MODAL TESTING OF A PARABOLIC DISH ANTENNA STRUCTURE
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1 MODAL TESTING OF A PARABOLIC DISH ANTENNA STRUCTURE Titus Gh. Cioaa, Ph.D. Pofesso, Head of Vibation Testing and Reseach Labaatoy Univesity "Politehnica" Timisoaa St. Soin Titel N Timisoaa ROMANIA cioaa@mec.utt.o Doin Simoiu, Ph.D.student. Vibation Testing and Reseach Labaatoy Univesity "Politehnica" Timisoaa St. Mihai Viteazul N Timisoaa ROMANIA simoiu@mec.utt.o ABSTRACT. In view to captue o to emit electomagnetic waves the paabolic dish antenna is installed in the fee atmospheic field, being dynamic loaded by wind flow aound its stuctue. As an elastic stuctue, light damped, andom excited by the wind foces, the esonant effects can easily occu. The esonant effects on the antenna stuctue can be dangeous fo its integity, and lage elative motions between LNC eceive and paabolic eflecto ae developed. These elative motions may also affect damatically the tansmission signal gain, especially in the micowave ange of tansmission. The pape pesent esults of expeimental modal testing of a 1.2 m dish paabolic antenna pefomed on a laboatoy stand, the antenna being placed fixed on a igid massive metallic bloc which is hinged by elastic belts, the mechanical system being a fee-fee one. The excitation of mechanical system, including the antenna elastic substuctue, is excited in a single point on the igid metallic bloc using an electodynamic shae, this allow all the modes of antenna stuctue to be excited, and modal paamete being estimated with high accuacy. Finally, ae analyzed the pactical solutions fo the vibation elative motion diminishing. NOMENCLATURE p th mode damped natual fequency, [ad./sec],σ th mode damping decay S th modal magnitude in a given point of the stuctue 1. CONSIDERATION ON PARABOLIC ANTENNA STRUCTRE VIBRATION AND ITS PERFORMANCE All cuent communication satellites ae eath-synchonous o geo-stationay. This means they cicle the eath in a specified obit, at the same speed as the eath itself. As a esult, the satellite, as the adiation souce, appea to stand still, so a eception station as a paabolic antenna (Figue 1) can collects the adiated enegy fom satellite. The paabolic antenna consist of a paabolic eflecto 1, a feed suppot system 2, and a azimuth/elevation suppot (3) which allows the geometic optics to be optimal adjusted: the waves ays coming paallel fom the satellite, and eflected by the paabolic suface eflecto 1, to be focused in the focal point F, whee the senso of the LNC (Low Noise Convete) is placed. In view, each a high signal gain (fo antenna tested of 42.5 dbi at GHz; Kathein CAS 120)_ the focal point F is set off fom paabola focal point F p with the angle 2α, α being the otation angle position of the paabola axis Ox against hoizontal axis OX
2 300 Figue 1 Illustation of paabolic offset antenna pinciple The focal position OF p o focal length 2f espect the paabola equation in plane Oxy y 2 4fx = 0 (1) so that fo the x=c, paabola depth and y=d/2, D being dish diamete, esults the focal length 2 D f = (2) 16c Figue 2 Illustation of ays path deviations unde extenal loads The above assumptions ae consistent fo the igid stuctue of all antenna components. But the antenna components ae elastic stuctues, which unde extenal static and dynamic loads induce stuctual defomations, occuing deviations in the ays paths and finally the deviation of the eflected waves to the focal point F. Consequence the enegetically flux of the waves on the LNC deceases and the eception gain being affected. So, unde a wind gust of speed v(t) on the paabolic suface Sp (Figue 2) aise dynamic pessue foces, of which components ae usually anged in the fequency band 0 to 45 Hz. The dynamic effects ae moe complex implying fluid (wind) stuctue inteaction, aising esonant effects tuned by the antenna natual fequencies, lied in the fequency domain of excitation. In this way, by the defomation of the antenna stuctue a given geometic point P o, an undefomed dish stuctue Sp moves at the time t in a vaiable position P. The
3 nomal suface constant vecto n0 in points P oi becomes in the position P a time vaiable diection vecto n,so that the fixed axis of eflection ay in point P o, of vecto 0 passing though the focal point F o coesponding to undefomed stuctue become a vaiable axis eflection ay of vecto, also of vaiable. In the same time the focal point F 0 moves in point F, the LNC also change its position and sensitivity axis of its ay flux senso. In this condition, some eflection ays do not each the focal aea of the senso and the antenna gain is deceased affecting its pefomance. In view to give solutions fo vibation antenna stuctue eduction is necessay to detemine the modal paametes by expeimental testing. 2. EXPERIMENTAL STAND FOR MODAL TESTING OF PARABOLIC ANTENNA STRUCTRE In Figue 3 is pesented the stand and setup fo laboatoy modal testing of the paabolic antenna, Kathein CAS 120. The antenna is mounted on a igid massive metallic bloc 4, hinged by fou elastic belts 5 to an auxiliay metallic suppot 6. The ensemble of two stuctues, antenna and the metallic bloc 4, hanged by the fou elastic belts 5, consist a vibation system of which all six igid body natual fequencies ae vey low, below 3 Hz. In case of exciting the vibation system with a shae 7 at excitation fequency of values highe than 5-7 Hz, the dynamic behavio of the system, now as an elastic one, is a fee-fee system. Figue 3. Expeimental setup fo modal testing of paabolic antenna Fo a igid stuctue, as 4, having vibation motions of low elongations, the distibution law of vibation is given by the displacements vecto w v of a point P along the stuctue (components w x, w y and w z along the axes of fixed efeence fame O o YZ) w = w 0 + φ (3) whee w0 and φ ae the vectos of the igid body motion: thee tanslations of an abitay point O (components of the vecto w 0 ) and thee otation vibation motions (components of the vecto φ ) espectively, = OP being the position vecto. If the point P belong to the elastic stuctue (antenna stuctue), at the low (3) it is attached the vecto of
4 defomation u d w = w 0 + φ + u d (4) The vibation of a stuctue is a cycling pocess of changing between the two enegetically foms: inetics and potential of defomation enegy, the vibations ae pesent only if the defomations exist. In this situation a dynamical discete model of vibation in defomations must to be of fom (2) [ M ]{ u& } [ K ]{ u } = [ M ]{ W& } e d + e d t o (5) whee {u} nx1 is the column matix vecto of the displacements fom defomation in N, which define the defomation of the elastic stuctue. The column vecto { W & O} is of the fom { W } { w,w,w, φ, φ φ } T = (6) O 0x 0y 0z x y, whose entes ae the tanslation and otation lows of igid tanspotation motions. The matix [M t ] must contain masses and inetial moments, deiving fom applying the discetization method. Now, consideing that the two stuctues constituted fom a system of mateial points, we can apply the mass cente low and momentum low, which, fo a fee-fee system (no extenal foces acting), ae of foms w&dm & = 0 (7) S and if point O is the mass cente position d wdm = o dt & (8) S the integal being applied to the whole stuctue. The two elations (7) and (8) leads, by fom (4), to six equations, of matix foms whee the matix [ ] { u } m[ H] { W } { o} D 6 N D N o = 6 1 (9) [] I [ 0] 1 [] 0 [] J [ H ] = (10) m is inetial matix of the undefomed stuctue divided by m, the mass of the stuctue, [J] being the matix of inetial moments, efeence fame Oxyz (O mass cente). Now by the fom (9) the equation (5) becomes of the fom 1 1 [ Me ] [ Mt ][ H] [ D] { u& d} + [ Ke]{ ud} = { 0} (11) m The mass m is of 1200 g fom which only 18.3g belong to antenna stuctues. In this condition the entes of matix poducts [M][H] -1 [D] divided by mass value of 1200 g can be neglected against the entes of mass matix [M e ], so that [ M ]{ u& } + [ K ]{ u } { 0} & (12) e d e d = the system of equations of antenna stuctue with its suppot clamped in the igid stuctue, fixed. By an foced excitation of low f(t) using a single shae (7), the motions of igid bloc induces to elastic stuctue of antenna an inetial excitation simila to a multipoint excitation. Applying this pocedue, all natual modes pesented in the fequency domain of testing ae excited. The fee-fee suspension pesents anothe impotant advantage: stuctue tested is isolated against envionmental petubations, which in modal testing of the low damping stuctues can alte damatically the esults. Fo the vey low damping stuctues the hamonically excitation fo modal testing is vey difficult to set a steady state at the fequencies closed to a given natual fequency. In this condition moe suitable is to obtain a esonant effect and to stop suddenly the excitation, the fee decay vibation occuing ae tuned by the given natual fequency. z
5 3. ALGORITHM FOR MODAL PARAMETRE ESTIMATION USING FREE DECAY VIBRATION SIGNALS Fo a linea dynamic model of a stuctue the fee decay motion lows ae descibed by the column vecto of modal summation whee the eigenvalues n λt { (t)} = { X} e + { X } = 1 λ and eigenvectos { }, { }, λ Equation (7) becoming of the fom, λt u e (13) X having the complex conjugate foms X λ = σ + jp ; { X } = { X } + j{ X }; { X } = { X } j{ X } (j = 1) R n λ = σ jp σt { u(t) } = 2 e A cos( p t + ϕ ) = 1 ϕ = I X R 1 2 I tan A = XR + XR The image in fequency domain of the low motion (13) is obtain applying Fouie tansfom, n jωt e dt = = 1 { X } { X } { } { } U ( ω) = u(t) +, (17) jω λ jω λ mathematically simila to the steady state esponse at an hamonic excitation of cicula fequency ω. Fo a measued signal ecoded on the stuctue in fee decay vibation motion the modal component coesponding to an excitation closed to the th mod (ω p ) can be appoximated by R I u ( ) + + σ + j( ω p ) I X 2 I (14) (15) (16) X + jx ω B jc (18) B +jc being a complex constant. In the complex plane OR e I m (Figue 4) the Nyquist plot of q (ω) is a cicle C, By a cicle fitting algoithm, as [1], can be estimated the damped natual fequency p and σ modal damping decay facto, solving the linea system of equation Figue 4 Nyquist plot aound a esonant pea epesentative point P(p ) of is given by the angle to efeence fame O x y and to the efeence fame OR e Im, by θ p p ( ωs ωi ) + σ[ ( ωs ω ) Ts ( ω ωi ) Ti ] = ω ( ωs ωi ) ( ω ω ) + σ [( ω ω ) T ( ω ω ) T ] = ω ( ω ω ) s s i is i i i s (14) whee ω i, ω and ω s ae the fequencies coesponding to thee epesentative points P(ω i ), P(ω ) and P(ω s ) on the cuve C, obtained by expeimental data, and 1 Ti = (15) tg( βi / 2) also obtained fom cuve C. The system (14) can be solved in a leastsquaes manne if ae taing in consideation moe than thee points data. Having detemined the two paametes p and σ the position of ω p β = 2.actg( ) (16) σ
6 whee θ = δ + β (17) Im( ω ) Ic () δ = actg( ) (18) R ( ω ) R () e Follows the constants A and B A = Rc ( ) R0( ) cos B = Ic () R0( ) sinθ and the two components of modal components θ (19) X = 2σ R () cos θ ; Y = 2σ R () sinθ (20) 0 0 Having two signals a 1 (t) and a 2 (t) ecoded simultaneously with two acceleometes Ac 1 and Ac 2 placed on the antenna suface in two diffeent points (Figue 3),- one Ac 2, of efeence, ept in the same location M 2 and second moved along the antenna suface in the points M 1 which define geometic antenna stuctue, we can obtain the complex modal shape data efeence point M 2 u X1 + jy1 R1 () j( θ1 θ1 ) j θ 0 + jv = = e = Se (21) X2 + jy2 R20() S defining the th modal magnitude in the point M 1. Fo N points which define the geomety of antenna stuctue it is obtained two stings of values S (i and θ (i (i=1,2,3, N), which define the th modal shape along to antenna suface. The modal shape can be nomalized (i Sn Sn = (i = 1,2,3,...N) (22) (i max(sn ) which is vey useful fo gaphic epesentation of the modal shapes.. (i 4. TESTING RESULTS AND CONCLUSIONS In view to get geneal infomation on pefomance of pocedues and data pocessed, in the pape ae given only a few esults of antenna modal testing. So, in the Table 1 ae pesented the values of the modal fequency f, and pecent damped decay atio of th mode accounted as mode by the algoithm, calculated by the foms σ f = p + σ2 ; ρ = 100, 2π 2 2 p + σ 2 Table 1 Values fo modal paametes of fist elastic mode extacted fom signals of the acceleomete placed in six points 1n --6n (Figue 3 ) on the paabolic antenna N. of point f [Hz] obtained fom signal ρ [%] obtained fom signal a1(t) a2(t) a1(t) a2(t) R1o() R2o() Sn (i 1 θ11 deg 1n n n n n n mean θ21 deg θ1 deg
7 Figue 5 a)-b) Fee decay time histoy of the signals a1(t) and a2(t) occuing by a tuned excitation close to a modal fequency c) the Fouie spectums and d) cuve fitting in Nyquist plot aound the peas P1 and P2. The values in table 1 wee obtained pocessing the fee decay signals, ecoded afte stopping the hamonic excitation (by shae 7) at the fequency closed to fist modal fequency at f 1 =9.728 Hz. The time histoy of the two signals loo as in Figue 5 a and b. The Fouie spectums of the two signals pesented in Figue 5c point out two peas P 1 and P 2 close to the fequency of 10 Hz. Applying the cuve fitting algoithm aound the two peas (Figue 5d) the esults ae pesented in the fist ow of Table 1 which point out negligible deviation between the fequency values (9.729 to 9.73 Hz) and low deviation between the values of pecent damped decay atio (0.635 to 0.655). The same small deviates ae obtained even one of the two signals has a vey low magnitude, as of the signals of acceleomete Ac 1 placed in the points 2n, 4n and 5n which point out that fist mode is a tosion mode aound a vetical axis closed to line of the thee points. The same good esult can be obtained by impulse excitation applying soft shocs in diffeent points of the antenna stuctue. The fee decay signals will contain a summation of modal components excited, as pointed out in Figue 7.a and b whee ae pesented the time histoies and Fouie spectums of two acceleometes,ac 1 being placed in the point 1n on the antenna stuctue and Ac 2 on the bac side of paabolic eflecto. In the Fouie spectums (Figue 6a) along the fequency ange 0 to 50 Hz appea nine distinct peas, P 1..P 9, belonging to the fist nine natual modes. In the Table 2 ae given the values of modal paametes extacted fom Figue 6 Vibation distibution in six points of antenna stuctue fo fist natual mode of fequency f1=9.783 Hz fee decay signals a1(t) and a2(t), using the same cicle fitting algoithm. And in this case of impulsive excitation, the deviations between the values of natual fequencies f extacted fom signal a 1 (t) (column 1) and fom signal a 2 (t) (column 2) ae negligible and between values of the pecent damped decay atios (columns 3 and 4) ae smalls, instead of lage diffeence between the magnitudes of modal components occuing. Modal atio S (column 7) vaiation being the same lage, between and
8 Figue 7 a) Fee decay time histoy of the signals a1(t) and a2(t) impulsive excited, and b) the Fouie spectums of two signals Table 2 Values of modal paametes extacted fom fee decay signals a1(t) and a2(t), ecoded at a impulsive excitation N. of mode f [Hz] obtained fom signal ρ [%] obtained fom signal R1o() R2o() S θ1 θ2 θ a1(t) a2(t) a1(t) a2(t) deg deg deg Fo vey low damped stuctues, as antenna, the phase diffeence between motions of two points modal ae close to values 0 0 o 180 0, poved by column 10 values. Using the above methods and test pocedues can be easy and of low pice, the modal shapes, which allow to be analyzed the optical deviations of the ays induced by defomations of antenna stuctue, and to be studied solution fo the vibation diminish. The wos stated in Vibation Testing and Reseach Laboatoy of Univesity of Timişoaa ae in pogess and the esults will be published in futue papes. REFERENCES [1]. Cioaa T.Gh. On a Cuve-Fitting Algoithm fo Modal Paamete Estimation, Poceedings of the 6th Intenational Modal Analysis Confeence, Kissimmee, Floida, Febuay 1988 [2]. Silaş Gh., Cioaa T. Gh. On the Dynamic Analysis of the Complex Stuctues by Vibation Testing of the Substuctues Rev. Roum. Sci. Techn Mec Appl Tome 39 N pp [3]. Cioaa T.Gh. Vibation testing of stuctues Lectues (in Romanian unpublished) Univ. Politehnica Timişoaa [4]. Simoiu D., Tocaciuc A., Cecetai pivind influenţa vântului asupa uno stuctui elastice. Consideaţii teoetice, Annals of the Oadea Univesity, fascicle of Management and Technological Engineeing, Vol. III (XIII), May 2004, pp 53 d, ISSN [5]. Simoiu D., Tocaciuc A., Cecetăi pivind influenţa vântului asupa uno stuctui elastice. Aplicaţie la o antenă paabolică, Annals of the Oadea Univesity, fascicle of Management and Technological Engineeing, Vol. III (XIII), May 2004, pp 53 c, ISSN
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