gor ith m which minimizes a s e t of measurements [l] cons i s t s o f c a l c u l a t i o n o f t h e most s u i t a b l e i nc re m en t f o r t h e argument (from af i n i t e se r i e s of d i s c r e t e v a l u es accep ted) f o r th e nex t s i ng le measurementbased on in format ion a l read y accumula ted i n d i g i t a l computer s t o r ag e concern ingbehavior of t h e r e l a t i o n be i n g s t u di e d . AS a r e s u l t , t h e d e n s i t y o f p o i n t sof measurement becomes uneven: i t i n c r e a s e s n e a r s e c t o r s o f h i g h i n f o rm a t io nvalue--extremes--and dec re as es between them.

The e f f e c t i v e n e s s o f t h e a l g o ri t h m w i t h v a r i a b l e s t e p i n t h e c ha ng e o fth e a rgument depends , o f course , on th e c la ss o f dependence be ing s t ud ie d .Machine modeling of an algo ri th m as ap pli cab le t o th e frequency response of ano s c i l l a t in g l in k wi th o ne d eg ree o f f reed om i n th e au d io f r equ en cy r an g e witho p t ima l ly se l ec t ed co n t ro l p a r ame te rs in c lu d ed in th e a lg o r ith m in d ica tedt h a t

1) Redu ction of t h e measurement t i m e (number of measurements) u si ng t h i sa lg o ri th m i n co mpari so n wi th e q u i d i s t an t d iv i s i o n o f th e a rg umen t wi th eq u a laccuracy o f th e sea r ch f o r th e ext reme i s

where n i s t h e number of terms i n t h e se r i e s o f d i s c r e t e v al ue s o f t h e i n c r e - Ement of t h e argument, Q i s t h e f i g u r e o f m e r i t o f t h e o b j e c t be i n g s t u d i e d ,A < 102 i s t h e r a t i o o f t h e ext reme v a lu e o f t h e fu n c t io n to t h e zone o f i n - se n s i t iv i t y , b elow which v a lu es o f th e fu n c t io n a r e n o t o f in t e r e s t and a r en o t us ed i n c a l c u l a t i o n , 1 < E < m i s t h e q u a l i t y p ar am e te r i n c l ud e d i n t h ea lgor i thm;

2 ) The re du ct io n i n measurement time using th e a lgor i thm suggested i n comp a r i so n t o a co n tin uo u s l in ea r ch an ge in f requency wi th cond i t ions o f equalaccuracy of s e a r ch f o r t h e e xt re me v a l u e o f t h e f u n c ti o n i s

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where A f i s t h e wid th o f th e t r an smiss io n b an d o f t h e an a ly z in g ch an n e l, F i sth e range of change of f requency, 6 i s t h e f i x e d e r r o r i n s e ar c h f o r t h e e xtreme. Fur thermore , when reproducing funct ions accord ing t o t h e a lgor i thmsuggest ed, th e displacement of th e extreme by frequency and i t s expansion,ar i s in g wi th a con t inuous change i n f requency , are e l im i na t ed i n p r i n c i p l e ,

Thus, f o r example, where E = m, A f = 1/3, Q = 2 0 , F = 2-104, 6 = 5*10-2,A = 1 0, t h e g a i n i n t im e Bt h e a lg or it hm , p a r t i c u l a r l y f o r a high-Q system with a narrow transmission band

lo4, which i n d ic a t e s th. e h ig h e f f e c t i v en ess o fi n t h e an a lyz in p ch an ne l . A s the number o f extrema1 maxima q i s i n c r eased ,t h e e f f e c t i v e n e s s of t h e a lgor i thm decreases approx imate ly in t h e r a t i o o f2q + 1.

REFERENCESLyubashevskiy, G . S., Yu. I . Matveyev and B. D . Yaratkovskiy, Tr. V I Vssesoyuznoy Akust . Konf.. S e k t s f y a B [Works of Sixth All-Union Acoustical Confer en ce , Sec t io n B] , Moscow, 1968.

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- -AUTOMATION OF PROCESSING OF EXPERIMENTAL DATA AND DEVELOPMENT

- . ._OF- - - .-RECOMMENDAT-IONS FOR SELECTION.- .OPTI-MAL VIBRATION-ABSORBING COATINGB. D . Tartakovskiy and B . A . Kanayev (Moscow)

Automation of t h e s el e ct i on of recommendations designed f o r improvement oft h e v i b r a t i o n - aco u s t i c a l p r o p e r t i e s o f an o b j ec t by v i b r a t i o n ab s o rp t i o n ont h e b a s i s o f d a t a f ro m ex pe r im ent a l s t u d i e s o f v i b r a t i o n s o f t h e o b j e c t gassumes knowledge of t he dependence of th e qua nt i t ie s c hara cter iz in g v ib ra tion (mean amplitude, mean square ampl i tude , e tc . ) on th e parameter s o f thevibr atio n-a bsor bin g coa ting s (VC) t o be used. For example, assuming i n thef i r s t approximation

( < A i ) and ( 2 : ) w e th e mean square ampl itude o f os c i l l a t io ns o f th e ob jec t bef o r e and a f t e r u s e o f t h e co a t i n g, n o i s t h e l o s s f a c t o r of t h e u nt r e a te d o bj e c t , n l i s t h e l o s s f a c t o r c rea t ed by t h e co a t i n g ) , we can ca l cu l a t e t h eparameter s o f th e coa t ing p rov id ing a f ix ed decrease i n mean square v ib r a t i onamplitude with minimum weight.

F or rods and p l a t es when ex c i ta t i on i s by a monochromatic s ignal o f f r e quency w, th e mean square v i br at io n ampli tude can be repres ented as

UJ(A2>=c k Z k rk - l

where m i s t h e s p e c i f i c mass, B i s t h e b e n d i n g r i g i d i t y , r\ i s t h e l o s s f a c t o r ,hk i s a co e f f i c i en t co rr es po nd in g t o t h e K-th r o o t o f t h e ch a r a c t e r i s t i cequa t ion . I t fo l lows f rom th e expres s ion fo r Z k t h a t i f Pk u, thenz K - 1 / q 2 B 2 ; where r( < i and pk = w, t h e v a l u e of (A2>- 1 / n 2 ~ 2 ,rom which

V C ) . = $ f = ( $ - ) ( I + ~ ) 9where B0 and B1 a r e t h e be nd in g r i g i d i t y o f t h e s t r u c t u r e b e f o re a nd a f t e rt r ea tment .

Let us now assume t h a t t h e spe ctrum of t h e modes o f t h e s t r u c t u r e e x c i t e dby t h e o s c i l l a t i n g fo r c e s i s very b road , and i n t h e f i r s t approximation w e cancons ider t h a t t he ampl i tudes o f th e modes exc i te d a re approx imate ly id en t i ca l .

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I n t h i s case w e can l i m i t ourse lves t o the power approx imat ion , con sider ingt h a t t h e change i n k i n e t i c e ne rg y o f t h e s t r u c t u r e i s i n v e r s e l y p r o p o r t i o n a lt o t he l o s s f a c t o r. We t h en o b ta in

The l o s s f a c t o r and be nd in g r i g i d i t y o f a two la ye r c oa ti ng depend ont h e r a t i o o f l a y e r t hi c k ne s s es o f t h e c o a t in g [1,2]. The dependence of (VC),o n r a t i o H2/H1 o f t h e t h ic k n e s s o f t h e i n t e r m e d ia t e l a y e r i n a t h r e e l a y e rs t r u c t u r e t o t h e t hi c kn e ss o f t h e p l a t e f o r va ri o us v a l ue s o f r e l a t i v ecoating weight i s shown on Figure 1 . The nature of t h e change i n f requencyc h a r a c t e r i s t i c s o f (VC) 2 and ( V C ) 3 as a f u nc t i on o f t h i ck n e ss r e l a t i o n s h i p i na two laye r coa t in g i s shown on Figure 2 . Kn o wled g e o f th ese r eg u la r i t i e sa l lows u s , on the ba s i s o f p rocess ing o f exper imenta l da ta , t o per form comp u t e r c a l c u l a t i o n o f t h e c o a t i n g p a ra m et e rs m ost f u l l y m ee ti ng t h e r e q u i r e men ts f o r r ed u c t io n i n v ib r a t i o n l ev e l o f th e s t r u c tu r e b e in g measured .

r P . 0 . 4 /

3 2


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10 lg (VC)qoI

Figure 2 .

REFERENCES1. Tartakovskiy , B . D . , "Longidut inal Rending Os ci l la t io ns of Compound RodsCons i s t ing of Rigid Layers I 1 Vi'ib24a-t.sii Shumy (F iz iehesk iye IssZedovm i ya ) [Vibra t ions and Nois e (Phys ic al I n v es t i g a t i o n s ) - -Col l e c t ion

o f Works], Nauka Press , Moscow, 1969.2 . Ross, E . , E . Ungar and E . M . Kerwin, "Damping o f P l a t e F l ex u r a l V i b r ations by Means o f Visc oe l as t i c Laminae ," i n the book: StructuraZ Damping,Pergamon Press , 1960.

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.. . - . IDENTIFICATION_-? PARAMETERS- OF . -. -= ~- . . . - - . - - -ANDOM OSCILLATIONS AND . = . ~ - - - . . . . . _ _ OF- _ _ _ I _ - - = PRECISION .BALL...BEARINGS. .A - E . Yu. Vitkute (Kaunas)

The r e as o ns f o r v i b r a t i o n o f b a l l b e a ri n gs a r e e r r o r s i n t h e s ha pe of t h ero l l in g su r f a ce co n s i s t i n g o f so - c a l l ed wav in ess , chang es i n lo ad s , chang esi n th i ck n ess o f lu b r i c an t l ay e r , wh ich cann ot be d esc r ib ed as d e t e r m i n i s t i cfunc t ions . Therefore , ca lc u l a t io n and s tudy of th e cor responding dynamics y s t e m r e q u i r e s t h a t p r o b a b i l i s t i c o r s t a t i s t i c a l m e t h o d s b e u s e d .

Within the f ramework of th e c or re la t i on theory o f f u n c t i o n s , t h e d e g r e eo f i n f l u e n c e o f i nd i v i d u a l f a c t o r s on t h e t o t a l v i b r a t i o n l e v e l can b e de t e rmined by so lv in g an id e n t i f i c a t io n p ro blem, i n which th e o p t ima l e s t ima teo f t h e o p e r a t o r o f a s ys te m i n a c l a s s o f l i n e a r o p e r a t o r s i s sought on t h eb a s i s o f t h e r e a c t i o n s o f i n p u t s and t h e o u tp u t .

D e te rm i na ti on o f t h e d i s t r i b u t i o n o f s y st em v a r i a b l e s as t h e f u l l c ha ra ct e r i s t i c o f sy s tem b eh av io r a t each moment i n time can be performed by s t a -t i s t i c a l t e s t i n g . A n al y ti c s tu dy o f the mathematical model i s rep laced byexper imenta t ion wi th a physica l model, reproduced us in g a d i g i t a l computer .

The p r i n c i p l e s t a g e s i n t h e s o l u t i o n a r e : a ) d e t e r m in a t io n of p r o b a b i l i s t i c models of ac tu al random proc ess es; b) gene rat i on of a sequence of randomp r oc e ss e s c or re sp on di ng i n t h e i r p r o b a b i l i t y c h a r a c t e r i s t i c s t o a c t u a l p ro cesses i n th e sy st em b e in g s tu d ied ; c ) co n v er s io n o f t h e seq uen ce in co r r e s - /Ep ond enc e t o t h e s t r u c t u r e o f t h e a c t u a l s y st em ; d) s t a t i s t i c a l p ro ce ss in g oft h e o u tp u t s i g n a l s p rod u ced .

D urin g i n v e s t i g a t i o n , a l l c h a r a c t e r i s t i c s o f t h e b e a r i ng a r e measured i nt h e washed s t a t e and a f t e r l u b r i c a t i o n . D e vi a ti o ns from t h e o r e t i c a l c i r c u l a r i t y o f t h e t r a c k s of t h e i n t e r n a l and e x t e r n a l b e a r in g r i n g s a r e recorded using Indiron.

The a c t u a l p r o ce s s e s a r e r e p la c e d w i th l a t t i c e f u n c t i o n s . The r e c o rd i n gin te rv a l and r e s u l t a n t sample sp ace a r e d e te rmin ed acco rd in g t o th e r eq u i r e men ts f o r u n b iased , co n s i s t en t and o p tima l e s t ima te s i n th e sen s e o f th e min imum v a ri a nc e o f e s t i ma t e s o f t h e p r o b a b i l i t y c h a r a c t e r i s t i c s - - t h e d i s t r i b u t i o n ,mathematica l expe cta t ion , co r r e l a t io n funct ion and sp ec t r a l densi t .y . When random inf lue nce s a r e formu lated by computer , Gauss ian whi te no ise i s u s e d a s t h e

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i n i t i a l p ro ce ss , p as se d t hr ou gh l i n e a r f i l t e r s . S t a t i s t i c a l i n t e r p r e t a t i o n o ft h e o u t p u t s i g n a l s i s performed us ing or di na ry methods f o r es t ima t ion of t h eparameters o f d i s t r i b u t i o n s .

The t r a ns fe r func t ions o f sys tems are determined: the er ro r i n manufactu re o f bea r ing r aces - - the va r i ab le d r ag to rque , t he e r r o r i n manufac tu red- -v i br a ti o n i n t h e a x i a l d i r e c ti o n .

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USE OF DIGITAL COMPUTER FOR DETERMINATION OF EFFECT OF VIBRATION-DAMPING- - .- . .. . . . . . OF- sTRUCTU-RECEMENT .~. OF RESONANT- FREQUENCIES .. .

V. B. Stepanov and V. D . Tartakovskiy (Moscow)The i n t e g r a l power r e l a t i o n s h i p s , r e l a t i n g t h e d i s p lacemen t o f r e s o n an t

f requenc ies o f a s t ru c t u r e when coa t in gs are ap p l i ed t o t h e change i n l o s sf a c t o r a r e s t u d i e d e xp e r im e n ta l ly , a ll ow i ng t h e l o s s f a c t o r t o be e s t i m a t edon t h e b a s i s o f t h e c a l c u l a t e d o r measured f r eq u en cy s h i f t , t h e l o s s f a c t o ri n t u r n c h a r ac t e r iz i n g t h e d ec re a se i n v i b r a t i o n l e v e l .

I n t h i s c o nn e ct io n , t h e d i s t r i b u t i o n o f v i b r a t i o n a m pl it u de s i n va r i o u smodes o f o s c i l l a t i o n o f t h e s t r u c t u r e b e f o r e and a f t e r ap p l i c a t i o n o f t h eco a t i n g i s s tu die d. The proce ss ing of measurement r e su l t s produced as t h edependence of o sc i l la t i n g ampli tude on f requency and poin t o f p lacement o fvibrometer was performed by d i g i t a l computer us ing an a lgor i th m al lowing thes t a t i s t i c a l mean va lues o f the r equ i re d parameter s , measurement e r r o r andc a l c u l a t i o n e r r o r , c on fi de nc e i n t e r v a l s f o r t h e t r u e v a l ue s o f p ar a me t e rs ,mean decrease i n v ib ra t io n le ve l and neces sary secondary ch ar ac te r i s t i c s o ft h e s t r u c t u r e t o be d e te r mi n ed .

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FACTORS DETERMINING VIBRATION ACOUSTICAL CHARACTERISTICS OF BALL BEARINGS_ _B . Y e . Bolotov and V . B . Marnin (Kuybyshev)

Resu l t s a r e p resen ted f rom a combined s tudy o f the in f luence o f the fo l - El ow ing i n t e r n a l f a c t o r s on t h e v i b r a t i o n c h a r a c t e r i s t i c s o f a b e a r in g . Radialc l ea r ance , c l e a r an ce o f bea r i n gs i n s ep a r a t o r h o l e s , f l a t spots and waves onco n t ac t i n g s u r f ace s o f i n t e r n a l and ex t e r n a l r i n g s and r o l l i n g b o d i e s,d i f fe r e n ce s i n s i z e s o f r o l l i n g b o di es .

The s tudies were performed using a commercial ly manufactured v ib ra t i onaco u s t i c a l ap p a r a t u s and a number of s pe ci al dev ices : i n s t a l l a t i o n s f o r m easurement o f v i b r a t i o n acce l e r a t i o n s of ind iv idua l bear ing e lements , a deviceana lyz ing v ib ra t i on acc e le ra t io ns o f th e assembled bear in g , a semiau tomat icd e vi c e f o r t e s t i n g w av in es s o f r o l l i n g b o d i e s.

Recommendations a r e p resen ted f o r decreas ing th e l ev e l o f v ib r a t i on o fgenera l and spec ia l purpose b a l l bear ings , and a new v i b ra t io n ac ou s t i ca lapparatus i s descr ibed , a l lowing v i b r a t io n f law de t ec t ion t o be Performed, in d i ca t i n g t h e co n c r e t e c aus e o f i n c r eas ed v i b r a t i o n s o f b ea r i n g s .

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STUDY OF NOISE FORMATION I N THROTTLING DEVICES FOR MEASUREMENT OF NOISE CHARAC~- . _ _TERISTICS OF FANSYe. Ya. Yudin and N . N . Severina (Moscow)

In th is work, w e s t u d i ed t h r o t t l e s o f t h e wash er -and -s cr een t y pe , u sed i naerodynamic t e s t s of f ans . The advan tage o f t h i s type over o t he r types o f chokes( e . g ., s l i d e g a t e s , b u t t e r f l y v a lv e s and o t h e r s) i s t h a t t h e y crea te a d i s t r i bu ted re s i s t an ce which does no t t w i s t t h e f lo w . They t h e r e f o r e do n o t c r ea t ead d i t i o n a l t u r b u l en ce w hi ch , p a r t i cu l a r l y i n an i n t ak e l i n e , may cau se ad di t i o n a l n o i s e f or m a ti o n i n t h e f an . F ur th er mo re , s t u d i e s o f n o i s e fo rm a ti o n i na i r l i n e elements (A. V. Tolmachev, Ye. Ya. Yudin) have demonstrated t h a t th el e a s t n o i s y e l emen ts are t h os e wi th c e n t r a l pl ac em en t o f t h e a p e r t u r e f o r a i rtra nsm issi on. Devices which compress t he f low through a s l i t aga ins t onew a l l o f t h e a i r l i n e ( s l i d e g a t e s , b u t t e r f l y v a l ve s c a u se a d d i t i o n a l (7-10 dbmore) noise gene rat i on.

The measurements described were performed using a method and experimental 3 0i n s t a l l a t i o n d e sc ri be d e a r l i e r , t h e d ia me te r o f a l l s c r een s was 150 m m . Anapp ara tus produced by t h e "Bruel and Kjaer" flrm was us ed . Measurements wereperfo rmed by the r e f le c t ed f i e l d method with minimum no i se l e ve l ( a t n i gh t ) .

The range of required change in a i r stream v e l o c i t y d u ri n g t h e t e s t s w

New Methods of Studying Noise and Vibration and Cybernetic Diagnosis of Machines and Mechanisms

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