717. Investigation of dynamic and precision characteristics of low frequency vibration measurement device

Transcription

1 717. Ivestigatio of dyamic ad precisio characteristics of low frequecy vibratio measuremet device V. Volkovas, E. Uldiskas, M. Eidukevičiūtė Techological Systems Diagostics Istitute, Kauas Uiversity of Techology Kauas, Lithuaia (Received 10 September 011; accepted 14 February 01) Abstract. The measuremet of low frequecy vibratio is importat i evaluatig vibratio i costructios of wid power statios, buildigs ad also i televisio ad retrasmit ateas. The paper presets a ew costructio of low frequecy vibratio measuremet device ad methodology of parameters optimizatio based o BEM model. Experimetal ivestigatio of device prototype shows the adequacy of the theoretical model which ca be used for sythesis of devices with desirable frequecy parameters. The measuremet precisio of the device is evaluated by accomplishig measuremet ucertaity aalysis. Keywords: low frequecy vibratio, measuremet device, ucertaity, precisio. 1. Itroductio Curretly the umber of uique costructio obects i the world like ad i Lithuaia is icreasig: multi-storey buildigs, wid power-statios, may bridges, frame costructios ad others. Some of the costructios after log exploitatio are iflueced by agig processes the evirometal effects such as wid, earthquakes may iduce low frequecy vibratios i obects or costructios. I order to esure their safety it is essetial to observe their parameters by measurig low frequecy vibratios. Low frequecy vibratio measuremets are used i such fields as: earthquake vibratios [1], wave measuremets o sea ice [], etc. The vibratios i various costructios are measured usig vibratio trasducers of Edevco [3], Brüel&Kær [4], Kistler [5] ad of other kow eterprises. Yet the aalysis of the products shows that they have specific characteristics: they are based o specific physical priciple of trasformig mechaical eergy ito other type of eergy; desiged for specific frequecy rage ad has omial precisio ad purpose. Their costructioal solutios ad physical priciples prevet usig simple method to sythesize vibratio trasducers for specific low frequecy vibratio rage with reliable measuremet ucertaity ad suitably extesive precisio. This paper suggests uified mechatroic system for low frequecy vibratio measuremet ad the costructio of adequate device, which theoretical groud is dyamics of multi degree of freedom vibratig system (i simple case of oe degree). The dyamic properties of the device ad possibilities of sythesis of its dyamics were examied, the compoets for measuremet ucertaity model were aalyzed ad the ucertaity model itself provided.. Measuremet scheme ad costructio of the device The device for low frequecy vibratio measuremet (Fig. 1) is costructed as mechatroic system, cosistig of mechaical subsystem (cosole beam with a seismic mass 4 at the ed) ad two iductive displacemet sesors 3, which are placed i x ad y directios of the correspodig orthogoal coordiate system. The measuremet of the obect s low frequecy vibratios is supplemeted with filter 5 ad secodary (e. g. recorder or aalyzer) equipmet 6. 5

2 The device is fasteed to the measured obect 7. This is sufficietly simple costructioal solutio, which allows warrat frequecy sythesis, is cheap ad has possibilities for the wide applicatio. They ca be widely used i various idustrial fields due to its simplicity, reliable work ad good electric characteristics. Fig. 1. The scheme of low frequecy vibratio measuremet of the obect: 1 device s housig; steel rod, actig as sprig; 3 oe of two ocotact displacemet sesors; 4 seismic mass; 5 aalog low frequecy filter; 6 sigal recorder ad aalyzer; 7 measured obect; F(t) force iduced of the obect s vibratios; d distace betwee seismic mass ad displacemet sesor Whe force F(t) iduces low frequecy vibratio i the obect 7, they are trasferred ito device s mechaical system. Iside it steel beam is built-i, which acts as a sprig with seismic mass 4, the latter begis to move accordig the law A(x, y). Displacemet sesors 3 are fasteed perpedicularly to the plae yoz i two directios x ad y. I that case we measure the displacemet of seismic mass x(t), which is the same rage as low frequecy vibratios y(t) i directio y. The measured sigal from displacemet sesors 3 through aalogue low frequecy filter 5 is trasmitted to sigal recorder ad aalyzer 6. The paper ivestigates the idepedet vibratio i metioed directios ad the followig is oly about oe x directio of vibratio. Fig.. The costructio of low frequecy vibratio measuremet device The costructio of the low frequecy vibratio measuremet device is show i Fig.. It cosists of housig 1 i which o the beam the seismic mass 4 is fasteed. The beam is fasteed o the cover 1. The fasteig elemets 7 of the displacemet sesor 3 are built-i the housig [4]. 3. Theoretical model of the device The trasverse vibratio of mechaical subsystem of the device is described usig equatio [7]: 53

3 4 x x EI + ρf = 0; u, 4 u t [ 0 L] (1) here x(u, t) trasverse vibratios i u coordiate of L legth (whe u = L, the mass m is at the ed of the beam); E Youg s modulus; I iertia momet of the rod cross-sectio; ρ the desity of rod material; F the area of rod cross-sectio. The margial coditios: x x u u= 0 x u = 0; = 0; u = 0 u = 0 3 x EI u = 0; x = m t 3 u = L u= L. () The solutio of the equatio (1) usig variables separatio method gives mechaical subsystem vibratio forms such as: Q ( u) = C1S ( ku) + CT ( ku) + C3U ( ku) + C4V ( ku), (3) ρfω EI here C i costats (i = 1,, 3, 4); S, T, U, V Krylov fuctios; k = 4 frequecy parameter (ω agular vibratio frequecy). The equatio (1) is embedded ito margial coditio (), the costats C i determiatio equatio system is obtaied from which we get frequecy. Its geeral form is: det( A ) = 0, (4) here A matrix, which cosists of coefficiets C i. The equatios (1) ad (4) describe the dyamics of the ivestigated trasducer mechaical subsystem. Accordig this model the atural frequecies ad vibratio forms are calculated ad also their chage is aalyzed with differet meaigs of parameters E, I, ρ, F ad m. The model parameters determie the precisio ad possibilities of the low frequecy vibratio measuremet. To esure them idetificatio procedure is used. The frequecy equatio of the mathematical model of the ivestigated subsystem is deoted as = 0, its E, I, ρ, F ad m, esurig required low frequecy vibratio measuremet, as ukow variables p 1, p, p 5, ad the vector of atural frequecies Ω. The frequecy equatio becomes: (p 1, p,, p 5, Ω ) = 0; = 1,, (5) 54

4 The first compoet of the atural frequecies vector Ω should be at least 7 times larger tha the biggest measured frequecy of low frequecy vibratio. Thus optimizatio task ca be solved by searchig optimal parameters p i. The aalytical form (5) of the frequecy equatios arrow the rage of possible mechaical subsystems costructios as sometimes it is impossible to compose frequecy equatios for complex mechaical systems. For frequecy equatios compositio also fiite elemets method ca be used i which the task of atural frequecies evaluatio is such [8]: det K Ω M = 0 (6) here K matrix of the system rigidity; M matrix of the system masses. Such access allows more extesive rage of solvable task as it does ot have limitatios metioed above, but it does possible usig structural optimizatio [9] to desig the device. Suppose for ivestigative mechaical subsystem there is a iformatio i the form of resoat frequecies ω, ad ω is 7 times larger tha maximum measured frequecy of low frequecy vibratio. It is kow that resoat frequecies determied experimetally will ot coicide with atural system frequecies due to eergy loss i mechaical systems ad measuremet errors. The equatio (5) becomes: (p 1, p,, p, ω ) = E ; = 1,,, (7) here E error due to mismatch betwee Ω ad ω. If resoat quatities are measured, the equatio (7) ca be expressed by equatio system of ukow parameters p i. Their idetificatio procedure is determied by iterative procedure of digital equatio solvig, where p 1 (0), p, (0), p 5 (0) are the iitial meaigs (evaluatios) of the ukow parameters. The right side of the equatio (7) will be expaded usig Taylor series accordig iitial evaluatio of parameters ad lets limit ourselves oly with liear members of the series. We will obtai such equatio: E = E + dp dp (8) p p 1 here idex (0) deotes iitial evaluatio of the parameters. By assumig that residual error E is equal to zero, we get: (0) E = dp dp; i = 1,,..., p p 1 (0) (0) Or, by usig least square method, total error E s ca be writte i quadratic form: E s e e = e + dp dp = 1 p1 p By requirig that Es 0, we get: 55

5 E p E = dp dp ( ) 1 p1 pi 1 p pi = = = 1 i 0 ; i = 1,,..., (9) Thus we have liear equatio system of ukow parameters variatio/chages (correctios) dp 1, dp,,dp. By determiig these correctios, the iitial evaluatio of the parameters p i (o) ca be improved like this: ( 1) ( 0 p p ) i = i + dpi; i = 1,,..., (10) The iterative calculatio is cotiued till the correctios of the parameters dp i will become sufficietly small. The resultat meaigs of the idetificatio procedure p i values are put ito mathematical model of the system. For practical egieerig aalysis the simplified model is used with lumped parameters of mechaical subsystem, i.e. mathematical model of the device aalyses dyamical respose x(t) of the mass m to the kiematic excitatio X(t). Dyamic model of the device mechaical system is composed of the mass attached to the beam (as the sprig i dyamical property) ad dampeer is provided i Fig. 3 [3]. The dampig is assumed as viscous. Accordig the Hooke s law, if the amplitude of the seismic mass vibratio is sufficietly small, the betwee displacemet ad vibratig system amplitude there is liear relatioship. The mass m elemet is supported by beam (as the sprig) of rigidity k, ad the movemet of the mass is damped by elemet, which has dampeig coefficiet c. The mass ca move oly i directio x respectively to the housig of vibratio measuremet device. Durig the actio, the housig ca achieve the acceleratio displacemet x(t) of the mass m. d dt X ad the output sigal is proportioal to the 56 Fig. 3. Mechaical model of low frequecy vibratio measuremet device The model is aalyzed as a system of oe degree of freedom. Accordig the Newto law we get: ma = kx cxɺ (11) Here a acceleratio of the mass m ad is expressed as: a = ɺɺ x X ɺɺ (1)

6 Isertig the expressio (1) ito equatio (11) we get approximate mathematical model of the simplified mechaical subsystem: mɺ x + cxɺ + kx = mxɺ (13) 5. The characteristics of the vibratio measuremet device Assume that desiged device should measure low frequecy vibratios i two directios x ad y i the frequecy rage of Hz, it also must have small exteral dimesios ad have adequate amplitude frequecy characteristics. For this purpose desig problem should be solved ad appropriate creatio ad dyamic sythesis methodology should be created. The methodology of the trasducer desig is based o vibratio trasfer fuctio ad its aalysis. I order for displacemet measuremet to correspod the required precisio, the device should be techological, high reliability ad good qualitative characteristics [6]. The latter requiremets ca be realized through these properties of the device: 1. Qualitative characteristics: sesitivity; precisio; liearity of characteristics; reproducibility (repeatability); reactio speed; absece of hysteresis loop; small iteral oise.. High reliability: log operatig age; resistace to evirometal ifluece (stability of characteristics o the ifluece of eviromet). 3. Techological parameters: overall dimesios ad mass; simplicity of costructio; low costs. Some parameters of the trasducer have much impact o measuremet result, ad the others small. The quality of device ad measuremet precisio is much depedet o the mai device s qualitative features. Thus it is importat to determie the mai criteria. For low frequecy measuremet systems few criteria ca be distiguished to be cosidered: amplitude-frequecy characteristic; atural frequecy of the mechaical subsystem of the device; measuremet rage; sesitivity; liearity of characteristic; ifluece of exteral parameters. Natural device frequecy ca be aalyzed as oe of criteria. By chagig mechaical parameters of the device: rigidity of the beam (sprig) ad dampig the measured low frequecy rage may be cotrolled. I our case the device should measure low frequecy vibratio i a rage of Hz. Fig. 4 shows the theoretical curves of device atural frequecy with differet dampig coefficiets. It ca be oticed that by chagig dampig ratio also curve chages ad appropriately chages measuremet rage. With small dampig i the system the measuremet rage decreases from 15 Hz to 10 Hz. I order to icrease measured frequecy rage system dampig should be elarged. Practical characteristics of the vibratio measuremet device are show i Fig 5. The theoretical results of parameters optimizatio of the vibratio measuremet device are preseted i Fig. 6 ad Fig. 7. From the last figure it is see that created device measure vibratio i frequecy rage of Hz ad amplitude-frequecy characteristic is liear eough. 6. Aalysis of device s measuremet ucertaity compoets The device s measuremet rage mostly depeds o mechaical subsystem costructio ad parameters of iductive sesor. The mai compoets are listed i the Fig. 8. The biggest 57

7 ifluece o the measuremet rage has amplitude-frequecy characteristic, which depeds o oe freedom degree vibratio system elemets: mass, rod (sprig) ad dampig coefficiet. Fig. 4. Theoretical frequecy characteristics of the amplitude ratio with differet dampig degrees Fig. 5. Measured characteristics of amplitude ratio for mechaical system with dampig ad without it Fig. 6. Relatioship betwee parameters (m ad L) of the device ad resoace frequecy 58

8 Fig. 7. Amplitude-frequecy characteristic of device with optimal parameters (m ad L) i frequecy rage of Hz Fig. 8. Vibratio measuremet ucertaity compoet diagram The classical ucertaity aalysis theory suggests the use of formula (14) for measuremet ucertaity expressio [10]: N N 1 N f f f u c ( y) = u ( xi ) + u( xi, x ). (14) x x x i= 1 i= 1 = i+ 1 i f Here u c (y) total ucertaity value; dx depicts ifluece factor of the iput variable, u(x i ) ucertaity compoet of the iput factor x i ad u(x i, x ) depicts the correlatio betwee the iput factors x i ad x. The measuremet ucertaity ca be used i trasducer desigig i two ways: by desigig sesor by specifyig its ucertaity; ad the other way would be to desig sesor, the calculate ucertaity. If ecessary, the trasducer has to be redesiged. I this case the biggest iput o the measuremet result ucertaity is caused by displacemet sesor which is built-i to the device. It s measuremet ucertaity is already evaluated durig calibratio, thus the fial value of the ucertaity depeds o the evirometal factors i which the device will be workig ad fial dimesios of its housig ad its material. 59

9 7. Coclusios The device measurig low frequecy vibratios (i the frequecy rage of 0 15 Hz) i two directios were costructed. The features which affect reliability ad precisio of the device were aalyzed. The aalysis showed that icreasig dampig allows to measure vibratios i the wider rage of frequecies, but the further ivestigatio has to be doe. Ackowledgmets The preseted work was fuded by the Grat (No. MIP-71/010) from the Research Coucil of Lithuaia. Refereces [1] How do seismologists measure earthquakes? ( [] Doble M. J., Mercer D. J. L., Meldrum D. T., Peppe O. C. Wave measuremets o sea ice: developmets i istrumetatio. Aals of Glaciology, Volume 44, Number 1, November 006, p [3] Athoy Chu. Accelerometer Selectio Based o Applicatios. ENDEVCO TP 91, 8 p. [4] Homepage of Bruel&Kear ( [5] Homepage of Kistler ( [6] Volkovas V., Slavickas S. ad others. Aalysis of vibratios of wid power statios ad their emitted oise levels. Report of Experimetal Work (i Lithuaia), Kauas: Kauas Uiversity of Techology, Techological Systems Diagostics Istitute, 005, 7 p. [7] Volkovas V., Joušas J., Jurkauskas J. Dyamics ad Diagostics of Rotatig Systems. Kauas: Techologia, 001, 95 p. (i Lithuaia). [8] Barauskas R. The Basics of Fiite Elemet Method. Kauas: Techologia, 1998, 375 p. (i Lithuaia). [9] A. Poviliois, A. Bargelis. Structural optimizatio i product desig process// MECHANIKA. ISSN Nr.1(81) p [10] ISO GUM 1995 Guide to the Expressio of Ucertaity i Measuremet. Editio 1993, Corrected ad Reprited 1995, Geeva, Switzerlad: Iteratioal Orgaizatio for Stadardisatio, 89 p. 60